[Paper Review] The Stückelberg Holographic Superconductors in Constant External Magnetic Field

Jian-Pin Wu|arXiv (Cornell University)|Jun 2, 2010

Black Holes and Theoretical Physics23 references3 citations

TL;DR

This paper investigates the Stückelberg holographic superconductor under a constant external magnetic field using numerical methods in an AdS4 black hole background. It finds that the magnetic field induces a discontinuous jump in the scalar condensate at the critical temperature even for $ c_4 = 1 $, indicating that the magnetic field strongly influences the phase transition, including tuning the order of the transition and altering critical behavior beyond the standard holographic superconductor model.

ABSTRACT

We investigate the Stückelberg holographic superconductor in present of the constant external magnetic field. We observe that a critical value of magnetic field exists as the cases in usual holographic superconductor. Furthermore, we find that the applied magnetic field strongly influence the phase transition of this model and have a jump in the condensate at the critical temperature even for $c_{4}=1$.

Motivation & Objective

• To study the behavior of the Stückelberg holographic superconductor in the presence of a constant external magnetic field.
• To investigate how the magnetic field influences the phase transition dynamics, including critical temperature and condensate formation.
• To explore the role of model parameters $ c_4 $ and $ c_\alpha $ in modifying the critical magnetic field and transition order.
• To determine whether the magnetic field induces a jump in the condensate at the critical temperature, especially for $ c_4 = 1 $.
• To analyze the numerical stability and physical consistency of the probe limit in the presence of a magnetic field.

Proposed method

• Formulates a generalized action in asymptotically AdS4 spacetime with a U(1) gauge field, a scalar field, and a Stückelberg mechanism via $ \mathcal{F}(\tilde{\Psi}) = \tilde{\Psi}^2 + c_\alpha \tilde{\Psi}^\alpha + c_4 \tilde{\Psi}^4 $.
• Uses the probe limit, assuming the gauge and scalar fields do not back-react on the black hole geometry, with $ L = 1 $ and $ m^2 = -2 $.
• Imposes boundary conditions: $ \Phi(r_+) = 0 $, regularity at the horizon, and $ \Psi_1 = 0 $ at infinity to define the condensate $ \langle \mathcal{O}_2 \rangle = \Psi_2 $.
• Solves the coupled equations of motion numerically: $ \Psi'' + \left( \frac{f'}{f} + \frac{2}{r} \right) \Psi' - \frac{m^2}{f} \Psi + \frac{\dot{\mathcal{F}}(\Psi)}{2f^2} \Phi^2 = 0 $ and $ \Phi'' + \frac{2}{r} \Phi' - \frac{\mathcal{F}(\Psi)}{f} \Phi = 0 $.
• Fixes charge density $ \rho = -1 $, varies $ c_4 $, $ c_\alpha $, and magnetic field $ H $, and computes $ T_c $ and $ \langle \mathcal{O}_2 \rangle $ as functions of temperature.
• Analyzes the critical magnetic field $ H_c $ and its dependence on $ c_4 $, $ c_\alpha $, and $ \alpha $, especially near the horizon censorship limit $ H_{\text{max}} = 0.47247 $ for $ M = 1 $.

Experimental results

Research questions

• RQ1 Does a critical magnetic field exist in the Stückelberg holographic superconductor model under a constant external magnetic field?
• RQ2 How does the magnetic field affect the critical temperature $ T_c $ and the scalar condensate $ \langle \mathcal{O}_2 \rangle $?
• RQ3 Does the magnetic field induce a jump in the condensate at the critical temperature, even when $ c_4 = 1 $?
• RQ4 How do the parameters $ c_4 $ and $ c_\alpha $ influence the critical magnetic field and the order of the phase transition?
• RQ5 What is the role of $ \alpha $ in the $ \mathcal{F}(\Psi) $ function in determining the appearance of condensate jumps under magnetic fields?

Key findings

• A critical magnetic field $ H_c $ exists in the Stückelberg holographic superconductor, analogous to standard holographic superconductors, with $ H_{\text{max}} = 0.47247 $ for $ M = 1 $.
• For $ c_4 = 1 $, the condensate exhibits a discontinuous jump at the critical temperature under an external magnetic field, a feature absent in the absence of the field.
• When $ c_4 \leq 1 $, the critical temperature $ T_c $ is independent of $ c_4 $ for a fixed magnetic field $ H $, but changes with $ c_4 $ when $ c_4 > 1 $.
• For $ c_4 = 0.7, 0.9, 1 $, the $ T_c $ vs. $ H $ curves are qualitatively similar, but the behavior diverges beyond $ T/T_c \in [0.55, 1] $ due to numerical instability at high fields.
• The magnetic field enhances the jump in the condensate, and this effect is more pronounced for larger $ c_\alpha $ and increasing $ \alpha $, especially when $ c_4 = 0 $.
• The external magnetic field strongly influences the phase transition, indicating that it can tune characteristic quantities like the coherence peak and fluctuation strength, similar to other model parameters.