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Published for SISSA by Springer

Received: December 7, 2016

Revised: March 3, 2017

Accepted: March 20, 2017

Published: March 28, 2017

Conductivities from attractors

Johanna Erdmenger,a,b Daniel Fernández,a,c Prieslei Goularta,d and Piotr Witkowskia

aMax-Planck-Institut für Physik (Werner-Heisenberg-Institut),

Föhringer Ring 6, D-80805 Munich, Germany
bInstitut für Theoretische Physik und Astrophysik, Julius-Maximilians-Universität Würzburg,

Am Hubland, 97074 Würzburg, Germany
cScience Institute, University of Iceland,

Dunhaga 3, 107 Reykjav́ık, Iceland
dInstituto de F́ısica Teórica, UNESP-Universidade Estadual Paulista,

R. Dr. Bento T. Ferraz 271, Bl. II, São Paulo 01140-070, SP, Brazil

E-mail: jke@mppmu.mpg.de, danielf@mppmu.mpg.de, prieslei@ift.unesp.br,

piotrw@mppmu.mpg.de

Abstract: In the context of applications of the AdS/CFT correspondence to condensed

matter physics, we compute conductivities for field theory duals of dyonic planar black holes

in 3+1-dimensional Einstein-Maxwell-dilaton theories at zero temperature. We combine

the near-horizon data obtained via Sen’s entropy function formalism with known expres-

sions for conductivities. In this way we express the conductivities in terms of the extremal

black hole charges. We apply our approach to three different examples for dilaton theories

for which the background geometry is not known explicitly. For a constant scalar poten-

tial, the thermoelectric conductivity explicitly scales as αxy ∼ N3/2, as expected. For the

same model, our approach yields a finite result for the heat conductivity κ/T ∝ N3/2 even

for T → 0.

Keywords: AdS-CFT Correspondence, Gauge-gravity correspondence, Holography and

condensed matter physics (AdS/CMT)

ArXiv ePrint: 1611.09381

Open Access, c© The Authors.

Article funded by SCOAP3.
doi:10.1007/JHEP03(2017)147

mailto:jke@mppmu.mpg.de
mailto:danielf@mppmu.mpg.de
mailto:prieslei@ift.unesp.br
mailto:piotrw@mppmu.mpg.de
https://arxiv.org/abs/1611.09381
http://dx.doi.org/10.1007/JHEP03(2017)147


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Contents

1 Introduction 1

2 Entropy function 4

3 Dyonic AdS-RN planar black hole 5

4 DC conductivity from horizon data 8

5 DC conductivities at T = 0 for Einstein-Maxwell-Dilaton theories 10

5.1 Massless scalar 14

5.2 Exponential couplings 16

5.3 Quadratic couplings 18

5.4 Heat conductivity κ/T 21

5.5 S-duality of the attractor equations and conductivities 22

6 Generalizations and conclusions 24

A Dilatonic black holes 25

B Numerical UV completion 27

1 Introduction

Holography provides a simple way of computing conductivities in models for condensed

matter systems. Applying electric fields and thermal gradients induces linear perturbations

about the black hole. The matrix of thermoelectric conductivities is obtained by solving

the linearized perturbation equations. The most commonly studied case in the literature

deals with electrically charged Anti-de Sitter-Reissner-Nordström (AdS-RN) black holes [1],

where the electric charge is the dual of the chemical potential of the field theory. In the

presence of chemical potential and magnetic field on the CFT side, the gravity dual contains

a dyonic charged black hole, i.e. a black hole with both electric and magnetic charges.

The simplest example is the dyonic AdS-RN planar black hole, for which the electric (σ),

thermoelectric (α, ᾱ) and heat conductivities (κ̄) are given by

σij =
ρ

B

(
0 1

−1 0

)
, αij = αij =

s

B

(
0 1

−1 0

)
, κij =

s2Tg2
4

B(ρ2g4
4 +B2)

(
B ρg2

4

−ρg2
4 B

)
. (1.1)

Here, ρ is the charge density, B the magnetic field, g4 the constant coupling to the field

strength and s the entropy density. These conductivities were computed in [2, 3] and are

valid for all temperatures.

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As the full dyonic AdS-RN solution is known, the conductivities (1.1) can be written

in terms of the charges of the black hole. For more involved applications to condensed mat-

ter systems, more elaborate supergravity theories are necessary. For instance, these may

contain several scalar fields coupling to the field strength with corresponding potentials.

In general, the full analytical gravity solutions for these theories are not known, and the

explicit computation of these conductivities at finite temperature is not possible. Here,

we give a prescription to compute these conductivities explicitly at zero temperature via

Sen’s entropy function method, even for theories whose full dyonic black hole solution is

not known. There are numerical investigations for many different backgrounds (see for

instance [4–6] and references therein), however these generically have the limitation that

it is very hard to study very low or zero temperature backgrounds numerically in the pres-

ence of a chemical potential or magnetic field. On the gravity side, the zero temperature

limit corresponds to taking extremal limit of black branes. Our present approach gives an

analytical result for the conductivities precisely in this limit, at least for the supergravity

actions considered here.

Recently, a novel holographic framework to compute DC conductivities was developed

in [3, 7–11]. Here, the data relevant for computing the response of the system can be

extracted solely from the horizon of the black hole. The expressions obtained in these

references are very general: given a theory with a known black hole solution, one can

express the conductivities in terms of the horizon values of the scalar and gauge fields, and

of the metric components. Holographic renormalization techniques [11] and a generalized

Stokes equation [3] were used to obtain the horizon data for dyonic backgrounds. Explicit

examples, in which the conductivities are written in terms of the charges of the black

hole, were given for Reissner-Nordström black holes and Q-lattices. However, analytical

expressions for the conductivities for the simplest bosonic supergravity theory with one

scalar field, i.e. the Einstein-Maxwell-dilaton (EMD) theory, are still unknown.

In this present paper, we add a new ingredient to the issues discussed above by finding

explicit analytical results for conductivities at zero temperature for dyonic black holes of

EMD theory. We achieve this by applying Sen’s entropy function method [12] to obtain

the near-horizon data for this system in the presence of a scalar potential. Sen’s method

consists of solving a set of algebraic equations, the attractor equations, which are obtained

from the entropy function and whose solution gives the values of the fields at the horizon.

The entropy function is constructed by evaluating the Lagrangian in the near-horizon region

of the black hole. It is extremized by the solution to the attractor equations, which results

in the entropy of the black hole. This method was originally applied for black holes with

spherical horizons, but it can easily be used for black holes with planar horizons. Sen’s

approach is not commonly used in the holographic context. As far as we know, the only

paper considering applications of the Sen’s entropy function method in the AdS/CMT

context is [13]. More generally, attractors were discussed in relation to the holographic

entanglement entropy in [14].

In this paper we focus on dyonic black holes, as these are examples for which obtain-

ing zero temperature results is possible but particularly hard: no closed-form solutions are

known and also numerical techniques experience problems at vanishing temperature. We

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note that holography of electrically charged dilatonic branes was studied in [15] for the

case of constant dilaton potential and for exponential coupling to the field strength. Later,

dyonic dilatonic branes were investigated in [16]. In the latter, the magnetic charge was in-

cluded via SL(2,R) rotations, which was achieved by including an axion to the theory [17].

More general dyonic black holes in the absence of an axion exist [18], and they are substan-

tially different from the ones obtained via SL(2,R) rotation. The most important difference

is that they have a Reissner-Nordström-like structure in the time and radial components

of the metric, with two horizons. This guarantees the existence of the extremal limit, in

contrast to the case of dyonic solutions generated via a SL(2,R) rotation: these have a

Schwarzschild-like structure in the time and radial components of the metric. This will be

crucial in this present work, in particular for the application of Sen’s method. There exist

also numerical investigations about the holography of charged dilatonic branes at finite

temperature [19] for several couplings of the scalar with the field strength.

The results that we obtain for the horizon data are written in terms of the charges

of the black holes and the coupling constants of the theory. These explicit results allow

us to analyze the behavior of the conductivities in field theory dual in terms of the rank

of the gauge group N , of the magnetic field applied and of the charge density. The Hall

conductivity is universal, and is the same as the case of AdS-Reissner-Nordström, i.e. it

reads σxy = ρ/B. When the potential is just a cosmological constant, we show that the

thermoelectric conductivity at zero temperature is given by

αxy =
1

3γ

√
Q
3B

N3/2 , (1.2)

withQ ≡ g2
4ρ the normalized charge density of the black hole and γ a dimensionless constant

which fixes the relation between metric and gauge field from supergravity. g4 is the gauge

theory coupling in the gravity action, which is chosen such that the Einstein-Hilbert and

gauge kinetic terms have the same scaling with N . The constant potential model is used

as a prototype for more involved dilaton theories. In particular, we consider theories

with exponential coupling to the field strengths and exponential scalar potentials, and a

theory with quadratic dilaton expansion in both the coupling to the field strengths and the

potential. We calculate the electric and thermoelectric conductivities for these theories as

well. For all these theories, we also compute the ratio between the heat conductivities and

the temperature under the assumption that this is a finite quantity at T = 0. Surprisingly,

κxx/T = κxy/T for the constant scalar potential model. Moreover, the scaling with N is

κxx/T = κxy/T ∼ N3/2, similarly to the other conductivities.

This paper is organized as follows: in section 2 we briefly review Sen’s entropy function

method. In section 3 we give an example of our approach by considering the well-known

AdS-RN black hole solution, and show that the near-horizon data which we obtain using

Sen’s entropy function coincides with the one directly obtained from the black hole solu-

tion. In section 4 we review the computation of conductivities in terms of horizon data,

highlighting the result which are the starting point for ourw subsequent evaluation of the

conductivities. Section 5 is devoted to the computation of the horizon data for some cases

of Einstein-Maxwell-dilaton theories. Moreover, we express the conductivities in terms of

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the extremal black hole charges and gauge coupling of the potential. We briefly analyse

the S-duality transformation properties of the attractor equations and conductivities. In

section 6 we conclude with an outlook. Appendix A contains details about the black hole

solutions to Einstein-Maxwell-dilaton theories in the spherical case. Appendix B contains

details about the numerical UV completions of some of the solutions considered.

2 Entropy function

The entropy function method was developed by Sen [12, 20], and was first applied for the

computation of the entropy and the study of the attractor mechanism [21, 22] for black

holes with spherical symmetry. As discussed below, it can be shown to be also valid for

black holes with planar symmetry. The near-horizon geometry of an extremal dyonic black

hole in four dimensions is AdS2×S2 [23], and in the planar limit this becomes AdS2×R2.

The presence of both electric and magnetic charges assures that the space will have such a

near-horizon geometry. When the black hole is only electrically (or magnetically) charged,

the planar solution may have Lifshitz scaling symmetry as near-horizon geometry, and

Sen’s formalism may not be applied for these cases.

For dyonic black holes, Sen’s method provides us with an efficient way to compute

the near-horizon data, i.e. the near-horizon metric, the scalar and gauge fields on the

horizon, and consequently the entropy density of these black holes. The starting point of

the formalism is to consider that the near-horizon geometry of the planar black hole1 is

AdS2 × R2, whose general form is written as

ds2 = v

(
−r2dt2 +

dr2

r2

)
+ wd~x2, (2.1)

where the constants v and w are the AdS2 radius and the R2 radius,2 respectively. The

curvature associated to the near-horizon metric (2.1) is

R = −2

v
. (2.2)

The scalar and vector fields are constants for this geometry and are written as

φs = us, F
(A)
rt = eA, F

(A)
θφ = BA, (2.3)

where eA and BA are related to the integrals of the magnetic and electric fluxes, which are

in turn related to the electric and magnetic charges, respectively. The attractor mechanism

states that the value of the scalars on the horizon of the extremal black hole is independent

of any asymptotic condition at infinity. This value is completely determined by the electric

and magnetic charges of the black hole. The function f(us, v, w, eA, pA) is defined as the

Lagrangian density
√
− det gL evaluated for the near-horizon geometry (2.1) and integrated

over the planar horizon variables [12, 20],

f(us, vi, eA, pA) =

∫
dxdy

√
− det gL. (2.4)

1Here we adapt the method to account for planar black holes. The original formalism deals with the

spherical case only.
2The volume of R2 is infinite, but we will only deal with finite densities in the whole paper.

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We extremize this function with respect to us, v, w and eA by

∂f

∂us
= 0,

∂f

∂v
= 0,

∂f

∂w
= 0,

∂f

∂eA
= QA, (2.5)

where the first equation is the equation of motion for the scalar, and the second and third

are the equations of motion for the metric. Next, one defines the entropy function

E(~u,~v,~e, ~q, ~p) ≡ 2π[eAQ
A − f(~u,~v,~e, ~p)]. (2.6)

The equations that extremize the entropy function are

∂E
∂us

= 0,
∂E
∂v

= 0,
∂E
∂w

= 0,
∂E
∂eA

= 0 , (2.7)

and are called the attractor equations. At the extremum, this new function equals the

entropy of the black hole

SBH = E(~u,~v,~e, ~q, ~p). (2.8)

In the context of planar black holes the horizon has infinite area, so we consider the entropy

density, since this is a finite quantity. The solutions of the equations (2.7) are the near-

horizon data that will be used later to compute the conductivities.

The attractor mechanism is independent of supersymmetry, and relies only on the

near horizon geometry [12], which is AdS2 × R2 in our case. In the study of spherical

black holes with AdS2 × S2 near-horizon geometry, the attractor mechanism is present

even after the inclusion of α′ corrections [12, 24–28]. It is now understood that the long

throat of AdS2 is the basis of the attractor phenomena [12, 29, 30]. Since the AdS2 × R2

near-horizon geometry is the starting point of the construction of Sen’s entropy function,

finding solutions to the attractor equations guarantees that the attractor mechanism exist

for the theories analyzed.

Notice that the entropy function method does not require the use of boundary terms.

It was shown in [31] that a similar formalism can be obtained by replacing the Einstein-

Hilbert term of the action by a boundary term, and the results obtained are exactly the

same as the ones obtained by using the original method. In this paper we will use the

original method developed by Sen.

3 Dyonic AdS-RN planar black hole

As one example of application of the entropy function we will consider the dyonic AdS-RN

case without scalars.3 As we know the planar black hole solution for this model, we can

compare the results obtained by applying Sen’s method with the ones obtained by taking

the near-horizon limit of this dyonic solution. We adopt the same units as in [1]. The

theory contains just a metric and a gauge field, and we consider also a constant potential.

The action is written as

S =

∫
d4x
√
−g
[

1

2κ2
4

(
R+

6

L2

)
− 1

4g2
4

FµνF
µν

]
. (3.1)

3We emphasize that there is no new result in this section. This is just an illustration of how one can

obtain near-horizon data using the entropy function method.

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Here, L is the AdS4 radius, and 2κ2
4 = 16πGN . The gauge coupling g4 is assumed to scale

as ∼ N−3/4. This ensures that the gravity and gauge kinetic terms have the same scaling

with N . The relation between the gravity coupling κ2, the AdS4 radius, and the rank N

of the gauge group of the field theory is given by [2]

2L2

κ2
4

=

√
2N3/2

6π
. (3.2)

The equations of motion admit the solution

ds2 =
L2

u2

(
−f(u)dt2 +

du2

f(u)

)
+
L2

u2
(dx2 + dy2), (3.3)

f(u) = 1−
(

1 +
u2

+µ
2 + u4

+B
2

γ2

)(
u

u+

)3

+

(
u2

+µ
2 + u4

+B
2

γ2

)(
u

u+

)4

, (3.4)

F =
µ

u+
du ∧ dt+Bdx ∧ dy. (3.5)

In this coordinate system the horizon of the black hole is located at u+, the asymptotic

region is achieved when u → ∞. B is the magnetic field and µ is the chemical potential.

The temperature and the constant γ2 are given by

T =
1

4πu+

(
3−

u2
+µ

2 + u4
+B

2

γ2

)
, γ2 =

2g2
4L

2

κ2
4

. (3.6)

Notice that, as g4 ∼ N−3/4, γ is independent of N . The black hole becomes extremal when

u2
+µ

2 + u4
+B

2

γ2
= 3. (3.7)

In quantum field theory, the charge density ρ is defined as the expectation value of the

charge operator, J t, which is the time component of a conserved current on flat space-time

(∂iJ
i = 0). The holographic dual of this current is the U(1) gauge field, and so the standard

holographic dictionary gives

ρ = 〈J t〉 =
δSon-shell

δAt,bdry
. (3.8)

This variation of the on-shell action with respect to the boundary value of the gauge field

At reads [9]

ρ =

√
−g
g2

4

F tr
∣∣∣∣
bdry

, (3.9)

where the “bdry” subscript denotes that the expression should evaluated at the boundary.

Substituting the RN action and gauge field, we obtain (see [1])

ρ =
2L2

κ2
4

µ

u+γ2
. (3.10)

Notice that the above expression is independent of the radius, even without taking the limit

r → boundary. This important feature is exhibited in a broader class of models, and we

shall comment on it later on. As we intend to make a comparison between the quantities

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computed from the full extremal solution with the ones obtained via the entropy function,

we write all of them in terms of the charge density, since this is the quantity that appears

in the entropy function. So, using (3.7), the value of u+ for the extremal black hole, is

given by the relation
1

u2
+

=

√
1

3γ2

(
B2 + g4

4ρ
2
)
. (3.11)

In order to see how the AdS2 × R2 near-horizon geometry arises from the full extremal

solution, we Taylor expand the function f(u) around the horizon:

f(u) ≈ f(u+) + (u− u+)f ′(u+) +
(u− u+)2

2
f ′′(u+), (3.12)

where the primes define derivatives with respect to u. The first term in the expansion is

zero due to the definition of the horizon of the black hole, and the second one is zero due

to the fact that we consider extremal black holes. Notice that for the extremal black hole

f ′′(u+) =
12

u2
+

. (3.13)

The metric (3.3) becomes

ds2 ≈ L2

u2
+

(
− 6

u2
+

(u− u+)2dt2 +
u2

+

6

du2

(u− u+)2

)
+
L2

u2
+

(dx2 + dy2). (3.14)

Defining new coordinates as

r ≡ u− u+, τ ≡
6

u2
+

t, (3.15)

and the metric and gauge field will be

ds2 ≈ L2

6

(
−r2dτ2 +

dr2

r2

)
+
L2

u2
+

(dx2 + dy2), (3.16)

F =
ρg2

4u
2
+

6
dr ∧ dτ +Bdx ∧ dy. (3.17)

This just shows that the AdS-RN planar black hole has AdS2×R2 near-horizon geometry.

We now show how these results are recovered within Sen’s entropy function formalism.

For this purpose, we define the near-horizon metric and the gauge field as

ds2 = v

(
−r2dτ2 +

dr2

r2

)
+ w(dx2 + dy2), (3.18)

F = e dr ∧ dτ +Bdx ∧ dy. (3.19)

For this background and fields, the Lagrangian reads

√
−gL =

1

κ2
4

(
−w +

3

L2
vw

)
+

w

2g2
4v
e2 − v

2g2
4w

B2, (3.20)

and the entropy function is just

E = 2π[eQ−
∫
dxdy

√
−gL]. (3.21)

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Computing derivatives with respect to the fields we obtain the attractor equations

e = g2
4

v

w
Q̃, (3.22)

w

v2
e2 +

1

w
B2 − 6g2

4

κ2
4L

2
w = 0, (3.23)

2g2
4

κ2
4

− e2

v
− v

w2
B2 − 6g2

4

κ2
4L

2
v = 0, (3.24)

where Q̃ = Q/Vol R2. The solution to the system is written as

e =
g2

4Q̃

2

√
γ2

3(g4
4Q̃

2 +B2)
, v =

L2

6
, w = L2

√
1

3γ2
(g4

4Q̃
2 +B2), (3.25)

where the definition of γ2 is given in (3.6). The constant Q̃ is a parameter in Sen’s entropy

function method that is proportional to the charge density ρ. In order to obtain the same

near-horizon we make the identification

ρ ≡ Q̃. (3.26)

Comparing the above results with (3.16) and (3.17), we see that we have obtained exactly

the same near-horizon metric and gauge fields via the entropy function. The extremal AdS-

RN theory is a particular example of the model with quadratic expansion in the dilaton

field, as we will discuss in subsection 5.3.

4 DC conductivity from horizon data

The computation of conductivities from horizon data was developed for theories exhibiting

charge dissipation. This analysis began in the context of massive gravity [7], and subse-

quently was performed in the context of Q-lattices [8, 32], where the axionic term in the

action is responsible for breaking translation symmetry. In all of these cases, the black

hole solution considered is electrically charged. The generalization to include a dyonic

black hole was recently developed in [3, 10]. Since the conductivity expressions obtained

there are the starting point for the analysis of this present paper, we review them here.

The theory considered in [3, 10] is given by4

S =

∫
d4x
√
−g
[

1

16πGN

(
R− 1

2
[(∂φ)2 + Φ(φ)

(
(∂χ1)2 + (∂χ2)2

)
]− V (φ)

)
− Z(φ)

4g2
4

F 2

]
,

(4.1)

where φ is a real scalar field and the couplings Φi(φ), Z(φ) and V (φ) depend only on the

real scalar. F is an Abelian field strength. We retain the explicit g2
4 dependence in the

action to be able to track the N dependence (recall that g4 ∼ N−3/4 in our case). At the

4In our conventions, the potential appears in the action with a minus sign, and we write the gauge

coupling g24 explicitly.

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moment the functions Φ, Z and V are arbitrary, but in order to guarantee that there exists

an asymptotically AdS solution, the potential must satisfy the conditions

V (0) = − 6

L2
, V ′(0) = 0, (4.2)

where the prime denotes derivative with respect to the argument (i.e field value). The

existence of an AdS solution ensures that this four-dimensional spacetime is dual to a

three-dimensional conformal field theory. At this point it is instructive to recall that

Z ′(0) = 0 (4.3)

ensures that the AdS-RN black hole with vanishing scalar is a solution to the EOM. This

describes a conformal field theory at finite chemical potential. However, (4.3) not essential

for the existence of a CFT dual [3].

The two fields χ1, χ2 in (4.1) are auxiliary scalars which may be switched on to

introduce a spatial inhomogeneity, for example by setting

χ1 = k1x, χ2 = k2y. (4.4)

For isotropic theories, the background takes the form

ds2 = −U dt2 + U−1dr2 + e2V (dx2 + dy2), (4.5)

A = a(r)dt−Bydx, (4.6)

k1 = k2 ≡ k , (4.7)

where U,V , a and φ are functions of r only. The form of the action (4.1) is very general,

and describes the Q-lattice models in particular [33]. The Einstein-Maxwell-dilaton theory,

which is the focus of this work, is obtained by taking Φ(φ) = 0.

The expressions for electric, thermoelectric and heat conductivities for dyonic black

holes are respectively given by5

σxx =
e2V k2Φ(2κ2

4g
4
4ρ

2 + 2κ2
4B

2Z2 + g2
4Ze

2V k2Φ)

4κ4
4g

4
4B

2ρ2 + (2κ2
4B

2Z + g2
4e

2V k2Φ)2

∣∣∣∣
r+

, (4.8)

σxy = 4κ2
4Bρ

κ2
4g

4
4ρ

2 + κ2
4B

2Z2 + g2
4Ze

2V k2Φ

4κ4
4g

4
4B

2ρ2 + (2κ2
4B

2Z + g2
4e

2V k2Φ)2

∣∣∣∣
r+

, (4.9)

αxx =
2κ2

4g
4
4sρe

2V k2Φ

4κ4
4g

4
4B

2ρ2 + (2κ2
4B

2Z + g2
4e

2V k2Φ)2

∣∣∣∣
r+

, (4.10)

αxy = 2κ2
4sB

2κ2
4g

4
4ρ

2 + 2κ2
4B

2Z2 + g2
4Ze

2V k2Φ

4κ4
4g

4
4B

2ρ2 + (2κ2
4B

2Z + g2
4e

2V k2Φ)2

∣∣∣∣
r+

. (4.11)

κ̄xx =
2κ4

4g
2
4s

2T (2κ2
4B

2Z + g2
4e

2V k2Φ)

4κ6
4g

4
4B

2ρ2 + κ2
4(2κ2

4B
2Z + g2

4e
2V k2Φ)2

∣∣∣∣
r+

, (4.12)

κ̄xy =
4κ4

4g
4
4s

2TρB

4κ4
4g

4
4B

2ρ2 + (2κ2
4B

2Z + g2
4e

2V k2Φ)2

∣∣∣∣
r+

. (4.13)

5The expressions given in [10] for these conductivities are obtained by setting 16πGN = 2κ2
4 → 1, g24 → 1.

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In these expressions, ρ is the charge density and s is the entropy density, which are

given by

ρ =
Ze2V a′

g2
4

∣∣∣∣
r+

, s =
4πe2V

16πGN

∣∣∣∣
r+

. (4.14)

The charge density is now evaluated at the horizon of the black hole, although it describes

a field theory (boundary) quantity. This is possible due to its radius independence (see [9]).

In short, this feature follows from the fact that the gauge equation of motion has the form

∂µ
(√
−gZ(φ)Fµt

)
= 0, (4.15)

which in translationally invariant set-ups implies the radial independence of
√
−gZ(φ)F rt.

Note that (3.9) is modified in presence of a dilatonic coupling, where it reads ρ =√
−gZ(φ)

g24
F tr.

The conductivity expressions given above are valid for all temperatures. They become

very simple when Φ(φ) = 0. In the next sections we will consider precisely this case. Then,

the off-diagonal non-zero electric and thermoelectric conductivities are always given by

σxy =
ρ

B
, αxy =

s

B
. (4.16)

In the next section we will derive the entropy density for some specific models in which

Φ(φ) = 0. The electric Hall conductivity, σxy as given by (4.14) is already expressed in

terms of the electric and magnetic charges of the black hole. Below we will find a similar

expression for αxy in terms of the parameters of the black hole, i.e. the electric charge,

the magnetic charge and the gauge coupling of the potential. This requires to calculate

the entropy density in particular. In subsection 5.4, we compute the ratio between the

heat conductivity and the temperature for all the theories analyzed, assuming that this is

non-zero at zero temperature.

5 DC conductivities at T = 0 for Einstein-Maxwell-Dilaton theories

Our strategy for obtaining explicit expressions for the conductivities will be following: first

we construct Sen’s entropy function for an EMD theory with coupling Z(φ) and potential

V (φ) unspecified. Then we use these functions to derive the general form of the attractor

equations for this class of theories. Then, in the subsections, we analyze the equations for

specific choices of Z(φ), V (φ) and use the solutions to compute conductivities.

The four-dimensional EMD theory with action

S =

∫
d4x
√
−g
[

1

2κ2
4

(
R− 1

2
∂µφ∂

µφ− V (φ)

)
− Z(φ)

4g2
4

FµνF
µν

]
, (5.1)

involves the metric gµν , a gauge field Aµ and a real scalar φ, which is the dilaton. As stated

before, this theory is obtained by taking Φ(φ) = 0 in (4.1). Using (2.1), (2.2) and (2.3), we

compute the Lagrangian in the near-horizon region, i.e.

√
−gL =

1

16πGN
(−2w − wvV (uD)) +

Z(uD)

2g2
4

(w
v
e2 − v

w
B2
)
, (5.2)

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where uD is the value of the dilaton field on the horizon of the black hole. The entropy

function (2.6) is then

E = 2π[eAQ
A −Vol R2√−gL]. (5.3)

The attractor equations for this system are

Q

Vol R2
− Z(uD)

g2
4

w

v
e = 0, (5.4)

Z(uD)

2g2
4

(
w

v2
e2 +

B2

w

)
+

w

16πGN
V (uD) = 0, (5.5)

2

16πGN
− Z(uD)

2g2
4

(
1

v
e2 +

v

w2
B2

)
+

v

16πGN
V (uD) = 0, (5.6)

− 1

2g2
4

∂Z(uD)

∂uD

(w
v
e2 − v

w
B2
)

+
wv

16πGN

∂V (uD)

∂uD
= 0. (5.7)

Using (5.4) we eliminate Q from (5.3), and obtain

E = 2πVol R2

[
1

(16πGN )
(2w + wvV (uD) +

Z(uD)

2g2
4

(w
v
e2 +

v

w
B2
)]
. (5.8)

We combine equations (5.5) and (5.6) in such a way as to obtain

V (uD) = −1

v
, (5.9)

Z(uD)

2g2
4

(
e2

v2
+
B2

w2

)
=

1

(16πGN )

1

v
, (5.10)

and replacing this in (5.3), we obtain

E =
4πwVol R2

16πGN
=
wVol R2

4GN
=

A

4GN
. (5.11)

which shows that this is in agreement with the Hawking formula.

We now aim at using equations (5.4), (5.5), (5.6), (5.7) to evaluate the conductivity

formulae given in the previous section, i.e. (4.8), (4.9), (4.10), (4.11), (4.12), (4.13). This

requires to establish a map between the metric elements and fields appearing in Sen’s

formalism with the metric and fields appearing in the conductivity formulae. One example,

as will be shown below, is the the ratio e/v. Both the electric field e and the AdS2 radius

v appears in Sen’s formalism. This ratio is related to the quantity a′(r) appearing in the

expressions for the conductivity through equation (5.21). We will clarify these issues in

what follows.

The field strength for the theory (4.1) is written in terms of the quantities a′(r) and

B such that

F = a′(r)dr ∧ dt+Bdx ∧ dy. (5.12)

This expression is valid anywhere in the bulk. Let us now consider the field strength and

the metric in the near-horizon region. Notice that the function U(r) appearing in (4.5) can

be Taylor expanded around the horizon of the black hole as

U(r) ≈ U(rH) + (r − rH)U′(rH) +
(r − rH)2

2
U′′(rH) +O(r3). (5.13)

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The first term in this expansion vanishes at the horizon by definition, and the linear term

vanishes for extremal black holes. So in the near-horizon region the metric is written as

ds2 = −(r − rH)2

2
U′′(rH)dt2 +

2

(r − rH)2U′′(rH)
dr2 + e2V (rH)(dx2 + dy2). (5.14)

In order to see how the AdS2 × R2 geometry emerges, we need to choose an appropriate

coordinate system. For our case this is

r − rH → ρ̃, t→ 2τ

U′′(rH)
, (5.15)

so that

ds2 =
2

U′′(rH)

(
−ρ̃2dτ2 +

dρ̃2

ρ̃2

)
+ e2V (rH)(dx2 + dy2), (5.16)

As we can see, the metric has AdS2 × R2 as its near horizon geometry, as expected. A

direct comparison with (2.1) shows that the term multiplying the AdS2 part of this metric

is identified with v, and the term multiplying the R2 part is identified with w. Also, under

the change of coordinates (5.15), the field strength changes, since it transforms as a tensor.

It becomes

F =
2a′(rH)

U′′(rH)
dρ̃ ∧ dτ +Bdx ∧ dy. (5.17)

Again, a direct comparison with (2.3) tells us that the (ρ̃τ) component of the field strength

is identified with e, and the angular part containing the magnetic field is the same. This

provides us with the quantities that map the horizon data obtained via Sen’s entropy func-

tion to the quantities appearing in the expressions (4.8), (4.9), (4.10), (4.11), (4.12), (4.13).

They are written as

v =
2

U′′(rH)
, w = e2V (rH), e =

2a′(rH)

U′′(rH)
= va′(rH). (5.18)

The main ingredient used in the computation of the conductivities from section 4 is

that the currents in the boundary theory are related to quantities in the bulk that do not

depend on the radial coordinate, see [9] and [10] and references therein. This means that

the boundary charge density for the theory (4.1) is given by the horizon expression

ρ =
Z(uD)wa′(rH)

g2
4

, (5.19)

which replaces (3.9). In our notation At is just a. This vanishes at the horizon but its first

derivative a′ does not, as can be seen from equation (B.7) in appendix B.

The last important quantity is the entropy density, which is computed by direct use

of Hawking formula. This is just

s =
4πw

16πGN
. (5.20)

Notice that v and w in these expressions come from the metric elements. Using the iden-

tification (5.18), we see that
e

v
= a′(rH), (5.21)

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so, by replacing this in the charge density and using the attractor equation (5.4), the charge

density is related to the quantity Q̃ in Sen’s entropy function as

ρ = Q̃. (5.22)

We note that this result coincides with (3.26). This provides a consistency check on the

approach presented above.

We see that we can obtain the AdS2 factor in the near horizon region only when the

linear term in the Taylor expansion (5.13) vanishes, i.e. U′(rH) = 0. As the temperature

of the black hole is directly proportional to this factor, T ∼ U′(rH), this requires that the

black hole be extremal in order to obtain the AdS2 × R2 near horizon geometry. For the

planar AdS-RN black hole reviewed in section 3, the extremality can be achieved even if

we set one of the charges to zero, i.e. it is not necessary to have a dyonic black hole to have

T = 0. This can be easily seen by setting the electric or the magnetic charge to zero in

the expression for the temperature in (3.6). But this is a feature of the Einstein-Maxwell

theory, and it is absent in the EMD theory, in which the coupling to the field strength

Z(φ) is non-constant. For black holes with spherical horizons, as shown in appendix A,

the magnetically charged spherical black hole has a temperature given by (A.5), which is

exactly the temperature of the Schwarzschild black hole. The only way to achieve T = 0

in this case is by having infinite mass M , which is physically unacceptable. This means

that the non-constant coupling Z(φ) made the magnetically (or electrically) charged black

hole for the EMD theory to have the same structure as the Schwarzschild black hole, with

just one horizon and no zero temperature limit, unlike the spherical Reissner-Nordström

black hole, which has two horizons and well defined zero temperature limit. But when

the spherical black holes of the EMD theory are dyonic, the black hole has two horizons

in the same way as the Reissner-Nordström black hole. Then, the temperature is given

by (A.12), and it is exactly zero if and only if the horizons coincide. In other words, the

black holes with spherical horizons for the EMD theories can only be extremal if it contains

electric and magnetic charge at the same time, i.e. it is dyonic. We would expect the same

behavior for planar black holes, but it turns out that the potential must be non-zero in

order to obtain regular planar black holes.6 If we turn on the potential, as is the case in this

paper, we may have a well defined extremal limit with AdS2 × R2 near-horizon geometry,

even for cases in which the black hole contains only electric charge for instance. This is

due to the fact that U′(rH) may be put to zero when the potential is non-trivial. But

having AdS2×R2 near-horizon geometry in the extremal limit does not guarantee that the

electric field or dilaton field are non-zero or finite in the extremal limit when evaluated on

the horizon. We must obtain the solutions to the attractors to show this explicitly. Since

DC conductivities are written in terms of fields evaluated on the horizon of the black hole,

its finiteness is directly related to the finiteness of these fields on the horizon. Here, in this

paper, we obtain non-trivial and finite fields on the horizon only when the black hole is

dyonic, so, the DC conductivities are also non-trivial and finite. This analysis also tells us

6Regularity of the solutions in the extremal limit is guaranteed if the attractor equations admit solutions,

since they are also solutions to the equations of motion with AdS2 × R2 near-horizon geometry.

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that the entropy function formalism only fails when the black hole does not contain AdS2

factor in the near-horizon limit, or when one of the fields on the horizon is infinite, but

both limitations are absent for the dyonic black holes considered in the paper.

With these results, we are now able to compute conductivities explicitly for differ-

ent theories. We will analyze three models separately. First we will write the solution

to the attractor equations and then we combine with the expressions for the conductivi-

ties (4.8), (4.9), (4.10) and (4.11).

5.1 Massless scalar

In this case the potential assumes the form

V (φ) = − 6

L2
, (5.23)

where we wrote V (φ) only to keep notation, since the potential is a constant and does

not really depend on the dilaton field φ. Equation (3.2) gives a map between the number

of colors in the gauge theory N and the constants in the gravity theory, i.e. Newton’s

constant GN and the AdS4 scale L. As the constant potential satisfies the requirement

given by equations (4.2), the EMD theory with such potential is dual to a three-dimensional

conformal field theory. Although this is the simplest potential one can consider, to the best

of our knowledge, a full dyonic black hole solution for this theory is not known, so writing

explicitly the conductivities of the previous section in terms of the black hole parameters

at finite temperature is not possible. In this subsection, we show that it is possible to solve

the attractor equations for this theory and consequently for the extremal case, we can

write the conductivities explicitly in terms of the black hole parameters for the extremal

case. Moreover, we also use equation (3.2) and write the thermoelectric conductivity αxy
in terms of the rank of the gauge group N .

The attractor equations for this theory admit the solution

Z(uD) = g2
4

Q̃

B
, e =

√
L2B

6(16πGN )Q̃
, v =

L2

6
, w =

√
L2(16πGN )Q̃B

6
. (5.24)

The solution is independent of the functional form of the coupling Z(uD). The entropy

density follows from the solution for w,

s = 4π

√
L2

6

Q̃B

(16πGN )
=

1

3γ

√
QB

3
N3/2 , (5.25)

where we defined Q ≡ g2
4ρ as the normalized charge density of the black hole and we

used (3.2) to rewrite s in order to show explicitly the scaling of this result with the rank

N of the gauge group. Note that Q is independent of N , indeed we have

ρ = Q̃ =
Q
g2

4

= Q 2L2

γ2κ2
4

=

√
2Q

6πγ2
N3/2 . (5.26)

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Replacing these results in (4.8), (4.9), (4.10) and (4.11) we arrive at our first results,

σxx = 0, σxy =
Q̃

B
=

√
2Q

6πγ2B
N3/2, (5.27)

αxx = 0, αxy = 4π

√
Q̃

6B

L2

16πGN
=

1

3γ

√
Q
3B

N3/2. (5.28)

Notice that we just rewrote (5.27) for completeness, since this is a general result for EMD

theories. The thermoelectric conductivity at zero temperature (5.28) for constant potential

is obtained for the first time here. We note that both thermoelectric and Hall conductivities

scale as N3/2, as is generically expected for theories in 2+1 dimensions [2].

If the dilaton is set to zero7 in the EMD theory that is being considered in this sec-

tion, then these conductivities should match the results computed for the AdS-RN black

hole (1.1), since the potential used here matches the cosmological constant of AdS (it is the

same potential as in section 3). In supergravity theories, the only other non-trivial cou-

pling, Z(φ), is generically an exponential of the type Z(φ) = eγ̃φ for an arbitrary constant

γ̃, so it reduces to Z(0) = 1 if the dilaton is set to zero. This comparison with section 3

can be used as a consistency check on our new results.

However, notice that in (1.1) the entropy density is written in terms of the charges, so

in order to make a comparison, we first need to express explicitly the conductivities of the

extremal AdS-RN case in terms of the charges too. Precisely, eqs. (5.19) and (5.20) give

the corresponding expressions for s and ρ, which can be evaluated for the horizon data

of (3.25), leading to the non-trivial result

σxy =
Q̃

B
=

√
2Q

6πγ2B
N3/2, (5.29)

αxy =
s

B
=

4π

B

L2

2κ2

√
1

3γ2

(
B2 + g4

4ρ
2
)

=
N3/2

3B
√

2γ

√
1

3
(B2 +Q2). (5.30)

On the other hand, if we set φ = 0, then the first equation in (5.24) implies the constraint

Q = B, (5.31)

which can be inserted into eqs. (5.29) and (5.30) to give

σxy =

√
2

6πγ2
N3/2, αxy =

N3/2

3
√

3γ
. (5.32)

Indeed, the same results are obtained by inserting (5.31) into the results of this section,

eqs. (5.27) and (5.28). In general terms, the EMD theory reduces to Einstein-Maxwell

theory (whose black hole solution is the AdS-RN solution) with an extra constraint on the

charges, eq. (5.31), which results from the dilaton equation of motion in the φ = 0 case.

From this analysis, it is natural to assume that the conductivities of the EMD theory scales

with the same powers of N as in the case of the conductivities of Einstein-Maxwell theory.

7The same argument applies if we set the dilaton to a constant, instead of zero. We consider only the

vanishing dilaton case here for simplicity.

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5.2 Exponential couplings

This model is defined by8

Z(φ) = eγφ, V (φ) = 2βe−δφ , (5.33)

which corresponds to a Liouville potential. Black holes with cylindrical symmetry for

this theory were considered in [34]. This theory was also considered on page 23 in [35]

for electrically charged black holes. For black holes with spherical horizons, the analysis

done so far were all numerical: there are no analytical result for conductivities of these

theories. As stated before, we will express the conductivities here analytically for the zero

temperature case.

As V ′(0) 6= 0, equations (4.2) are not fully satisfied, so this system does not admit

AdS4 vacuum solution. However, there are some interesting examples of top-down theories

that are very difficult to treat analytically, even with the present formalism. For instance,

if V (φ) = − 6
L2 cosh (φ/

√
3), Z(φ) = 1/cosh (φ

√
3), which is a string-theory inspired model,

the attractor equations cannot be solved analytically due to the high powers of the variables

in the algebraic equations. The example of this subsection is used as an approximation of

these theories, provided that the value of scalar field on the horizon is large, so that the

hyperbolic function can be approximated by an exponential. We shall obtain results for

exponential potential and coupling and use them for cases in which the condition of large

values for the scalar is satisfied.

In this model, γ and δ are parameters defining the theory, and the constant β must

be related to the AdS4 radius, as will be done in the end of this subsection. As discussed

in [34, 35], the case γδ = 1 is of special interest since the associated models arise within

string theory. The formulae we write are valid for general values of γ and δ. Using (5.4)

and (5.7) we have

e

v
=

g2
4Q̃

weγuD
, (5.34)

e2

v2
=
B2

w2
− 4βδ

γ

g2
4

(16πGN )
e−(δ+γ)uD . (5.35)

Replacing the last equation in (5.5) we have

B2

w2
= −2β

g2
4

(16πGN )

(
1− δ

γ

)
e−(δ+γ)uD . (5.36)

Combining these three equations, we obtain as solution to the attractor equations

euD =

[
g4

4Q̃
2

B2

(γ − δ)
(γ + δ)

] 1
2γ

, (5.37)

e =
g3

4Q̃

B

√
(γ − δ)

−2β(16πG)γ

[
g4

4Q̃
2

B2

(γ − δ)
(γ + δ)

] δ−3γ
4γ

, (5.38)

8The parameter γ here is an arbitrary constant. This must not be confused with the parameter γ of

equation (3.6), which will not appear in the remaining part of this paper.

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v = − 1

2β

[
g4

4Q̃
2

B2

(γ − δ)
(γ + δ)

] δ
2γ

, (5.39)

w = B

√
(16πG)γ

−2βg2
4 (γ − δ)

[
g4

4Q̃
2

B2

(γ − δ)
(γ + δ)

] δ+γ
4γ

. (5.40)

The entropy density is given by

s = 4πB

√
γ

−2βg2
4(16πG) (γ − δ)

[
g4

4Q̃
2

B2

(γ − δ)
(γ + δ)

] δ+γ
4γ

. (5.41)

Although the simple relation (3.2) does not readily work in this set-up we can provide an

argument that a similar dependence on the rank of the gauge group holds for that case.

First, let us observe that for dimensional reasons the full potential should have form

Vf (φ) = − 6

L2
AdS4

f(φ) (5.42)

where f is a dimensionless function. Our approximate potential (5.33) has a similar form

with dimensionful constant β and dimensionless function exp(−δφ). Then the approxima-

tion mentioned when introducing this model is in fact f(φ|hor) ≈ exp(−δφ|hor) for some

values of φ|hor the dilaton at the horizon, which leaves the dimensionful constant unchanged.

That leads to the conclusion that β ∼ −1/L2, and the entropy density will have the same

dependence in N as in the previous model, i.e. s ∝ N3/2. It is worth mentioning that

in the argument above we only identify the AdS4 scale with the parameter of our model,

while the AdS2 scale remains dynamically determined. Purely from dimensional reasons,

one would expect relation such as

LAdS2 = F
(
Q̃/B, . . .

)
LAdS4 (5.43)

to hold, with dots standing for the parameters of the model. This is indeed the case, if one

compares the above with (5.39) it matches the expectations based on dimensional analysis,

and provides further evidence for the argument given above.

For the model of this subsection, we study the cases in which δ = 1/γ, and the nonzero

conductivities of (4.16) are given by

σxy =
Q̃

B
, αxy =

4πγ√
−2βg2

4(16πG) (γ2 − 1)

[
g4

4Q̃
2

B2

(γ2 − 1)

(γ2 + 1)

] γ2+1

4γ2

. (5.44)

As discussed in [35], the cases γ =
√

3 and γ = 1 are of special interest: γ = 1 arises from

string theory in four dimensions with a vector arising from the NS sector, while γ =
√

3

arises from a Kaluza-Klein reduction. In our case, for γ =
√

3 our conductivity results

reduce to

σxy =
Q̃

B
, αxy =

2π√
(16πG)

√
− 3

β

1

g
2/3
4 21/3

(
Q̃

B

) 2
3

. (5.45)

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0.0 0.2 0.4 0.6 0.8 1.0

0

1

2

3

4

(a) Dilaton φ profile along the bulk.

r̃

φ

0.0 0.2 0.4 0.6 0.8 1.0
0

5

10

15

20

25

(b) Gauge field At profile along the bulk.

r̃

At

Figure 1. Result of RG flow for the case γ = −
√

3, δ = −1/
√

3. In this case, the magnetic field

needs to be B � Q in order for the coupling Z(φ) to be well-approximated by an exponential near

the horizon. We chose B/Q = 100 for this plot. The horizon is at r̃ = 1, the boundary at r̃ = 0.

On the other hand, for γ = 1 we have

σxy =
Q̃

B
, αxy = 2π

g4√
(16πGN )

Q̃

B

√
− 1

β
. (5.46)

For these two special values of γ, which correspond to top-town models, we numerically

computed the bulk solution that connects this solution in the IR to AdS4 in the UV, in

order to check that the attractor mechanism provides the appropriate solution. For more

details on the numerical UV completion of the solutions, see appendix B. Note that the

analytical continuation of the coupling and potential (B.1) usually requires us to select

appropriate values of B and Q such that B � Q, in order to produce a large dilaton at

the horizon. However, when both γ and δ are close to -1, the dilaton becomes large at the

horizon for regular values of the ratio B/Q. The plots are presented in figures 1 and 2.

5.3 Quadratic couplings

This is a simple bottom-up model where both potential and gauge coupling are second

order polynomials, namely

Z(φ) = 1 +
α

2
φ2, V (φ) = − 6

L2
+

β

2L2
φ2. (5.47)

This case was investigated in [19] and may be viewed as an approximation of more com-

plicated top-down models in case where the scalar is small near the horizon. Since the

constants are arbitrary, we expect this model to capture universal features. First we note

that (5.7) reduces to a linear equation in uD, which is the value of the dilaton on the

horizon,

−
αuD

(
2B2

w2 − 2e2

v2

)
g2

4

− 8βuD
16πGL2

= 0 . (5.48)

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0.0 0.2 0.4 0.6 0.8 1.0

0

1

2

3

4

5

6

7

r̃

φ

(a) Dilaton φ profile along the bulk.

0.0 0.2 0.4 0.6 0.8 1.0
0.0

0.1

0.2

0.3

0.4

r̃

At

(b) Gauge field At profile along the bulk.

Figure 2. Result of RG flow for the case γ ' −1, δ ' −1 and B/Q = 1. In this case, the magnetic

field is not required to be large. The horizon is at r̃ = 1, the boundary at r̃ = 0.

Therefore there are two possible solutions. One of them is uD = 0, in which case the

other three attractor equations are solved by the dyonic AdS-Reissner-Nordström geometry

without scalar, as in the example of section 3. A further solution is obtained by assuming

uD 6= 0. Equation (5.4) and (5.7) give

e

v
=

g2
4Q̃

wZ(uD)
, (5.49)

1

2g2
4

e2

v2
=

1

2g2
4

B2

w2
+
β

α

1

(16πG)L2
. (5.50)

We use (5.50) to eliminate the term containing e from (5.10) and write

B2

g2
4w

2
Z(uD) =

1

L2(16πG)

(
6− β

α

)
− β

L2(16πG)
u2
D. (5.51)

We also use (5.49) to eliminate e in (5.50) and write

1

g2
4w

2

(
g4

4Q̃
2

Z(uD)2
−B2

)
=

2β

α

1

L2(16πG)
. (5.52)

Combining equations (5.51) and (5.52) we can write the quadratic equation for u2
D,

u4
D +

u2
D

α

(
1 +

β

(β − 6α)

g4
4Q̃

2

B2

)
+

1

α2

(
1 +

g4
4Q̃

2

B2

(β − 6α)

(β + 6α)

)
= 0. (5.53)

The solution for this equation is

1 +
α

2
u2
D = −g

4
4Q̃

2

B2

β

(3α− 2β)
±

√√√√−3 +
g2

4Q̃
2

B2(β + 6α)

(
(24α− 6β) +

g2
4Q̃

2

B2

β2

(6α+ β)

)
.

(5.54)

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This is the solution to uD written in terms of the charges, found by solving the attractor

equations. The expressions for w, v and e written in terms of the charges as provided by the

attractor equations are rather lengthy, and we don’t write them explicitly here. Instead,

we write them in terms of u2
D, i.e.

e =
v

w

g2
4Q̃(

1 + α
2u

2
D

) , (5.55)

v =
L2(

6− β
2u

2
D

) , (5.56)

w =

√
L2(16πG)B2α

g2
4

(
1 + α

2u
2
D

)
6α− β(1 + αu2

D)
. (5.57)

The entropy density is given by

s = 4πB

√
L2α

g2
4(16πG)

(1 + α
2u

2
D)

6α− β(1 + αu2
D)
. (5.58)

From this result we express the conductivities in terms of the black hole parameters as

σxx = 0, σxy =
Q̃

B
, (5.59)

αxx = 0, αxy = 4π

√
L2α

g2
4(16πG)

(
1 + α

2u
2
D

)
6α− β(1 + αu2

D)
, (5.60)

where again uD is obtained from equation (5.54).

It is natural to ask about the stability of these solutions. In order to tackle this

question we perform a simple analysis: it is likely that possible instabilities are triggered

by the scalar sector. We therefore calculate the equation for scalar fluctuations on the

dyonic AdS-RN background, and then compare the effective mass to the Breitenlohner-

Freedman (BF) bound. Since the near-horizon geometry is translationally invariant in

t, x, y directions, we investigate an Ansatz for the scalar of the form

φ(r, t, x, y) = R(r)e−iωt+k1x+k2y (5.61)

subject to

�AdS2×R2φ− αφ−
2κ2

4g2
4

β

L2
FµνF

µνφ = 0. (5.62)

From this equation, following the logic of [36], we extract the effective mass generated by

the scalar potential and interaction with the background electromagnetic field,

m2
eff =

B2(6α+ β) + g4
4Q̃

2(β − 6α)

6
(
B2 + g4

4Q̃
2
) . (5.63)

The BF bound states that geometry is unstable against scalar perturbations if their mass

satisfies:

m2
eff ≤ −

1

4
. (5.64)

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Two consequences follow from this: first, the possible phase transition is driven by the

electric field in the bulk, which is dual to the chemical potential of the boundary theory,

while the magnetic field tends to stabilize the uncondensed phase by increasing the effective

mass. Second, this instability does not occur for all models (i.e not for all values of α, β):

in the limit

lim
Q̃→∞

m2
eff =

1

6
(β − 6α), (5.65)

a violation of the BF bound (5.64) requires that

6α− β > 3

2
. (5.66)

If this is satisfied, the instability occurs at

Q̃2 =
B2
(
6α+ β + 3

2

)
g2

4

(
6α− β − 3

2

) . (5.67)

This superficial analysis indicates that the model may exhibit a possible quantum critical

phase transition (i.e. a phase transition at T = 0). Instabilities due to fluctuations in

the non-scalar sector are also conceivable. It is also not clear what happens if the con-

dition (5.66) is not satisfied. This question would require a more sophisticated analysis

which we leave for future work. It would also be interesting to investigate the possible

connections with the work of [19] and [37], where the phase structure of similar models is

investigated at finite temperature and without magnetic field.

5.4 Heat conductivity κ/T

The approach presented in this paper may also be applied to the heat conductivity κ. In

particular, we confirm the result of [38] in which a bound on κ/T was derived. In that

paper, it was argued that the heat conductivity is always non-zero at finite temperature, so

long as the dilaton potential is bounded from below. By applying the method advocated

in this paper, we also find finite results for κ/T even for T → 0.

In order to obtain the ratio κ/T within the formalism considered here, we take Φ = 0

in (4.12). This gives
κ̄xx
T

=
s2Z

g2
4

(
ρ2 + B2Z2

g44

) . (5.68)

In the same way, we we take Φ = 0 in (4.13), and write the ratio

κ̄xy
T

=
ρ

B

s2(
ρ2 + B2Z2

g44

) . (5.69)

For general EMD theory as the one given by (5.1), the dimensionless ratio κ/T is strictly

finite at finite temperature (T > 0) if the dilaton potential V (φ) is bounded from below [38].

A question that remains to be answered is whether this ratio continues to be finite as

T → 0. Here, we assume that this is the case, and then write κ/T for the theories

studied in subsections 5.1 and 5.2. We do not write the results for κ/T for the model of

subsection 5.3 explicitly, since the resulting expressions are lengthy and not so enlightening.

However they can be easily found following the same procedure.

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• Massless scalar: for the massless scalar, the term inside the parentheses in the de-

nominator of (5.68) and (5.69) can be simplified using the solution to the attractor

equations (5.24), and the ratio κ/T is written as

κ̄xx
T

=
κ̄xy
T

=
s2

2ρB
, (5.70)

where s and ρ are the entropy and charge densities at zero temperature, which are

given by (5.24) and (5.25). Surprisingly, after using the solution to the attractor

equations (5.24), the values for the ratios κ̄xx/T and κ̄xy/T coincide at T = 0,

κ̄xx
T

=
κ̄xy
T

=
(4π)2

12

L2

(16πG)
=

π

9
√

2
N3/2. (5.71)

Unlike the electric and thermoelectric conductivities (5.27) and (5.28), the final result

does not depend on the electric and magnetic charges of the black hole.

• Exponential couplings: for the exponential coupling model, we take γδ = 1 and

use (5.37) to show that

κ̄xx
T

=
s2

2ρB

(γ2 + 1)

γ2
,
κ̄xy
T

=
s2

2ρB

√
(γ2 + 1)(γ2 − 1)

γ2
. (5.72)

Notice that the entropy density squared is given by

s2 =
(4π)2B2γ2

−2βg2
4(16πG) (γ2 − 1)

[
g4

4Q̃
2

B2

(γ2 − 1)

(γ2 + 1)

] γ2+1

2γ2

, (5.73)

and it is non-zero for γ = 1,
√

3. Different from the massless scalar case, the ratios

κ̄xx/T and κ̄xy/T for this model are not the same at T = 0. For γ =
√

3 we have

κ̄xx
T

=
(4π)2

(−2β)(16πG)

(
g2

4ρ

4B

)1/3

,
κ̄xy
T

=

√
2(4π)2

(−2β)(16πG)

(
g2

4ρ

4B

)1/3

. (5.74)

For γ = 1, we have
κ̄xx
T

=
(4π)2

(−2β)(16πG)

g2
4ρ

2B
,
κ̄xy
T

= 0. (5.75)

We note that the numerical values we obtain for κ/T differ from those obtained in [38],

since in that paper, L2/κ2 ∝ N2 is used instead of (3.2) which is the appropriate expression

for a 2+1-dimensional boundary theory.

5.5 S-duality of the attractor equations and conductivities

In [3], the authors derive conductivities from a set of generalized Stokes equations at

the horizon. For a 3+1-dimensional gravity dual, these equations are invariant under S-

duality. Here we examine the S-duality transformation properties of the attractor equations

considered in the sections above, restricting our attention to theories whose potential is

even and which satisfy Z(−φ) = Z(φ)−1.

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The S-duality transformations are given by

Fµν → Z(φ)
ε̃µνρσ

2
√
−g

Fρσ, φ→ −φ, (5.76)

where ε̃µνρσ is the totally antisymmetric Levi-Civita symbol, with ε̃trxy = 1. If we focus

on horizon of the black hole, then the S-duality transformation results in

− e

v2
→ − 1

vw
Z(uD)B,

B

w2
→ − 1

vw
Z(uD)e, Z(uD)→ 1

Z(uD)
. (5.77)

The S-duality transformation of the charge density ρ is obtained from its definition within

EMD theory, from which we obtain

ρ =
√
−gZ(φ)

g2
4

F tr →
√
−g 1

Z(φ)g2
4

Z(φ)
ε̃trxy√
−g

Fxy =
B

g2
4

. (5.78)

Using Q̃ = ρ we see directly that the transformation for the magnetic field in (5.77) is given

by B → −g2
4ρ. Applying the transformations (5.77) and (5.78) to the attractor equations,

we see that (5.5), (5.6) and (5.7) are invariant, and (5.4) gives a trivial identity. This shows

that the attractor equations are invariant under S-duality transformations. However note

that S-duality is an invariance of the equations of motion, but not of the action, so we may

expect that the entropy function (2.6) is not invariant under S-duality either. In fact, as

was pointed out in reference [20], applying an S-duality transformation gives rise to a new

entropy function in the attractor formalism, which might generate new attractor equations.

Nevertheless, the extremization of both the initial and the S-dual entropy functions yields

the same black hole entropy, which shows that the black hole entropy is invariant under

S-duality.

The result (5.25) for the entropy density contains a square root of the product of

the charges. This comes from the fact that the attractor equations involve squares. For

investigating the S-duality properties of our explicit expressions for the entropy and for

the conductivities, we have to take care of the signs carefully when taking square roots

of quadratic expressions. As an example, the result for the thermoelectric conductivity

of (5.28) should be written as

αxy = sgn(B) 4π

√
L2|Q̃|
6|B|

16πG . (5.79)

Then, the transformed conductivities are given by

σxy → −
B

g4
4Q̃

, αxy → −
4π

g2
4Q̃

√
Q̃B

6

L2

(16πG)
, (5.80)

with Q̃ → ρ. Performing the transformation once more, we see that the conductivities

transform as

σxy → σxy, αxy → −αxy , (5.81)

which is precisely what authors of [3] find.

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The result for κ/T is independent of the absolute values of charges for the massless

scalar model, but again its sign depends on signs of charges. This implies that after

transforming the charges once we have

κ̄xx → κ̄xx, κ̄xy → −κ̄xy . (5.82)

Transforming the charges again results in

κ̄xx → κ̄xx, κ̄xy → κ̄xy , (5.83)

which again is consistent with general transformation laws of [3].

6 Generalizations and conclusions

In this paper we computed conductivities at zero temperature for EMD theories. The

expressions we obtained are written in terms of the extremal black hole parameters, and we

showed that the off-diagonal components of the electric and thermoelectric conductivities

scale as

σxy ∼ N3/2, αxy ∼ N3/2 (6.1)

for a constant potential, where N is the rank of the gauge group of the conformal field

theory dual to the EMD theory. We argued that this should also be the case for theories

with other kinds of potential. We briefly discussed that in the T = 0 limit the EMD presents

different phases, which may be related to quantum phase transitions. All the results were

obtained by applying Sen’s entropy function method in the AdS/CMT context. We also

computed κxx/T and κxy/T for EMD theories assuming that these ratios are finite at

T = 0. For a constant potential, κxx/T is equal to κxy/T , and they also scale as

κxx
T

=
κxy
T
∼ N3/2. (6.2)

Explicit analytical zero-temperature expressions are expected to be very useful in par-

ticular for universality arguments in AdS/CMT, see for instance [5, 6] and references

therein. We expect that the results of this paper may be generalized to more involved

geometries relevant in that context.

Although we investigated the simplest EMD theories, generalizations of this approach

are indeed possible. The two most immediate ones are the following: one direction is to

generalize the equations for conductivities (4.8), (4.9), (4.10), (4.11), (4.12), (4.13) to take

into account multiple scalar fields and gauge fields coupled in the non-minimal way. In

the context of AdS/CFT it is useful to consider maximally supersymmetric supergravities,

and including more scalars and field strengths will guarantee that we can handle the four-

dimensional N = 8 gauged supergravity. Solving the attractor equations for the planar

case, the conductivities of the three-dimensional CFT can be expressed explicitly in terms of

the black hole parameters and of N , just as we did in the paper for simpler cases. Another

possibility is to generalize the computation of the conductivities to higher-dimensional

supergravity theories. This means that the gravity theory will have a Chern-Simons term

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in odd dimensions, which may complicate the computation. The entropy function method

is valid only for even-dimensional gravity theories, but it is possible to perform dimensional

reduction down to even dimensions, treat the even dimensional theory as an effective theory,

and then compute the horizon data, in the same spirit as [13]. That would allow to use the

entropy function method to obtain the conductivities at zero temperature also for these

cases, and in particular for the case most studied in the literature, which is the supergravity

dual of four-dimensional N = 4 Super Yang-Mills theory.

Yet another possibility is to extend the superficial analysis of the phase transition con-

sidered in subsection 5.3. It would be interesting to look not only at the scalar instabilities

but also at the full linearized EMD system on both backgrounds. Moreover, as we saw

in 5.3, the scalar instability occurs only for particular values of parameters α, β, but the

coexistence of phases with and without scalar fields seems to happen also for other values

of those parameters. Exploring this phase diagram would probably require knowledge of

the full solution, and therefore one would probably be forced to invoke some numerical

methods. Apart from this scalar condensation it seems possible that there may be another

phase transition in the regime of vanishing magnetic field, which may lead to a scaling

geometry in the far IR.

Our results imply that the attractor mechanism is a very important ingredient in the

computation of zero temperature conductivities, as well as in the study of quantum phase

transitions.

Acknowledgments

The authors thank Yago Bea, René Meyer, Horatiu Nastase and Nick Poovuttikul for useful

discussions. We also thank the referee of this paper for very constructive suggestions. DF

was supported by an Alexander von Humboldt Foundation fellowship. PG is grateful for

the hospitality of the Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), where

the largest part of this work was carried out. The work of PG is supported by FAPESP

grant 2013/00140-7 and 2015/17441-5.

A Dilatonic black holes

In the introduction we claimed that it was necessary to consider a dyonic black hole solution

in order to obtain a well-defined zero temperature solution. This analysis is based on

spherical black holes, and it must also hold for the planar case. The black hole solutions

for EMD theory are listed below.

• Magnetically charged black holes:

This solution was obtained by Gibbons and Maeda [39] and later it was also obtained

independently by Garfinkle, Horowitz and Strominger [40]. It is written as:

ds2 = −
(

1− 2M

r

)
dt2 +

dr2(
1− 2M

r

) + r

(
r − e−2φ0P 2

M

)
dΩ2

2, (A.1)

e−2φ = e−2φ0

(
1− e−2φ0P 2

Mr

)
, (A.2)

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Frt = 0, (A.3)

Fθφ = P sin θ, (A.4)

T =
1

8πM
, (A.5)

where M is the mass of the black hole, P is the magnetic charge and φ0 is the asymptotic

value of the dilaton at infinity. The dilaton charge is given by an integral over a two-sphere

at spatial infinity9

Σ =
1

4π

∫
dΣµ∇µφ = −e

−2φ0P 2

M
. (A.6)

Because this is a negative quantity, the dilaton contributes with a long-range, attractive

force between black holes. The surface r = 2M is a regular horizon, and the singularity

is at r = Σ. We can obtain the electrically charged solution by applying electric-magnetic

(S-duality) transformations, i.e.

F ′µν =
1

2
√
−g

e−2φε̃µνρσFρσ, φ
′ = −φ, (A.7)

where ε̃µνρσ is the totally antisymmetric Levi-Civita symbol. The gtt and grr part of the

metric is exactly the same as in the Schwarzschild solution, and consequently, the temper-

ature is given by (A.5). There is no classical zero temperature limit for this solution. The

inclusion of the electric charge via SL(2, R) rotation [17] gives the same temperature as the

magnetically charged solution, so it does not have a well-defined classical zero temperature

limit either.

• Dyonic black holes:

The most general dyonic solution was obtained very recently [41]. This solution contains

five independent parameters, which generalizes the one that was given in [18], which con-

tains four independent parameters. The non-extremal solution is written as

ds2 = −e−λdt2 + eλdr2 + C2(r)dΩ2
2, (A.8)

e−λ =
(r − r1)(r − r2)

(r + d0)(r + d1)
, C2(r) = (r + d0)(r + d1), (A.9)

e2φ = e2φ0 r + d1

r + d0
, (A.10)

Frt =
e2φ0Q

(r + d0)2
, Fθφ = P sin θ, (A.11)

T =
1

4π

(r2 − r1)

(r2 + d0)(r1 + d0)
, (A.12)

with

d0 =
−(r1 + r2)±

√
(r1 − r2)2 + 8e2φ0Q2

2
, (A.13)

d1 =
−(r1 + r2)±

√
(r1 − r2)2 + 8e−2φ0P 2

2
. (A.14)

9The dilaton charge Σ should not be confused with the two-sphere surface element dΣµ.

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This solution contains five independent parameters: the electric charge Q, the magnetic

charge P , the value of the dilaton at infinity φ0, and two integration constants, r1 and

r2. The boundary conditions must be imposed on r1 and r2. The singularity is located at

rS = −d0 when d0 > d1, or at rS = −d1 when d1 > d0. Just like the Reissner-Nordström

black hole, this solution contains an outer and an inner horizon, written as r2 and r1

respectively, and the temperature (A.12) is zero when r1 = r2 ≡ r0. Changing coordinates

to ρ = r − r0, the extremal solution is then written as

ds2 = −e−2Udt2 + e2U (dρ2 + ρ2dΩ2
2),

e2U =

(
1±
√

2eφ0Q

ρ

)(
1±
√

2e−φ0P

ρ

)
, (A.15)

e2φ = e2φ0 (ρ±
√

2e−φ0P )

(ρ±
√

2eφ0Q)
, (A.16)

Frt =
e2φ0Q

(ρ±
√

2eφ0Q)2
, Fθφ = P sin θ. (A.17)

Here, we have used isotropic coordinates, i.e. ρ2 = x2
1 + x2

2 + x2
3, and consequently dρ2 +

ρ2dΩ2
2 = d~x2. The horizon is located at ρ = 0. We see that there exist a very well defined

zero temperature limit for the dyonic black holes of the EMD theory, which is not the case

for the electrically (or magnetically) charged black hole. There also exists a massless limit

for the dyonic solution, which is discussed in [42].

The entropy function method works for extremal BPS and non-BPS black holes, and

that is why it is important to consider dyonic black hole solutions, as we did in the text.

B Numerical UV completion

In this paper, we work under the assumption that the IR solution provided by the attractor

equations can be completed towards the UV, in such a way that a full bulk solution displays

AdS4 asymptotics. For the case studied in section 5.2, we provided an explicit check of this

statement by computing the full solution numerically. Here we give some details regarding

the computation behind that verification, which constructs an RG flow from the IR solution

up to the conformal boundary. For the sake of completeness, the codes used to obtain results

of this appendix are available on-line at https://github.com/picalke/Conductivities2016.

The routines were written and executed in Wolfram Mathematica 11.

It is not possible to directly extrapolate the choice (5.33) for the couplings V (φ),

Z(φ) at the horizon, because exponential couplings do not behave near the boundary in

a way compatible with AdS asymptotics. Therefore, in order to obtain a full solution

that connects with these expressions, we need to find a way to analytically continue these

expressions. To this end, we consider the following choice of functions:

V (φ) = 4β cosh(−δφ), Z(φ) =
1

2 cosh(−γφ)
(B.1)

When δ = −1/
√

3 and γ = −
√

3, this is the top-down model presented in [43]. Also,

β = −3/2 is needed in order for the potential to asymptote to the cosmological constant

– 27 –

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0.0 0.2 0.4 0.6 0.8 1.0
-60

-50

-40

-30

-20

-10

0

r̃

R

Figure 3. Ricci curvature of the metric for the case γ = −
√

3, δ = −1/
√

3 and B/Q = 100.

of AdS when the dilaton vanishes. Assuming a large value of the dilaton at the horizon

uD � 1, they are well approximated by (5.33), as we will see below.

In order to find the full black hole solution, we adopt the same radial ansatz as in [44]:

ds2 =
1

g(r̃)2

[
−f(r̃)G(r̃)dt2 +

g′(r̃)2G(r̃)

f(r̃)
dr̃2 + dx2 + dy2

]
, (B.2)

A = At(r̃)dt+
B

2
(x dy − y dx), φ = φ(r̃). (B.3)

Using the gauge freedom of redefining the radial coordinate, r̃ → R(r̃), we can fix the

gxx, gyy components by choosing

g(r̃) =
1− (1− r̃)2

r+
, (B.4)

where r+ is a free constant. The radial coordinate r̃ that is fixed by this particular choice

relates to the coordinate ρ̃ in (5.16) by r̃ = 1−
√
ρ̃ in the near-horizon region. The horizon

is located at r̃ = 1 (the conformal boundary lies at r̃ = 0) and in this region, generically

f(r̃) ∼ f (2)(r̃− 1)2. However, the temperature of such a black hole is T ∝ f (2). Therefore,

in order to find an extremal solution, we impose the expansions

f(r̃) = f (4)(r̃ − 1)4 +O
(
(r̃ − 1)6

)
(B.5)

G(r̃) = G(0) +O
(
(r̃ − 1)2

)
(B.6)

At(r̃) = A
(2)
t (r̃ − 1)2 +O

(
(r̃ − 1)4

)
(B.7)

φ(r̃) = φ(0) +O
(
(r̃ − 1)2

)
, (B.8)

where φ(0) = uD. In order to have AdS2 × R2, we need to choose f (4) = −1/r+. The

solution is determined by the free parameters {β, γ, δ, B, r+}. For certain regions of the

parameter space (γ, δ), there is a unique regular extremal solution, otherwise no extremal

solution can be found. Of course, those are the cases we are interested in.10 In these cases,

10Exploring the entire parameter regime is beyond the scope of this paper.

– 28 –



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all the coefficients of the expansion are fixed by the equations of motion, confirming that

the solution is completely determined by the electric charge and magnetic field, as expected

from the attractor mechanism. The other lowest order coefficients are given by

A
(2)
t = −

√
cosh (γφ(0))

√
γ + δ tanh (δφ(0))/ tanh (γφ(0))

−2βγ cosh (δφ(0))
(B.9)

B2 =
8r4

+β

γ

(
δ

sinh (δφ(0))

tanh (γφ(0))
− γ cosh (δφ(0))

)
(B.10)

G(0) = − 1

4βr+ cosh (δφ(0))
(B.11)

In order to make a connection to the solution in eqs. (5.37) to (5.40), we need to make the

replacement cosh (δφ(0)) → 1/2eδφ
(0)
, cosh (γφ(0)) → 1/2eγφ

(0)
, which is valid as long as

δφ(0) � 0, γφ(0) � 0. Also, these expressions are written in terms of the free constant r+,

which is related to the charge Q by r4
+ = γB2

2β(δ−γ)

(
Q2

B2
γ−δ
γ+δ

) δ+γ
2γ

.

The equations of motion derived for this ansatz can be integrated numerically using

standard shooting techniques, and setting boundary conditions at the horizon. We get four

dynamical equations and a constraint. If the former hold, then the constraint equation must

be satisfied everywhere provided it is satisfied at the horizon. It serves as a check of the

numerics to solve the dynamical equations only and check that the constraint vanishes

automatically.

At the boundary, the numerical solution gives a vanishing φ, while the metric asymp-

totes to

ds2 =
1

r̃2

[
λ2r2

+

4
dt2 + dr̃2 +

r2
+

4

(
dx2 + dy2

)]
, (B.12)

where λ is a constant determined by the numerical integration which depends non-trivially

on the parameters of the solution. Performing the rescalings r̃ → λr+
2 r̂ and (x, y)→ λ(x̂, ŷ),

we recognize the familiar form of the AdS4 metric. Such a rescaling corresponds to a

dynamically generated AdS scale, a generic feature of holographic RG flows.

Open Access. This article is distributed under the terms of the Creative Commons

Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in

any medium, provided the original author(s) and source are credited.

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	Introduction
	Entropy function
	Dyonic AdS-RN planar black hole
	DC conductivity from horizon data
	DC conductivities at T=0 for Einstein-Maxwell-Dilaton theories
	Massless scalar
	Exponential couplings
	Quadratic couplings
	Heat conductivity kappa/T 
	S-duality of the attractor equations and conductivities

	Generalizations and conclusions
	Dilatonic black holes
	Numerical UV completion