[Paper Review] Holographic superconductor in hyperscaling violation geometry with Maxwell-dilaton coupling

Shao-Jun Zhang, Qiyuan Pan+1 more|arXiv (Cornell University)|Nov 5, 2015

Black Holes and Theoretical Physics49 references3 citations

TL;DR

This study investigates s-wave holographic superconductors in hyperscaling violating spacetimes with explicit Maxwell-dilaton coupling, revealing that such coupling significantly alters superconducting properties: critical temperature decreases with increasing hyperscaling violation exponent θ, and the universal gap ratio ωg/Tc ≈ 8 is strongly violated, especially at small or large θ. The results contrast sharply with previous uncoupled models, indicating that Maxwell-dilaton coupling is crucial for accurate modeling of charged hyperscaling violating black branes.

ABSTRACT

We re-investigate the holographic superconductor in hyperscaling violation geometry by considering the coupling between the probed Maxwell field and the background dilaton. We find that the phenomenon of superconductivity still exists, but with properties affected by such a coupling. The critical temperature decreases as the hyperscaling violation exponent is increased. The influence of the dynamical exponent on the critical temperature becomes complicated which depends on the mass of the probed scalar field and the hyperscaling violation exponent. The results of the frequency gap show a large deviation from the expected universal relation.

Motivation & Objective

• To investigate the impact of Maxwell-dilaton coupling on holographic superconductivity in hyperscaling violating black brane geometries.
• To determine how the hyperscaling violation exponent θ and dynamical exponent z affect the critical temperature Tc and condensate formation.
• To assess whether the universal gap ratio ωg/Tc ≈ 8 is preserved under such coupling, especially in extreme θ regimes.
• To compare results with previous uncoupled models and identify distinguishing features for model discrimination.

Proposed method

• Adopt a bulk action including Einstein-Maxwell-dilaton gravity with a charged scalar field to model the holographic superconductor.
• Use numerical integration and the Sturm-Liouville analytical method to solve the equations of motion for scalar and gauge field perturbations.
• Construct a hyperscaling violating black brane background with metric functions dependent on z (dynamical exponent) and θ (hyperscaling violation exponent).
• Compute the frequency-dependent conductivity σ(ω) to identify superconducting gap and probe the onset of condensation.
• Define the gap frequency ωg as the minimum of |σ(ω)| to quantify the superconducting gap.
• Vary parameters θ, z, and scalar field mass m to analyze their influence on Tc and ωg.

Experimental results

Research questions

• RQ1 How does Maxwell-dilaton coupling affect the critical temperature Tc in hyperscaling violating holographic superconductors?
• RQ2 What is the role of the hyperscaling violation exponent θ in determining the condensate formation, and how does it differ from the uncoupled case?
• RQ3 How does the dynamical exponent z influence Tc, and does this dependence depend on m and θ?
• RQ4 Does the universal gap ratio ωg/Tc ≈ 8 still hold when Maxwell-dilaton coupling is included?
• RQ5 What physical interpretation can be given to the non-monotonic behavior of Im[σ(ω)] at small θ, which diverges linearly with frequency?

Key findings

• The critical temperature Tc decreases monotonically with increasing hyperscaling violation exponent θ, indicating that stronger hyperscaling violation suppresses superconducting condensation.
• The influence of the dynamical exponent z on Tc is non-monotonic and depends on both the scalar field mass m and θ: for m²rH⁻²θ/d = 0, Tc increases with z, but for lighter scalars, a critical θc exists where the trend reverses.
• The superconducting gap frequency ωg increases with θ for fixed z, and decreases with increasing z for fixed θ, consistent with previous uncoupled models.
• The universal gap ratio ωg/Tc ≈ 8 is significantly violated, especially for very small or very large θ, indicating strong deviation from BCS-like behavior.
• The imaginary part of the conductivity Im[σ(ω)] diverges linearly with frequency at small θ, a novel behavior not seen in uncoupled models, whose physical origin remains unclear.
• The critical exponent of the condensate remains at the mean-field value of 1/2, confirming second-order phase transition behavior despite coupling modifications.