What is the dual Ginzburg-Landau theory for holographic superconductors?

Holographic superconductors are holographic duals of superconductors. Macroscopically, a superconductor should be described by the Ginzburg-Landau (GL) theory. There is ample evidence that the holographic superconductors are described by the standard GL theory, but the exact form of the dual GL theory is little known. We identify the dual GL theory for a class of bulk 5-dimensional holographic superconductors, where numerical coefficients are obtained exactly.

💡 Research Summary

The paper addresses a fundamental question in the holographic approach to condensed‑matter physics: what is the precise Ginzburg‑Landau (GL) effective theory that is dual to a class of five‑dimensional holographic superconductors? The authors consider a bulk theory consisting of Einstein gravity with a negative cosmological constant, a Maxwell field, and a complex scalar field. The scalar mass is tuned to saturate the Breitenlohner‑Freedman bound (m² = −4 in units where the AdS radius L = 1), which gives a scaling dimension Δ₊ = 2 for the dual operator. At the critical chemical potential μ_c = 2 the scalar admits an analytic solution Ψ ∝ −u/(1+u) (with u = (r₀/r)²), allowing the authors to perform a systematic expansion in the deviation ε ≡ μ − μ_c.

The analysis proceeds in several stages. First, the authors work in the probe limit (large N_c, negligible back‑reaction) and write down the bulk equations of motion for the gauge field A_M and the scalar Ψ. Near the AdS boundary they identify the source and expectation values of the dual current J_μ and order‑parameter operator O. They also introduce the necessary counterterms, in particular a logarithmic Maxwell counterterm, to cancel UV divergences.

Using the exact critical‑point solution, they compute a variety of physical quantities both above and below the transition. In the high‑temperature (normal) phase they study linear perturbations δΨ ∝ e^{iq·x} and obtain the order‑parameter response function χ(q) = 4/(q² − 2ε), the correlation length ξ² = 1/(1 − 2ε), and the static susceptibility χ_T = 2 − ε. These results match the mean‑field predictions of the standard GL theory, confirming that the holographic model reproduces the expected critical exponents.

Next, they turn on a magnetic field B and solve the linearized scalar equation in the presence of the background gauge potential A_y = Bx. The spatial part reduces to a Landau‑level problem, and the lowest Landau level (n = 0) yields the upper critical magnetic field B_{c2} = 1 − ξ². This is precisely the GL relation B_{c2}=Φ₀/(2πξ²) once the appropriate units are restored. The authors also discuss the GL parameter κ = (1/√2)√(B_{c2}/B_c) and show that the sign and magnitude of the bulk higher‑order couplings determine whether the system is Type I (κ < 1/√2) or Type II (κ > 1/√2).

A central achievement of the paper is the derivation of the full GL free‑energy functional from the bulk theory. By evaluating the on‑shell bulk action, including the counterterms, and performing the ε‑expansion, they obtain
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