The Emergence of Superconducting Systems in Anti-de Sitter Space

Abstract

In this article, we investigate the mathematical relationship between a (3+1) dimensional gravity model inside Anti-de Sitter space $\mathrm{Ad}{\mathrm{S}}_4$, and a (2+1) dimensional superconducting system on the asymptotically flat boundary of $\mathrm{Ad}{\mathrm{S}}_4$ (in the absence of gravity). We consider a simple case of the Type II superconducting model (in terms of Ginzburg-Landau theory) with an external perpendicular magnetic field $\mathbf{H}$. An interaction potential $V(r,\psi )=\alpha (T)|\psi {|}^2/r^2+\chi |\psi {|}^2/L^2+\beta |\psi {|}^4/(2r^k)$ is introduced within the Lagrangian system. This provides more flexibility within the model, when the superconducting system is close to the transition temperature $T_c$. Overall, our result demonstrates that the two Ginzburg-Landau differential equations can be directly deduced from Einstein's theory of general relativity.

10 pages, 2 figures

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