A note on holographic superconductors with Weyl Corrections

TL;DR

This paper analytically investigates holographic superconductors with Weyl corrections, deriving critical temperatures and exponents, and confirming numerical results with explicit formulas in the probe limit.

Contribution

It provides analytical expressions for critical temperatures and exponents in holographic superconductors with Weyl corrections, matching previous numerical findings.

Findings

Critical temperature for conformal dimension 3: T_c ≈ 0.170845 ρ^{1/3}

Critical temperature for conformal dimension 1: T_c ≈ 0.21408 ρ^{1/3}

Critical exponent β = 1/2

Abstract

We study analytical properties of the holographic superconductors with Weyl corrections. We describe the phenomena in the probe limit neglecting backreaction of the space-time. We observe that for the conformal dimension $\triangle_{+}=3$, the minimum value of the critical temperature $T^{Min}_c$ at which condensation sets, can be obtained directly from the equations of motion as $T^{Min}_c\approx 0.170845\sqrt[3]{\rho}$, which is in very good agreement with the numerical value $T^{Min}_c=0.170\sqrt[3]{\rho}$ [Phys.Lett.B697:153-158,2011]. This value of $T^{Min}_c$ corresponds to the value of the Weyl's coupling $\gamma=-0.06$ in table (1) of [Phys.Lett.B697:153-158,2011]. We calculate the $T^{Min}_c\approx 0.21408\sqrt[3]{\rho}$ for another Weyl's coupling $\gamma=0.02$ and the the conformal dimension $\triangle_{-}=1$. Further, we show that the critical exponent is $\beta=1/2$. We observe that there is a linear relation between the charge density $\rho$ and the chemical potential difference $\mu-\mu_c$ qualitatively matches the numerical curves.