A Rotating Holographic Superconductor

TL;DR

This work studies spontaneous symmetry breaking in 3+1 dimensional rotating, charged AdS black holes, revealing a boundary $2+1$-D rotating holographic superconductor on $\mathbb{R}\times S^2$. By seeking marginal modes of a charged scalar in Kerr-Newman-AdS$_4$, the authors derive separable radial and angular equations and map the problem to a phase diagram showing that the condensation temperature $T_c$ decreases with rotation; in some regimes a finite rotation destroys superconductivity, analogous to a critical magnetic field. In the planar limit near the north pole, rotation induces a London field $B_L = \dfrac{2m}{\sqrt{\Xi}\, e}\Omega_\infty$, reproducing London’s magneto-rotation coupling and yielding localized condensate droplets. The results illuminate how rotation and geometry influence holographic superconductivity, imply a new branch of stationary hairy black holes, and point to future work on backreaction and vortex formation to realize full London screening in the dual theory.

Abstract

In this paper we initiate the study of SSB in 3+1 dimensional rotating, charged, asymptotically AdS black holes. The theory living on their boundary, R x S^2, has the interpretation of a 2+1 dimensional rotating holographic superconductor. We study the appearance of a marginal mode of the condensate as the temperature is decreased. We find that the transition temperature depends on the rotation. At temperatures just below T_c, the transition temperature at zero rotation, there exists a critical value of the rotation, which destroys the superconducting order. This behaviour is analogous to the emergence of a critical applied magnetic field and we show that the superconductor in fact produces the expected London field in the planar limit.

Figures (5)

• Figure 1: Lowest angular AdS spheroidal harmonics. Regions with high density of the condensate are shown lighter and regions of low density are shown darker. Both panels correspond to a rotation parameter of $\alpha=0.9$. On the left panel, we have chosen a positive mass-squared term, while on the right the mass-squared term is negative.
• Figure 2: a) Plot of effective potential barrier around Kerr-Newman-AdS black hole for charged scalar waves with $m^2_\Psi=4$. In the chosen coordinates, the horizon is at $r_*=-\infty$ and asymptotic infinity is on the right at $r_*=0$. b) Square well toy model. The depth of the well is $U$ and its width is $w$. AdS boundary conditions are equivalent to putting a reflecting barrier at the origin.
• Figure 3: Plot of effective potential barrier for the $\lambda$ value corresponding to $\ell=0$ and $eL=10, m^2_\Psi=4$. The potential is given as a function of the deformation parameter $\alpha=aL^{-1}$ at constant temperature $T_0$. We clearly see that the potential well disappears as one increases the rotation.
• Figure 4: Examples of lowest ( i.e. no nodes) marginal modes for condensation of ${\cal O}_1$. We take $eL=8$, $m^2L^2=4$ and $\alpha=0,0.7,0.9$ for the solid, dashed and dot-dashed modes respectively. In the Schrödinger representation these are the radial wavefunctions of the zero-energy bound states.
• Figure 5: Critical temperature for the condensation of ${\cal O}_1$ at $q_e=1$ as a function of $\alpha=a/L$. Left panel for $m_\Psi^2 L^2=-2$ and right panel for $m_\Psi^2 L^2 = 4$. Superconductivity is increasingly suppressed as the rotation is increased.