Holographic Superconductors and Symmetry Breaking

A holographic superconductor is not identified by a charged scalar profile alone. The boundary source for the charged operator must vanish, a regular hairy black-brane branch must exist, and its thermodynamic and electromagnetic response must be compared with the normal branch in a fixed ensemble. These tests establish spontaneous breaking in the stated large-$N$ model; they do not by themselves identify the model with an electronic superconductor.

Required background. Chemical Potential and Charged Black Branes supplies the finite-density ensemble and horizon gauge convention. AdS Black Branes and Holographic Thermodynamics supplies Euclidean regularity and free-energy comparison.

Helpful background. Symmetry Realization and Order Parameters distinguishes a source from an order parameter. Nambu–Gor’kov Green Functions and Anomalous Propagators and Superfluid Order and Phase Stiffness give the microscopic many-body language. Gauge-Invariant Meissner Response and Superfluid Weight explains which response distinguishes a superfluid pole from electromagnetic superconductivity.

A charged scalar instability

Consider the bottom-up Einstein–Maxwell–scalar model

$$S=\frac{1}{2\kappa^2}\int d^{d+1}x\sqrt{\lvert g\rvert}\left[R+\frac{d(d-1)}{L^2} -\frac{L^2}{4g_F^2}F_{ab}F^{ab}+\lvert D\Psi\rvert^2-m^2\lvert\Psi\rvert^2\right], \qquad D_a=\nabla_a-iqA_a .$$

The normal state is a charged black brane with $\Psi=0$. A static normalizable perturbation obeys

$$\left[D_aD^a+m^2\right]\Psi=0,$$

regular at the future horizon and source-free at the AdS boundary. The electrostatic term lowers the effective mass. Near an extremal horizon with an $\mathrm{AdS}_2$ throat, a useful diagnostic is violation of the infrared Breitenlohner–Freedman bound,

$$m_{\mathrm{eff}}^2L_2^2=m^2L_2^2-q^2 e_2^2< -\frac14 .$$

This is a sufficient local signal in the simple model, not a replacement for the global eigenvalue problem. The first temperature at which the source-free static mode exists is the linear onset $T_c$. This mechanism was isolated by Gubser 2008, §§ II–III.

Near the boundary, with standard quantization,

$$\Psi(z)=z^{d-\Delta}\psi_{(s)}+z^\Delta\psi_{(v)}+\cdots, \qquad \Delta(\Delta-d)=m^2L^2 .$$

Spontaneous breaking requires $\psi_{(s)}=0$ and $\langle\mathcal O\rangle\propto\psi_{(v)}\ne0$. When both quantizations are allowed, which coefficient is the source must be declared; changing that choice changes the boundary theory.

From the zero mode to the preferred phase

The zero mode only locates a bifurcation. Below $T_c$, solve the coupled nonlinear equations for $g_{ab}$, $A_t$, and $\Psi$, impose regularity at the horizon, and impose the same chemical potential and scalar source on the normal and hairy branches. In the grand-canonical ensemble the preferred saddle minimizes

$$\Omega=T I_E^{\mathrm{ren}}[\text{saddle}] .$$

Thus the relevant comparison is $\Delta\Omega=\Omega_{\mathrm{hairy}}-\Omega_{\mathrm{normal}}$ at common $(T,\mu,\psi_{(s)})$. A condensate curve without this comparison can describe a metastable or subdominant branch.

For the simplest $d=3$ model, the source-free condensate turns on continuously and the hairy branch has lower free energy below $T_c$ Hartnoll, Herzog, and Horowitz 2008. That statement is model-specific: scalar potentials, backreaction, and boundary conditions can change the order and even the existence of the transition.

Optical response as the first application

Perturb the homogeneous phase by $\delta A_x(z)e^{-i\omega t}$, including any metric fluctuation with which it mixes. Impose an infalling condition at the horizon. If

$$A_x(z)=A_x^{(0)}+z^{d-2}A_x^{(1)}+\cdots,$$

then, after holographic renormalization and with the Maxwell normalization above,

$$G_R^{J_xJ_x}(\omega,0)=\frac{d-2}{2\kappa^2g_F^2}L^{d-1} \frac{A_x^{(1)}}{A_x^{(0)}}+G_{\mathrm{contact}}, \qquad \sigma(\omega)=\frac{G_R^{J_xJ_x}(\omega,0)-G_R^{J_xJ_x}(0,0)}{i\omega}.$$

The subtraction displays the convention for the nondissipative contact term. In the broken phase,

$$\operatorname{Im}\sigma(\omega)\sim\frac{\rho_s}{\omega}$$

implies a $\pi\rho_s\delta(\omega)$ contribution to $\operatorname{Re}\sigma$. In a boundary theory where the $U(1)$ is global, this is superfluid weight. Calling it a Meissner response requires coupling the current to a dynamical boundary photon and computing the corresponding transverse screening.

Adversarial source and quantization checks

Repeat the calculation with a small nonzero $\psi_{(s)}$. The condensate is then explicitly induced, the would-be Goldstone mode is gapped, and a nonzero $\psi_{(v)}$ no longer proves spontaneous breaking. Repeat it again under the admissible alternate quantization. If the critical mode or thermodynamic ordering changes, the conclusion belongs to the specified boundary condition rather than to the bulk potential alone.

The defensible claim is therefore: this bulk theory realizes a source-free charged instability and a preferred broken phase with a stated response in a stated ensemble. It does not establish microscopic pairing, an electron charge, a phonon mechanism, or quantitative agreement with a material.

Exercises

For $m^2L^2=-2$ in $\mathrm{AdS}_4$, find the two possible operator dimensions.

Solution

The equation $\Delta(\Delta-3)=-2$ gives $\Delta=1$ and $\Delta=2$. Both are admissible; a source-free condition must specify which coefficient is treated as the source.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Gubser, Steven S. “Breaking an Abelian Gauge Symmetry Near a Black Hole Horizon.” Physical Review D 78, 065034 (2008). DOI.
• Hartnoll, Sean A., Christopher P. Herzog, and Gary T. Horowitz. “Building a Holographic Superconductor.” Physical Review Letters 101, 031601 (2008). DOI.