A Holographic Superconductor in an External Magnetic Field

We study a system of a complex charged scalar coupled to a Reissner–Nordström 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.