A Holographic Superconductor in an External Magnetic Field

TL;DR

The paper investigates a holographic superconductor in AdS/CFT with an external magnetic field by embedding a charged scalar in a dyonic Reissner-Nordström–AdS black hole while treating the scalar as a perturbation. The x-profile of the condensate maps exactly to a quantum harmonic oscillator, yielding a stripe-like condensate whose width decreases with increasing magnetic field $B=2h\alpha^2$, and which vanishes in the strong-field limit, signaling a Meissner-like effect within the probe-limit. The authors use a perturbative analysis and numerical shooting to establish the existence and properties of the condensate, defining dimensionless observables $\tilde{T}$, $B/\tilde{T}^2$, and $\rho/\tilde{T}^2$ that characterize the condensation region. The work provides a tractable, solvable limit that clarifies how magnetic flux confines holographic superconductors and offers a stepping stone toward fully backreacted, spatially inhomogeneous phases in AdS/CFT superconductivity.

Abstract

We study a system of a complex charged scalar coupled to a Reissner-Nordstrom black hole in 3+1 dimensional anti-de Sitter spacetime, neglecting back-reaction. With suitable boundary conditions, the cases of a neutral and purely electric black hole have been studied in various limits and were shown to yield key elements of superconductivity in the dual 2+1 dimensional field theory, forming a condensate below a critical temperature. By adding magnetic charge to the black hole, we immerse the superconductor into an external magnetic field. We show that a family of condensates can form and we examine their structure. For finite magnetic field, they are localized in one dimension with a profile that is exactly solvable, since it maps to the quantum harmonic oscillator. As the magnetic field increases, the condensate shrinks in size, which is reminiscent of the Meissner effect.