[Paper Review] Inhomogeneous magnetic field in AdS/CFT superconductor

Wen-Yu Wen|ArXiv.org|May 11, 2008

Black Holes and Theoretical Physics15 references17 citations

TL;DR

This paper constructs a holographic model of $(2+1)$-dimensional superconductors in the presence of an inhomogeneous magnetic field using the AdS/CFT correspondence. By coupling a charged scalar field and Maxwell field in a planar AdS black hole background with probe magnetic fields, it identifies both type I and type II superconductor phases, with a novel type-changing transition near the critical temperature due to temperature-dependent order parameter scaling.

ABSTRACT

We study the holographically dual description of superconductor in (2+1)-dimensions in the presence of inhomogeneous magnetic field and observe that there exists type I and type II superconductor. A new feature of type changing is observed for type I superconductor near critical temperature.

Motivation & Objective

• To study the holographic dual of a $(2+1)$-dimensional superconductor under inhomogeneous magnetic fields using the AdS/CFT correspondence.
• To investigate whether the system exhibits type I and type II superconducting behavior, as predicted by Ginzburg-Landau theory.
• To explore the novel phenomenon of type changing in superconductors near the critical temperature $T_c$.
• To examine the role of the order parameter's temperature dependence in modifying the effective Ginzburg-Landau parameter $\tilde{\kappa}$.
• To assess the impact of partial back-reaction on the metric due to spatially varying fields in a simplified model.

Proposed method

• Adopt a probe limit where the electromagnetic field is introduced in the matter sector without back-reacting on the gravity sector, using a planar AdS-Schwarzschild black hole metric.
• Introduce an inhomogeneous magnetic field via a spatially dependent gauge field $A_y = a(x)$, with $H = \kappa \dot{a}(x)$, to model spatially varying magnetic flux.
• Use a factorized ansatz $\Psi(r,x) = \psi(r)s(x)$ to decouple radial and spatial dependence in the scalar and gauge field equations.
• Derive coupled differential equations for $s(x)$ and $a(x)$ from the Ginzburg-Landau-type action with $m^2L^2 = -2$, leading to effective equations at the boundary ($r \to \infty$).
• Introduce a simplified back-reaction correction to the metric $g_{xx} \sim r^2(1 - \frac{2\psi_2}{\psi_1 r} + \cdots)$ to account for spatial variation of $s(x)$ and $a(x)$, preserving planar symmetry.
• Define effective parameters $\tilde{g} = \frac{g\psi_1^2}{L^4}$ and $\tilde{\kappa} = \frac{\kappa}{\sqrt{2}\psi_1}$ to map the system to a standard GL model with temperature-dependent $\tilde{\kappa}$.

Experimental results

Research questions

• RQ1 Can a holographic superconductor model in $(2+1)$ dimensions exhibit both type I and type II behavior under an inhomogeneous magnetic field?
• RQ2 Does the effective Ginzburg-Landau parameter $\tilde{\kappa}$ vary with temperature, leading to a type transition in a system initially classified as type I?
• RQ3 How does the spatial variation of the order parameter $s(x)$ and magnetic field $H(x)$ behave in the presence of an inhomogeneous field?
• RQ4 What is the impact of partial metric back-reaction on the consistency and convergence of the solution for spatially varying fields?
• RQ5 Can a simplified probe model with non-back-reacting gravity still capture key features of unconventional superconductors, such as vortex formation and flux penetration?

Key findings

• The model successfully realizes both type I and type II superconducting phases in the presence of an inhomogeneous magnetic field, consistent with Ginzburg-Landau theory.
• A novel type-changing transition is observed near the critical temperature $T_c$, where a system initially classified as type I ($\tilde{\kappa} \ll 1$) evolves into a type II superconductor due to the temperature dependence of $\psi_1 \propto (1 - T/T_c)^{1/2}$.
• The order parameter $s(x)$ exhibits vortex-like structure in the type II phase, with magnetic flux penetrating the vortex core, as shown in Figure 1.
• The effective Ginzburg-Landau parameter $\tilde{\kappa}$ is found to depend on temperature through $\psi_1$, leading to a continuous transition from type I to type II behavior as $T \to T_c^-$.
• A naive metric correction $g_{xx} \sim r^2(1 - \frac{2\psi_2}{\psi_1 r} + \cdots)$ is proposed to account for spatial dependence, improving consistency of the equations without full back-reaction.
• Despite the absence of full back-reaction, the model captures essential features of superconductivity, including vortex formation and flux penetration, suggesting its viability as a phenomenological model for strongly correlated systems.