The emergence of superconducting systems in Anti-de Sitter space

In this article, we investigate the mathematical relationship between a (3+1) dimensional gravity model inside Anti-de Sitter space AdS₄, and a (2+1) dimensional superconducting system on the asymptotically flat boundary of AdS₄ (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 H. An interaction potential V(r,ψ)=α(T)|ψ|²/r²+χ|ψ|²/L²+β|ψ|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 Ginzburg-Landau differential equations can be directly deduced from Einstein's theory of general relativity.