Characteristic length of a Holographic Superconductor with $d$-wave gap

Abstract: After the discovery of the $s$-wave and $p$-wave holographic superconductors, holographic models of $d$-wave superconductor have also been constructed recently. We study analytically the perturbation of the dual gravity theory to calculate the superconducting coherence length $ξ$ of the $d$-wave holographic superconductor near the superconducting phase transition point. The superconducting coherence length $ξ$ divergents as $(1-T/T_c)^{-1/2}$ near the critical temperature $T_c$. We also obtain the magnetic penetration depth $λ\propto(T_c-T)^{-1/2}$ by adding a small external homogeneous magnetic field. The results agree with the $s$-wave and $p$-wave models, which are also the same as the Ginzburg-Landau theory.

• The paper presents analytical derivations of the coherence length and magnetic penetration depth, showing a divergence with exponent -1/2 near the superconducting transition.
• It employs a tensor field model and linear perturbations within the AdS/CFT framework to establish universal critical behavior similar to s-wave and p-wave superconductors.
• The findings reinforce the mean-field Ginzburg-Landau framework in holographic models and pave the way for exploring quantum corrections and inhomogeneous configurations.

Characteristic Lengths in Holographic d-Wave Superconductors

Introduction

This work investigates the critical properties of the d-wave holographic superconductor, focusing on two characteristic length scales: the superconducting coherence length and the magnetic penetration depth. Utilizing the @@@@1@@@@ correspondence, the authors analytically determine the mean-field critical exponents for these quantities near the superconducting phase transition. The study uses the tensor field model for the d-wave condensate developed in prior literature, which provides a bottom-up phenomenological realization of a boundary d-wave superconductor via bulk classical gravity.

Holographic d-Wave Superconductor Model

The authors consider the d-wave holographic superconductor model as formulated in [23, 24], where a symmetric, traceless charged spin-2 field minimally couples to a U(1) gauge field in a Schwarzschild-AdS background. The Lagrangian is constructed to be ghost-free and to reduce to known consistent forms in appropriate limits, though the generality of higher-spin consistency is only approximate, and the model is studied in the probe limit to ensure well-posedness.

The ansatz for the dual gravity fields is chosen to reproduce a d-wave order parameter at the AdS boundary. The critical temperature is defined as the point at which the spin-2 field condenses spontaneously for fixed chemical potential. Near the boundary, the order parameter behaves as $(1 - T/T_c)^{1/2}$, consistent with mean-field theory.

Analytical Computation of the Coherence Length

The superconducting coherence length, $\xi$, is studied via linear perturbations of the bulk fields around the background solution close to $T_c$. The analysis proceeds with a perturbative expansion in $\epsilon = 1 - T/T_c$, focusing on the lowest nontrivial order.

By considering the pole structure of the two-point correlation function of the order parameter, the coherence length is related to the inverse mass of the associated mode. A perturbative solution delivers the leading divergence:

$\xi \propto (1 - T/T_c)^{-1/2}$

This result matches the Ginzburg-Landau prediction and is identical to previous findings for s-wave and p-wave holographic superconductors. The calculation is robust under variations in the tensor mass parameter, as the Hermiticity of the governing operator is preserved for $m^2 > 0$.

London Equation and Magnetic Penetration Depth

A homogeneous external magnetic field is introduced via a small perturbation of the vector potential. Solving the decoupled equation for the magnetic perturbation in the probe limit, the induced current on the boundary is found to be proportional to the vector potential, yielding a London equation for the dual field theory. The superfluid density near $T_c$ satisfies $n_s \sim T_c - T$, and thus the magnetic penetration depth $\lambda$ diverges as

$\lambda \propto (T_c - T)^{-1/2}$

Once again, this matches the Ginzburg-Landau mean-field expectation. The analysis confirms that the d-wave holographic superconductor shares this universal critical behavior with its s-wave and p-wave counterparts.

Theoretical and Practical Implications

The primary implication is the universality of the mean-field critical exponents for coherence length and penetration depth in the context of classical AdS/CFT holographic superconductor models. The large-$N$ limit suppresses quantum fluctuations; as a result, spontaneous breaking of continuous symmetry in 2+1D is allowed, circumventing the Coleman-Mermin-Wagner theorem. This mean-field structure contrasts with some holographic models (including those manifesting Berezinskii-Kosterlitz-Thouless transitions) where non-mean-field behavior and strong IR fluctuations have been realized.

The results solidify the predictive power of the holographic approach for superconductors with anisotropic gap structures, reinforcing its relevance for strongly-coupled models of high-$T_c$ phenomena. Practically, understanding the universality and its breakdown is critical for extending such models toward more realistic condensed matter settings, particularly in regimes where quantum corrections or backreaction come into play.

Future Directions

Potential avenues include higher-order quantum corrections to the order parameter expectation value in the d-wave context, inspired by recent one-loop analyses in s-wave backgrounds. The study of vortex lattice solutions and inhomogeneous configurations, especially under strong magnetic fields and in the presence of topological defects, is directly motivated by the analytical groundwork provided here. Additionally, elucidating the precise conditions under which mean-field exponents are violated in holographic models remains a subject for further theoretical investigation.

Conclusion

The analysis rigorously demonstrates that both the superconducting coherence length $\xi$ and the magnetic penetration depth $\lambda$ of the d-wave holographic superconductor model diverge with the mean-field exponent $-1/2$ near $T_c$. This result aligns with phenomenological Ginzburg-Landau predictions and previous holographic models of lower-spin order. The findings underscore the mean-field character of criticality in the large-$N$ AdS/CFT framework and establish both the robustness and the boundaries of this behavior for anisotropic holographic superconductors (1006.5483).