Supercurrent: Vector Hair for an AdS Black Hole

TL;DR

The paper investigates a holographic AdS$_4$ black hole with a charged scalar that models a superconducting boundary theory, and it introduces a DC supercurrent by turning on a spatial gauge component $A_x$ without an external electric field. By solving the coupled equations in the probe limit, the authors reveal a rich phase structure in the $(T/\mu, S_x/\mu)$ plane, including a special point where the superconducting transition changes from second order to first order, and they show that a nontrivial current can coexist with superconductivity as vector hair on the black hole. The results are validated for two boundary quantizations, $\Psi_1=0$ and $\Psi_2=0$, and are closely tied to free-energy comparisons that produce swallow-tailed diagrams and critical currents $J_x^{(c)}$. The work provides qualitative concordance with condensed-matter supercurrent phenomenology and proposes extensions to backreaction, magnetic fields, and potential string/M-theory embeddings, highlighting the holographic regime's utility for exploring strongly coupled superconducting dynamics.

Abstract

In arXiv:0803.3295 [hep-th] a holographic black hole solution is discussed which exhibits a superconductor like transition. In the superconducting phase the black holes show infinite DC conductivity. This gives rise to the possibility of deforming the solutions by turning on a time independent current (supercurrent), without any electric field. This type of deformation does not exist for normal (non-superconducting) black holes, due to the no-hair theorems. In this paper we have studied such a supercurrent solution and the associated phase diagram. Interestingly, we have found a "special point" (critical point) in the phase diagram where the second order superconducting phase transition becomes first order. Supercurrent in superconducting materials is a well studied phenomenon in condensed matter systems. We have found some qualitative agreement with known results.

Figures (16)

• Figure 1: Phase diagram in $S_x,T$ plane showing critical point, first order and second order transition. For $T<T_{sp}$ the phase transition is first order. The dotted line is the extension of second order transition line.
• Figure 2: Zero mode of $\psi$ at $\mu=\mu_c$ with a normalization $\psi=1$ at the horizon.
• Figure 3: Plots of $\psi$ and $A_t$ at $1/mu \approx 0.105, 0.079$. $\mu$ is increasing from below.
• Figure 4: Nature of solution for $\frac{1}{\mu} \approx 0.174$ and $\frac{S_x}{\mu} \approx 0.369$
• Figure 5: Nature of solution for $\frac{1}{\mu} \approx 0.087$ and $\frac{S_x}{\mu} \approx 0.609$