The dual Ginzburg-Landau theory for a holographic superconductor: Finite coupling corrections

The holographic superconductor is the holographic dual of superconductors. We recently identified the dual Ginzburg-Landau (GL) theory for a class of bulk 5-dimensional holographic superconductors (arXiv:2207.07182 [hep-th]). However, the result is the strong coupling limit or the large-$N_c$ limit. A natural question is how the dual GL theory changes at finite coupling. We identify the dual GL theory for a minimal holographic superconductor at finite coupling (Gauss-Bonnet holographic superconductor), where numerical coefficients are obtained exactly. The GL parameter $κ$ increases at finite coupling, namely the system approaches a more Type-II superconductor like material. We also point out two potential problems in previous works: (1) the “naive” AdS/CFT dictionary, and (2) the condensate determined only from the GL potential terms. As a result, the condensate increases at finite coupling unlike common folklore.

💡 Research Summary

This paper investigates how finite ’t Hooft coupling corrections affect the dual Ginzburg‑Landau (GL) description of a holographic superconductor. The authors focus on a minimal five‑dimensional holographic superconductor placed on a Gauss‑Bonnet (GB) black‑hole background, which introduces the first non‑trivial higher‑derivative (α′) correction parameterized by λ_GB ≪ 1. While previous work identified the GL theory only in the strong‑coupling (large‑N_c) limit, this study derives the full GL free energy, kinetic coefficients, and all relevant physical quantities at linear order in λ_GB.

The analysis begins by reviewing the GB black‑hole solution, emphasizing the need to keep the boundary metric Minkowski by choosing the normalization factor N_GB≈1−½λ_GB. The Hawking temperature receives an O(λ_GB) shift, and the thermodynamic quantities are expressed accordingly. The bulk matter sector consists of a Maxwell field and a charged scalar with mass saturating the Breitenlohner‑Freedman bound. In the probe limit (no back‑reaction), the authors solve the equations of motion separately in the normal phase (Ψ=0, A_t=μ(1−u)) and the superconducting phase (Ψ≠0). They locate the critical chemical potential μ_c and critical temperature T_c, finding μ_c=2+(10−12 ln 2)λ_GB, i.e. the critical point moves to higher μ at finite coupling.

A central technical contribution is the derivation of the correct AdS/CFT dictionary for the GB background. The scalar field asymptotics acquire a prefactor N_GB: Ψ≈J z ln z−N_GB ψ z, with N_GB=1−½λ_GB. This modifies the relation between the bulk coefficient ψ and the boundary condensate, correcting the “naive” dictionary used in many earlier works. The authors show that neglecting this factor leads to qualitatively opposite conclusions for most observables.

Using the corrected dictionary, they construct the GL free energy f = c₀|D_iψ|² − a₀ ε_μ|ψ|² + (b₀/2)|ψ|⁴ + (1/4)μ_m F_{ij}² − (ψJ*+c.c.), where the coefficients are c₀ = ¼