#### Thermoelectric conductivities at finite magnetic field and the Nernst effect

Received: March
Thermoelectric conductivities at finite magnetic field and the Nernst effect
Keun-Young Kim 0 1 2 4 5
Kyung Kiu Kim 0 1 2 4 5
Yunseok Seo 0 1 2 3 5
Sang-Jin Sin 0 1 2 3 5
0 We consider a class
1 Seoul 133-791 , Korea
2 Gwangju 500-712 , Korea
3 Department of Physics, Hanyang University
4 School of Physics and Chemistry, Gwangju Institute of Science and Technology
5 Open Access , c The Authors
We study the thermoelectric conductivities of a strongly correlated system in the presence of a magnetic field by the gauge/gravity duality. of Einstein-Maxwell-Dilaton theories with axion fields imposing momentum relaxation. General analytic formulas for the direct current (DC) conductivities and the Nernst signal are derived in terms of the black hole horizon data. For an explicit model study, we analyse in detail the dyonic black hole modified by momentum relaxation. In this model, for small momentum relaxation, the Nernst signal shows a bell-shaped dependence on the magnetic field, which is a feature of the normal phase of cuprates. We compute all alternating current (AC) electric, thermoelectric, and thermal conductivities by numerical analysis and confirm that their zero frequency limits precisely reproduce our analytic DC formulas, which is a non-trivial consistency check of our methods. We discuss the momentum relaxation effects on the conductivities including cyclotron resonance poles. respondence
and; the; Nernst; effect; Holography and condensed matter physics (AdS/CMT); Gauge-gravity cor-
1 Introduction
2 General analytic DC conductivities at finite magnetic field 2.1 Nernst effect
3 Example: dyonic black branes with momentum relaxation 3.1 3.2 4.1
5 Conclusions
Introduction
Model with massless axions
3.3 DC conductivities: Hall angle and Nernst effect
4 Numerical AC conductivities Equations of motion and on-shell action Numerical method 4.3 AC conductivities and the cyclotron poles
Strongly coupled electron systems show many interesting phases such as non-Fermi liquid,
high Tc superconductor and pseudo gap phase. Some of the most important and basic
theoretical method to compute conductivities to explain and guide experiments. However,
due to strong coupling, the perturbative analysis of quantum field theory does not work
and we don’t have a reliable systematic tool to compute them.
Gauge/gravity duality is an approach for such strong coupling problems, and it has
been developed as a method for conductivity [1, 2]. Some early works treated systems
which have translation invariance. However, at finite charge density the direct current
(DC) conductivities in such systems are infinite. To solve this problem, it is essential to
introduce momentum relaxation. For this, several ideas have been proposed.
The most straightforward way is to impose inhomogeneous boundary conditions of the
bulk fields to break translation invariance [3–8]. Massive gravity models studied in [9–
13] give mass terms to gravitons, which break the spatial diffeomorphisms (not the radial
and temporal ones), and consequently break momentum conservation of the boundary
field theory with translational invariance unbroken [11]. Holographic Q-lattice models and
those with massless linear dilaton/axion fields [14–23] take the advantage of a continuous
global symmetry of the bulk theory. Some other models utilise a Bianchi VII0 symmetry
to construct black holes dual to helical lattices [24–27]. All these models yield finite DC
conductivities as desired.
However, all models with momentum relaxation, except [23], did not include the
magnetic field. Since the transport properties at finite magnetic field such as the quantum Hall
effect, the Nernst effect, and the Hall angle, are also important basic probes for strongly
correlated electron system, it is timely and essential to develop the methods for them in the
presence of momentum relaxation. Indeed, the holographic analysis on conductivities at
finite magnetic field was one of the pioneering themes opening up the AdS/CMT (condensed
matter theory) [28–30]. The purpose of our paper is to extend them by implementing
momenturm relaxation holographically.1 This paper is also a companion of [19, 32, 33], where
thoroughly in the absence of magnetic field.
We consider a general class of Einstein-Maxwell-dilaton theories with axion fields
imfollowing the method developed in [34]. Based on these formulas we discuss the model
independent features of the Nernst signal. Notice that the Nernst signal (2.55)
is zero in the holographic model without momentum relaxation, since the electric
conductivity is infinite. Thus, momentum relaxation is essential for the Nernst effect. The Nernst
signal has interesting properties which could support the existence of the quantum critical
point (QCP). As we approach to the QCP or the superconducting domain, the strength
of the Nernst signal becomes stronger and shows a non-linear dependence on the
magnetic field, which is different from the expectation based on the Fermi liquid theory [35].2
See [29, 31], for pioneering works on the Nernst effect by the holographic approach and
the magnetohydrodynamics with a small impurity effect. We deal with similar topics by
means of a general class of holographic models encoding momentum relaxation, where we
assume that momentum relaxation is related to finite impurity density, which could be
large. Note, however, that this relation is not proven yet.
After discussions on a class of models, we study in detail the dyonic black hole
background [28–30], modified by the specific axion fields introduced in [16]. We numerically
their zero frequency limits agree to the DC formulas that we have derived analytically. It
recovers the results in [28] if momentum relaxation vanishes. We discuss the momentum
relaxation effect on the conductivities including the cyclotron resonance poles, which was
first observed in [30].
This paper is organized as follows. In section 2, we consider a general class of
EinsteinMaxwell-dilaton theories with axion fields and derive general formulas for the DC electric,
thermoelectric, and thermal conductivity at finite magnetic field as well as the Nernst
1In [31], momentum relaxation is introduced perturbatively in the hydrodynamic limit.
2These anomalous behavior may be described by a liquid of quantized vortices and anti-vortices in
non-superconducting phase. See [37] for a speculative point of view.
signal. In section 3, as an explicit example, we analyse the dyonic black brane with the axion
hair and discuss the Hall angle and the Nernst effect. In section 4, we continue our analysis
on the model introduced in section 3. We compute the AC electric, thermoelectric, and
thermal conductivity numerically. The momentum relaxation effect on AC conductivities
and the cyclotron resonance poles are discussed. We compare the zero frequency limit of
our numerical AC conductivities with the DC analytic formulas derived in section 3. In
section 5 we conclude.
Note added.
While this work was near completion, we noticed the appearance of [38–40]
which have some overlap with ours. [38] deals with a massive gravity model at finite
magnetic field. [39] considers the same class of models as ours. [40] obtains general expressions
for conductivities at finite magnetic field using the memory matrix formalism.
General analytic DC conductivities at finite magnetic field
the presence of a magnetic field, from a general class of Einstein-Maxwell-Dilaton theories
S =
sented in [23]. Here we compute all the other conductivities as well.
We employ the
method developed in [8, 34], but a finite magnetic field poses some technical subtlety. We
will explain how to treat it. The action (2.1) yields equations of motion:
RMN − 2 gMN L− 2 (∂M φ)(∂N φ)− 2 i=1,2
1 X Φi(φ)(∂M χi)(∂N χi)−
1 X ∂Φi (∂χi)2 −
− 4
1 F 2 ∂Z(φ)
= 0 ,
= 0 ,
where, L is the Lagrangian density of the action (2.1).
we take the gauge potential as
To study the system at finite chemical potential with a background magnetic field (B),
We choose the axion fields
A = AM dxM = a(r)dt +
(xdy − ydx) .
which break translational invariance and can give rise to momentum relaxation [14, 15]. A
metric anasatz consistent with the choice (2.6) and (2.7) is
ds2 = GMN dxM dxN = −U (r)dt2 +
dr2 + ev1(r)dx2 + ev2(r)dy2,
v1(r) = v2(r) =: v(r) ,
−gZ(φ)F rt = Z(φ)ev(r)a0(r) ,
which is identified with the number density in the boundary field theory. The axion
equation (2.4) is trivially satisfied. The Einstein equation (2.2) and scalar equation (2.5) become
v00(r) = − 2
e−2v(r)
U 00(r) =
+ ev(r)Z(φ) 2β2Φ(φ) + ev(r)U (r) v0(r)2 − φ0(r)2
if we choose
the background becomes the AdS-dynonic black hole geometry with the momentum
relaxation. We will discuss it in section 3 in detail.
To compute the conductivities for the general background, we consider small
perturbations around the background obtained by (2.11)–(2.13)
δAi = t δfi(1)(r) + δai(r) ,
δGti = t δfi(2)(r) + δgti(r) ,
δfi(1)(r) = −Ei + ζia(r) ,
spite of the explicit t dependence in (2.15) and (2.16) all equations of motion of fluctuations
turn out to be time-independent, which is the reason to introduce the specific forms of (2.19)
and (2.20). Furthermore, the electric current and heat current can be computed as the
boundary (r → ∞) values of J i(r) and Qi(r),
J i(r) = Z(φ)√−g F ir(r)
− a(r)J i(r) ,
Our task is to plug the solutions of the fluctuation equations into (2.21) and (2.23)
Therefore, the current at the boundary is given by
0 = ∂M Z(φ)√−g F iM
= ∂r Z(φ)√−g F ir + ∂t Z(φ)√−g F it
= ∂rJ i − B ij e−v(r)ζj Z(φ) .
J i(∞) = J i(rh) + B ij ζj
Z ∞
Next, let us turn to the heat current, Qi, (2.23). It is convenient to start with the derivative
− a0(r)J i − a(r)J i0
= B ij Ej e−v(r)
− 2B ij ζj a(r)e−v(r)Z(φ) .
After using the Einstein equations for fluctuations with the ansatz (2.17)–(2.18),
2B2Z(φ)e−2v(r) + 2β2Φ(φ)e−v(r) + U (r) v02(r) − φ02(r)
2B2Z(φ)e−2v(r) + 2β2Φ(φ)e−v(r) + U (r) v02(r) − φ02(r)
−Bρe−v(r)δgtx −BZ(φ)U (r)δa0x(r)
where Qi(rh) and J i(rh) are functions at horizon, which can be further simplified by the
regularity condition at the black hole horizon [8, 34]
J i(∞) = J i(rh) + B ij ζj Σ1 ,
Q (∞) = Qi(rh) + B ij Ej Σ1 + B ij ζj Σ2 ,
i
Ei ln(r − rh) + · · · ,
δgti(r) ∼ δgt(ih) + O (r − rh) + · · · ,
δgri(r) ∼ e−v(rh) δgt(ih)
δχi(r) ∼ χi(h) + O (r − rh) + · · · .
Thus, the boundary currents yield
Q (∞) = −4πT δij δgt(jh) + B ij Ej Σ1 + B ij ζj Σ2 ,
i
using the equations of motion (2.27). The near horizon expansion of the last two equations
in (2.27) gives
Bρe−vh δgt(yh) − BZhEy − ρEx − 4πevh T ζx ,
− Bρe−vh δgt(xh) − BZhEx − ρEy − 4πevh T ζy ,
we end up with a relatively simple expression for the heat current;
dr0e−v(r0)Z φ(r0) − 2B ij ζj
Z ∞
Z ∞
≡ Qi(rh) + B ij Ej Σ1 + B ij ζj Σ2 .
In summary, we have two boundary currents:
− β2ρevh Φh Ei − B(ρ2 + B2Zh2 + β2evh ZhΦh) ij Ej
+ 4πT evh (B2Zh + β2evh Φh) ζi − 4πT evh Bρ ij ζj ,
(ii) The thermoelectric conductivity:
(iii) The thermal conductivity:
∂J i(∞)
∂Ej
1 ∂J i(∞)
1 ∂Qi(∞)
1 ∂Qi(∞)
∂Ej
= −(ρ δike−vh + B ike−vh Zh)
∂Ej
= −(ρ δike−vh + B ike−vh Zh) T
= −4πδik δgt(kh) + ij B
= −4πδik δgt(kh) + ij B
∂Ej
where we put hats on the conductivities to distinguish them from the ones where
magnetization current are taken out [29]. More explicitly, with (2.36), the general DC conductivity
formulas are given as follows.
(i) The electric conductivity and Hall conductivity:
Z ∞
Z ∞
current and the energy magnetization current, which should be subtracted [29]. In
particular, in the case of the dyonic black hole in section 3.2,3 Σ1 = MB and Σ2 = 2(ME−μM) ,
B
where M is the magnetization and ME is the energy magnetization. The relation between
− T
− T
which are expressed in terms of the black hole horizon data.
Nernst effect
The thermoelectric conductivites play an important role in understanding high Tc
superconductors. In the presence of a magnetic field, a transverse electric field can be generated
by a transverse or longitudinal thermal gradient. The former is called the ‘Seebeck’ effect
and the latter is called the ‘Nernst’ effect.
The electric current, J~, can be written in terms of the external electric field and the
thermal gradient as follows;
Based on the definition of the Nernst effect, the Nernst signal (eN ) is defined as
The Nernst signal in cuprates shows different features from conventional metals, so it is
one of the important observables in understanding high Tc superconductors. For example,
in conventional metals the Nernst signal is linear in B, while in the normal state of a
cuprate it is bell-shaped as a function of B. See, for example, figure 12 in [35]. At a fixed
B, the Nernst signal increases as temperature decreases in the normal state of a cuprate
and, in turn, near the superconducting phase transition the Nernst signal becomes much
stronger than conventional metals as shown in figure 20 in [35].
Now we have the general formulas for the DC transport coefficients, (2.50) and (2.51),
we can compute a general Nernst signal (2.55)
eN =
ρ4 + 2evh β2ρ2ZhΦh + Zh2(B2ρ2 + e2vh β4Φ2 ) ,
h
3The magnetization and energy magnetization current in a general setup were computed in [39].
which is expressed in terms of the black hole horizon data. By playing with the parameters,
more realistic models showing aforementioned cuprate-like properties and furthermore in
understanding the physics of strongly correlated systems.
There are two comments on general features of the Nernst signal (2.56). First, it is
In this regime, relevant to the quantum critical point, the Nernst signal is proportional to
Example: dyonic black branes with momentum relaxation
Model with massless axions
As an explicit model, we consider the Einstein-Maxwell system with massless axions. The
action is given by (2.1) with the following choices
where L is the AdS radius which will be set to be 1 from now on. Adding the
GibbonsHawking term, we start with
S0 =
4 √
−g R + 6 − 4
− 2
1 F 2
1 X2 (∂χI )2
3 √
the extrinsic curvature tensor KMN
N ∇(P nQ), where n is the
outward
Sc =
dx3√
1 X2 γμν ∂μχI ∂ν χI ,
5In our case, nM = 0, 0, 0, 1/pU (r) . See (3.7).
and the finite renormalized on-shell action is
Sren = lim (S0 + Sc)on-shell .
Since the boundary terms do not change the equations of motion, the equations (2.2)–(2.5)
are valid and yield, with (3.1),
RMN = 2 gMN R+6− 41 F 2 − 12 XI=21(∂χI )2 +
1
2 I
1 X ∂M χI ∂N χI + 2 FM P FNP ,
∇M F MN = 0 ,
We want to find a solution of the equations of motion, describing a system at finite
mentum relaxation. It turns out the dyonic black brane solution modified by the axion
hair (2.7) does the job. I.e.
ds2 = −U (r)dt2 +
dr2 + r2(dx2 + dy2) ,
U (r) = r2 − 2 − r
B = qmrh ,
where rh is the location of the horizon and
m0 = rh3 1 +
Thermodynamics
To obtain a thermodynamic potential for this black brane solution, we compute the
onperiod is the inverse temperature
SE = −iSren ,
where SE is the Euclidean action. By a regularity condition at the black brane horizon the
temperature of the system is given by the Hawking temperature,
T =
U 0(rh) =
3rh −
and the entropy density is given by the area of the horizon,
Plugging the solution (3.7) into the Euclidean renormalised action (3.4), we have the
SE =
(−m0 − β2rh + qm2rh) ≡ T
namic variables as
W =
is obtained by using the relation
which is derived as follows. We want to compute one-point functions of the boundary
metric is of the following form
ds2 = N2dr2 + γμν (dxμ + V μdr)(dxν + V ν dr) ,
define ‘conjugate momenta’ of the fields as
−γ Kμν − Kγμν − 2γμν + Gμν [γ] − 2 ∂μχI ∂ν χI +
Thus, the one point functions are
hOI i = rl→im∞ √
which yield (3.14).
Notice that the pressure is not equal to hTxxi since
P = hTxxi + rhβ2 − rhqm2 .
The magnetization (M) and the energy magnetization (ME ) are
M = −
= −qm ,
ME = −
1 − r2
1 − r
δAy = x δB + μδBE r
δE = T δs + μδρ − rhδ(β2) − MδB ,
δW = −MδB − sδT − rhδ(β2) − ρδμ ,
We find the first law of thermodynamics
and the variation of (3.18).
DC conductivities: Hall angle and Nernst effect
In this section we study the DC conductivities of the dyonic black brane with momentum
relaxation. Because we have derived the general formulas in section 2, we only need to
plug model-dependent information (3.1) and (3.7) into (2.50)–(2.52).
The electric conductivities yield
where temperature dependence drops out and we recover the results expected on general
grounds from Lorentz invariance, agreeing with [28]. In the limit B → 0, the expressions
6For details we refer to appendix C of [29].
which reproduces the result in [16, 19]. The thermoelectric and thermal conductivities read
Therefore, we make plots of conductivities in figure 1, where we scaled the variables by T
increases, the electric and thermoelectric conductivity generally increases while its overall
2-dimensional shape does not change qualitatively. However, the thermal conductivities
strange metal phase, we are interested in the temperature dependence of the Hall angle,
which is proportional to 1/T 2. In our case, we numerically found that the Hall angle ranges
between 1/T 0 and 1/T 1 (figure 2(b)). In the large T regime, the Hall angle always scales
as 1/T . It can be seen also from the formula (3.29), where if T is large compared to the
other scales, rH ∼ T so tan θH ∼ 1/T .
The Nernst signal (2.56) yields
eN =
As discussed in section 2.1, the Nernst signal is linear in B in conventional metals while
it becomes bell-like in the normal state of cuprates [35]. To see whether our model can
capture this feature, we make a three dimensional plot of the Nernst signal as a function of
The blue line is almost straight while the green and red ones are bell-like. Therefore, we
find that our system shows the transition from the normal metal (blue line) to cuprate-like
red curves are similar to figure 12 in [35] and it was proposed that the bell shape can be
explained by the dynamics of the vortex liquid in non-superconducting phase [36, 37]. It
was also interpreted as an evidence of the pseudo-gap phase [35]. If we take this point of
view, our model might be relevant to the pseudo-gap phase.
One may wonder if the blue line is qualitatively similar to the green and red ones
for higher B/T 2. As B/T 2 increases the blue line reaches the maximum and has almost
plateaued out, which is not bell-shaped. Furthermore, the numerical value of B/T 2 at the
(red, green, blue)
Numerical AC conductivities
So far we have discussed the DC conductivities. In this section, we will consider the AC
conductivities. To be concrete, we continue to investigate the model in the previous section,
namely, the dyonic black brane with the momentum relaxation by the axion fields.
Equations of motion and on-shell action
should not be changed by this gauge choice.7 Indeed, we will show the zero frequency limit
section 2. It serves as a good consistency check of our numerical method.
To compute conductivities holographically it is consistent to turn on linear fluctuations
where r2 in the metric fluctuation (4.2) is introduced to make the asymptotic solution of
the linearized equations for the Fourier components are given as follows.
7There is a subtle issue on the gauge choice and holographic renormalization. For more details we refer
− Einstein equations:
− Maxwell equations:
− Scalar equations:
− r4 −
− h0t0i = 0 ,
Among these eight equations, only six are independent.
Near the black hole horizon (r → 1)8 the solutions are expanded as
hti = (r − 1)ν±+1 ht(iI) + ht(iII)(r − 1) + · · · ,
ai = (r − 1)ν± a(I) + ai(II)(r − 1) + · · · ,
i
ψi = (r − 1)ν± ψ(I) + ψi(II)(r − 1) + · · · ,
i
out that the 4 parameters ai(I) and ψ(I) may be chosen to be independent since ht(iI) and
i
all higher power coefficients can be determined by them.
hti = ht(i0) +
ai = ai(0) +
where the leading terms ht(i0), a
i(0), and ψi(0) are independent constants, which fix ht(i2), ψ(2)
i
leading terms, hti , ai(0), and ψi(0). The leading terms play the role of sources for the
(0)
Expanding the renormalized action (3.4) around the dyonic black brane background
and using the equations of motion, we obtain a quadratic on-shell action:
Sr(e2n) = lim
4r3 − pU (r)
contributions from the horizon, which is the prescription for the retarded Green’s
function [41]. In particular, with the spatially homogeneous ansatz (4.1)–(4.3), the quadratic
action in momentum space yields
Sr(e2n) =
hti − 2m0h¯t(i0)ht(i0) + a¯
where V2 is the two dimensional spatial volume R dxdy. The argument of the variables with
to obtain complex two point functions [41].
The on-shell action (4.11) is nothing but the generating functional for the two-point
point functions from the first two terms in (4.11). The other three terms are nontrivial
linearity of the equations (4.4)–(4.7) makes it easy to find out a linear relation between
{ i , ht(i3), ψ(3)} and {ai(0), ht(i0), ψ(0)}. In the following subsection we will explain how to
a(1)
find such a relationship numerically in a more general setup and apply it to our case.
Numerical method
A systematic numerical method with multi fields and constraints was developed in [19]
based on [42, 43]. We summarize it briefly for the present case and refer to [19, 33] for
around a background. Suppose that they satisfy a set of coupled N second order differential
equations and the fluctuation fields depend on only t and r:
(r → ∞). For example, p = 2 in (4.2).
Φa(r) = (r − 1)νa± ϕa + ϕ˜a(r − 1) + · · · ,
(outgoing) boundary condition. In order to compute the retarded Green’s function we
choose the incoming boundary condition [41]. This choice reduces the number of
independent parameter from 2N to N . There may be further reductions by Nc if there are Nc
ϕa1 ϕa2 ϕa3 . . . ϕaN−Nc
Every column vector ϕaˆi yields a solution with the incoming boundary condition, denoted
(near boundary) ,
Φian(r) ≡ Φˆia(r)cˆi → Sˆacˆi + · · · +
i
(near boundary) ,
with real constants cˆi. We want to identify Sˆacˆi with the independent sources J a but if
i
there are constraints, it is not possible since a > ˆi. However, in this case, there may be Nc
other solutions corresponding to some residual gauge transformations [19, 33]
(near boundary) ,
solution together with the ingoing solutions.
are two sets of additional constant solutions of the equations of motion (4.4)–(4.7)
hti = ht0i ,
ai = −
Sa¯i =
y =
Therefore, the most general solution reads
Φian(r) + Φca(r) ≡ J a + · · · +
where we defined J a and Ra. For arbitrary sources J a we can always find cI
= 10
cI = (S−1)I aJ a,
Ra = OaI cI = OaI (S−1)I bJ b.
Sr(e2n) =
J¯aAabJ b + J¯aBabRb ,
The general on-shell quadratic action in terms of the sources and the responses can be
the action (4.11) is the case with:
J a = hhtt(yx0) ,
Ra = hhtt(yx3) ,
into the action (4.22) we have
which yields the retarded Green’s function
Sr(e2n) =
V2 Z ∞ dω J¯a Aab + BacOcI (S−1)I b J b,
Gab = Aab + BacOcI (S−1)I b .
In summary, to compute the retarded Green’s function, we need four square N × N
matrices, A, B, S and O. The matrices A and B can be read off from the action (4.22), which
is given by the on-shell expansion near the boundary. The matrices S and O are obtained
by solving a set of the differential equations. Part of them comes from the solutions with
incoming boundary conditions and the others may be related to the constraints. Notice
that the Green’s functions do not depend on the choice of initial conditions (4.14).
In the case of the dyonic black branes in section 4.1, we may construct a 6 × 6 matrix
of the retarded Green’s function. We will focus on the 4 × 4 submatrix corresponding to
the electric current J i and the ht(i0) is dual to the energy-momentum tensor T ti. From the
linear response theory, we have the following relation between the response functions and
the i direction. Notice that the electric and heat current here contain the contribution
of magnetization, so we use the conductivities with hat ((2.37)–(2.40)). By taking into
account diffeomorphism invariance [1, 2, 19], (4.28) can be expressed as
From (4.27) and (4.29) with the magnetization subtraction (2.50)–(2.52), the conductivities
are expressed in terms of the retarded Green’s functions as follows
i(GiTjT −GiTjT (ω=0)−μ(GiJjT +GiTjJ −μGiJjJ )) − T
−(∇j T )/T
AC conductivities and the cyclotron poles
In this section we present our numerical results. Some examples of the AC electric
conductivity are shown in figure 5 and 8; the thermoelectric conductivity is in figure 6; and
the thermal conductivity is in figure 7.
Before discussing the AC nature of the conductivities, we first examine the DC limit
in section 3.3. The comparisons are shown in figure 4, where all conductivities are plotted
lines were drawn by the analytic expressions, (3.23), (3.24), (3.25)–(3.28) and the red dots
serves as a supporting evidence for the validity of our analytic and numerical methods.
Indeed this agreement is not so trivial in technical perspective. In the DC computation we
turned on hri and read off the physics from the horizon data while in the AC computation
Let us turn to the AC properties of the conductivities.
Figure 5, 6, and 7 show
given by the analytic formulas (3.23), (3.24), (3.25)–(3.28). They agree to the numerical
curves become flatter, which is expected from stronger momentum relaxation. Notice that
because the gauge field for B (2.6) breaks translation invariance in the same way as the
axion fields do.
There is a peak in the curves in figures 5–7, which is related to the cyclotron resonance
of as arising from interactions between the positively charged current and the negatively
charged current of the fluid, which are counter-circulating. In the hydrodynamic regime
increases. It is indirectly shown in the plots since all curves become flat, which may reflect
frequencies as B increases, which is consistent with the hydrodynamic analysis (4.32).
same symmetry is preserved. It is demonstrated in figure 9(d)(e)(f); the figure (d) and (f)
position of the poles to the negative imaginary direction. This implies the width of the
and also fits well to the hydrodynamic analysis (black dashed line) for small magnetic field.
in figure 10 and additional similar numerical data for different parameters, we found the
following relation at small B
ω∗ = ωc − iγ = ωc0 + c1β2B − i(γ0 + c2β2) ,
0, 2, 3, 4 (gray, red, green blue). The dotted lines are the results by the hydrodynamic analysis (4.32).
figure 10(a) are linear to B. However, at large magnetic field B we numerically found the
In the presence of dissipation, the cyclotron frequency was shown [29] to be changed as
suspect that the analysis in [29], where B/T 2
imaginary part in (4.34) is consistent with the hydrodynamic calculation [29]. However,
1 is assumed, is valid in the limit c1B is
small. We leave this issue and the analytic justification of the specific form (4.34) for a
future project.
Conclusions
In this paper, we have computed the electric, thermoelectric, and thermal conductivity
at finite magnetic field by means of the gauge/gravity duality. First, by considering a
general class of Einstein-Maxwell-Dilaton theories with axion fields imposing momentum
relaxation, we have derived the analytic DC conductivities, which are expressed in terms
of the black hole horizon data. As an explicit model we have studied the dyonic black
hole modified by a momentum relaxation effect. The background solution is analytically
obtained and the AC electric, thermoelectric, and thermal conductivity were numerically
computed. The zero frequency limit of the numerical AC conductivities agree to the DC
formulas. This is a non-trivial consistency check of our analytic and numerical methods
to compute conductivities. Our numerical method can be applied to other cases in which
multiple transport coefficients need to be computed at the same time.
The Nernst signal, the Hall angle, and the cyclotron resonance pole were discussed
following [28–30]. Our general analytic formulas of the Nernst signal can be used to build
a realistic model and to investigate the universal properties of the model. In particular,
in the dyonic black hole case, the Nernst signal is a bell-shaped curve as a function of the
magnetic field, if momentum relaxation is small. It is similar to an experimental result
in the normal state of cuprates [35], which was speculatively explained by a vortex-liquid
effect [36, 37]. For large momentum relaxation the Nernst signal is proportional to the
magnetic field, which is a typical property of conventional metals. The Hall angle for the
dyonic black hole was computed explicitly. The Hall angle ranges between 1/T 0 and 1/T 1
and scales as 1/T for large T . However, in the strange metal phase, it was known that the
detail both numerically and analytically.
It is important to compare our AC conductivities with the general expressions based
on the memory matrix formalism [40, 44]. It would be also interesting to compare our AC
conductivity results with [45], where the AC electric conductivities have been studied at
finite magnetic field in the probe brane set up, focusing on the transport at quantum Hall
linear-T resistivity, so it is worthwhile to start with the models having that property and
then investigate the Hall angle and the Nernst effect in those models.
Acknowledgments
The work of KYK and KKK was supported by Basic Science Research Program through
the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT
& Future Planning (NRF-2014R1A1A1003220). The work of SS and YS was supported by
Mid-career Researcher Program through the National Research Foundation of Korea (NRF)
grant No. NRF-2013R1A2A2A05004846. YS was also supported in part by Basic Science
Research Program through NRF grant No. NRF-2012R1A1A2040881. We acknowledge
the hospitality at APCTP (“Aspects of Holography”, Jul. 2014) and Orthodox Academy
of Crete (“Quantum field theory, string theory and condensed matter physics”, Sep. 2014),
where part of this work was done.
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.
J. Phys. A 42 (2009) 343001 [arXiv:0904.1975] [INSPIRE].
JHEP 07 (2012) 168 [arXiv:1204.0519] [INSPIRE].
JHEP 11 (2012) 102 [arXiv:1209.1098] [INSPIRE].
[arXiv:1302.6586] [INSPIRE].
gravity, JHEP 11 (2013) 006 [arXiv:1309.4580] [INSPIRE].
lattices, JHEP 01 (2015) 035 [arXiv:1409.6875] [INSPIRE].
Phys. Rev. D 88 (2013) 086003 [arXiv:1306.5792] [INSPIRE].
Phys. Rev. D 88 (2013) 106004 [arXiv:1308.4970] [INSPIRE].
Phys. Rev. Lett. 112 (2014) 071602 [arXiv:1310.3832] [INSPIRE].
in momentum dissipating holography, arXiv:1411.6631 [INSPIRE].
[arXiv:1311.3292] [INSPIRE].
JHEP 06 (2014) 007 [arXiv:1401.5077] [INSPIRE].
JHEP 05 (2014) 101 [arXiv:1311.5157] [INSPIRE].
JHEP 04 (2014) 181 [arXiv:1401.5436] [INSPIRE].
[arXiv:1406.4870] [INSPIRE].
holographic model, JHEP 12 (2014) 170 [arXiv:1409.8346] [INSPIRE].
JHEP 05 (2012) 054 [arXiv:1202.4458] [INSPIRE].
JHEP 07 (2012) 129 [arXiv:1204.3008] [INSPIRE].
JHEP 07 (2014) 083 [arXiv:1404.5027] [INSPIRE].
[21] N. Iizuka and K. Maeda, Study of anisotropic black branes in asymptotically anti-de Sitter,
anisotropic holographic insulators, JHEP 05 (2015) 094 [arXiv:1501.07615] [INSPIRE].
quantum phase transitions in condensed matter and in dyonic black holes,
Phys. Rev. B 76 (2007) 144502 [arXiv:0706.3215] [INSPIRE].
resonance, Phys. Rev. D 76 (2007) 106012 [arXiv:0706.3228] [INSPIRE].
Phys. Rev. D 77 (2008) 106009 [arXiv:0801.1693] [INSPIRE].
relaxation, JHEP 04 (2015) 152 [arXiv:1501.00446] [INSPIRE].
[34] A. Donos and J.P. Gauntlett, Thermoelectric DC conductivities from black hole horizons,
[36] P.W. Anderson, Dynamics of the vortex fluid in cuprate superconductors: the Nernst effect,
field and disorder, arXiv:1502.02631 [INSPIRE].
arXiv:1502.03789 [INSPIRE].
[38] A. Amoretti and D. Musso, Universal formulae for thermoelectric transport with magnetic
[39] M. Blake, A. Donos and N. Lohitsiri, Magnetothermoelectric response from holography,
[40] A. Lucas and S. Sachdev, Memory matrix theory of magnetotransport in strange metals,
Phys. Rev. B 91 (2015) 195122 [arXiv:1502.04704] [INSPIRE].
recipe and applications, JHEP 09 (2002) 042 [hep-th/0205051] [INSPIRE].
[1] S.A. Hartnoll , Lectures on holographic methods for condensed matter physics, Class. Quant. Grav . 26 ( 2009 ) 224002 [arXiv:0903.3246] [INSPIRE].
[2] C.P. Herzog , Lectures on holographic superfluidity and superconductivity , [3] G.T. Horowitz , J.E. Santos and D. Tong , Optical conductivity with holographic lattices , [4] G.T. Horowitz , J.E. Santos and D. Tong , Further evidence for lattice-induced scaling , [5] G.T. Horowitz and J.E. Santos , General relativity and the cuprates , JHEP 06 ( 2013 ) 087 [6] Y. Ling , C. Niu , J.-P. Wu and Z.-Y. Xian , Holographic lattice in Einstein-Maxwell -dilaton [7] P. Chesler , A. Lucas and S. Sachdev , Conformal field theories in a periodic potential: results from holography and field theory , Phys. Rev. D 89 ( 2014 ) 026005 [arXiv:1308.0329] [8] A. Donos and J.P. Gauntlett , The thermoelectric properties of inhomogeneous holographic [9] D. Vegh , Holography without translational symmetry , arXiv:1301 .0537 [INSPIRE].
[10] R.A. Davison , Momentum relaxation in holographic massive gravity , [11] M. Blake and D. Tong , Universal resistivity from holographic massive gravity , [12] M. Blake , D. Tong and D. Vegh , Holographic lattices give the graviton an effective mass , [13] A. Amoretti , A. Braggio , N. Magnoli and D. Musso , Bounds on charge and heat diffusivities [14] A. Donos and J.P. Gauntlett , Holographic Q-lattices , JHEP 04 ( 2014 ) 040 [15] A. Donos and J.P. Gauntlett , Novel metals and insulators from holography , [16] T. Andrade and B. Withers , A simple holographic model of momentum relaxation , [17] B. Gout ´eraux, Charge transport in holography with momentum dissipation , [18] M. Taylor and W. Woodhead , Inhomogeneity simplified, Eur. Phys. J. C 74 ( 2014 ) 3176 [19] K.-Y. Kim , K.K. Kim , Y. Seo and S.-J. Sin , Coherent/incoherent metal transition in a [20] Y. Bardoux , M.M. Caldarelli and C. Charmousis , Shaping black holes with free fields , [22] L. Cheng , X.- H. Ge and S.-J. Sin , Anisotropic plasma at finite U(1) chemical potential, [23] M. Blake and A. Donos , Quantum critical transport and the Hall angle , Phys. Rev. Lett . 114 ( 2015 ) 021601 [arXiv:1406.1659] [INSPIRE].
[24] A. Donos and S.A. Hartnoll , Interaction-driven localization in holography, Nature Phys . 9 ( 2013 ) 649 [arXiv:1212.2998] [INSPIRE].
[25] A. Donos , B. Gout ´eraux and E. Kiritsis , Holographic metals and insulators with helical symmetry , JHEP 09 ( 2014 ) 038 [arXiv:1406.6351] [INSPIRE].
[26] A. Donos , J.P. Gauntlett and C. Pantelidou , Conformal field theories in d = 4 with a helical twist , Phys. Rev. D 91 ( 2015 ) 066003 [arXiv:1412.3446] [INSPIRE].
[27] J. Erdmenger , B. Herwerth , S. Klug , R. Meyer and K. Schalm , S-wave superconductivity in [28] S.A. Hartnoll and P. Kovtun , Hall conductivity from dyonic black holes , [29] S.A. Hartnoll , P.K. Kovtun , M. Mu¨ller and S. Sachdev , Theory of the Nernst effect near [30] S.A. Hartnoll and C.P. Herzog , Ohm's law at strong coupling: S duality and the cyclotron [31] S.A. Hartnoll and C.P. Herzog , Impure AdS /CFT correspondence, [32] K.-Y. Kim , K.K. Kim and M. Park , A simple holographic superconductor with momentum [33] K.-Y. Kim , K.K. Kim , Y. Seo and S.-J. Sin , Gauge invariance and holographic [35] Y. Wang , L. Li , and N.P. Ong , Nernst effect in high-Tc superconductors , [37] P.W. Anderson , Bose fluids above Tc: incompressible vortex fluids and “supersolidity”, [41] D.T. Son and A.O. Starinets , Minkowski space correlators in AdS/CFT correspondence: [42] I. Amado , M. Kaminski and K. Landsteiner , Hydrodynamics of holographic superconductors, [43] M. Kaminski , K. Landsteiner , J. Mas , J.P. Shock and J. Tarrio , Holographic operator mixing and quasinormal modes on the brane , JHEP 02 ( 2010 ) 021 [arXiv:0911.3610] [INSPIRE].
[44] A. Lucas , Conductivity of a strange metal: from holography to memory functions , JHEP 03 ( 2015 ) 071 [arXiv:1501.05656] [INSPIRE].
[45] J. Alanen , E. Keski-Vakkuri , P. Kraus and V. Suur-Uski , AC transport at holographic quantum Hall transitions , JHEP 11 ( 2009 ) 014 [arXiv:0905.4538] [INSPIRE].
[46] S. Sachdev , Nonzero-temperature transport near fractional quantum Hall critical points , Phys. Rev . B 57 ( 1998 ) 7157 [cond -mat/9709243].