Floquet superconductor in holography

We study nonequilibrium steady states in a holographic superconductor under time periodic driving by an external rotating electric field. We obtain the dynamical phase diagram. The superconducting phase transition is of first or second order depending on the amplitude and frequency of the external source. The rotating electric field decreases the superconducting transition temperature. The system can also exhibit a first order transition inside the superconducting phase. It is suggested that this transition exists all the way down to zero temperature. The existence of a nonequilibrium thermodynamic potential for such steady solutions is also discussed from the holographic point of view. The current induced by the electric field is decomposed into normal and superconducting components, and this makes it clear that the superconducting one dominates in low temperatures.

Figure 1

Phase diagram. The space of $(A/\mu ,\mathrm{\Omega }/\mu )$ is separated into three regions A, B and C. In each region, the superconducting order parameter $⟨O_2⟩$ behaves qualitatively differently as a function of temperature $T$. On the line labeled “quantum critical transition,” a $T=0$ first order quantum phase transition is expected.

Figure 2

Panels (a)–(c): Temperature dependence of the superconducting order parameter $\sqrt{⟨O_2⟩}/\mu$ at $A/\mu =0.25$, 0.4, 0.53. Panel (d): The $A/\mu$-dependence of the branching temperature where $⟨O_2⟩$ starts to become nonzero. This does not depend on $\mathrm{\Omega }$ for fixed $A$.

Figure 3

Violation of the integrability condition $\partial \rho /\partial |\overrightarrow{A}|=\partial J_{\parallel }/\partial \mu$ evaluated when $A/T=20.94$ and $\mathrm{\Omega }/T=6.283$. The chemical potential $\mu$ is varied for a fixed temperature $T$. The inset shows the scalar condensate for these parameters.

Figure 4

Contour plot of the first order transition temperature $T_c^{1\mathrm{st}}$. On the upper right boundary of the region C, the transition temperature reaches zero. In region A, there is no first order transition. For the second order transition temperature, see Fig. 2.

Figure 5

Panel (a): Current $J$. Panel (b): Joule heating $q$. Both panels are plotted for $A/\mu =0.53$, $\mathrm{\Omega }/\mu =0,0.1,0.13,0.15,0.18$. Dashed lines are normal phase values.

Figure 6

Panel (a): The conductivity ${\sigma }_n$. Panel (b): The superfluid density ${\rho }_s$. Both panels are plotted for $A/\mu =0.53$ and $\mathrm{\Omega }/\mu =0,0.1,0.13,0.15,0.18$. Horizontal dashed lines in each panel indicate the normal phase values ${\sigma }_n=1$ and ${\rho }_s=0$.

Figure 7

Panels (a) and (b): Typical profiles of the free energy measured from the normal phase.

Figure 8

Panel (a): The critical chemical potential ${\mu }_c^{\text{1st}}$ for the first order phase transition. Panel (b): The critical temperature $T_c^{\text{1st}}$ for the first order phase transition.

Figure 9

Violation of the integrability conditions, evaluated when $A/T=20.94$, $\mathrm{\Omega }/T=6.283$ and ${\mathrm{\Xi }}_1=0$, which are the same parameters as in Fig. 3. The ${\mathrm{\Xi }}_1$-derivative is calculated around the trivial scalar source background. The divergence to the right is due to the superconducting phase transition and therefore is inessential.