The gravity dual of a p-wave superconductor

TL;DR

The paper presents a gravity dual for a $p$-wave superconductor in an $AdS_4$-Schwarzschild background by solving Einstein–Yang–Mills equations with a vector order parameter. The resulting phase is anisotropic, featuring a directional gap and a distinct low-energy electromagnetic response, with $ ilde{\sigma}_{yy}$ showing a gap scale $ ilde ho^{1/2}$ and a narrow Drude-like peak in $ ilde{\sigma}_{xx}$. Quasinormal-mode analysis reveals that the $p$-wave backgrounds are dynamically stable near $T_c$ against $p+ip$ perturbations, while $p+ip$ configurations are unstable and tend to evolve toward the $p$-wave phase, suggesting the $p$-wave state is thermodynamically preferred at low $T$. The work provides a controlled, strongly coupled laboratory for exploring nodal-like behavior and anisotropic superconductivity in a holographic setting, with potential insights for real materials and extensions to backreaction and momentum-dependent response.

Abstract

We construct black hole solutions to the Yang-Mills equations in an AdS_4-Schwarzschild background which exhibit superconductivity. What makes these backgrounds p-wave superconductors is that the order parameter is a vector, and the conductivities are strongly anisotropic in a manner that is suggestive of a gap with nodes. The low-lying excitations of the normal state have a relaxation time which grows rapidly as the temperature decreases, consistent with the absence of impurity scattering. A numerical exploration of quasinormal modes close to the transition temperature suggests that p-wave backgrounds are stable against perturbations analogous to turning on a p+ip gap, whereas p+ip-wave configurations are unstable against turning into pure p-wave backgrounds.

Figures (7)

• Figure 1: A superconducting condensate floats above a black hole horizon because of a balance of gravitational and electrostatic forces. The condensate carries a finite fraction of the total charge density, so there is more electric flux above the condensate than there is right at the horizon. A massive charged particle, labeled $\psi_+$, may be driven upward by the electrostatic force, but because of the warped geometry of $AdS_4$, its trajectory cannot reach the boundary. So $\psi_+$ must participate in the condensate if it doesn't fall into the horizon. The frequency-dependent conductivity can be found by calculating an on-shell amplitude for a photon propagating straight down into the geometry.
• Figure 2: Each point along the contours plotted represents a solution to the non-linear boundary value problem specified by (\ref{['YMbackground']}), (\ref{['FarT']}), and (\ref{['NearT']}). Points on the line labeled "normal" are RNAdS solutions, and if charge density is held fixed, temperature rises as one moves to the left. Points on the curve labeled "superconducting" break the abelian gauge symmetry generated by $U(1)_3$. Points on the other curves also break the abelian gauge symmetry but are expected to be unstable. The point where the superconducting solutions join onto the normal solutions is labeled $T_c$ because the simplest scenario is for there to be a second order phase transition at this point.
• Figure 3: The fraction $\tilde{\rho}_s/\tilde{\rho}$ of the charge carried by the superconducting condensate and the order parameter $\tilde{W}_1$ are plotted against the rescaled temperature $T/\sqrt{\tilde{\rho}}$. At $T_c$, $\tilde{\rho}_s/\tilde{\rho}$ vanishes linearly, while $\tilde{W}_1$ vanishes as $\sqrt{T_c-T}$.
• Figure 4: Rescaled conductivities $\tilde{\sigma}_{xx}$ and $\tilde{\sigma}_{yy}$ as functions of frequency. The dotted curves are the best fits of the Drude model prediction (\ref{['SigmaDrude']}) to $\mathop{\rm Re}\nolimits\tilde{\sigma}_{xx}(\omega)$ at low $\omega$.
• Figure 5: Temperature-dependent quantities and approximate fits, as explained in (\ref{['SeveralFits']}) and the surrounding text. We have defined $\tilde{\sigma}_{xx,0} = \lim_{\omega\to 0} \mathop{\rm Re}\nolimits\tilde{\sigma}_{xx}(\omega)$, $\tilde{\sigma}_{yy,0} = \lim_{\omega\to 0} \mathop{\rm Re}\nolimits\tilde{\sigma}_{yy}(\omega)$, $\mathop{\rm Res}_{\omega = 0} \mathop{\rm Im}\nolimits\tilde{\sigma}_{xx}/\sqrt{\tilde{\rho}} = \lim_{\omega \to 0} {\omega \over \sqrt{\tilde{\rho}}} \mathop{\rm Im}\nolimits\tilde{\sigma}_{xx}(\omega)$, $\mathop{\rm Res}_{\omega = 0} \mathop{\rm Im}\nolimits\tilde{\sigma}_{yy}/\sqrt{\tilde{\rho}} = \lim_{\omega \to 0} {\omega \over \sqrt{\tilde{\rho}}} \mathop{\rm Im}\nolimits\tilde{\sigma}_{yy}(\omega)$, and $\mathop{\rm Res}_{\omega = \omega_0} \mathop{\rm Im}\nolimits\tilde{\sigma}_{xx}/\sqrt{\tilde{\rho}} = \lim_{\omega \to \omega_0} {\omega-\omega_0 \over \sqrt{\tilde{\rho}}} \mathop{\rm Im}\nolimits\tilde{\sigma}_{xx}(\omega)$.