1003/1003.5064.md

Holography of Charged Dilaton Black Holes in General Dimensions

Chiang-Mei Chen¹ and Da-Wei Pang²

¹ Department of Physics and Center for Mathematics and Theoretical Physics,
National Central University, Chungli 320, Taiwan

² Center for Quantum Spacetime, Sogang University,
Seoul 121-742, Korea

cmchen@phy.ncu.edu.tw, pangdw@sogang.ac.kr

Abstract

We study several aspects of charged dilaton black holes with planar symmetry in $(d+2)$ -dimensional spacetime, generalizing the four-dimensional results investigated in arXiv:0911.3586 [hep-th]. We revisit the exact solutions with both zero and finite temperature and discuss the thermodynamics of the near-extremal black holes. We calculate the AC conductivity in the zero-temperature background by solving the corresponding Schrödinger equation and find that the AC conductivity behaves like $\omega^\delta$ , where the exponent $\delta$ is determined by the dilaton coupling $\alpha$ and the spacetime dimension parameter $d$ . Moreover, we also study the Gauss-Bonnet corrections to $\eta/s$ in a five-dimensional finite-temperature background.# Contents

1	Introduction	1
2	The solution	3
3	Thermodynamics	5
4	Conductivity in zero-temperature backgrounds	8
4.1	Near horizon analysis . . . . .	10
4.2	Asymptotic analysis . . . . .	10
4.3	Matching . . . . .	11
4.4	Conductivity . . . . .	11
5	Gauss-Bonnet corrections to $\backslash eta/s$ at finite temperature	13
6	Summary and discussion	16
A	Solving the Schrödinger equation	17
	References	18

1 Introduction

The AdS/CFT correspondence [1, 2] has been shown to be a useful tool for studying the dynamics of a strongly coupled field theory, since many physical properties of field theory can be derived by its weakly coupled and tractable dual gravitational description. In recent years numerous applications for the AdS/CFT correspondence, such as in QCD etc., have been explored in detail. In particular, the investigation of certain condensed matter systems, namely the AdS/CMP correspondence, has accelerated quickly in the past years. Some excellent reviews have appeared [3].

In order to study the gravity dual of a condensed matter system at finite temperature, weneed to consider a suitably corresponding black hole for the background spacetimes. One particular class of interesting backgrounds is the charged dilaton black holes [4, 5, 6, 7] of the Einstein-Maxwell-dilaton theory in which the dilaton field is exponentially coupled with the gauge field, $e^{2\alpha\phi}F^2$ . A specific property of this type of black hole is that the Bekenstein-Hawking entropy vanishes at the extremal limit for any value of the non-zero dilaton coupling, $\alpha \neq 0$ , therefore the higher curvature corrections are crucial. In addition, the charged dilaton black holes with a Liouville potential were studied, e.g. in [8], and the results suggest that their AdS generalizations may provide interesting holographic descriptions of certain condensed matter systems.

Recently, the holography of charged dilaton black holes in $\text{AdS}_4$ with planar symmetry was extensively investigated in [9]. It turns out that the near horizon geometry was Lifshitz-like with a dynamical exponent $z$ determined by the dilaton coupling. The global solution was constructed via numerical methods, and the attractor behavior was also discussed. The authors also examined the thermodynamics of near extremal black holes and computed the AC conductivity in a zero-temperature background. For related work on charged dilaton black holes see [10, 11, 12, 13].

In this paper we generalize the work of [9] in four dimensional spacetime to arbitrary $(d + 2)$ -dimensions. For a practical application to a specific system in condensed matter physics, the value of $d$ is given. For example, one should choose $d = 2$ to study the $(2 + 1)$ -dimensional layered systems. However, our physical spacetime may be higher dimensional with tiny extra dimensions, and more spacetime dimensions might be holographically generated when extra adjoint fields are involved. Therefore, it is of interest to consider the generalization to various dimensions and try to find the universal behavior.

By considering a $(d + 2)$ -dimensional Einstein-Maxwell-dilaton action with dilaton coupling of the form $e^{2\alpha\phi}$ , in both zero and finite temperatures, we obtain particular exact scaling solutions which are expected to be the near horizon geometries of the considered black hole backgrounds in $\text{AdS}_{d+2}$ . The zero-temperature solution is still Lifshitz-like and the dynamical exponent is determined by $\alpha$ and the spacetime dimension. The thermodynamics of the near extremal black holes is also studied. Furthermore, we compute the AC conductivity in the zero-temperature background. We can transform the corresponding equation of motion into a Schrödinger form with an effective potential of the form $V(z) = c/z^2$ and then determine the frequency dependence of the AC conductivityas $\text{Re}(\sigma) \sim \omega^\delta$ . Here both constants $c$ and $\delta$ are determined by $\alpha$ and $d$ . Moreover, we compute the Gauss-Bonnet corrections to $\eta/s$ in a five-dimensional finite temperature background, and the result agrees with the well-known AdS counterpart when the dynamical exponent $z \rightarrow 1$ .

The rest of the paper is organized as follows. In Section 2 we present the exact solutions in $(d+2)$ -dimensional spacetime. The thermodynamics of the near extremal black holes is discussed in Section 3, and the AC conductivity is calculated in Section 4. The derivation of the Gauss-Bonnet corrections to $\eta/s$ in a five-dimensional finite temperature background is given in Section 5. A summary and discussion will be given in the final part.

2 The solution

In this section we will exhibit the exact solutions, including both zero and finite temperature cases. Consider the following Einstein-Maxwell-dilaton action in $(d+2)$ -dimensional spacetime with a negative cosmological constant:

$$S=\frac{1}{16\pi G_{d+2}}\int d^{d+2}x\sqrt{-g}(R-2\mathrm{\Lambda }-2{\mathrm{\partial }}_{\mu }\varphi {\mathrm{\partial }}^{\mu }\varphi -e^{2\alpha \varphi }{F}_{\mu \nu }{F}^{\mu \nu }),(2.1)S\; =\; \backslash frac\{1\}\{16\backslash pi\; G\_\{d+2\}\}\; \backslash int\; d^\{d+2\}x\; \backslash sqrt\{-g\}\; (R\; -\; 2\backslash Lambda\; -\; 2\backslash partial\_\backslash mu\backslash phi\backslash partial^\backslash mu\backslash phi\; -\; e^\{2\backslash alpha\backslash phi\}\; F\_\{\backslash mu\backslash nu\}F^\{\backslash mu\backslash nu\}),\; \backslash quad\; (2.1)$$

the corresponding equations of motion can be summarized as follows:

The cosmological constant is related to the AdS radius $L$ by

$$\mathrm{\Lambda }=-\frac{d(d+1)}{2L^2},(2.3)\backslash Lambda\; =\; -\backslash frac\{d(d+1)\}\{2L^2\},\; \backslash quad\; (2.3)$$

and we will set $L = 1$ for simplicity in the following. The dependence on $L$ can be recovered simply by a dimensional analysis.

Let us take the following ansatz for the planar symmetric metric and gauge potential:

$$d{s}^2=-a^2(r)d{t}^2+\frac{d{r}^2}{a^2(r)}+b^2(r)\sum _{i=1}^{d}d{x}_i^2,A=A_t(r)dt,(2.4)ds^2\; =\; -a^2(r)dt^2\; +\; \backslash frac\{dr^2\}\{a^2(r)\}\; +\; b^2(r)\; \backslash sum\_\{i=1\}^d\; dx\_i^2,\; \backslash quad\; A\; =\; A\_t(r)dt,\; \backslash quad\; (2.4)$$then the solution for the gauge field is

$$F_{tr}=q_e{b}^{-d}{e}^{-2\alpha \varphi },(2.5)F\_\{tr\}\; =\; q\_e\; b^\{-d\}\; e^\{-2\backslash alpha\backslash phi\},\; \backslash quad\; (2.5)$$

and the other equations of motion can be reformulated as

$${\varphi }^{\mathrm{\prime }2}+\frac{d}{2}\frac{b^{\mathrm{\prime }\mathrm{\prime }}}{b}=0,(2.6)\backslash phi\text{'}^2\; +\; \backslash frac\{d\}\{2\}\; \backslash frac\{b\text{'}\text{'}\}\{b\}\; =\; 0,\; \backslash quad\; (2.6)$$

$$(a^2{b}^{2d-2}{)}^{\mathrm{\prime }\mathrm{\prime }}-2(d-2)a{b}^{d-2}{b}^{\mathrm{\prime }}(a{b}^{d-1}{)}^{\mathrm{\prime }}+4\mathrm{\Lambda }{b}^{2d-2}=0,(2.7)(a^2\; b^\{2d-2\})\text{'}\text{'}\; -\; 2(d-2)ab^\{d-2\}b\text{'}(ab^\{d-1\})\text{'}\; +\; 4\backslash Lambda\; b^\{2d-2\}\; =\; 0,\; \backslash quad\; (2.7)$$

$$(a^2{b}^d{\varphi }^{\mathrm{\prime }}{)}^{\mathrm{\prime }}+\alpha q_e^2{b}^{-d}{e}^{-2\alpha \varphi }=0,(2.8)(a^2\; b^d\; \backslash phi\text{'})\text{'}\; +\; \backslash alpha\; q\_e^2\; b^\{-d\}\; e^\{-2\backslash alpha\backslash phi\}\; =\; 0,\; \backslash quad\; (2.8)$$

together with the first order constraint coming from the $rr$ -component of the Einstein equation

$$da{a}^{\mathrm{\prime }}{b}^{\mathrm{\prime }}{b}^{\mathrm{\prime }}+\frac{1}{2}d(d-1)a^2{b}^{\mathrm{\prime }2}+\mathrm{\Lambda }{b}^2={\varphi }^{\mathrm{\prime }2}{a}^2{b}^2-q_e^2{b}^{-2d+2}{e}^{-2\alpha \varphi }\mathrm{.}(2.9)daa\text{'}b\text{'}b\text{'}\; +\; \backslash frac\{1\}\{2\}d(d-1)a^2b\text{'}^2\; +\; \backslash Lambda\; b^2\; =\; \backslash phi\text{'}^2\; a^2\; b^2\; -\; q\_e^2\; b^\{-2d+2\}\; e^\{-2\backslash alpha\backslash phi\}.\; \backslash quad\; (2.9)$$

In order to find the near-horizon solution, we define the new variable $w = r - r_H$ where $r_H$ denotes the radius of the horizon, and the scaling solution should be like

$$a(w)=a_0{w}^{\gamma },b(w)=b_0{w}^{\beta },\varphi (w)=-k_0\ln w,(2.10)a(w)\; =\; a\_0\; w^\backslash gamma,\; \backslash quad\; b(w)\; =\; b\_0\; w^\backslash beta,\; \backslash quad\; \backslash phi(w)\; =\; -k\_0\; \backslash ln\; w,\; \backslash quad\; (2.10)$$

where $a_0, b_0$ and $k_0$ are constants. After some algebra we can find the following zero-temperature solution (fixing $b_0 = 1$ by rescaling $x_i$ ):

The corresponding finite-temperature solution can be easily generalized,

$$d{s}^2=-a^2(w)f(w)d{t}^2+\frac{d{w}^2}{a^2(w)f(w)}+b^2(w)\sum _{i=1}^{d}d{x}_i^2,f(w)=1-\frac{w_0^{d\beta +1}}{w^{d\beta +1}},(2.12)ds^2\; =\; -a^2(w)f(w)dt^2\; +\; \backslash frac\{dw^2\}\{a^2(w)f(w)\}\; +\; b^2(w)\; \backslash sum\_\{i=1\}^d\; dx\_i^2,\; \backslash quad\; f(w)\; =\; 1\; -\; \backslash frac\{w\_0^\{d\backslash beta+1\}\}\{w^\{d\backslash beta+1\}\},\; \backslash quad\; (2.12)$$

with the other fields and parameters remaining invariant.

Here are some comments on these solutions:

• • The zero temperature solutions with Lifshitz-like scaling symmetry has already been obtained in [14], and the above solutions reduce to those obtained in [9] when $d = 2$ .- • The near-horizon metric takes a Lifshitz-like form with anisotropic scaling [15] whose dynamical exponent is $z = 1/\beta$ . However, it cannot be treated as the genuine gravity dual of the Lifshitz fixed-point, since the scaling symmetry is broken by the non-trivial dilaton.¹
• • Following the spirit of [9], here we are interested in finding asymptotically $\text{AdS}{d+2}$ solutions, the near horizon geometries of which are either (2.10) for zero temperature or (2.12) for finite temperature (both are exact solutions). From (2.11) we can see that the charge parameter $q_e$ is fixed, while in the asymptotically $\text{AdS}{d+2}$ case $q_e$ is related to the number density in the dual field theory.

3 Thermodynamics

We will discuss the thermodynamics of the finite-temperature solution in this section. First we recall the finite-temperature solution,

$$d{s}^2=-a_0^2{w}^{2}f(w)d{t}^2+\frac{d{w}^2}{a_0^2{w}^{2}f(w)}+w^{2\beta }\sum _{i=1}^{d}d{x}_i^2,f(w)=1-\frac{w_0^{d\beta +1}}{w^{d\beta +1}}\mathrm{.}(3.1)ds^2\; =\; -a\_0^2\; w^2\; f(w)\; dt^2\; +\; \backslash frac\{dw^2\}\{a\_0^2\; w^2\; f(w)\}\; +\; w^\{2\backslash beta\}\; \backslash sum\_\{i=1\}^d\; dx\_i^2,\; \backslash quad\; f(w)\; =\; 1\; -\; \backslash frac\{w\_0^\{d\backslash beta+1\}\}\{w^\{d\backslash beta+1\}\}.\; \backslash quad\; (3.1)$$

As $w \rightarrow \infty$ , this solution reduces to the original scaling solution. Since the scaling solution (2.11) corresponds to the near horizon of an extremal black hole, it can be expected that the finite-temperature solution corresponds to the near-horizon region of a near-extremal black hole.

The temperature of the black hole is given by

$$T=\frac{(\beta d+1)a_0^2}{4\pi }{w}_0,(3.2)T\; =\; \backslash frac\{(\backslash beta\; d\; +\; 1)a\_0^2\}\{4\backslash pi\}\; w\_0,\; \backslash quad\; (3.2)$$

and the entropy density is

$$s=\frac{1}{4}{b}^d(w){\mid }_{w=w_0}=\frac{1}{4}{w}_0^{\beta d}\sim T^{\beta d}\mathrm{.}(3.3)s\; =\; \backslash frac\{1\}\{4\}\; b^d(w)\; \backslash Big|\_\{w=w\_0\}\; =\; \backslash frac\{1\}\{4\}\; w\_0^\{\backslash beta\; d\}\; \backslash sim\; T^\{\backslash beta\; d\}.\; \backslash quad\; (3.3)$$

For a charged black hole, the entropy density can be expressed as a function of the temperature $T$ and the chemical potential $\mu$ . Since the dimensions of $T$ and $\mu$ are

---

¹Related work on Lifshitz black holes is listed in [16]$\dim T = \dim \mu = [M]$ , the entropy density of a slightly non-extremal black hole in $(d+2)$ -dimensions is

$$s\sim T^{\beta d}{\mu }^{d-\beta d}(3.4)s\; \backslash sim\; T^\{\backslash beta\; d\}\; \backslash mu^\{d-\backslash beta\; d\}\; \backslash quad\; (3.4)$$

by dimensional analysis. The entropy density can also be obtained by the standard Euclidean path integral, which gives

$$s=aC{T}^{\beta d}{\mu }^{d-\beta d},C\sim L^d\mathrm{/}{G}_{d+2}\mathrm{.}(3.5)s\; =\; a\; C\; T^\{\backslash beta\; d\}\; \backslash mu^\{d-\backslash beta\; d\},\; \backslash quad\; C\; \backslash sim\; L^d\; /\; G\_\{d+2\}.\; \backslash quad\; (3.5)$$

Here $G_{d+2}$ denotes the $(d+2)$ -dimensional Newton constant, and the coefficient $a$ depends on $\alpha$ and the asymptotic value of the dilaton $\phi_0$ . Here the specific heat,

$$C_v=T{\left(\frac{ds}{dT}\right)}_{\mu }=aC\beta d{T}^{\beta d}{\mu }^{d-\beta d},(3.6)C\_v\; =\; T\; \backslash left(\; \backslash frac\{ds\}\{dT\}\; \backslash right)\_\backslash mu\; =\; a\; C\; \backslash beta\; d\; T^\{\backslash beta\; d\}\; \backslash mu^\{d-\backslash beta\; d\},\; \backslash quad\; (3.6)$$

is always positive. The other thermodynamical quantities can be obtained by the entropy density via the Gibbs-Duhem relation $s dT - dP + n d\mu = 0$ . Here $P$ and $n$ are the pressure and number density. Keeping $\mu$ fixed and performing the integration gives

$$P=\int sdT=\frac{a}{\beta d+1}C{\mu }^{d-\beta d}{T}^{\beta d+1}+P_0(\mu ),(3.7)P\; =\; \backslash int\; s\; dT\; =\; \backslash frac\{a\}\{\backslash beta\; d\; +\; 1\}\; C\; \backslash mu^\{d-\backslash beta\; d\}\; T^\{\backslash beta\; d+1\}\; +\; P\_0(\backslash mu),\; \backslash quad\; (3.7)$$

where $P_0(\mu)$ is a temperature independent integration constant which can be fixed by dimensional analysis:

$$P_0(\mu )=bC{e}^{(d+1)\alpha {\varphi }_0}{\mu }^{d+1}\mathrm{.}(3.8)P\_0(\backslash mu)\; =\; b\; C\; e^\{(d+1)\backslash alpha\backslash phi\_0\}\; \backslash mu^\{d+1\}.\; \backslash quad\; (3.8)$$

Then from the relation

$$dP=aC{\mu }^{d-\beta d}{T}^{\beta d}dT+\frac{(d-\beta d)}{\beta d+1}aC{\mu }^{d-1-\beta d}{T}^{\beta d+1}d\mu +bC(d+1)e^{(d+1)\alpha {\varphi }_0}{\mu }^{d}d\mu ,(3.9)dP\; =\; a\; C\; \backslash mu^\{d-\backslash beta\; d\}\; T^\{\backslash beta\; d\}\; dT\; +\; \backslash frac\{(d-\backslash beta\; d)\}\{\backslash beta\; d\; +\; 1\}\; a\; C\; \backslash mu^\{d-1-\backslash beta\; d\}\; T^\{\backslash beta\; d+1\}\; d\backslash mu\; +\; b\; C\; (d+1)\; e^\{(d+1)\backslash alpha\backslash phi\_0\}\; \backslash mu^d\; d\backslash mu,\; \backslash quad\; (3.9)$$

one can identify the number density $n$ as

$$n=\frac{(d-\beta d)}{\beta d+1}aC{\mu }^{d-1-\beta d}{T}^{\beta d+1}+bC(d+1)e^{(d+1)\alpha {\varphi }_0}{\mu }^d\mathrm{.}(3.10)n\; =\; \backslash frac\{(d-\backslash beta\; d)\}\{\backslash beta\; d\; +\; 1\}\; a\; C\; \backslash mu^\{d-1-\backslash beta\; d\}\; T^\{\backslash beta\; d+1\}\; +\; b\; C\; (d+1)\; e^\{(d+1)\backslash alpha\backslash phi\_0\}\; \backslash mu^d.\; \backslash quad\; (3.10)$$

Finally, the energy density is determined by the relation $\rho = Ts + \mu n - P$ , which gives

$$\rho =\frac{d}{\beta d+1}aC{\mu }^{d-\beta d}{T}^{\beta d+1}+dbC{e}^{(d+1)\alpha {\varphi }_0}{\mu }^{d+1}\mathrm{.}(3.11)\backslash rho\; =\; \backslash frac\{d\}\{\backslash beta\; d\; +\; 1\}\; a\; C\; \backslash mu^\{d-\backslash beta\; d\}\; T^\{\backslash beta\; d+1\}\; +\; d\; b\; C\; e^\{(d+1)\backslash alpha\backslash phi\_0\}\; \backslash mu^\{d+1\}.\; \backslash quad\; (3.11)$$

The equation of state of this near-extremal system is

$$P=\frac{1}{d}\rho \mathrm{.}(3.12)P\; =\; \backslash frac\{1\}\{d\}\; \backslash rho.\; \backslash quad\; (3.12)$$The susceptibility is given by

$$\chi \equiv {\left(\frac{\mathrm{\partial }n}{\mathrm{\partial }\mu }\right)}_T=\frac{(d-1-\beta d)(d-\beta d)}{\beta d+1}aC{T}^{\beta d+1}{\mu }^{d-2-\beta d}+d(d+1)bC{e}^{(d+1)\alpha {\varphi }_0}{\mu }^{d-1},(3.13)\backslash chi\; \backslash equiv\; \backslash left(\; \backslash frac\{\backslash partial\; n\}\{\backslash partial\; \backslash mu\}\; \backslash right)\_T\; =\; \backslash frac\{(d-1-\backslash beta\; d)(d-\backslash beta\; d)\}\{\backslash beta\; d+1\}\; a\; C\; T^\{\backslash beta\; d+1\}\; \backslash mu^\{d-2-\backslash beta\; d\}\; +\; d(d+1)\; b\; C\; e^\{(d+1)\backslash alpha\backslash phi\_0\}\; \backslash mu^\{d-1\},\; \backslash quad\; (3.13)$$

which is positive when $T \ll \mu$ . Notice that the first term becomes negative when $d-1-\beta d < 0$ , i.e. $\beta > (d-1)/d$ . As emphasized in [9], the formulae in this section are valid when $T \ll \mu$ . Furthermore, whether $\chi$ changes sign as the temperature increases, signalling a phase transition, requires one to explore beyond the regime $T \ll \mu$ .

Rather than constructing the global solution which is asymptotically $\text{AdS}_{d+2}$ , we focus on some qualitative behavior of the asymptotic solution. Similar to the four-dimensional example [9], the bulk solutions related by a rescaling of coordinates should be treated as being distinct with different chemical potential. Therefore, all solutions can be obtained by a suitable rescaling and shift in the dilaton from a particular simple solution, e.g. $q_e = 1$ and $\phi_0 = 0$ . The rescaling is given by

$$r\to \lambda r,(t,x_i)\to {\lambda }^{-1}(t,x_i)\mathrm{.}(3.14)r\; \backslash rightarrow\; \backslash lambda\; r,\; \backslash quad\; (t,\; x\_i)\; \backslash rightarrow\; \backslash lambda^\{-1\}(t,\; x\_i).\; \backslash quad\; (3.14)$$

Furthermore, from the equations of motion we can see that the metric and $\phi - \phi_0$ only depend on $q_e^2 e^{-2\alpha\phi_0}$ .

Reconsidering the equations of motion (2.6)–(2.8) and the constraint (2.9), we can see that for an asymptotically $\text{AdS}_{d+2}$ solution, the metric and dilaton must take the following form:

where the ellipses denote terms that are subdominant at large $r$ . The parameter $\rho$ is the energy density of the black hole, and $e_1$ is a constant depending on $L$ . Under the rescaling of (3.14), the corresponding rescaling of the energy density and charge parameter should be

$$\rho \to {\lambda }^{d+1}\rho ,q_e\to {\lambda }^d{q}_e\mathrm{.}(3.16)\backslash rho\; \backslash rightarrow\; \backslash lambda^\{d+1\}\; \backslash rho,\; \backslash quad\; q\_e\; \backslash rightarrow\; \backslash lambda^d\; q\_e.\; \backslash quad\; (3.16)$$

This implies the following relation:

$$\rho =D_1(q_e{e}^{-\alpha {\varphi }_0}{)}^{\frac{d+1}{d}},(3.17)\backslash rho\; =\; D\_1\; (q\_e\; e^\{-\backslash alpha\backslash phi\_0\})^\{\backslash frac\{d+1\}\{d\}\},\; \backslash quad\; (3.17)$$where $D_1$ is an $\alpha$ dependent parameter. Similarly, the chemical potential,

$$\mu ={\int }_{r_h}^{\mathrm{\infty }}\frac{q_e}{b^d(r)}{e}^{-2\alpha \varphi }dr,(3.18)\backslash mu\; =\; \backslash int\_\{r\_h\}^\{\backslash infty\}\; \backslash frac\{q\_e\}\{b^d(r)\}\; e^\{-2\backslash alpha\backslash phi\}\; dr,\; \backslash quad\; (3.18)$$

can be determined by a similar rescaling argument as

$$\mu =D_2(q_e{e}^{-\alpha {\varphi }_0}{)}^{\frac{1}{d}}{e}^{-\alpha {\varphi }_0},(3.19)\backslash mu\; =\; D\_2\; (q\_e\; e^\{-\backslash alpha\backslash phi\_0\})^\{\backslash frac\{1\}\{d\}\}\; e^\{-\backslash alpha\backslash phi\_0\},\; \backslash quad\; (3.19)$$

where $D_2$ is also an $\alpha$ dependent parameter. This gives

$$\rho =D_3{e}^{(d+1)\alpha {\varphi }_0}{\mu }^{d+1},(3.20)\backslash rho\; =\; D\_3\; e^\{(d+1)\backslash alpha\backslash phi\_0\}\; \backslash mu^\{d+1\},\; \backslash quad\; (3.20)$$

which agrees with the second term in (3.11).

4 Conductivity in zero-temperature backgrounds

We will calculate the conductivity $\sigma$ in the $(d+2)$ -dimensional extremal black hole background, generalizing the result obtained in [9]. A useful formulation was proposed in [17], which stated that after introducing a perturbative gauge field $A_x(r, t)$ , the corresponding equation of motion for $A_x$ could be recast in a Schrödinger-like form

$$-A_{x,zz}+V(z)A_x={\omega }^2{A}_x,(4.1)-A\_\{x,zz\}\; +\; V(z)A\_x\; =\; \backslash omega^2\; A\_x,\; \backslash quad\; (4.1)$$

where $z$ is a redefinition of the radial variable $r$ . Then by studying scattering with ingoing boundary condition at the horizon, the conductivity is determined in terms of the reflection coefficient

$$\sigma (\omega )=\frac{1-\mathcal{R}}{1+\mathcal{R}}\mathrm{.}(4.2)\backslash sigma(\backslash omega)\; =\; \backslash frac\{1\; -\; \backslash mathcal\{R\}\}\{1\; +\; \backslash mathcal\{R\}\}.\; \backslash quad\; (4.2)$$

It has been pointed out in [17] that such a formulation could be generalized to higher-dimensional cases.

In the following we will perform the calculations in our $(d+2)$ -dimensional extremal background. Our task is still to find the equation for $A_x$ and cast it in the form of (4.1). Consider the general metric of the form

$$d{s}^2=-g(r)e^{-\chi (r)}d{t}^2+\frac{d{r}^2}{g(r)}+r^2\sum _{i=1}^{d}d{x}_i^2,(4.3)ds^2\; =\; -g(r)e^\{-\backslash chi(r)\}dt^2\; +\; \backslash frac\{dr^2\}\{g(r)\}\; +\; r^2\; \backslash sum\_\{i=1\}^d\; dx\_i^2,\; \backslash quad\; (4.3)$$which is extensively adopted in the discussions of holographic superconductors [18]. The gauge field, including the perturbation, is given by

$$A=A_t(r)dt+{\tilde{A}}_x(r)e^{-i\omega t}dx\mathrm{.}(4.4)A\; =\; A\_t(r)dt\; +\; \backslash tilde\{A\}\_x(r)e^\{-i\backslash omega\; t\}dx.\; \backslash quad\; (4.4)$$

Consider the Lagrangian of the following form:

$$\mathcal{L}=\cdots -2(\mathrm{\nabla }\varphi {)}^2-\frac{1}{4}{f}^2(\varphi )F_{\mu \nu }{F}^{\mu \nu }+\dots ,(4.5)\backslash mathcal\{L\}\; =\; \backslash dots\; -\; 2(\backslash nabla\backslash phi)^2\; -\; \backslash frac\{1\}\{4\}f^2(\backslash phi)F\_\{\backslash mu\backslash nu\}F^\{\backslash mu\backslash nu\}\; +\; \backslash dots,\; \backslash quad\; (4.5)$$

where the gauge coupling is $f^2(\phi)$ . The $t$ -component of the Maxwell equation determines the background $A_t$ ,

$${\mathrm{\partial }}_r{A}_t=q_e{f}^{-2}(\varphi )r^{-d}{e}^{-\chi \mathrm{/}2},(4.6)\backslash partial\_r\; A\_t\; =\; q\_e\; f^\{-2\}(\backslash phi)\; r^\{-d\}\; e^\{-\backslash chi/2\},\; \backslash quad\; (4.6)$$

and the $x$ -component equation,

Notice that $g_{tx}$ should be turned on at the same order as the gauge field perturbation and we denote $g_{tx}(t, r) = e^{-i\omega t} \tilde{g}_{tx}(r)$ . Furthermore, the $rx$ -component of the Einstein equations gives

$${\mathrm{\partial }}_r{\tilde{g}}_{tx}-\frac{2}{r}{\tilde{g}}_{tx}=-f^2(\varphi ){\tilde{A}}_x{\mathrm{\partial }}_r{A}_t\mathrm{.}(4.8)\backslash partial\_r\; \backslash tilde\{g\}\_\{tx\}\; -\; \backslash frac\{2\}\{r\}\; \backslash tilde\{g\}\_\{tx\}\; =\; -f^2(\backslash phi)\; \backslash tilde\{A\}\_x\; \backslash partial\_r\; A\_t.\; \backslash quad\; (4.8)$$

Substituting (4.8) into (4.7), we can obtain

$${\mathrm{\partial }}_r(f^2(\varphi )r^{d-2}g{e}^{-\chi \mathrm{/}2}{\mathrm{\partial }}_r{\tilde{A}}_x)+{\omega }^2{f}^2(\varphi )r^{d-2}{g}^{-1}{e}^{\chi \mathrm{/}2}{\tilde{A}}_x-f^4(\varphi )r^{d-2}{e}^{\chi \mathrm{/}2}({\mathrm{\partial }}_r{A}_t{)}^2{\tilde{A}}_x=0,(4.9)\backslash partial\_r\; \backslash left(\; f^2(\backslash phi)\; r^\{d-2\}\; g\; e^\{-\backslash chi/2\}\; \backslash partial\_r\; \backslash tilde\{A\}\_x\; \backslash right)\; +\; \backslash omega^2\; f^2(\backslash phi)\; r^\{d-2\}\; g^\{-1\}\; e^\{\backslash chi/2\}\; \backslash tilde\{A\}\_x\; -\; f^4(\backslash phi)\; r^\{d-2\}\; e^\{\backslash chi/2\}\; (\backslash partial\_r\; A\_t)^2\; \backslash tilde\{A\}\_x\; =\; 0,\; \backslash quad\; (4.9)$$

which agrees with (3.7) of [9] for $d = 2$ . The background $\partial_r A_t$ is given by (4.6). By taking a new coordinate $z$ and a new wavefunction $\Psi$ ,

$${\mathrm{\partial }}_z=e^{-\chi \mathrm{/}2}g{\mathrm{\partial }}_r,\mathrm{\Psi }=f(\varphi )r^{\frac{d-2}{2}}{\tilde{A}}_x\mathrm{.}(4.10)\backslash partial\_z\; =\; e^\{-\backslash chi/2\}\; g\; \backslash partial\_r,\; \backslash quad\; \backslash Psi\; =\; f(\backslash phi)\; r^\{\backslash frac\{d-2\}\{2\}\}\; \backslash tilde\{A\}\_x.\; \backslash quad\; (4.10)$$

this equation becomes a Schrödinger equation:

$$-{\mathrm{\Psi }}^{\mathrm{\prime }\mathrm{\prime }}+V(z)\mathrm{\Psi }={\omega }^2\mathrm{\Psi },(4.11)-\backslash Psi\text{'}\text{'}\; +\; V(z)\backslash Psi\; =\; \backslash omega^2\; \backslash Psi,\; \backslash quad\; (4.11)$$

where the potential is

$$V(z)=f^{-1}(\varphi )r^{-\frac{d-2}{2}}{\mathrm{\partial }}_z^2(f(\varphi )r^{\frac{d-2}{2}})+q_e^2{f}^{-2}(\varphi )r^{-2d}g{e}^{-\chi }\mathrm{.}(4.12)V(z)\; =\; f^\{-1\}(\backslash phi)\; r^\{-\backslash frac\{d-2\}\{2\}\}\; \backslash partial\_z^2\; \backslash left(\; f(\backslash phi)\; r^\{\backslash frac\{d-2\}\{2\}\}\; \backslash right)\; +\; q\_e^2\; f^\{-2\}(\backslash phi)\; r^\{-2d\}\; g\; e^\{-\backslash chi\}.\; \backslash quad\; (4.12)$$

Here prime stands for the derivative with respect to $z$ , and it agrees with (3.15) of [9] once again when $d = 2$ .## 4.1 Near horizon analysis

Recall the metric of the zero-temperature background,

$$d{s}^2=-a_0^2{w}^{2}d{t}^2+\frac{d{w}^2}{a_0^2{w}^2}+w^{2\beta }\sum _{i=1}^{d}d{x}_i^2\mathrm{.}(4.13)ds^2\; =\; -a\_0^2\; w^2\; dt^2\; +\; \backslash frac\{dw^2\}\{a\_0^2\; w^2\}\; +\; w^\{2\backslash beta\}\; \backslash sum\_\{i=1\}^d\; dx\_i^2.\; \backslash quad\; (4.13)$$

Comparing with (4.3), we can obtain

$$r=w^{\beta },g(r)={\beta }^2{a}_0^2{r}^2,e^{-\chi (r)}=\frac{1}{{\beta }^2}{r}^{\frac{2}{\beta }-2}\mathrm{.}(4.14)r\; =\; w^\backslash beta,\; \backslash quad\; g(r)\; =\; \backslash beta^2\; a\_0^2\; r^2,\; \backslash quad\; e^\{-\backslash chi(r)\}\; =\; \backslash frac\{1\}\{\backslash beta^2\}\; r^\{\backslash frac\{2\}\{\backslash beta\}-2\}.\; \backslash quad\; (4.14)$$

Then

$$\frac{\mathrm{\partial }r}{\mathrm{\partial }z}=e^{-\chi \mathrm{/}2}g=\beta a_0^2{r}^{\frac{1}{\beta }+1}\Rightarrow z=-\frac{1}{a_0^{2}w}\mathrm{.}(4.15)\backslash frac\{\backslash partial\; r\}\{\backslash partial\; z\}\; =\; e^\{-\backslash chi/2\}\; g\; =\; \backslash beta\; a\_0^2\; r^\{\backslash frac\{1\}\{\backslash beta\}+1\}\; \backslash Rightarrow\; z\; =\; -\backslash frac\{1\}\{a\_0^2\; w\}.\; \backslash quad\; (4.15)$$

Here the gauge coupling is

$$f(\varphi )=2\exp (\alpha \varphi )=\frac{2}{w^{\beta d}}\mathrm{.}(4.16)f(\backslash phi)\; =\; 2\; \backslash exp(\backslash alpha\backslash phi)\; =\; \backslash frac\{2\}\{w^\{\backslash beta\; d\}\}.\; \backslash quad\; (4.16)$$

Plugging (4.15) and (4.16) into (4.12), we obtain the following expression for $V(z)$

$$V_0(z)=\frac{c_0}{z^2},(4.17)V\_0(z)\; =\; \backslash frac\{c\_0\}\{z^2\},\; \backslash quad\; (4.17)$$

where

$$c_0=\frac{(d+2{)}^2}{4}{\beta }^2-\frac{d+2}{2}\beta +\frac{q_e^2}{4a_0^2}\mathrm{.}(4.18)c\_0\; =\; \backslash frac\{(d+2)^2\}\{4\}\; \backslash beta^2\; -\; \backslash frac\{d+2\}\{2\}\; \backslash beta\; +\; \backslash frac\{q\_e^2\}\{4a\_0^2\}.\; \backslash quad\; (4.18)$$

It can be seen that for a general $(d+2)$ -dimensional extremal charged dilaton black holes, the equation of the gauge field perturbation $\tilde{A}_x$ can always be transformed into a Schrödinger equation. Furthermore, the effective potential takes a universal form $V_0(z) = c_0/z^2$ , where $c_0$ is a constant which is determined by $\alpha$ and $d$ .

The technique of solving the Schrödinger equation with specific potential is summarized in the Appendix. The ingoing mode solution is

$$\mathrm{\Psi }(z)=C_0^{(\text{in})}\sqrt{-\frac{\pi \omega z}{2}}{H}_{{\nu }_0}^{(1)}(-\omega z)\sim C_0^{(\text{in})}{e}^{-i(\omega z+\frac{1}{2}{\nu }_0\pi +\frac{1}{2}\pi )},(4.19)\backslash Psi(z)\; =\; C\_0^\{(\backslash text\{in\})\}\; \backslash sqrt\{-\backslash frac\{\backslash pi\backslash omega\; z\}\{2\}\}\; H\_\{\backslash nu\_0\}^\{(1)\}(-\backslash omega\; z)\; \backslash sim\; C\_0^\{(\backslash text\{in\})\}\; e^\{-i(\backslash omega\; z\; +\; \backslash frac\{1\}\{2\}\backslash nu\_0\backslash pi\; +\; \backslash frac\{1\}\{2\}\backslash pi)\},\; \backslash quad\; (4.19)$$

where $\nu_0^2 = c_0 + 1/4$ .

4.2 Asymptotic analysis

For the asymptotic solution we have

$$\chi =0,g=r^2,f=f({\varphi }_0),(4.20)\backslash chi\; =\; 0,\; \backslash quad\; g\; =\; r^2,\; \backslash quad\; f\; =\; f(\backslash phi\_0),\; \backslash quad\; (4.20)$$so

$$\frac{\mathrm{\partial }r}{\mathrm{\partial }z}=r^2\Rightarrow z=-\frac{1}{r}\mathrm{.}(4.21)\backslash frac\{\backslash partial\; r\}\{\backslash partial\; z\}\; =\; r^2\; \backslash quad\; \backslash Rightarrow\; \backslash quad\; z\; =\; -\backslash frac\{1\}\{r\}.\; \backslash quad\; (4.21)$$

Unlike the $d = 2$ case in [9], a crucial difference is that the effective potential cannot be neglected at the asymptotic boundary:

$$V_{\mathrm{\infty }}(z)=\frac{c_{\mathrm{\infty }}}{z^2},c_{\mathrm{\infty }}=\frac{d(d-2)}{4}\mathrm{.}(4.22)V\_\backslash infty(z)\; =\; \backslash frac\{c\_\backslash infty\}\{z^2\},\; \backslash quad\; c\_\backslash infty\; =\; \backslash frac\{d(d-2)\}\{4\}.\; \backslash quad\; (4.22)$$

The general solution is (refer to the Appendix for the details)

where $\nu_\infty = \sqrt{c_\infty + 1/4} = (d-1)/2$ .

4.3 Matching

In order to match the coefficients in the near horizon and asymptotic analysis, we should take the small $\omega$ limit to extrapolate the near horizon and asymptotic wavefunctions to an intermediate region of small $-\omega z$ . From the near horizon side ( $-z \gg 1$ ), we have

$$\mathrm{\Psi }(z)=C_0^{(\text{in})}\sqrt{-\frac{\pi \omega z}{2}}{H}_{{\nu }_0}^{(1)}(-\omega z)\to (-\omega z{)}^{\frac{1}{2}-{\nu }_0},(4.24)\backslash Psi(z)\; =\; C\_0^\{(\backslash text\{in\})\}\; \backslash sqrt\{-\backslash frac\{\backslash pi\; \backslash omega\; z\}\{2\}\}\; H\_\{\backslash nu\_0\}^\{(1)\}(-\backslash omega\; z)\; \backslash rightarrow\; (-\backslash omega\; z)^\{\backslash frac\{1\}\{2\}\; -\; \backslash nu\_0\},\; \backslash quad\; (4.24)$$

and from the asymptotic side it is just (4.23). The frequency dependence can be neglected in the intermediate region. Therefore the $\omega$ -dependence of the essential combination of coefficients can be determined:

$$C_{\mathrm{\infty }}^{(1)}-C_{\mathrm{\infty }}^{(2)}\sim {\omega }^{{\nu }_{\mathrm{\infty }}-{\nu }_0}\mathrm{.}(4.25)C\_\backslash infty^\{(1)\}\; -\; C\_\backslash infty^\{(2)\}\; \backslash sim\; \backslash omega^\{\backslash nu\_\backslash infty\; -\; \backslash nu\_0\}.\; \backslash quad\; (4.25)$$

4.4 Conductivity

Next we calculate the conductivity in a general $(d+2)$ -dimensional spacetime. It can be seen that the asymptotic form of $\tilde{A}_x$ is

$${\tilde{A}}_x={\tilde{A}}_x^{(0)}+\frac{{\tilde{A}}_x^{(1)}}{r^{d-1}},(4.26)\backslash tilde\{A\}\_x\; =\; \backslash tilde\{A\}\_x^\{(0)\}\; +\; \backslash frac\{\backslash tilde\{A\}\_x^\{(1)\}\}\{r^\{d-1\}\},\; \backslash quad\; (4.26)$$and the conductivity takes the following form:

$$\sigma =-i\frac{d-1}{\omega }{f}^2({\varphi }_0)\frac{{\tilde{A}}_x^{(1)}}{{\tilde{A}}_x^{(0)}},(4.27)\backslash sigma\; =\; -i\; \backslash frac\{d-1\}\{\backslash omega\}\; f^2(\backslash phi\_0)\; \backslash frac\{\backslash tilde\{A\}\_x^\{(1)\}\}\{\backslash tilde\{A\}\_x^\{(0)\}\},\; \backslash quad\; (4.27)$$

where $f(\phi_0)$ denotes the asymptotic value of the gauge coupling. Therefore

$$(d-1)f^2({\varphi }_0)({\tilde{A}}_x^{(1)\ast }{\tilde{A}}_x^{(0)}-{\tilde{A}}_x^{(1)}{\tilde{A}}_x^{(0)\ast })=-2i\omega {\mid {\tilde{A}}_x^{(0)}\mid }^2\mathrm{Re}\sigma ,(4.28)(d-1)f^2(\backslash phi\_0)\; \backslash left(\; \backslash tilde\{A\}\_x^\{(1)*\}\; \backslash tilde\{A\}\_x^\{(0)\}\; -\; \backslash tilde\{A\}\_x^\{(1)\}\; \backslash tilde\{A\}\_x^\{(0)*\}\; \backslash right)\; =\; -2i\backslash omega\; \backslash left|\; \backslash tilde\{A\}\_x^\{(0)\}\; \backslash right|^2\; \backslash operatorname\{Re\}\; \backslash sigma,\; \backslash quad\; (4.28)$$

and the asymptotic form of $\Psi$ can be written as

$$\mathrm{\Psi }=f({\varphi }_0)({\tilde{A}}_x^{(0)}{r}^{\frac{d-2}{2}}+{\tilde{A}}_x^{(1)}{r}^{-\frac{d}{2}})\mathrm{.}(4.29)\backslash Psi\; =\; f(\backslash phi\_0)\; \backslash left(\; \backslash tilde\{A\}\_x^\{(0)\}\; r^\{\backslash frac\{d-2\}\{2\}\}\; +\; \backslash tilde\{A\}\_x^\{(1)\}\; r^\{-\backslash frac\{d\}\{2\}\}\; \backslash right).\; \backslash quad\; (4.29)$$

Then we can obtain the conserved flux at the boundary:

Substituting (4.28) and noting that $\partial r / \partial z = r^2$ at the boundary, we have

$$\mathcal{F}=2\omega {\mid {\tilde{A}}_x^{(0)}\mid }^2\mathrm{Re}\sigma \mathrm{.}(4.31)\backslash mathcal\{F\}\; =\; 2\backslash omega\; \backslash left|\; \backslash tilde\{A\}\_x^\{(0)\}\; \backslash right|^2\; \backslash operatorname\{Re\}\; \backslash sigma.\; \backslash quad\; (4.31)$$

Notice that $\Psi \sim r^{d/2-1} \tilde{A}_x^{(0)} \sim (-z)^{1-d/2} \tilde{A}_x^{(0)}$ , then from the results (4.23) and (4.25) we have

$${\tilde{A}}_x^{(0)}=-i(2{)}^{{\nu }_{\mathrm{\infty }}}(C_{\mathrm{\infty }}^{(1)}-C_{\mathrm{\infty }}^{(2)}){\omega }^{\frac{1}{2}-{\nu }_{\mathrm{\infty }}}\sim {\omega }^{\frac{1}{2}-{\nu }_0}\mathrm{.}(4.32)\backslash tilde\{A\}\_x^\{(0)\}\; =\; -i(2)^\{\backslash nu\_\backslash infty\}\; (C\_\backslash infty^\{(1)\}\; -\; C\_\backslash infty^\{(2)\})\; \backslash omega^\{\backslash frac\{1\}\{2\}-\backslash nu\_\backslash infty\}\; \backslash sim\; \backslash omega^\{\backslash frac\{1\}\{2\}-\backslash nu\_0\}.\; \backslash quad\; (4.32)$$

By evaluating the conserved flux at the horizon, we can easily check that

$$\mathcal{F}\sim \omega ,(4.33)\backslash mathcal\{F\}\; \backslash sim\; \backslash omega,\; \backslash quad\; (4.33)$$

and thus, combining with the result (4.31), the real part of the conductivity is

$$\mathrm{Re}\sigma \sim {\omega }^{\delta },\delta =2{\nu }_0-1.(4.34)\backslash operatorname\{Re\}\; \backslash sigma\; \backslash sim\; \backslash omega^\backslash delta,\; \backslash quad\; \backslash delta\; =\; 2\backslash nu\_0\; -\; 1.\; \backslash quad\; (4.34)$$

The exponent $\delta$ has the same expression as the $d = 2$ case in [9], but the value of $\nu_0$ generically depends on the spacetime dimension and also on the dilaton coupling $\alpha$ .## 5 Gauss-Bonnet corrections to $\eta/s$ at finite temperature

In this section we will discuss the Gauss-Bonnet corrections to $\eta/s$ at finite temperature in five dimensions. One remarkable progress in the AdS/CFT correspondence is the calculation of the ratio of shear viscosity over the entropy density in the dual gravity side. It has been found that $\eta/s = 1/4\pi$ for a large class of CFTs with Einstein gravity duals in the large $N$ limit. Therefore, it was conjectured that $1/4\pi$ is a universal lower bound for all materials, which is the so-called Kovtun-Son-Starinets (KSS) bound [20]. However, in [21, 22, 23] it was observed that in $R^2$ gravity such a lower bound was violated, and a new lower bound $4/25\pi$ was proposed by considering the causality of the dual field theory.

It was argued in [24] that the shear viscosity is fully determined by the effective coupling of the transverse gravitons on the horizon. This was confirmed in [25] via the scalar membrane paradigm and in [26] by calculating the on-shell action of the transverse gravitons. However, the full solutions were still used in the actual calculations. Recently, $\eta/s$ with higher derivative corrections was revisited for various examples in [27]. They calculated $\eta/s$ in the presence of higher order corrections by making use of the near horizon data only. It turned out that the results agreed with those obtained in the previous literature. An efficient method for computing the zero frequency limit of transport coefficients in strongly coupled field theories described holographically by higher derivative gravity theories was proposed in [29].

Here we calculate $\eta/s$ for black holes in five-dimensional Gauss-Bonnet gravity. Since the charged dilaton black holes have vanishing entropy at extremality, we shall not consider the zero-temperature limit. We adopt the formalism proposed in [28], where a three-dimensional effective metric $\tilde{g}_{\mu\nu}$ was introduced and the transverse gravitons were minimally coupled to this new effective metric. The action in this new formalism can take a covariant form. Similar discussions on this issue were also presented in [30].

Consider a tensor perturbation $h_{xy} = h_{xy}(t, u, z)$ , where $u$ is the radial coordinate in which the horizon is located at $u = 1$ , and the momentum of the perturbation points along the $z$ -axis. If the transverse gravitons can be decoupled from other perturbations,the effective bulk action of the transverse gravitons can be written in a general form:

$$S=\frac{V_{x,y}}{16\pi G_5}(-\frac{1}{2})\int d^{3}x\sqrt{-\tilde{g}}[\tilde{K}(u){\tilde{g}}^{MN}{\tilde{\mathrm{\nabla }}}_M\tilde{\varphi }{\tilde{\mathrm{\nabla }}}_N\tilde{\varphi }+m^2{\tilde{\varphi }}^2],(5.1)S\; =\; \backslash frac\{V\_\{x,y\}\}\{16\backslash pi\; G\_5\}\; \backslash left(-\backslash frac\{1\}\{2\}\backslash right)\; \backslash int\; d^3x\; \backslash sqrt\{-\backslash tilde\{g\}\}\; \backslash left[\; \backslash tilde\{K\}(u)\; \backslash tilde\{g\}^\{MN\}\; \backslash tilde\{\backslash nabla\}\_M\; \backslash tilde\{\backslash phi\}\; \backslash tilde\{\backslash nabla\}\_N\; \backslash tilde\{\backslash phi\}\; +\; m^2\; \backslash tilde\{\backslash phi\}^2\; \backslash right],\; \backslash quad\; (5.1)$$

up to some total derivatives, where $\tilde{\phi} = h_y^x$ can be expanded as $\tilde{\phi}(t, u, z) = \tilde{\phi}(u) e^{-i\omega t + ipz}$ . Here $\tilde{g}{MN}$ , $M, N = t, u, z$ is a three-dimensional effective metric, $m$ is an effective mass and $\tilde{\nabla}M$ is the covariant derivative using $\tilde{g}{MN}$ . Notice that $\tilde{\phi}$ is a scalar in the three dimensions $t, u, z$ , while it is not a scalar in the whole five dimensions. The three-dimensional effective action itself is general covariant, and $\tilde{K}(u)$ is a scalar under general coordinate transformations. In the following we will use $g{\mu\nu}$ to denote the whole five-dimensional background.

The action of the transverse gravitons in momentum space can be written explicitly as follows

$$S=\frac{V_{x,y}}{16\pi G_5}(-\frac{1}{2})\int \frac{d\omega dp}{(2\pi {)}^2}du\sqrt{-\tilde{g}}[\tilde{K}(u)({\tilde{g}}^{uu}{\tilde{\varphi }}^{\mathrm{\prime }}{\tilde{\varphi }}^{\mathrm{\prime }}+{\omega }^2{\tilde{g}}^{tt}{\tilde{\varphi }}^2+p^2{\tilde{g}}^{zz}{\tilde{\varphi }}^2)+m^2{\tilde{\varphi }}^2],(5.2)S\; =\; \backslash frac\{V\_\{x,y\}\}\{16\backslash pi\; G\_5\}\; \backslash left(-\backslash frac\{1\}\{2\}\backslash right)\; \backslash int\; \backslash frac\{d\backslash omega\; dp\}\{(2\backslash pi)^2\}\; du\; \backslash sqrt\{-\backslash tilde\{g\}\}\; \backslash left[\; \backslash tilde\{K\}(u)\; \backslash left(\; \backslash tilde\{g\}^\{uu\}\; \backslash tilde\{\backslash phi\}\text{'}\; \backslash tilde\{\backslash phi\}\text{'}\; +\; \backslash omega^2\; \backslash tilde\{g\}^\{tt\}\; \backslash tilde\{\backslash phi\}^2\; +\; p^2\; \backslash tilde\{g\}^\{zz\}\; \backslash tilde\{\backslash phi\}^2\; \backslash right)\; +\; m^2\; \backslash tilde\{\backslash phi\}^2\; \backslash right],\; \backslash quad\; (5.2)$$

where

and the prime denotes the derivative with respect to $u$ . Following [28], $\eta$ is given by

$$\eta =\frac{1}{16\pi G_5}{[\sqrt{{\tilde{g}}_{zz}}\tilde{K}(u)]}_{u=1}\mathrm{.}(5.4)\backslash eta\; =\; \backslash frac\{1\}\{16\backslash pi\; G\_5\}\; \backslash left[\; \backslash sqrt\{\backslash tilde\{g\}\_\{zz\}\}\; \backslash tilde\{K\}(u)\; \backslash right]\_\{u=1\}.\; \backslash quad\; (5.4)$$

Next, consider a general background

$$d{s}^2=-g(u)(1-u)d{t}^2+\frac{d{u}^2}{h(u)(1-u)}+\frac{r_0^2}{u^{\kappa }}(d{x}^2+d{y}^2+d{z}^2),(5.5)ds^2\; =\; -g(u)(1-u)dt^2\; +\; \backslash frac\{du^2\}\{h(u)(1-u)\}\; +\; \backslash frac\{r\_0^2\}\{u^\backslash kappa\}(dx^2\; +\; dy^2\; +\; dz^2),\; \backslash quad\; (5.5)$$

where $g(u)$ and $h(u)$ are regular functions at the horizon $u = 1$ and $\kappa$ is a parameter. It turns out that the effective action of the transverse gravitons can be written in the form of (5.2) with the effective three-dimensional metric

$${\tilde{g}}^{uu}=(1+\frac{{\lambda }_{\text{GB}}}{2}\frac{\kappa g_{tt}^{\mathrm{\prime }}{g}^{uu}}{u{g}_{tt}})g^{uu},(5.6)\backslash tilde\{g\}^\{uu\}\; =\; \backslash left(\; 1\; +\; \backslash frac\{\backslash lambda\_\{\backslash text\{GB\}\}\}\{2\}\; \backslash frac\{\backslash kappa\; g\text{'}\_\{tt\}\; g^\{uu\}\}\{u\; g\_\{tt\}\}\; \backslash right)\; g^\{uu\},\; \backslash quad\; (5.6)$$

$${\tilde{g}}^{tt}=[1+\frac{{\lambda }_{\text{GB}}}{2}(\frac{\kappa g^{\mathrm{\prime }uu}}{u}-\frac{({\kappa }^2+2\kappa )g^{uu}}{u^2})]g^{tt},(5.7)\backslash tilde\{g\}^\{tt\}\; =\; \backslash left[\; 1\; +\; \backslash frac\{\backslash lambda\_\{\backslash text\{GB\}\}\}\{2\}\; \backslash left(\; \backslash frac\{\backslash kappa\; g\text{'}^\{uu\}\}\{u\}\; -\; \backslash frac\{(\backslash kappa^2\; +\; 2\backslash kappa)g^\{uu\}\}\{u^2\}\; \backslash right)\; \backslash right]\; g^\{tt\},\; \backslash quad\; (5.7)$$

$${\tilde{g}}^{zz}=[1+\frac{{\lambda }_{\text{GB}}}{2}(\frac{g_{tt}^{\mathrm{\prime }}{g}^{uu}}{g_{tt}^2}-\frac{g_{tt}^{\mathrm{\prime }}{g}^{\mathrm{\prime }uu}}{g_{tt}}-\frac{2g^{uu}{g}_{tt}^{\mathrm{\prime }\mathrm{\prime }}}{g_{tt}})]g^{zz}\mathrm{.}(5.8)\backslash tilde\{g\}^\{zz\}\; =\; \backslash left[\; 1\; +\; \backslash frac\{\backslash lambda\_\{\backslash text\{GB\}\}\}\{2\}\; \backslash left(\; \backslash frac\{g\text{'}\_\{tt\}\; g^\{uu\}\}\{g\_\{tt\}^2\}\; -\; \backslash frac\{g\text{'}\_\{tt\}\; g\text{'}^\{uu\}\}\{g\_\{tt\}\}\; -\; \backslash frac\{2g^\{uu\}\; g\text{'}\text{'}\_\{tt\}\}\{g\_\{tt\}\}\; \backslash right)\; \backslash right]\; g^\{zz\}.\; \backslash quad\; (5.8)$$In fact, the effective action of the transverse gravitons can also be written as

$$S=\frac{1}{16\pi G_5}(-\frac{1}{2})\int d^{5}x\sqrt{-g}{\widehat{g}}^{\mu \nu }{\mathrm{\partial }}_{\mu }\tilde{\varphi }{\mathrm{\partial }}_{\nu }\tilde{\varphi },(5.9)S\; =\; \backslash frac\{1\}\{16\backslash pi\; G\_5\}\; \backslash left(\; -\backslash frac\{1\}\{2\}\; \backslash right)\; \backslash int\; d^5x\; \backslash sqrt\{-g\}\; \backslash hat\{g\}^\{\backslash mu\backslash nu\}\; \backslash partial\_\backslash mu\; \backslash tilde\{\backslash phi\}\; \backslash partial\_\backslash nu\; \backslash tilde\{\backslash phi\},\; \backslash quad\; (5.9)$$

where the new metric integrated Gauss-Bonnet correction is given by $\hat{g}^{\mu\nu} = \tilde{g}^{\mu\nu}$ for $\mu, \nu = t, u, z$ and $\hat{g}^{\mu\nu} = g^{\mu\nu}$ for $\mu, \nu = x, y$ . Then the coupling can be computed by $\tilde{K}(u) = \sqrt{-g}/\sqrt{-\tilde{g}}$ . After a straightforward calculation one can finally derive the following expression:

$$\frac{\eta }{s}=\frac{1}{4\pi }[1-\frac{\kappa }{2}{\lambda }_{\text{GB}}h(1)],(5.10)\backslash frac\{\backslash eta\}\{s\}\; =\; \backslash frac\{1\}\{4\backslash pi\}\; \backslash left[\; 1\; -\; \backslash frac\{\backslash kappa\}\{2\}\; \backslash lambda\_\{\backslash text\{GB\}\}\; h(1)\; \backslash right],\; \backslash quad\; (5.10)$$

where we have used the fact that the Bekenstein-Hawking formula still holds in Gauss-Bonnet gravity. Notice that in order to obtain corrections to $\eta/s$ at the leading order of $\lambda_{\text{GB}}$ , it is sufficient to work in the original background (5.5).

Recall the five-dimensional black hole solution

$$d{s}^2=-a^2(w)f(w)d{t}^2+\frac{d{w}^2}{a^2(w)f(w)}+b^2(w)(d{x}^2+d{y}^2+d{z}^2),f(w)=1-\frac{w_0^{3\beta +1}}{w^{3\beta +1}},(5.11)ds^2\; =\; -a^2(w)\; f(w)\; dt^2\; +\; \backslash frac\{dw^2\}\{a^2(w)\; f(w)\}\; +\; b^2(w)\; (dx^2\; +\; dy^2\; +\; dz^2),\; \backslash quad\; f(w)\; =\; 1\; -\; \backslash frac\{w\_0^\{3\backslash beta+1\}\}\{w^\{3\backslash beta+1\}\},\; \backslash quad\; (5.11)$$

we can take the following coordinate transformation:

$${\left(\frac{w_0}{w}\right)}^{3\beta +1}=u^2,(5.12)\backslash left(\; \backslash frac\{w\_0\}\{w\}\; \backslash right)^\{3\backslash beta+1\}\; =\; u^2,\; \backslash quad\; (5.12)$$

to convert the black hole metric into the form of (5.5) with

Now substituting all the relevant data into (5.10), we can arrive at

$$\frac{\eta }{s}=\frac{1}{4\pi }(1-\frac{12\beta }{2\beta +1}{\lambda }_{\text{GB}})\mathrm{.}(5.14)\backslash frac\{\backslash eta\}\{s\}\; =\; \backslash frac\{1\}\{4\backslash pi\}\; \backslash left(\; 1\; -\; \backslash frac\{12\backslash beta\}\{2\backslash beta+1\}\; \backslash lambda\_\{\backslash text\{GB\}\}\; \backslash right).\; \backslash quad\; (5.14)$$

Notice that $\beta \rightarrow 1$ , that is, in the relativistic limit, it reproduces the well-known result obtained in [22].

It has been verified that in certain charged black hole backgrounds, the charge parameter $q_e$ also contributes to the corrections to $\eta/s$ [31, 32, 33, 34, 35, 36]. However, it seems that our result does not have any dependence on $q_e$ . This may be understood as follows: in [27] the near horizon configuration for charged black holes contained the charge parameter$q_e$ , while here in the near horizon metric, the charge parameter $q_e$ is fixed only by the parameter $\alpha$ after choosing $b_0 = 1$ . Then by restoring the explicit dependence of $b_0$ in the metric, the near horizon data should contain the charge parameter $q_e$ . Therefore one can expect that the explicit $q_e$ dependence in the corrections to $\eta/s$ might be recovered.

6 Summary and discussion

In this paper we study general $(d+2)$ -dimensional charged dilaton black hole with planar symmetry obtained in [14], generalizing the investigations in [9]. Rather than treating these black holes as global solutions, here we consider them to be the near horizon solutions of a generic black hole with $\text{AdS}_{d+2}$ asymptotic geometry. We discuss the thermodynamics of the near-extremal black holes, and we calculate the AC conductivity in the zero-temperature background. We find that the AC conductivity behaves as $\omega^\delta$ , where $\delta$ is a constant determined by the parameter $\alpha$ in the gauge coupling and $d$ . When $d = 2$ , we reproduce the result obtained in [9]. We also calculate the Gauss-Bonnet corrections to $\eta/s$ in a five-dimensional finite-temperature background. The result reduces to the previously known result in the relativistic limit. However, unlike other works studying the higher order corrections to $\eta/s$ for charged black holes, our result does not depend on the charge parameter $q_e$ . This may be due to the fact that the charge parameter is fixed by $\alpha$ and $d$ after choosing a specific value for $b_0$ , thus the near horizon configuration does not contain information about $q_e$ explicitly. The $q_e$ dependence of the corrections to $\eta/s$ might be recovered by restoring the explicit dependence of $b_0$ in the metric.

One further generalization is to discuss the case of a dyonic black hole, which carries both electric and magnetic charges. One can expect that such solutions possess Lifshitz-like near horizon geometry and an $\text{AdS}_{d+2}$ asymptotic geometry. It would be interesting to study the thermodynamics and transport coefficients, such as the Hall conductivity [37], in the presence of the magnetic field.

There have been several interesting papers investigating non-Fermi liquid states in an RN-AdS black hole background [38, 39, 40, 41]. The asymptotic geometry is $\text{AdS}_{d+2}$ and the near horizon geometry contains an $\text{AdS}_2$ part, which plays a central role in the investigations. It would be worthwhile to generalize their considerations to the solutions discussed here. Note that now we have a Lifshitz-like near horizon geometry instead, andin principle we can still calculate the corresponding correlation functions by making use of the matching technique. We expect to study such fascinating topics in the future.

Acknowledgements

DWP would like to thank Rene Meyer for helpful discussions. The work of CMC was supported by the National Science Council of the R.O.C. under the grant NSC 96-2112-M-008-006-MY3 and in part by the National Center of Theoretical Sciences (NCTS). The work of DWP was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) through the Center for Quantum Spacetime (CQUeST) of Sogang University with grant number 2005-0049409.

A Solving the Schrödinger equation

The Schrödinger equation with a $z^{-2}$ potential

$$-{\mathrm{\Psi }}^{\mathrm{\prime }\mathrm{\prime }}+V(z)\mathrm{\Psi }={\omega }^2\mathrm{\Psi },V(z)=\frac{c}{z^2},(\text{A.1})-\backslash Psi\text{'}\text{'}\; +\; V(z)\backslash Psi\; =\; \backslash omega^2\backslash Psi,\; \backslash quad\; V(z)\; =\; \backslash frac\{c\}\{z^2\},\; \backslash quad\; (\backslash text\{A.1\})$$

can be transformed, by introducing a new variable (the range of $z$ is $-\infty < z < 0$ )

$$\mathrm{\Psi }(z)={\chi }_0\sqrt{-\omega z}\chi (z),(\text{A.2})\backslash Psi(z)\; =\; \backslash chi\_0\backslash sqrt\{-\backslash omega\; z\}\backslash chi(z),\; \backslash quad\; (\backslash text\{A.2\})$$

to the Bessel equation:

$$z^2{\mathrm{\partial }}_z^2\chi +z{\mathrm{\partial }}_z\chi +({\omega }^2{z}^2-{\nu }^2)\chi =0,{\nu }^2=c+\frac{1}{4}\mathrm{.}(\text{A.3})z^2\backslash partial\_z^2\backslash chi\; +\; z\backslash partial\_z\backslash chi\; +\; (\backslash omega^2z^2\; -\; \backslash nu^2)\backslash chi\; =\; 0,\; \backslash quad\; \backslash nu^2\; =\; c\; +\; \backslash frac\{1\}\{4\}.\; \backslash quad\; (\backslash text\{A.3\})$$

The solutions are the Hankel functions

$$\chi (z)=C_1{H}_{\nu }^{(1)}(-\omega z)+C_2{H}_{\nu }^{(2)}(-\omega z)\mathrm{.}(\text{A.4})\backslash chi(z)\; =\; C\_1H\_\backslash nu^\{(1)\}(-\backslash omega\; z)\; +\; C\_2H\_\backslash nu^\{(2)\}(-\backslash omega\; z).\; \backslash quad\; (\backslash text\{A.4\})$$

The approximative formulae for the Hankel functions are [42]

$$H_{\nu }^{(1)}(-\omega z)\to -i\frac{\mathrm{\Gamma }(\nu )}{\pi }{\left(\frac{-\omega z}{2}\right)}^{-\nu },-\omega z\to 0,(\text{A.5})H\_\backslash nu^\{(1)\}(-\backslash omega\; z)\; \backslash rightarrow\; -i\backslash frac\{\backslash Gamma(\backslash nu)\}\{\backslash pi\}\backslash left(\backslash frac\{-\backslash omega\; z\}\{2\}\backslash right)^\{-\backslash nu\},\; \backslash quad\; -\backslash omega\; z\; \backslash rightarrow\; 0,\; \backslash quad\; (\backslash text\{A.5\})$$

$$H_{\nu }^{(2)}(-\omega z)\to i\frac{\mathrm{\Gamma }(\nu )}{\pi }{\left(\frac{-\omega z}{2}\right)}^{-\nu },-\omega z\to 0,(\text{A.6})H\_\backslash nu^\{(2)\}(-\backslash omega\; z)\; \backslash rightarrow\; i\backslash frac\{\backslash Gamma(\backslash nu)\}\{\backslash pi\}\backslash left(\backslash frac\{-\backslash omega\; z\}\{2\}\backslash right)^\{-\backslash nu\},\; \backslash quad\; -\backslash omega\; z\; \backslash rightarrow\; 0,\; \backslash quad\; (\backslash text\{A.6\})$$

$$H_{\nu }^{(1)}(-\omega z)\sim \sqrt{-\frac{2}{\pi \omega z}}{e}^{-i(\omega z+\frac{1}{2}\nu \pi +\frac{1}{2}\pi )},-\omega z\sim \mathrm{\infty },(\text{A.7})H\_\backslash nu^\{(1)\}(-\backslash omega\; z)\; \backslash sim\; \backslash sqrt\{-\backslash frac\{2\}\{\backslash pi\backslash omega\; z\}\}e^\{-i(\backslash omega\; z\; +\; \backslash frac\{1\}\{2\}\backslash nu\backslash pi\; +\; \backslash frac\{1\}\{2\}\backslash pi)\},\; \backslash quad\; -\backslash omega\; z\; \backslash sim\; \backslash infty,\; \backslash quad\; (\backslash text\{A.7\})$$

$$H_{\nu }^{(2)}(-\omega z)\sim \sqrt{-\frac{2}{\pi \omega z}}{e}^{i(\omega z+\frac{1}{2}\nu \pi +\frac{1}{2}\pi )},-\omega z\sim \mathrm{\infty }\mathrm{.}(\text{A.8})H\_\backslash nu^\{(2)\}(-\backslash omega\; z)\; \backslash sim\; \backslash sqrt\{-\backslash frac\{2\}\{\backslash pi\backslash omega\; z\}\}e^\{i(\backslash omega\; z\; +\; \backslash frac\{1\}\{2\}\backslash nu\backslash pi\; +\; \backslash frac\{1\}\{2\}\backslash pi)\},\; \backslash quad\; -\backslash omega\; z\; \backslash sim\; \backslash infty.\; \backslash quad\; (\backslash text\{A.8\})$$## References

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