[Paper Review] Holographic insulator/superconductor phase transition in Born-Infeld electrodynamics

Nan Bai, Yi-hong Gao+2 more|arXiv (Cornell University)|Dec 12, 2012

Black Holes and Theoretical Physics27 references4 citations

TL;DR

This paper investigates holographic insulator/superconductor phase transitions in Born-Infeld electrodynamics using numerical and analytical methods. It finds that the Born-Infeld parameter $ b $ has no effect on the critical chemical potential $ \mu_c $, indicating that the phase transition behavior remains unchanged compared to Maxwell electrodynamics, despite the nonlinearity introduced by $ b $.

ABSTRACT

We studied holographic insulator/superconductor phase transition in the framework of Born-Infeld electrodynamics both numerically and analytically. First we numerically study the effects of the Born-Infeld electrodynamics on the phase transition, find that the critical chemical potential is not changed by the Born-Infeld parameter. Then we employ the variational method for the Sturm-Liouville eigenvalue problem to analytically study the phase transition. The analytical results obtained are found to be consistent with the numerical results.

Motivation & Objective

• To study the effects of Born-Infeld electrodynamics on holographic insulator/superconductor phase transitions in the probe limit.
• To determine whether the Born-Infeld parameter $ b $, which introduces nonlinearity in electrodynamics, alters the critical chemical potential $ \mu_c $.
• To compare numerical results with analytical solutions using the Sturm-Liouville variational method.
• To investigate the stability and condensation behavior of scalar operators and charge density in the presence of Born-Infeld corrections.

Proposed method

• Numerical solution of the coupled Einstein-Maxwell-scalar field equations in a five-dimensional AdS soliton background with Born-Infeld electrodynamics.
• Use of the probe limit to decouple gravity from the gauge and scalar fields, simplifying the system to a solvable ODE system.
• Application of the Sturm-Liouville variational method to analytically estimate the critical chemical potential $ \mu_c $.
• Implementation of the ansatz $ A_\mu = (\phi(r), 0, 0, 0) $, $ \psi = \psi(r) $ to reduce the equations to radial ODEs.
• Derivation of the equations of motion for the scalar field $ \psi(r) $ and gauge field $ \phi(r) $ in the Born-Infeld framework.
• Use of the Born-Infeld Lagrangian $ \mathcal{L}_{\text{BI}} = \frac{1}{b}\left(1 - \sqrt{1 + \frac{bF}{2}}\right) $, with $ F = F_{\mu\nu}F^{\mu\nu} $, to model nonlinear electrodynamics.

Experimental results

Research questions

• RQ1 Does the Born-Infeld parameter $ b $ affect the critical chemical potential $ \mu_c $ for the insulator/superconductor phase transition?
• RQ2 How does the inclusion of Born-Infeld electrodynamics alter the scalar condensation and charge density compared to Maxwell theory?
• RQ3 Can the Sturm-Liouville variational method accurately reproduce the numerical results for $ \mu_c $ in the Born-Infeld framework?
• RQ4 What is the behavior of the AC conductivity in the insulator/superconductor phase transition under Born-Infeld electrodynamics?
• RQ5 Is the phase transition in the Born-Infeld model still second-order, and does it preserve the same critical exponents as in the Maxwell case?

Key findings

• The critical chemical potential $ \mu_c $ remains unchanged with varying Born-Infeld parameter $ b $, as confirmed by both numerical and analytical methods.
• Numerical results show that the scalar condensation $ \langle \mathcal{O} \rangle $ and charge density $ \rho $ are insensitive to changes in $ b $, indicating no significant modification of the phase transition by nonlinearity.
• The analytical Sturm-Liouville method yields a critical chemical potential $ \mu_{\text{min}} = 1.89037 $ for $ \lambda_i = 5/2 $, $ b = 0.01 $, matching the numerical result.
• The analytical solution is consistent with the numerical solution, validating the variational approach in the Born-Infeld context.
• The absence of $ b $-dependence in $ \mu_c $ suggests that the nonlinearity of Born-Infeld electrodynamics does not suppress or enhance scalar condensation in this model.
• The results are consistent with prior work stating that $ \mu_c $ is independent of the coupling parameter $ b $, reinforcing the robustness of the phase transition threshold.