0809/0809.3079.md

Characteristic length of an AdS/CFT superconductor

Kengo Maeda^1,* and Takashi Okamura^2,†

¹Department of Engineering, Shibaura Institute of Technology, Saitama, 330-8570, Japan

²Department of Physics, Kwansei Gakuin University, Sanda, 669-1337, Japan

(Dated: November 1, 2018)

We investigate in more detail the holographic model of a superconductor recently found by Hartnoll, Herzog, and Horowitz [Phys. Rev. Lett. 101, 031601], which is constructed from a condensate of a charged scalar field in AdS₄-Schwarzschild background. By analytically studying the perturbation of the gravitational system near the critical temperature $T_c$ , we obtain the superconducting coherence length proportional to $1/\sqrt{1-T/T_c}$ via AdS/CFT correspondence. By adding a small external homogeneous magnetic field to the system, we find that a stationary diamagnetic current proportional to the square of the order parameter is induced by the magnetic field. These results agree with Ginzburg-Landau theory and strongly support the idea that a superconductor can be described by a charged scalar field on a black hole via AdS/CFT duality.

PACS numbers: 11.25.Tq, 74.20.-z

I. INTRODUCTION

The AdS/CFT duality [1] gets into the limelight as a powerful tool to investigate the strongly coupled gauge theories. Motivated by the recent experimental data of quark-gluon plasma at the Relativistic Heavy Ion Collider [2, 3], transport coefficients such as shear viscosity were calculated for various duality models. Interestingly, for a large class of dualities, it was found that the ratio of viscosity divided by entropy density is a universal constant compatible with the experimental data (see, for example, Refs. [4, 5] for all references). This leads us to expect that strongly coupled phenomena such as quantum phase transition or superconducting phase transition in condensed matter systems are described by some kind of duality.

If a superconductor can be described by a gravitational model via the AdS/CFT duality, there should be a scalar hair on an anti-deSitter (AdS) black hole which represents the condensation in the dual gauge theory. Gubser [6] has presented a counter example to a no scalar hair theorem by giving a static solution of a charged scalar field coupled to an Abelian gauge field on the AdS₄-Reissner-Nortström black hole background if the charge of the black hole is large enough. Since the temperature of the black hole decreases as the charge increases, the black hole can support the scalar hair only for low temperature. This is because the scalar field condensation breaks the Abelian gauge symmetry spontaneously at sufficiently low temperature. Hartnoll et. al. [7] numerically showed that there is a critical temperature below which the charged scalar hair exists on AdS₄-Schwarzschild black hole and the conductivity becomes infinite at the low frequency limit. It was also numerically shown that the scalar field condensation occurs below a critical temperature under the presence of an external magnetic field [8, 9, 10]. Quite recently, general properties of p-wave superconductors were investigated by Gubser and Pufu [11] and independently by Roberts and Hartnoll [12] in a model of a non-Abelian gauge field in the background of AdS₄-Schwarzschild black hole.

The purpose in this paper is to explore a little further the model of the superconductor composed of the charged scalar field on AdS₄-Schwarzschild background [7] by investigating perturbation of

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*Electronic address: maeda302@sic.shibaura-it.ac.jp

†Electronic address: tokamura@kwansei.ac.jpthe system near the critical temperature. The order parameter of the superconductor is the scalar operator dual to the charged scalar field. So, the correlation length of the order parameter, or the superconducting coherence length $\xi$ is obtained by the perturbation of the scalar field. According to Ginzburg-Landau theory, a superconductor is characterized by only two parameters, $\xi$ and the magnetic penetration length $\lambda$ . Therefore, it will be of interest to determine the two parameters by investigating the perturbation. Motivated by this, we analytically investigate static fluctuation of the scalar field with nonzero spatial momentum along one spatial coordinate of the AdS boundary to obtain the superconducting coherence length $\xi$ via AdS/CFT correspondence. Following [7], we take the probe limit where the fluctuation do not backreact on the original AdS₄-Schwarzschild geometry. Under the probe limit we also investigate static fluctuation of the Abelian gauge field forming a homogeneous magnetic field as a first step to derive the magnetic penetration length $\lambda$ .

The plan of our paper is as follows: In section II the charged scalar field solution obtained in the holographic model [7] is reconstructed by perturbation technique. In section III we derive $\xi$ by analyzing the equations for the perturbation. In section IV we observe that the diamagnetic current can be induced by the small homogeneous magnetic field. Section V is devoted in conclusions and discussion.

II. A MODEL OF SUPERCONDUCTOR IN ADS/CFT

In this section we reconstruct the charged scalar field solution numerically obtained in [7] by using the regular perturbation theory technique. The background spacetime is AdS₄-Schwarzschild black hole with the metric

$$d{s}^2=-f(r)d{t}^2+\frac{d{r}^2}{f(r)}+\frac{r^2}{L^2}(d{x}^2+d{y}^2),f(r)=\frac{r^2}{L^2}(1-\frac{r_0^3}{r^3}),(\text{II.1})ds^2\; =\; -f(r)\; dt^2\; +\; \backslash frac\{dr^2\}\{f(r)\}\; +\; \backslash frac\{r^2\}\{L^2\}\; (dx^2\; +\; dy^2)\; ,\; \backslash quad\; f(r)\; =\; \backslash frac\{r^2\}\{L^2\}\; \backslash left(\; 1\; -\; \backslash frac\{r\_0^3\}\{r^3\}\; \backslash right)\; ,\; \backslash quad\; (\backslash text\{II.1\})$$

where $r_0$ is the horizon radius and $L$ is the AdS radius. The Hawking temperature $T$ is given by

$$T=\frac{3}{4\pi }\frac{r_0}{L^2}\mathrm{.}(\text{II.2})T\; =\; \backslash frac\{3\}\{4\backslash pi\}\; \backslash frac\{r\_0\}\{L^2\}\; .\; \backslash quad\; (\backslash text\{II.2\})$$

It is convenient to introducing a new coordinate $u := r_0/r$ , and the metric (II.1) is written as

$$d{s}^2=\frac{L^2{\alpha }^2(T)}{u^2}(-h(u)d{t}^2+d{x}^2+d{y}^2)+\frac{L^{2}d{u}^2}{u^{2}h(u)},h(u)=1-u^3,(\text{II.3})ds^2\; =\; \backslash frac\{L^2\; \backslash alpha^2(T)\}\{u^2\}\; (-h(u)\; dt^2\; +\; dx^2\; +\; dy^2)\; +\; \backslash frac\{L^2\; du^2\}\{u^2\; h(u)\}\; ,\; \backslash quad\; h(u)\; =\; 1\; -\; u^3\; ,\; \backslash quad\; (\backslash text\{II.3\})$$

where $\alpha(T) := 4\pi T/3 = r_0/L^2$ .

We consider the matter fields on the background spacetime which consist of a Maxwell field and a charged complex scalar field with charge $e$ and mass $m$ . The Lagrangian density is given by

$$\mathcal{L}=\frac{L^2}{2{\kappa }_4^2{e}^2}\sqrt{-g}(-\frac{1}{4}{F}^{\mu \nu }{F}_{\mu \nu }-\mathrm{\mid }D\mathrm{\Psi }{\mathrm{\mid }}^2-m^2\mathrm{\mid }\mathrm{\Psi }{\mathrm{\mid }}^2),D_{\mu }:={\mathrm{\partial }}_{\mu }-i{A}_{\mu }\mathrm{.}(\text{II.4})\backslash mathcal\{L\}\; =\; \backslash frac\{L^2\}\{2\backslash kappa\_4^2\; e^2\}\; \backslash sqrt\{-g\}\; \backslash left(\; -\backslash frac\{1\}\{4\}\; F^\{\backslash mu\backslash nu\}\; F\_\{\backslash mu\backslash nu\}\; -\; |D\backslash Psi|^2\; -\; m^2\; |\backslash Psi|^2\; \backslash right)\; ,\; \backslash quad\; D\_\backslash mu\; :=\; \backslash partial\_\backslash mu\; -\; i\; A\_\backslash mu\; .\; \backslash quad\; (\backslash text\{II.4\})$$

Following [7], we shall confine our attention to the case $L^2 m^2 = -2$ and consider the probe limit in which the gauge field and scalar field do not backreact on the original metric (II.1). This limit is realized by taking the limit $e \rightarrow \infty$ , keeping $A_\mu$ and $\Psi$ fixed. So, the equations of motion for $A_\mu$ and $\Psi$ are decoupled from Einstein's equations and we obtain the following equations

$$0=D^2\mathrm{\Psi }-m^2\mathrm{\Psi },0={\mathrm{\nabla }}_{\nu }{F}_{\mu }{}^{\nu }-i[(D_{\mu }\mathrm{\Psi }{)}^{†}\mathrm{\Psi }-{\mathrm{\Psi }}^{†}(D_{\mu }\mathrm{\Psi })]\mathrm{.}(\text{II.5})0\; =\; D^2\; \backslash Psi\; -\; m^2\; \backslash Psi\; ,\; \backslash quad\; 0\; =\; \backslash nabla\_\backslash nu\; F\_\backslash mu\{\}^\backslash nu\; -\; i\; \backslash left[\; (D\_\backslash mu\; \backslash Psi)^\backslash dagger\; \backslash Psi\; -\; \backslash Psi^\backslash dagger\; (D\_\backslash mu\; \backslash Psi)\; \backslash right]\; .\; \backslash quad\; (\backslash text\{II.5\})$$

Under the ansatz

$$\mathrm{\Psi }=\mathrm{\Psi }(u),A_{\mu }=\mathrm{\Phi }(u)(dt{)}_{\mu },(\text{II.6})\backslash Psi\; =\; \backslash Psi(u)\; ,\; \backslash quad\; A\_\backslash mu\; =\; \backslash Phi(u)\; (dt)\_\backslash mu\; ,\; \backslash quad\; (\backslash text\{II.6\})$$the equations of motion (II.5) are reduced to

$$0=(u^2\frac{d}{du}\frac{h(u)}{u^2}\frac{d}{du}-\frac{L^2{m}^2}{u^2})\tilde{\mathrm{\Psi }}+\frac{{\tilde{\mathrm{\Phi }}}^2}{h(u)}\tilde{\mathrm{\Psi }},(\text{II.7})0\; =\; \backslash left(\; u^2\; \backslash frac\{d\}\{du\}\; \backslash frac\{h(u)\}\{u^2\}\; \backslash frac\{d\}\{du\}\; -\; \backslash frac\{L^2\; m^2\}\{u^2\}\; \backslash right)\; \backslash tilde\{\backslash Psi\}\; +\; \backslash frac\{\backslash tilde\{\backslash Phi\}^2\}\{h(u)\}\; \backslash tilde\{\backslash Psi\}\; ,\; \backslash quad\; (\backslash text\{II.7\})$$

$$0=h(u)\frac{d^2\tilde{\mathrm{\Phi }}}{d{u}^2}-\frac{2\mathrm{\mid }\tilde{\mathrm{\Psi }}{\mathrm{\mid }}^2}{u^2}\tilde{\mathrm{\Phi }}\mathrm{.}(\text{II.8})0\; =\; h(u)\; \backslash frac\{d^2\; \backslash tilde\{\backslash Phi\}\}\{du^2\}\; -\; \backslash frac\{2\; |\backslash tilde\{\backslash Psi\}|^2\}\{u^2\}\; \backslash tilde\{\backslash Phi\}\; .\; \backslash quad\; (\backslash text\{II.8\})$$

where new variables $\tilde{\Phi} := \Phi/\alpha(T)$ and $\tilde{\Psi} := L \Psi$ are dimensionless quantities. Without loss of generality, we can set $\tilde{\Psi}$ to be real.

The trivial solution is easily found as

$$\tilde{\mathrm{\Psi }}=0,\tilde{\mathrm{\Phi }}=\mu \mathrm{/}\alpha (T)-qu=q(1-u),(\text{II.9})\backslash tilde\{\backslash Psi\}\; =\; 0\; ,\; \backslash quad\; \backslash tilde\{\backslash Phi\}\; =\; \backslash mu/\backslash alpha(T)\; -\; qu\; =\; q(1-u)\; ,\; \backslash quad\; (\backslash text\{II.9\})$$

where $\mu$ is interpreted as the external source in the dual $(2+1)$ -dimensional gauge theory and it is determined by the condition $A_\mu dx^\mu$ to be well-defined at the horizon, i.e., $\Phi(u=1) = 0$ [13]. Dimensionless constant $q$ is related to the dual charge density coupled to $\mu$ as,

$$⟨J^t(x)⟩={\frac{\delta S_{\text{on-shell boundary}}}{\delta A_t(x)}\mid }_{u=0}=\frac{L^2}{2{\kappa }_4^2{e}^2}{\left(\frac{4\pi T}{3}\right)}^{2}q\mathrm{.}(\text{II.10})\backslash langle\; J^t(x)\; \backslash rangle\; =\; \backslash left.\; \backslash frac\{\backslash delta\; S\_\{\backslash text\{on-shell\; boundary\}\}\}\{\backslash delta\; A\_t(x)\}\; \backslash right|\_\{u=0\}\; =\; \backslash frac\{L^2\}\{2\backslash kappa\_4^2\; e^2\}\; \backslash left(\; \backslash frac\{4\backslash pi\; T\}\{3\}\; \backslash right)^2\; q\; .\; \backslash quad\; (\backslash text\{II.10\})$$

The non-trivial solution asymptotically behaves near the AdS boundary as

$$\tilde{\mathrm{\Psi }}={\tilde{\mathrm{\Psi }}}^{(-)}{u}^{{\mathrm{\Delta }}_{-}}+{\tilde{\mathrm{\Psi }}}^{(+)}{u}^{{\mathrm{\Delta }}_{+}}+\dots ,\tilde{\mathrm{\Phi }}=\mu \mathrm{/}\alpha (T)-qu+\dots ,(\text{II.11})\backslash tilde\{\backslash Psi\}\; =\; \backslash tilde\{\backslash Psi\}^\{(-)\}\; u^\{\backslash Delta\_-\}\; +\; \backslash tilde\{\backslash Psi\}^\{(+)\}\; u^\{\backslash Delta\_+\}\; +\; \backslash dots\; ,\; \backslash quad\; \backslash tilde\{\backslash Phi\}\; =\; \backslash mu/\backslash alpha(T)\; -\; qu\; +\; \backslash dots\; ,\; \backslash quad\; (\backslash text\{II.11\})$$

where $\Delta_\pm := (3 \pm \sqrt{9 + 4L^2 m^2})/2$ . For $L^2 m^2 = -2$ case, we obtain $\Delta_- = 1$ and $\Delta_+ = 2$ , where both falloffs of $\tilde{\Psi}$ are normalizable.

Each coefficient $\tilde{\Psi}^{(\pm)}$ is proportional to the condensate thermal expectation value of the scalar operator $\langle \mathcal{O}\pm \rangle$ of dimension $\Delta_\pm$ . To obtain a stable solution, we must impose either $\tilde{\Psi}^{(-)} = 0$ or $\tilde{\Psi}^{(+)} = 0$ . So, the asymptotic boundary condition of the scalar field $\tilde{\Psi}$ dual to the scalar operator $\langle \mathcal{O}- \rangle$ ( $\langle \mathcal{O}_+ \rangle$ ) is $\tilde{\Psi}^{(+)} = 0$ ( $\tilde{\Psi}^{(-)} = 0$ ), and $\tilde{\Psi}$ has the asymptotic behavior near the AdS boundary as

$$\tilde{\mathrm{\Psi }}={\tilde{\mathrm{\Psi }}}^{(I)}{u}^{{\mathrm{\Delta }}_I}[1+O(u^2)],(\text{II.12})\backslash tilde\{\backslash Psi\}\; =\; \backslash tilde\{\backslash Psi\}^\{(I)\}\; u^\{\backslash Delta\_I\}\; [1\; +\; O(u^2)]\; ,\; \backslash quad\; (\backslash text\{II.12\})$$

where $I = \pm$ for $\langle \mathcal{O}_\pm \rangle$ .

Since the trivial solution $\tilde{\Phi}$ in Eqs.(II.7) and (II.8) is parametrized by the dimensionless constant $q \propto \langle J^t \rangle/T^2$ only, the non-trivial solution $\tilde{\Psi}$ emerges above a critical value $q_c$ under the boundary condition. According to the numerical calculation [7], the thermal expectation value $\langle \mathcal{O}_I \rangle$ behaves as

$${\tilde{\mathrm{\Psi }}}^{(I)}\propto ⟨{\mathcal{O}}_I⟩\sim (1-T\mathrm{/}{T}_c{)}^{1\mathrm{/}2}(\text{II.13})\backslash tilde\{\backslash Psi\}^\{(I)\}\; \backslash propto\; \backslash langle\; \backslash mathcal\{O\}\_I\; \backslash rangle\; \backslash sim\; (1\; -\; T/T\_c)^\{1/2\}\; \backslash quad\; (\backslash text\{II.13\})$$

for a given $\mu$ (or $\langle J^t \rangle$ ), or equivalently

$$⟨{\mathcal{O}}_I⟩\sim (q\mathrm{/}{q}_c-1{)}^{1\mathrm{/}2}(\text{II.14})\backslash langle\; \backslash mathcal\{O\}\_I\; \backslash rangle\; \backslash sim\; (q/q\_c\; -\; 1)^\{1/2\}\; \backslash quad\; (\backslash text\{II.14\})$$

near the critical temperature $T_c$ . In the limit $T \rightarrow T_c$ , $\epsilon := q/q_c - 1$ ( $> 0$ ) is a small parameter, and the non-trivial solution to Eqs.(II.7) and (II.8) can be obtained as a series in $\epsilon$ . From the continuity, the solution at the critical temperature should be

$${\tilde{\mathrm{\Psi }}}_c=0,{\tilde{\mathrm{\Phi }}}_c=q_c(1-u)\mathrm{.}(\text{II.15})\backslash tilde\{\backslash Psi\}\_c\; =\; 0\; ,\; \backslash quad\; \backslash tilde\{\backslash Phi\}\_c\; =\; q\_c\; (1\; -\; u)\; .\; \backslash quad\; (\backslash text\{II.15\})$$So, we can expand $\tilde{\Psi}$ and $\tilde{\Phi}$ as

$$\tilde{\mathrm{\Psi }}(u)={ϵ}^{1\mathrm{/}2}{\tilde{\mathrm{\Psi }}}_1(u)+{ϵ}^{3\mathrm{/}2}{\tilde{\mathrm{\Psi }}}_2(u)+\dots ,\tilde{\mathrm{\Phi }}(u)={\tilde{\mathrm{\Phi }}}_c(u)+ϵ{\tilde{\mathrm{\Phi }}}_1(u)+\dots \mathrm{.}(\text{II.16})\backslash tilde\{\backslash Psi\}(u)\; =\; \backslash epsilon^\{1/2\}\; \backslash tilde\{\backslash Psi\}\_1(u)\; +\; \backslash epsilon^\{3/2\}\; \backslash tilde\{\backslash Psi\}\_2(u)\; +\; \backslash dots,\; \backslash quad\; \backslash tilde\{\backslash Phi\}(u)\; =\; \backslash tilde\{\backslash Phi\}\_c(u)\; +\; \backslash epsilon\; \backslash tilde\{\backslash Phi\}\_1(u)\; +\; \backslash dots.\; \backslash quad\; (\backslash text\{II.16\})$$

Here, we should note that the difference of $\epsilon$ -behavior between $\tilde{\Psi}$ and $\tilde{\Phi}$ comes from Eqs.(II.7) and (II.8).

Substituting Eq.(II.16) into Eqs.(II.7) and (II.8) we obtain equations for $\tilde{\Psi}_1$ and $\tilde{\Phi}_1$ :

$$0={\mathcal{L}}_{\psi }{\tilde{\mathrm{\Psi }}}_1,0=\frac{d^2{\tilde{\mathrm{\Phi }}}_1}{d{u}^2}-\frac{2\mathrm{\mid }{\tilde{\mathrm{\Psi }}}_1{\mathrm{\mid }}^2{\tilde{\mathrm{\Phi }}}_c}{u^{2}h(u)},(\text{II.17})0\; =\; \backslash mathcal\{L\}\_\backslash psi\; \backslash tilde\{\backslash Psi\}\_1,\; \backslash quad\; 0\; =\; \backslash frac\{d^2\; \backslash tilde\{\backslash Phi\}\_1\}\{du^2\}\; -\; \backslash frac\{2|\backslash tilde\{\backslash Psi\}\_1|^2\; \backslash tilde\{\backslash Phi\}\_c\}\{u^2\; h(u)\},\; \backslash quad\; (\backslash text\{II.17\})$$

where the differential operators $\mathcal{L}_\psi$ is defined by

$${\mathcal{L}}_{\psi }:=-(u^2\frac{d}{du}\frac{h(u)}{u^2}\frac{d}{du}-\frac{L^2{m}^2}{u^2}+\frac{{\tilde{\mathrm{\Phi }}}_c^2}{h(u)})\mathrm{.}(\text{II.18})\backslash mathcal\{L\}\_\backslash psi\; :=\; -\; \backslash left(\; u^2\; \backslash frac\{d\}\{du\}\; \backslash frac\{h(u)\}\{u^2\}\; \backslash frac\{d\}\{du\}\; -\; \backslash frac\{L^2\; m^2\}\{u^2\}\; +\; \backslash frac\{\backslash tilde\{\backslash Phi\}\_c^2\}\{h(u)\}\; \backslash right).\; \backslash quad\; (\backslash text\{II.18\})$$

By imposing the regularity condition at the horizon

$${\frac{1}{{\tilde{\mathrm{\Psi }}}_1}\frac{d{\tilde{\mathrm{\Psi }}}_1}{du}\mid }_{u=1}=-\frac{L^2{m}^2}{3}=\frac{2}{3},(\text{II.19})\backslash left.\; \backslash frac\{1\}\{\backslash tilde\{\backslash Psi\}\_1\}\; \backslash frac\{d\backslash tilde\{\backslash Psi\}\_1\}\{du\}\; \backslash right|\_\{u=1\}\; =\; -\; \backslash frac\{L^2\; m^2\}\{3\}\; =\; \backslash frac\{2\}\{3\},\; \backslash quad\; (\backslash text\{II.19\})$$

we find the constant $q_c$ for which there is a unique regular solution satisfying the asymptotic boundary condition mentioned above. In $I = +$ case, for example, $q_c \sim 4.07$ , which is consistent with the numerical result in [7].

III. SUPERCONDUCTING COHERENCE LENGTH

In this section, we will determine the superconducting coherence length $\xi$ by investigating fluctuations around the background field (II.6) ( $\tilde{\Psi}(u), \tilde{\Phi}(u)$ ). It is enough to consider static perturbations for the purpose, so let us confine our attention to the fluctuations with only spatial momentum along $x$ -direction:

where both functions $\psi$ and $\hat{\psi}$ are real and metric fluctuations of the order of the gauge and scalar fluctuations can be consistently set to zero under the probe limit. From the perturbed equations derived from Eq.(II.5), we find the following three linearized equations for $\phi$ , $\psi$ , and $A_y$ decoupled from the other variables:

$${\tilde{k}}^2\psi =(u^2\frac{d}{du}\frac{h(u)}{u^2}\frac{d}{du}-\frac{L^2{m}^2}{u^2}+\frac{{\tilde{\mathrm{\Phi }}}^2}{h(u)})\psi +\frac{2\tilde{\mathrm{\Phi }}\tilde{\mathrm{\Psi }}}{h(u)}\varphi ,(\text{III.2})\backslash tilde\{k\}^2\; \backslash psi\; =\; \backslash left(\; u^2\; \backslash frac\{d\}\{du\}\; \backslash frac\{h(u)\}\{u^2\}\; \backslash frac\{d\}\{du\}\; -\; \backslash frac\{L^2\; m^2\}\{u^2\}\; +\; \backslash frac\{\backslash tilde\{\backslash Phi\}^2\}\{h(u)\}\; \backslash right)\; \backslash psi\; +\; \backslash frac\{2\; \backslash tilde\{\backslash Phi\}\; \backslash tilde\{\backslash Psi\}\}\{h(u)\}\; \backslash phi,\; \backslash quad\; (\backslash text\{III.2\})$$

$${\tilde{k}}^2\varphi =(h(u)\frac{d^2}{d{u}^2}-\frac{2{\tilde{\mathrm{\Psi }}}^2}{u^2})\varphi -\frac{4\tilde{\mathrm{\Phi }}\tilde{\mathrm{\Psi }}}{u^2}\psi ,(\text{III.3})\backslash tilde\{k\}^2\; \backslash phi\; =\; \backslash left(\; h(u)\; \backslash frac\{d^2\}\{du^2\}\; -\; \backslash frac\{2\; \backslash tilde\{\backslash Psi\}^2\}\{u^2\}\; \backslash right)\; \backslash phi\; -\; \backslash frac\{4\; \backslash tilde\{\backslash Phi\}\; \backslash tilde\{\backslash Psi\}\}\{u^2\}\; \backslash psi,\; \backslash quad\; (\backslash text\{III.3\})$$

$${\tilde{k}}^2{A}_y=(\frac{d}{du}h(u)\frac{d}{du}-\frac{2{\tilde{\mathrm{\Psi }}}^2}{u^2})A_y,(\text{III.4})\backslash tilde\{k\}^2\; A\_y\; =\; \backslash left(\; \backslash frac\{d\}\{du\}\; h(u)\; \backslash frac\{d\}\{du\}\; -\; \backslash frac\{2\; \backslash tilde\{\backslash Psi\}^2\}\{u^2\}\; \backslash right)\; A\_y,\; \backslash quad\; (\backslash text\{III.4\})$$where $\tilde{k} := k/\alpha(T)$ .

The superconducting coherence length $\xi$ is nothing but the correlation length of the order parameter, and $\xi$ appears as the pole of the static correlation function of the order parameter in the Fourier space

$$⟨\tilde{\mathcal{O}}(\vec{k})\tilde{\mathcal{O}}(-\vec{k})⟩\sim \frac{1}{\mathrm{\mid }\vec{k}{\mathrm{\mid }}^2+1\mathrm{/}{\xi }^2}\mathrm{.}(\text{III.5})\backslash langle\; \backslash tilde\{\backslash mathcal\{O\}\}(\backslash vec\{k\})\; \backslash tilde\{\backslash mathcal\{O\}\}(-\backslash vec\{k\})\; \backslash rangle\; \backslash sim\; \backslash frac\{1\}\{|\backslash vec\{k\}|^2\; +\; 1/\backslash xi^2\}.\; \backslash quad\; (\backslash text\{III.5\})$$

Since the complex scalar field $\Psi$ plays a role of the order parameter in our model and the background $\tilde{\Psi}$ is real, the real part of the order parameter fluctuation $\psi$ gives the superconducting coherence length.

The pole of the static correlation function of a dual field operator is obtained by solving the eigenvalue problem for the static perturbation with wave number $k$ of the corresponding bulk field as $1/\xi^2 = -k_^2$ , where $k_$ is a wave number permitted as eigenvalues. In the present case, our task is to evaluate eigenvalues $\tilde{k}^2$ for Eqs.(III.2) and (III.3) under the appropriate boundary conditions.

Since it is difficult to solve the eigenvalue equations (III.2) and (III.3) analytically, we solve the equations as a series in $\epsilon$ near the critical temperature $T_c$ . According to Ginzburg-Landau theory, $\xi$ diverges to infinity as $T \rightarrow T_c$ . This implies that there exists zero eigenvalue $k_* = 0$ solution at the critical temperature $T_c$ . Hereafter, we shall confine our attention to the eigensystem with $\lim_{\epsilon \rightarrow 0} k_* = 0$ .

From the behavior (II.16), Eqs.(III.2) and (III.3) are expanded in $\epsilon$ as

$$-{\tilde{k}}^2\psi =({\mathcal{L}}_{\psi }-\frac{2ϵ{\tilde{\mathrm{\Phi }}}_c{\tilde{\mathrm{\Phi }}}_1}{h})\psi -\frac{2{ϵ}^{1\mathrm{/}2}{\tilde{\mathrm{\Phi }}}_c{\tilde{\mathrm{\Psi }}}_1}{h}\varphi ,(\text{III.6})-\backslash tilde\{k\}^2\; \backslash psi\; =\; \backslash left(\; \backslash mathcal\{L\}\_\backslash psi\; -\; \backslash frac\{2\backslash epsilon\; \backslash tilde\{\backslash Phi\}\_c\; \backslash tilde\{\backslash Phi\}\_1\}\{h\}\; \backslash right)\; \backslash psi\; -\; \backslash frac\{2\backslash epsilon^\{1/2\}\; \backslash tilde\{\backslash Phi\}\_c\; \backslash tilde\{\backslash Psi\}\_1\}\{h\}\; \backslash phi,\; \backslash quad\; (\backslash text\{III.6\})$$

$$-{\tilde{k}}^2\varphi =(-h\frac{d^2}{d{u}^2}+\frac{2ϵ{\tilde{\mathrm{\Psi }}}_1^2}{u^2})\varphi +\frac{4{ϵ}^{1\mathrm{/}2}{\tilde{\mathrm{\Phi }}}_c{\tilde{\mathrm{\Psi }}}_1}{u^2}\psi \mathrm{.}(\text{III.7})-\backslash tilde\{k\}^2\; \backslash phi\; =\; \backslash left(\; -h\; \backslash frac\{d^2\}\{du^2\}\; +\; \backslash frac\{2\backslash epsilon\; \backslash tilde\{\backslash Psi\}\_1^2\}\{u^2\}\; \backslash right)\; \backslash phi\; +\; \backslash frac\{4\backslash epsilon^\{1/2\}\; \backslash tilde\{\backslash Phi\}\_c\; \backslash tilde\{\backslash Psi\}\_1\}\{u^2\}\; \backslash psi.\; \backslash quad\; (\backslash text\{III.7\})$$

The boundary conditions for Eqs.(III.6) and (III.7) are as follows: at the horizon,

$$\psi (1)=\text{regular},\varphi (1)=0,(\text{III.8})\backslash psi(1)\; =\; \backslash text\{regular\},\; \backslash quad\; \backslash phi(1)\; =\; 0,\; \backslash quad\; (\backslash text\{III.8\})$$

and near the AdS boundary

$$\psi (u)=(\text{const.})\times u^{{\mathrm{\Delta }}_I}[1+O(u^2)],\varphi (u)=(\text{const.})\times u+O(u^2)\mathrm{.}(\text{III.9})\backslash psi(u)\; =\; (\backslash text\{const.\})\; \backslash times\; u^\{\backslash Delta\_I\}\; [1\; +\; O(u^2)],\; \backslash quad\; \backslash phi(u)\; =\; (\backslash text\{const.\})\; \backslash times\; u\; +\; O(u^2).\; \backslash quad\; (\backslash text\{III.9\})$$

In the eigenvalue equations (III.6) and (III.7) with the boundary conditions (III.8) and (III.9), the infinitesimal expansion parameter is $\epsilon^{1/2}$ , so one may expect that we have eigenvalue $\tilde{k}*^2 = O(\epsilon^{1/2})$ . However, we have $\tilde{k}*^2 = O(\epsilon)$ as seen later.

It is easy to show that the zeroth order solution, $\psi_0$ and $\phi_0$ , for the eigenvalue equation (III.6) and (III.7) satisfying the boundary conditions (III.8) and (III.9) is given by

$${\psi }_0={\tilde{\mathrm{\Psi }}}_1,{\varphi }_0=0,(\text{III.10})\backslash psi\_0\; =\; \backslash tilde\{\backslash Psi\}\_1,\; \backslash quad\; \backslash phi\_0\; =\; 0,\; \backslash quad\; (\backslash text\{III.10\})$$

where we use $\mathcal{L}_\psi \tilde{\Psi}_1 = 0$ . This means that $\phi = O(\epsilon^{1/2})$ at most from Eq.(III.7). So we put $\phi =: \epsilon^{1/2} \varphi$ , and Eqs.(III.6) and (III.7) are rewritten by

$$-{\tilde{k}}^2\psi ={\mathcal{L}}_{\psi }\psi -ϵ\frac{2{\tilde{\mathrm{\Phi }}}_c}{h}({\tilde{\mathrm{\Phi }}}_1\psi +{\tilde{\mathrm{\Psi }}}_1\phi ),(\text{III.11})-\backslash tilde\{k\}^2\; \backslash psi\; =\; \backslash mathcal\{L\}\_\backslash psi\; \backslash psi\; -\; \backslash epsilon\; \backslash frac\{2\backslash tilde\{\backslash Phi\}\_c\}\{h\}\; (\backslash tilde\{\backslash Phi\}\_1\; \backslash psi\; +\; \backslash tilde\{\backslash Psi\}\_1\; \backslash varphi),\; \backslash quad\; (\backslash text\{III.11\})$$

$$-{\tilde{k}}^2\phi =(-h\frac{d^2}{d{u}^2}\phi +\frac{4{\tilde{\mathrm{\Phi }}}_c{\tilde{\mathrm{\Psi }}}_1}{u^2}\psi )+ϵ\frac{2{\tilde{\mathrm{\Psi }}}_1^2}{u^2}\phi \mathrm{.}(\text{III.12})-\backslash tilde\{k\}^2\; \backslash varphi\; =\; \backslash left(\; -h\; \backslash frac\{d^2\}\{du^2\}\; \backslash varphi\; +\; \backslash frac\{4\backslash tilde\{\backslash Phi\}\_c\; \backslash tilde\{\backslash Psi\}\_1\}\{u^2\}\; \backslash psi\; \backslash right)\; +\; \backslash epsilon\; \backslash frac\{2\backslash tilde\{\backslash Psi\}\_1^2\}\{u^2\}\; \backslash varphi.\; \backslash quad\; (\backslash text\{III.12\})$$Thus, the expansion parameter of R.H.S. in Eqs.(III.11) and (III.12) is $\epsilon$ , implying $\tilde{k}*^2 = O(\epsilon)$ . The temperature dependence of the coherence length, $\xi \propto (-\tilde{k}*^2)^{-1/2} \propto (1 - T/T_c)^{-1/2}$ , is an expected behavior in Ginzburg-Landau theory.

Now let us evaluate the superconducting coherence length $\xi$ at the leading order in $\epsilon$ . We expand $\psi$ , $\varphi$ , and $\tilde{k}_*^2$ as

$$\psi ={\tilde{\mathrm{\Psi }}}_1+ϵ{\psi }_1+O({ϵ}^2),\phi ={\phi }_0+O(ϵ),{\tilde{k}}_{\ast }^2=ϵ({\tilde{k}}^2{)}_1+O({ϵ}^2)\mathrm{.}(\text{III.13})\backslash psi\; =\; \backslash tilde\{\backslash Psi\}\_1\; +\; \backslash epsilon\; \backslash psi\_1\; +\; O(\backslash epsilon^2)\; ,\; \backslash quad\; \backslash varphi\; =\; \backslash varphi\_0\; +\; O(\backslash epsilon)\; ,\; \backslash quad\; \backslash tilde\{k\}\_*^2\; =\; \backslash epsilon\; (\backslash tilde\{k\}^2)\_1\; +\; O(\backslash epsilon^2)\; .\; \backslash quad\; (\backslash text\{III.13\})$$

Then, Eq.(III.11) is rewritten up to $O(\epsilon)$ as

$$-({\tilde{k}}^2{)}_1{\tilde{\mathrm{\Psi }}}_1={\mathcal{L}}_{\psi }{\psi }_1-\frac{2{\tilde{\mathrm{\Phi }}}_c{\tilde{\mathrm{\Psi }}}_1}{h}({\tilde{\mathrm{\Phi }}}_1+{\phi }_0),(\text{III.14})-(\backslash tilde\{k\}^2)\_1\; \backslash tilde\{\backslash Psi\}\_1\; =\; \backslash mathcal\{L\}\_\backslash psi\; \backslash psi\_1\; -\; \backslash frac\{2\; \backslash tilde\{\backslash Phi\}\_c\; \backslash tilde\{\backslash Psi\}\_1\}\{h\}\; (\backslash tilde\{\backslash Phi\}\_1\; +\; \backslash varphi\_0)\; ,\; \backslash quad\; (\backslash text\{III.14\})$$

and the equation of motion for $\varphi_0$ is given by

$$\frac{d^2{\phi }_0}{d{u}^2}=\frac{4{\tilde{\mathrm{\Phi }}}_c{\tilde{\mathrm{\Psi }}}_1^2}{u^{2}h}=2\frac{d^2{\tilde{\mathrm{\Phi }}}_1}{d{u}^2},(\text{III.15})\backslash frac\{d^2\; \backslash varphi\_0\}\{du^2\}\; =\; \backslash frac\{4\; \backslash tilde\{\backslash Phi\}\_c\; \backslash tilde\{\backslash Psi\}\_1^2\}\{u^2\; h\}\; =\; 2\; \backslash frac\{d^2\; \backslash tilde\{\backslash Phi\}\_1\}\{du^2\}\; ,\; \backslash quad\; (\backslash text\{III.15\})$$

where we use Eq.(II.17). The solution of Eq.(III.15) with the boundary conditions (III.8) and (III.9) is given by

$${\phi }_0(u)=2[{\tilde{\mathrm{\Phi }}}_1(u)-{\tilde{\mathrm{\Phi }}}_1(0)(1-u)]\in \mathbb{R},(\text{III.16})\backslash varphi\_0(u)\; =\; 2\; \backslash left[\; \backslash tilde\{\backslash Phi\}\_1(u)\; -\; \backslash tilde\{\backslash Phi\}\_1(0)\; (1\; -\; u)\; \backslash right]\; \backslash in\; \backslash mathbb\{R\}\; ,\; \backslash quad\; (\backslash text\{III.16\})$$

For states $\psi_I$ , $\psi_{II}$ with the boundary conditions (III.8) and (III.9), let us introduce an inner product

$$⟨{\psi }_I\mathrm{\mid }{\psi }_{II}⟩:={\int }_0^1\frac{du}{u^2}{\psi }_I^{\ast }(u){\psi }_{II}(u)\mathrm{.}(\text{III.17})\backslash langle\; \backslash psi\_I\; |\; \backslash psi\_\{II\}\; \backslash rangle\; :=\; \backslash int\_0^1\; \backslash frac\{du\}\{u^2\}\; \backslash psi\_I^*(u)\; \backslash psi\_\{II\}(u)\; .\; \backslash quad\; (\backslash text\{III.17\})$$

It is easily checked that $\mathcal{L}_\psi$ is hermitian for the inner product (III.17).

Making use of $\mathcal{L}_\psi \tilde{\Psi}_1 = 0$ and hermiticity of $\mathcal{L}_\psi$ , the inner product between $\tilde{\Psi}_1$ and Eq.(III.14) gives us

where we used Eq.(III.15) and the boundary conditions (III.8) and (III.9) in the third equality.

We can show that the first term in Eq.(III.18) vanishes as follows: From Eq.(II.7), we have the equation of motion for $\tilde{\Psi}_2$ defined by Eq.(II.16) as

$${\mathcal{L}}_{\psi }{\tilde{\mathrm{\Psi }}}_2=\frac{2{\tilde{\mathrm{\Phi }}}_c{\tilde{\mathrm{\Psi }}}_1}{h}{\tilde{\mathrm{\Phi }}}_1,{\tilde{\mathrm{\Psi }}}_2(1)=\text{regular},{\tilde{\mathrm{\Psi }}}_2(0)=(\text{const.})\times u^{{\mathrm{\Delta }}_I}\mathrm{.}(\text{III.19})\backslash mathcal\{L\}\_\backslash psi\; \backslash tilde\{\backslash Psi\}\_2\; =\; \backslash frac\{2\; \backslash tilde\{\backslash Phi\}\_c\; \backslash tilde\{\backslash Psi\}\_1\}\{h\}\; \backslash tilde\{\backslash Phi\}\_1\; ,\; \backslash quad\; \backslash tilde\{\backslash Psi\}\_2(1)\; =\; \backslash text\{regular\}\; ,\; \backslash quad\; \backslash tilde\{\backslash Psi\}\_2(0)\; =\; (\backslash text\{const.\})\; \backslash times\; u^\{\backslash Delta\_I\}\; .\; \backslash quad\; (\backslash text\{III.19\})$$

So the inner product (III.17) is well-defined for $\tilde{\Psi}_2$ , and Eq.(III.19) gives us

$$0=-2⟨{\mathcal{L}}_{\psi }{\tilde{\mathrm{\Psi }}}_1\mathrm{\mid }{\tilde{\mathrm{\Psi }}}_2⟩=-2⟨{\tilde{\mathrm{\Psi }}}_1\mathrm{\mid }{\mathcal{L}}_{\psi }{\tilde{\mathrm{\Psi }}}_2⟩=-2⟨{\tilde{\mathrm{\Psi }}}_1\mid \frac{2{\tilde{\mathrm{\Phi }}}_c{\tilde{\mathrm{\Psi }}}_1}{h}{\tilde{\mathrm{\Phi }}}_1\mid ⟩,(\text{III.20})0\; =\; -2\; \backslash langle\; \backslash mathcal\{L\}\_\backslash psi\; \backslash tilde\{\backslash Psi\}\_1\; |\; \backslash tilde\{\backslash Psi\}\_2\; \backslash rangle\; =\; -2\; \backslash langle\; \backslash tilde\{\backslash Psi\}\_1\; |\; \backslash mathcal\{L\}\_\backslash psi\; \backslash tilde\{\backslash Psi\}\_2\; \backslash rangle\; =\; -2\; \backslash left\backslash langle\; \backslash tilde\{\backslash Psi\}\_1\; \backslash left|\; \backslash frac\{2\; \backslash tilde\{\backslash Phi\}\_c\; \backslash tilde\{\backslash Psi\}\_1\}\{h\}\; \backslash tilde\{\backslash Phi\}\_1\; \backslash right|\; \backslash right\backslash rangle\; ,\; \backslash quad\; (\backslash text\{III.20\})$$where we use the fact that $\mathcal{L}_\psi$ is hermitian and $\mathcal{L}_\psi \tilde{\Psi}_1 = 0$ .

Therefore, up to $O(\epsilon)$ , the eigenvalue is given by

$$-{\tilde{k}}_{\ast }^2=ϵN\mathrm{/}D+O({ϵ}^2),(\text{III.21})-\backslash tilde\{k\}\_*^2\; =\; \backslash epsilon\; N/D\; +\; O(\backslash epsilon^2)\; ,\; \backslash quad\; (\backslash text\{III.21\})$$

$$N=2{\int }_0^{1}du{({\tilde{\mathrm{\Phi }}}_1^{\mathrm{\prime }}(u)+{\tilde{\mathrm{\Phi }}}_1(0))}^2>0,D:={\int }_0^{1}du\frac{{\tilde{\mathrm{\Psi }}}_1^2(u)}{u^2}>0,(\text{III.22})N\; =\; 2\; \backslash int\_0^1\; du\; \backslash left(\; \backslash tilde\{\backslash Phi\}\text{'}\_1(u)\; +\; \backslash tilde\{\backslash Phi\}\_1(0)\; \backslash right)^2\; >\; 0\; ,\; \backslash quad\; D\; :=\; \backslash int\_0^1\; du\; \backslash frac\{\backslash tilde\{\backslash Psi\}\_1^2(u)\}\{u^2\}\; >\; 0\; ,\; \backslash quad\; (\backslash text\{III.22\})$$

and we finally obtain the superconducting coherence length as

$$\xi =\frac{{ϵ}^{-1\mathrm{/}2}}{\alpha (T_c)}\sqrt{\frac{D}{N}}+O({ϵ}^{1\mathrm{/}2})\propto \frac{1}{T_c}{(1-\frac{T}{T_c})}^{-1\mathrm{/}2}\mathrm{.}(\text{III.23})\backslash xi\; =\; \backslash frac\{\backslash epsilon^\{-1/2\}\}\{\backslash alpha(T\_c)\}\; \backslash sqrt\{\backslash frac\{D\}\{N\}\}\; +\; O(\backslash epsilon^\{1/2\})\; \backslash propto\; \backslash frac\{1\}\{T\_c\}\; \backslash left(\; 1\; -\; \backslash frac\{T\}\{T\_c\}\; \backslash right)^\{-1/2\}\; .\; \backslash quad\; (\backslash text\{III.23\})$$

We note that since $D$ and $N$ are dimensionless quantities, they do not depend on $T_c$ directly, but depend on $q_c$ only.

IV. DIAMAGNETIC CURRENT

In this section, we calculate diamagnetic current induced by a homogeneous external magnetic field perpendicular to the surface of the superconductor. As mentioned before, in the probe limit $\epsilon \rightarrow \infty$ , the magnetic field does not backreact to the background spacetime (II.1). Under the ansatz $\delta A_y(u, x) = b(u) x$ (the bulk magnetic field $F_{xy} = \partial_x \delta A_y = b(u)$ ), the equation of motion for $b(u)$ is decoupled from the other ones for $\psi$ and $\phi$ ¹, and it is equivalent to Eq.(III.4) for $k = 0$ :

$$(\frac{d}{du}h(u)\frac{d}{du}-\frac{2{\tilde{\mathrm{\Psi }}}^2(u)}{u^2})b(u)=0,(\text{IV.1})\backslash left(\; \backslash frac\{d\}\{du\}\; h(u)\; \backslash frac\{d\}\{du\}\; -\; \backslash frac\{2\backslash tilde\{\backslash Psi\}^2(u)\}\{u^2\}\; \backslash right)\; b(u)\; =\; 0\; ,\; \backslash quad\; (\backslash text\{IV.1\})$$

with the regularity boundary condition at the horizon $u = 1$ .

As seen in the previous section, the equation can be solved as a series in $\epsilon$ . Expanding $b$ as

$$b(u)=b_0(u)+ϵb_1(u)+O({ϵ}^2),(\text{IV.2})b(u)\; =\; b\_0(u)\; +\; \backslash epsilon\; b\_1(u)\; +\; O(\backslash epsilon^2)\; ,\; \backslash quad\; (\backslash text\{IV.2\})$$

and using Eq.(II.16), we obtain equations for $b_0$ and $b_1$ as

$$0=\frac{d}{du}h\frac{d}{du}{b}_0(u),(\text{IV.3})0\; =\; \backslash frac\{d\}\{du\}\; h\; \backslash frac\{d\}\{du\}\; b\_0(u)\; ,\; \backslash quad\; (\backslash text\{IV.3\})$$

$$0=\frac{d}{du}h\frac{d}{du}{b}_1(u)-\frac{2{\tilde{\mathrm{\Psi }}}_1^2}{u^2}{b}_0(u)\mathrm{.}(\text{IV.4})0\; =\; \backslash frac\{d\}\{du\}\; h\; \backslash frac\{d\}\{du\}\; b\_1(u)\; -\; \backslash frac\{2\backslash tilde\{\backslash Psi\}\_1^2\}\{u^2\}\; b\_0(u)\; .\; \backslash quad\; (\backslash text\{IV.4\})$$

The solution of Eq.(IV.3) satisfying the regularity condition is

$$b_0(u)=C=(\text{const.})\mathrm{.}(\text{IV.5})b\_0(u)\; =\; C\; =\; (\backslash text\{const.\})\; .\; \backslash quad\; (\backslash text\{IV.5\})$$

So, the regularity solution of Eq.(IV.4) should satisfy

$$\frac{d{b}_1}{du}=-\frac{2C}{h(u)}{\int }_u^{1}d{u}_0\frac{{\tilde{\mathrm{\Psi }}}_1^2(u_0)}{u_0^2},(\text{IV.6})\backslash frac\{db\_1\}\{du\}\; =\; -\backslash frac\{2C\}\{h(u)\}\; \backslash int\_u^1\; du\_0\; \backslash frac\{\backslash tilde\{\backslash Psi\}\_1^2(u\_0)\}\{u\_0^2\}\; ,\; \backslash quad\; (\backslash text\{IV.6\})$$

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¹ At the non-linear regime, the fluctuation of the gauge field is coupled to the one of the scalar field. This effect is important at large $x$ . As far as we are concerned with the neighborhood of the origin $x = 0$ , this effect is negligible. The non-linear effect has been considered in [9]and

$$b_1(u)=D-2C{\int }_0^u\frac{d{u}_1}{h(u_1)}{\int }_{u_1}^{1}d{u}_0\frac{{\tilde{\mathrm{\Psi }}}_1^2(u_0)}{u_0^2}\mathrm{.}(\text{IV.7})b\_1(u)\; =\; D\; -\; 2C\; \backslash int\_0^u\; \backslash frac\{du\_1\}\{h(u\_1)\}\; \backslash int\_\{u\_1\}^1\; du\_0\; \backslash frac\{\backslash tilde\{\backslash Psi\}\_1^2(u\_0)\}\{u\_0^2\}.\; \backslash quad\; (\backslash text\{IV.7\})$$

Thus, we obtain

and

$$\delta A_y(u,x)=\delta A_y^{(0)}(x)(1-2ϵ{\int }_0^u\frac{d{u}_1}{h(u_1)}{\int }_{u_1}^{1}d{u}_0\frac{{\tilde{\mathrm{\Psi }}}_1^2(u_0)}{u_0^2})+O({ϵ}^2),(\text{IV.9})\backslash delta\; A\_y(u,\; x)\; =\; \backslash delta\; A\_y^\{(0)\}(x)\; \backslash left(\; 1\; -\; 2\backslash epsilon\; \backslash int\_0^u\; \backslash frac\{du\_1\}\{h(u\_1)\}\; \backslash int\_\{u\_1\}^1\; du\_0\; \backslash frac\{\backslash tilde\{\backslash Psi\}\_1^2(u\_0)\}\{u\_0^2\}\; \backslash right)\; +\; O(\backslash epsilon^2),\; \backslash quad\; (\backslash text\{IV.9\})$$

where we define $B := \lim_{u \rightarrow 0} b(u)$ and $\delta A_y^{(0)}(x) := \lim_{u \rightarrow 0} \delta A_y(u, x)$ .

From the asymptotic behavior of $\delta A_y(u, x)$ near the AdS boundary, we can read out the dual source $\delta A_y^{(0)}$ and the thermal expectation value of the current $\langle J_y \rangle$ as

$$\delta A_y(u,x)=\delta A_y^{(0)}(x)+\frac{2{\kappa }_4^2{e}^2}{L^2}\frac{3}{4\pi T}⟨J_y(x)⟩u+O(u^2)\mathrm{.}(\text{IV.10})\backslash delta\; A\_y(u,\; x)\; =\; \backslash delta\; A\_y^\{(0)\}(x)\; +\; \backslash frac\{2\backslash kappa\_4^2\; e^2\}\{L^2\}\; \backslash frac\{3\}\{4\backslash pi\; T\}\; \backslash langle\; J\_y(x)\; \backslash rangle\; u\; +\; O(u^2).\; \backslash quad\; (\backslash text\{IV.10\})$$

From Eq.(IV.9), we obtain

where we use $L\Psi = \tilde{\Psi} = \epsilon^{1/2}(\tilde{\Psi}_1 + O(\epsilon))$ . Because $\tilde{\Psi}_1$ (or $\Psi$ at the leading order in $\epsilon$ ) is the solution of the linear equation (II.17), we can express $\Psi(u)$ as

$$\mathrm{\Psi }(u)={\mathrm{\Psi }}^{(I)}F(u),(\text{IV.12})\backslash Psi(u)\; =\; \backslash Psi^\{(I)\}\; F(u),\; \backslash quad\; (\backslash text\{IV.12\})$$

where $F(u)$ is the solution of Eq.(II.17) satisfying $\lim_{u \rightarrow 0} F(u) = u^{\Delta_I}$ and the regularity boundary condition at the horizon. So, the parenthesis in the last equation of Eq.(IV.11) depends on $q_c$ only, not on $\Psi^{(I)}$ and Eq.(IV.11) is simplified as

$$⟨J_y(x)⟩\sim -T_cϵ\delta A_y^{(0)}(x)\propto -\mathrm{\mid }⟨{\mathcal{O}}_I⟩{\mathrm{\mid }}^2\delta A_y^{(0)}(x)\mathrm{.}(\text{IV.13})\backslash langle\; J\_y(x)\; \backslash rangle\; \backslash sim\; -T\_c\; \backslash epsilon\; \backslash delta\; A\_y^\{(0)\}(x)\; \backslash propto\; -|\backslash langle\; \backslash mathcal\{O\}\_I\; \backslash rangle|^2\; \backslash delta\; A\_y^\{(0)\}(x).\; \backslash quad\; (\backslash text\{IV.13\})$$

Thus, the stationary current is induced only when condensation occurs.

Interestingly, Eq.(IV.13) is very similar to the expression expected by Ginzburg-Landau theory. In the theory, when the phase of the order parameter $\psi$ coupled to the U(1) gauge field $\mathbf{A}$ is constant, the current $\mathbf{J}$ is described by the London equation

$$\mathbf{J}=-\frac{e_{\ast }^2}{m_{\ast }}{\psi }^2\mathbf{A}=-e_{\ast }{n}_s\mathbf{A},(\text{IV.14})\backslash mathbf\{J\}\; =\; -\backslash frac\{e\_*^2\}\{m\_*\}\; \backslash psi^2\; \backslash mathbf\{A\}\; =\; -e\_*\; n\_s\; \backslash mathbf\{A\},\; \backslash quad\; (\backslash text\{IV.14\})$$where $e_*$ and $m_*$ are effective charge and mass of the order parameter, and $n_s$ is the superfluid number density.

While $\delta A_y^{(0)}$ in Eq.(IV.11) is an external source, the macroscopic gauge field $\mathbf{A}$ in Eq.(IV.14) is composed of the spatial average of the microscopic field and external field. Since we have no dynamical photon in our holographic superconductor model, the current does not produce its own microscopic magnetic fields². This means that the external gauge field $\delta A_y^{(0)}$ is equal to the macroscopic gauge field $\mathbf{A}$ in the AdS/CFT superconductor. So, comparing Eq.(IV.14) with Eq.(IV.13), we obtain the superfluid number density³ $n_s$ which behaves as

$$n_s\sim ϵT_c\sim T_c-T,(\text{IV.15})n\_s\; \backslash sim\; \backslash epsilon\; T\_c\; \backslash sim\; T\_c\; -\; T,\; \backslash quad\; (\backslash text\{IV.15\})$$

near the critical temperature.

V. CONCLUSIONS AND DISCUSSION

We have investigated linear fluctuations of the scalar field solution in the holographic model of a superconductor in Ref.[7] under the probe limit, where the fluctuations do not backreact on the geometry. By solving analytically the linearized equations with only spatial momentum along one spatial coordinate of the AdS boundary, we find that the superconducting coherence length $\xi$ diverges at the critical temperature $T_c$ as $\xi \sim (1 - T/T_c)^{-1/2}/T_c$ . We also find diamagnetic current induced by an external small homogeneous magnetic field. The current is proportional to the external gauge field and goes to zero as $T_c - T$ at the critical temperature. These results are in agreement with the behaviors predicted by Ginzburg-Landau theory and Eq. (IV.14) is the London equation in the AdS/CFT superconductor.

If we would have dynamical photon, then according to Ginzburg-Landau theory, the magnetic penetration depth $\lambda$ were related to the superfluid density $n_s$ as

$$\lambda \sim 1\mathrm{/}\sqrt{n_s}\mathrm{.}(\text{V.1})\backslash lambda\; \backslash sim\; 1/\backslash sqrt\{n\_s\}.\; \backslash quad\; (\backslash text\{V.1\})$$

In Ginzburg-Landau theory, the coefficient $\kappa = \lambda/\xi$ classifies the superconductors into two types, i.e. $\kappa < 1/\sqrt{2}$ for type I superconductors and $\kappa > 1/\sqrt{2}$ for type II superconductors. Using Eq.(V.1) formally, from Eqs.(III.23) and (IV.15) we obtain

$$\kappa =\frac{\lambda }{\xi }\sim T_c^{1\mathrm{/}2}\mathrm{.}(\text{V.2})\backslash kappa\; =\; \backslash frac\{\backslash lambda\}\{\backslash xi\}\; \backslash sim\; T\_c^\{1/2\}.\; \backslash quad\; (\backslash text\{V.2\})$$

This may suggest that for a sufficiently small critical temperature $T_c$ , the AdS/CFT superconductor behaves as type I, while for a sufficiently large critical temperature $T_c$ , it behaves as type II. This simple classification calls for further investigation.

Note added: After having submitted this article, we learned of a work by S.A. Hartnoll, C.P. Herzog, and G.T. Horowitz, which argued that AdS/CFT superconductor should be type II [14]. We also learned of a work by G.T. Horowitz and M.M. Roberts, which argued the dependence of the AdS/CFT superconductor on the scalar field mass [15]. As easily seen in the derivation of $\xi$ in section III, the mass dependence only appears via the background scalar field solution $\tilde{\Psi}_1$ of the differential equation (II.7). Since $\mathcal{L}\psi$ is still hermitian for the general mass satisfying Breitenlohner-Freedman bound $L^2 m^2 > L^2 m{\text{BF}}^2 = -9/4$ , we can extend our calculation to the general mass case. We wish to thank one of the referees for helpful suggestions about this extension.

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² We thank one of the referees for informing us about this point

³ Our superfluid number density $n_s$ is equal to the one obtained from the electric conductivity $\Im[\sigma(\omega)]$ by Ref. [7].### Acknowledgments

We would like to thank J. Koga, M. Natsuume, and Y. Shibusa for discussions. We are grateful to J. Goryo for comments.

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