#### Coherent/incoherent metal transition in a holographic model

Keun-Young Kim
0
3
Kyung Kiu Kim
0
3
Yunseok Seo
0
1
Sang-Jin Sin
0
2
0
Open Access
,
c The Authors
1
Research Institute for Natural Science, Hanyang University
2
Department of Physics, Hanyang University
3
School of Physics and Chemistry, Gwangju Institute of Science and Technology
We study AC electric (), thermoelectric (), and thermal () conductivities in a holographic model, which is based on 3+1 dimensional Einstein-Maxwell-scalar action. There is momentum relaxation due to massless scalar fields linear to spatial coordinate. The model has three field theory parameters: temperature (T ), chemical potential (), and effective impurity (). At low frequencies, if < , all three AC conductivities (, , ) exhibit a Drude peak modified by pair creation contribution (coherent metal). The parameters of this modified Drude peak are obtained analytically. In particular, if the relaxation time of electric conductivity approaches to 23/2 and the modified Drude peak becomes a standard Drude peak. If > the shape of peak deviates from the Drude form (incoherent metal). At intermediate frequencies (T < < ), we have analysed numerical data of three conductivities (, , ) for a wide variety of parameters, searching for scaling laws, which are expected from either experimental results on cuprates superconductors or some holographic models. In the model we study, we find no clear signs of scaling behaviour.
1 Introduction 2 3 4
Introduction
AdS-RN black branes with scalar sources
General action
AdS-RN black brane
General numerical methods with constraint
Electric/thermal/thermoelectric AC conductivities
Green functions and Transport coefficients
Optical conductivity and coherent/incoherent metal
Thermoelectric and thermal conductivity
Holographic methods (gauge/gravity duality) provide novel tools to study many properties
of strongly correlated systems by analysing the corresponding higher dimensional gravity
theories [14]. In particular it gives a new way of computing transport coefficients such as
viscosity, relaxation time, and electric/thermal conductivities as well as various equilibrium
thermodynamic quantities. In this paper we focus on electric, thermoelectric, and thermal
conductivities of strongly coupled systems by holographic methods.
The early works on this subject, holographic conductivity, have dealt with the systems
with translation invariance [2]. However, any system with finite charge density and
translation invariance will exhibit an infinite electric DC conductivity. The reason is
straightforward: a constant electric field will accelerate charges indefinitely because there is no
momentum dissipation, which is implied by translation invariance. Real condensed
matter systems will not have translation symmetry. It is broken by a background lattice or
impurities. To remedy this infinite conductivity problem, there have been a number of
proposals to introduce the momentum dissipation effect in the framework of holography.
They fall into two classes: models with inhomogeneous boundary conditions (IBC) and
homogeneous boundary conditions (HBC).1
In IBC models, one gives some bulk fields inhomogeneous boundary conditions breaking
translation invariance explicitly [914]. One may introduce a spatially modulated scalar
1There is an earlier conceptually different idea. It considers a model of a small number (Nf ) of charged
degrees of freedom in a bath of a large number (Nc) of neutral degrees of freedom. If Nf is parametrically
much smaller than Nc, the momentum of charged degrees of freedom can be absorbed into a bath. For
example, see [58].
black hole of Einstein-Maxwell-scalar system, we may understand the translation symmetry
breaking by the Ward identity (2.13)
hTi = F hJ i + hOi ,
In HBC models, one does not impose explicit inhomogeneous boundary conditions,
but find a way to break translation invariance effectively. A few models have been studied.
Massive gravity approach [1518] introduces mass terms for some gravitons. It breaks bulk
diffeomorphism invariance and consequently violates the conservation of the stress-energy
tensor in the boundary field theory. Some models exploit a continuous global symmetry of
the bulk theory [14, 19, 20], where, for example, the global phase of a complex scalar field
breaks translational invariance.2 In [23], a simple model with massless scalar fields linear
in spatial coordinate, breaking translation symmetry, was introduced.3 It was extended
further in [24, 25].
On a technical level, IBC models require to solve complicated coupled partial
differential equations (PDE) because of explicit inhomogeneous boundary condition. An advantage
of HBC models is that they allow to deal with coupled ordinary differential equations (ODE)
because the stress tensor still remains independent of field theory directions and all bulk
fields can be treated as functions of the holographic direction. This technical advantage
enables us to analyse a model more easily and extensively. Thus it will make possible
more analytic and universal understanding on momentum dissipation mechanism at strong
coupling, even though its microscopic field theory interpretation is unclear yet.
In this paper, we study AC electric, thermoelectric, and thermal conductivities of a
HBC model proposed in [23], focusing on a Drude nature at low frequencies and scaling laws
at intermediate frequencies. The model we study is based on the Einstein-Maxwell-scalar
coordinate are considered so that translation symmetry is broken. Because they enter the
and gauge field still can be homogeneous in field theory direction. Furthermore, to have
isotropic bulk fields the identical scalar field is introduced for every field theory spatial
direction. In this model, the DC electric conductivity [23], thermoelectric and thermal
conductivity [26] were computed analytically and our focus is on AC conductivities. AC
electric conductivity was also studied in [25] and here we analyse it in greater detail as well
as thermoelectric and thermal conductivities. For AC conductivities in other HBC models
including massive gravity models we refer to [14, 2729].
At low frequencies, the Drude peak of electric conductivity has been observed in many
holographic models with momentum dissipation. For example see [912].
2Some of these models may be related to IBC models [19]. In a similar spirit, there are models utilising
a Bianchi VII0 symmetry to construct black holes dual to helical lattices [21, 22].
3This model may be understood also based on [19]. A single massless complex scalar with constant
in (2.6) of [19] gives rise to a massless axion linear in the x1 direction.
a quasi-particle picture. However, it was shown that this Drude-like peak can be realised
even when there is no quasi-particle picture at strong coupling if the translation symmetry
is broken weakly [30]. In this context, metal without quasi-particle can be divided into two
classes: coherent metal with a Drude peak and incoherent metal without a Drude peak [31].
However, since our model is based on AdS-RN black brane solution, there will be a term
containing the contribution from pair production affected by net charge density, which we
controlling the strength of the translation symmetry breaking, we may investigate how
coherent/incoherent metal phase is realised.4 Indeed, In our model, we find that when
and a modified Drude form is reduced to a standard Drude from. Also we confirm the
sum rule is satisfied for both cases, Drude and non-Drude. For thermoelectric and thermal
conductivities, qualitatively the same results are obtained.
potential, it was shown experimentally that certain high temperature superconductors in
the normal phase exhibit scaling law
models in a following modified form.
behaviours have been produced while in [14, 19, 25] no scaling law has been observed. In
our model we have analysed electric, thermoelectric, and thermal conductivities in a wide
range of parameters for both scaling laws (1.4) and (1.5). However it seems that there is
no robust scaling law, which agrees to the conclusion in [25].
From holographic perspective, the computation of electric, thermoelectric, and thermal
conductivities are related to the Dynamics of three bulk fields fluctuations (metric, gauge,
scalar fields). Their dynamics are determined by equations of motion, a system of second
order coupled ODEs. From the on-shell quadratic action for these fluctuations we can read
off the retarded Greens functions relevant to three conductivities. In the case that many
bulk fields are coupled, the computation of the holographic retarded Greens functions
is not very straightforward. To facilitate solving this important problems we introduce a
4The same question was addressed based on analytic DC conductivities in [22, 24].
systematic numerical method following [33, 34] adapted to our purpose. This method, used
to compute conductivities in this paper, can be applied to other models and problems. It
will be useful especially when many fields are coupled and the system has constraint coming
from the residual gauge symmetry.
This paper is organised as follows. In section 2, after reviewing Einstein-Maxwell
theory with massless scalar fields in general, we focus on a specific ground state solution to
introduce momentum relaxation. To set up the stage for AC conductivities, we summarise
equations for small fluctuations of relevant metric, gauge and scalar fields around the
ground state. In section 3, we present a general numerical method to compute retarded
Greens functions when many fields are coupled. By using this method, in section 4, we
compute AC electric, thermoelectric, and thermal conductivities. At low frequencies we
focus on the shape of the peak, Drude or non-Drude, and at intermediate frequencies we
search for possible scaling laws. In section 5 we conclude.
AdS-RN black branes with scalar sources
In this section we briefly review the holographic model of momentum relaxation studied
in [23]. We summarize essential minimum to set up stage for our study, AC conductivities,
and refer to [23, 25] for more details and extensions.
General action
boundary M
Let us start with the Einstein-Maxwell action on a four dimensional manifold M with
SEM =
4
1 F 2
2
3
and the cosmological constant l are equal to 1 . The second term is the Gibbons-Hawking
term required for a well defined variational problem with Dirichlet boundary conditions.
extrinsic curvature. In order to have a momentum relaxation effect, we include two free
massless scalars
4
g 2
1 X2 (I )2 .
RMN =
1 F 2 2
1 X2 (I )2 +
1 X M I N I +
M F MN = 0 ,
Given the solutions of these equations of motion, the holographically renormalised action
(Sren) [35] is obtained by the on-shell action of
where Sc is the counter term
Sc =
dx3
1 X2 I I
coordinate system
For a general understanding of Sren, it is useful to employ the Fefferman-Graham
Near the boundary the solutions are expanded as
ds2 =
g = g(0) + 2g(2) + 3g(3) + ,
I = (0) + 2(2) + 3(3) + ,
I I I
where leading terms g(0), A(0), (0) are chosen to be functions of the boundary coordinates
I
(incoming) boundary condition at the horizon.
With small fluctuations, the renormalisation on shell action up to linear order in
fluctuations reads
Sr(e1n) =
g(0)
OI respectively. Their expectation values are
The constraint (2.9) in terms of the one point function (2.11) yields the Ward identities
hT i = hOI iI(0) + F(0)hJ i ,
which correspond to the invariance of the renormalised action under a U(1) transformation
AdS-RN black brane
We want to study the field theory at finite charge density and finite temperature with
momentum dissipation. A gravity dual will be a charged black brane solution with broken
translation symmetry. Indeed the equations (2.3)(2.4) admit the following solutions [36]
ds2 = GMN dxM dxN = f (r)dt2 +
m0 = r03 1 +
4r02 2r02
These analytic solutions have been reported in [36] and explored further in the context of
functions, metric and gauge field are not, thanks to equal contributions from two scalars for
two spatial coordinates. However, with only one scalar field, the solutions are anisotopic
and this case has been studied in [37, 38].
dual field theory:
from which, r0 yields
T =
r0 =
3r0
the parameter which controls momentum relaxation. The parameter m0 obtained by the
density. In summary, for solutions (2.14)(2.17), one point function (2.11) is
T tti = 2m0 ,
T xxi = hT yyi = m0 ,
hO1i = 0 ,
Now we want to study the responses of this system for small perturbations. In
particular we are interested in the electric conductivity, which is related to the boundary
current operators J~. Because of rotational symmetry in x-y space, it is enough to consider
Jx. Since this operator is dual to the bulk gauge fields Ax, we consider a following linear
fluctuation around the background
The fluctuation is chosen to be independent of x and y. It is allowed since all the background
fields entering the equations of motion are independent of x and y. The gauge field
fluctuaand all the other fluctuations can be decoupled. Since we will work in momentum space,
that it goes to constant as r goes to infinity.
In momentum space, the linearised equations around the background are
h0t0x =0 ,
which are obtained from (2.3)(2.4). Among these four equations, only three are
independent.8 We need to solve these equations satisfying two boundary conditions: incoming
boundary conditions at the black hole horizon and the Dirichlet boundary conditions at
ax = a(x0) +
8The equations (2.24)(2.27) may be decoupled in terms of gauge invariant combinations [23]. The
equation governing electric conductivity turns out to be the same as the one in the massive gravity model [16],
while the equations for thermal/thermoelectric conductivities are different.
and the DC electric conductivity [23] is
Sr(e2n) = lim
r 2
which is derived from
which was computed at the horizon (not at the boundary) by rewriting the DC conductivity
technique using r-independent quantity is in line with [17, 39], but does not work for finite
the following section.
General numerical methods with constraint
The analytic method used in [23] is efficient to obtain the DC electric conductivity.
However, to compute AC electric conductivity together with AC thermal/thermoelectric
conductivity we have to resort to a numerical method. Since the conductivities are related
to the retarded Greens functions through the Kubo formula, we need to obtain an
action (generating functional) including two sources. A natural holographic starting point
is the on-shell renormalised action to quadratic order in fluctuation fields [2, 40, 41]. In
momentum space the on-shell action with the fluctuations (2.21)(2.23) reads
htx
where V2 is the two dimensional spatial volume R dxdy.
Notice that the boundary term at the horizon is deleted according to the prescription
to the retarded green function [40]. The boundary values of the fields are interpreted
as the sources of some dual field theory operators, so we may readily read off the two
point functions from the first two terms in (3.1), while the other three terms look not
straightforward. However, thanks to linearity of equations (2.24)(2.27), we can always
linear relation the action is reduced to the schematic form as follows.
Sr(e2n) =
a little bit to be more succinct and economical.9
a = 1, 2, , N ,
Sr(e2n) = lim
r 2
ddk hak(r)Aab(r, k)bk(r) + ak(r)Bab(r, k)rbk(r)i ,
A and B are regular matrices of order N . The renormalized action (3.5) is assumed to
contain all the counter terms. For example, see (3.1) for an action and (2.24)(2.27) for a
system of equations.
When the differential equations are second order we need to give 2N boundary
conthem J a) are at the boundary. For numerical integration, we have to convert the boundary
value problem to the initial value problem, by considering 2N canonical initial data at
the event horizon. We solve the initial value problem for N independent initial value set
and judiciously combine N set of solutions such that final value of solution is identical to
the boundary value we have chosen. The procedure will be independent of the chose of
canonical initial data as we will show below.
where the index a includes components of higher spin fields. rq is multiplied such that the
general on-shell quadratic action in momentum space has the form of
9In some cases the equations may be separable in terms of master fields. However, our method applies
to any number of coupled fields straightforwardly and we dont need to try to figure out master fields.
a(r) = (r 1)a (a + a(r 1) + )
where we omitted the subscript k for simplicity and correspond to incoming/outgoing
boundary conditions. To compute the retarded Greens function we choose the incoming
boundary condition [40], fixing N initial conditions. The other N initial conditions
correincoming boundary condition, a determines a through horizon-regularity condition so
that we can determine the solution completely.
For example, ia may be chosen as
1 1 1 . . . 1
1 . . . 1
is expanded as10
where Sia denote the leading terms of i-th solution and Oia are the sub-leading term
dewritten as regular matrices of order N , where the superscript a runs for row index and the
subscript i runs for column index.
The general solution is a linear combination of them: let
(near boundary) ,
In this case, the second term of (3.5) may be written as
J a = Siaci .
with real constants cis. We need to choose ci such that the combined source term matches
the boundary value J a:
no constraint related to the diffeomorphic invariance, it can be done simply by noting that
ci is expressed in terms of J a
Bac(r, k)(crc1Ock)(S1)ibi J b + := [Cab(r, k)] J b + .
the independence of the choice of the initial condition (3.7), because the different choice
right multiplications in the solutions: S SR, O OR.
With (3.10) and (3.12) the final boundary action yields
Sr(e2n) =
1 Z
where we reinserted the subscript k. Since matrices A, C are independent of J , the retarded
Greens function is
GaRb = Aab(, k) + Cab(, k) .
example, see g(2) and (2) in (2.8). Those terms can be taken care of by counter terms and we dont write
I
them here to focus on essential ideas.
Notice that for one field case without mass term, this is the well known structure of the
of the coefficients of the subleading term and the leading term.
In summary, to compute the retarded Greens function we need four square matrices
of order N (the number of fields): A, B, S, O. A and B can be read off from the boundary
action (3.5). S and O are given from the solution of a set of differential equations. We have
to solve N times with 2N independent initial conditions to construct regular matrices of
A + C. The
precise form of C is shown in (3.12).
Our story so far is for the system without constraint. In actual calculation, Einstein
the space spanned by {S~1, , S~N1}. Then,
equation always contains constraint equations (CE) due to the residual diffeomophism
invariance of the linearised equation of motion. We describe how to fix this complication
for the case of one CE for notational convenience. Generalization to two or more CE is
straightforward. At the horizon, the CE relates the (initial) field values. For example, the
last component of ~j can be determined by other components. So the space of initial value
vectors (IVV) is N-1 dimensional subspace. Solving differential equation using such N-1
IVV gives, of course, only N-1 solutions. However, when we give boundary conditions, we
formally assign N boundary values Ji. As a consequence, eq. (3.10) is not invertible! To
N1
X S~j cj + S~0 c0 = J~.
Now this equation is invertible. We can find proper c0 and cis for any choice of initial
condition vectors to fit the given J~. Notice that S~0 does not generate true degree of freedom
since it is along the diffeomorphism orbit direction. It is not hard to see why S~0 actually
satisfies the on-shell condition: this happens since the residual gauge transformation leaves
is not a true solution satisfying in falling conditions, it can serve as an element of a basis
of the on-shell J space and on shell condition.
that in this proof of basis independence, we used the fact that the differential equations
define a linear map U (r) : ia Uba(r)ib, so that the evolution operator U (r) is a left
multiplication (acting on upper index a), while basis change is a right multiplication acting
on index i, so that initial data ia at the horizon and the final solution Sia at the boundary
is multiplied by the same matrix R.
In order to check the validity of our numerical method and code, we computed AC
the figure 6 of [2]. It is a nontrivial consistency check of our method since the plot in [2]
corresponding to delta functions
has been obtained by solving a single equation of the gauge field ax, while we have solved
coupled equations of ax and gtx. Of course if the coupled equations can be decoupled
as shown in [2] there is no point of solving coupled equations. However, because this
decoupling is not always possible it is important to develop a systematic and efficient
method for coupled fields cases. In addition to the agreement of figure 1 to figure 6 of [2]
3 4r02
K = r0
figure 1(c) from figure 1(b).
Electric/thermal/thermoelectric AC conductivities
we want to attack our main problem, AC conductivity with moment dissipation generated
can read off the conductivities from the action (3.1). To closely follow the general methods
presented in section 3 we rewrite the action as
s(re2n)
Sr(e2n) = lim
r
Z d ha(r)Aab(r, )b(r) + a(r)Bab(r, )rb(r)i ,
B =
f (r) 0
0 r2f (r)
To compute the matrix C in (3.12) we have to solve the equations (2.24)(2.27), which
we rewrite here setting r0 = 1:
Since only three equations are independent we may solve any three of them. Near the black
hole horizon (r 1) the solutions are expanded as
r4
h0t0x =0 ,
htx = (r 1)+1(ht(xI) + ht(xII)(r 1) + ),
ax = (r 1) (a(xI) + a(xII)(r 1) + ),
= (r 1) ((I) + (II)(r 1) + )
htx = ht(x0) +
ax = a(x0) +
With incoming boundary condition and initial values (3.7) at horizon we numerically
integrate the equations from the horizon.
For our equations there is one subtlety caused by a symmetry of the system. Analysing
a complete basis to reconstruct a general boundary value vector J~. However, there is a
S~0 = (a0x, ht0x, 0)T = (0, 1, i/)T .
Notice that S~0 satisfies the equations of motion, as we mentioned in earlier section. Since
S~0 happened to be r-independent, it is equivalent to formally add a constant solutions
ax = 0 ,
htx = ht0x ,
introduced in [46]. It is interesting that (4.10) is similar to (3.8) of [19], which is a condition
imposed at infinity to extract the gauge-invariant conductivity.11 They first choose initial
data such that the final solution lies on the gauge orbit and then using the gauge
transfor i ht(x0) = 0 is nothing but the equation
of the gauge orbit (in the space of leading component of solutions) passing (a0x, 0, 0).
one can calculate electric conductivity by a(x1)/a(x0). In this approach, one can calculate
only electric conductivity and shooting is cumbersome.
simultaneously. Again, we emphasise that true physical degree of freedom is 2 dimension
and S~0 does not generate a true physical degree of freedom.
Green functions and Transport coefficients
Having computed numerically the matrices S and O, we may construct a 3 3 matrix of
retarded Greens function. We will focus on the 2 2 submatrix corresponding to a(x0) and
ht(x0) in (4.8). Since a(x0) is dual to U(1) current Jx and ht(x0) is dual to energy-momentum
tensor Ttx the Greens function matrix may be written as
11We thank Aristomenis Donos and Jerome Gauntlett for pointing out this similarity.
where we introduced the second term for notational simplicity. From the linear response
theory, we have the following relation between the response functions and the sources:
We want to relate the Greens functions (4.11) to phenomenological transport
coefficients. Our goal is to study the electric, thermal, thermoelectric conductivities defined as
(xT )/T
temperature gradient. By taking into account a diffeomorphism invariance [2, 3], (4.13)
can be expressed as
corresponding to a delta function in (a)
transition to incoherent metal is manifest.
are the analytic DC values (4.16).
Comparing (4.14) and (4.12) we have
response theory: one has to subtract static term whenever the latter is non-zero [47].
This is related to the fact that for the causal Green functions, we have to apply source
fields only in the past not in future. Without such subtraction, we would have unphysical
term [27]. In summary, we numerically compute G11, G12, G21, G22 by (3.14) and combine
them as (4.15) for physical conductivities.
electric conductivity respectively. In figure 4 and 5 we analyse the conductivity at small
maximum value of the peak (DC conductivities) decreases. In this variation, we checked
that the area of the real part of the conductivity does not change numerically. The area
inferred from the imaginary part of the conductivity. This is an example of a sum rule and
we have confirmed it for various parameters. Numerical DC conductivities agree to the
analytic result (2.29)
There are two issues on finite frequency regime: one is Drude-like peaks at small
frequency and the other is possible scaling laws at intermediate frequency regime. Let us
is broken weakly and we expect to have a Drude form according to [30]. For large values
much these peaks can resemble the Drude model, Let us examine the Ward identity. At
the level of fluctuation the Ward identity (2.13) is
Comparing with the Drude model
12We thanks the referee to point this out.
, it turns out that hOi hpxi,
our model is based on AdS-RN black brane solution, there will be a contribution from pair
K = r0
r0 =
which is defined in (2.19).
effect. For T
time (4.24) yields
The expression (4.24) is not very illuminating so we make a plot of the relaxation time
We compare numerical data (blue dotted lines) with a Drude model (red solid
To check the validity of our analytic expression of the Drude model (4.19) with
parameters (4.20) and (4.24), we have made numerical plots for a wide range of parameters
and compared with (4.19). Figure 5(a,d) and (c,f) are examples showing a good agreement
of numerical data to (4.19) and deviation from (4.19) respectively.14 Blue dotted lines are
13This form of Drude conductivity implicitly appeared in [48] with shifted pole due to the magnetic field.
14Similar plots were obtained independently by Blaise Gouteraux and Richard Davison and presented at
the workshop, Holographic methods and application, Iceland, August, 2014.
numerical data and red solid curves are the analytic expression (4.19). In figure 5 (c,f) if
fitting curve is slightly improved, but it is still deviated from (4.19). In these examples,
peak is not a Drude form. It is a concrete realisation of coherent/incoherent transition
Next, we want to investigate the scaling property in the intermediate frequency regime.
superconductors in the normal phase exhibits scaling law
models with momentum dissipation. In models studied in [9, 10, 12] modified scalings (4.28)
have been reported while in [14, 19, 25] no scaling law have been observed. With our model
we have analysed several cases for a wide range of parameters to search a scaling behaviour.
However it seems that there is no robust scaling law.
-5.0 -4.5 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5
robust scaling law under change of parameters. These plots are presented to show how the constant
shift C in (4.28) can improve the scaling behaviour.
do not see a scaling behaviour of the form (4.27).
Now we want to investigate if there is a modified scaling law motivated by previous
holographic models [9, 10, 12, 16].
which is shown in figure 7. Interestingly, in this case, the constants B and C in (4.28)
determined. However, this approximate scaling behaviour is not precise15 and robust under
change of parameters. We present figure 7 to show how the constant shift C in (4.28) can
improve the scaling behaviour of figure 6(c) even though it is not an evidence of a scaling
behaviour. After numerical analysis with a wide variety of parameters and cases we do not
see a scaling behaviour of the form (4.28), which agrees to the conclusion in [25].
Thermoelectric and thermal conductivity
analytically computed in [26]
In order to discuss the Wiedemann-Franz law, we compute the ratio of the DC thermal
conductivity to the DC electric conductivity as follows
where we took low temperature limit in the last expression since the Wiedemann-Franz law
L =
but the numerical values are different from the Fermi-liquid case, as expected in a
nonFermi liquid, see e.g. [50].
At small frequencies, like electric conductivity, thermoelectric and thermal
conductivbe obtained analytically by using the hydrodynamics results in [4345]. Figure 9 shows an
excellent agreement of numerical data to (4.34), where the blue dots are numerical values
are fitting curves of the from (4.34).
that three relaxation time are almost the same. This is because in all three cases the Drude
frequencies, we do not see any scaling law unlike [10].
Conclusions
based on the 3+1 dimensional Einstein-Maxwell-scalar action. Momentum is dissipated
due to massless scalar fields linear to every spatial coordinate.
purity. There are two more free parameters in the model: temperature (T ) and chemical
given analytically. They depend on only holographic direction because the scalar field
gauge, and scalar fields) relevant for three conductivities can be chosen to be functions of
only the holographic direction, so the computations can be done by coupled ODEs rather
16We thanks the referee for pointing this out.
independent constants, but the numerical values are different from the Fermi-liquid case.
We presented a concrete realisation of coherent/incoherent transition induced by impurity
conductivities show a modified Drude peak. For example, for electric conductivity,
with different parameter values. For example, the relaxation times are different for three
below (2.20). If we can use massive one the constant plateaux will disappear.
some holographic models [9, 10]. but we find no robust scaling law, which agrees to the
conclusion in [14, 19, 25]. In [51] a mechanism to engineer scaling laws was provided, where
translation symmetry is not broken. It would be interesting to generalize it to our case.
by Ward identities, and once electric conductivity is given the other two are algebraically
determined [2, 3]. In our model the relationship between them are more complicated,
involving the background scalar fields. It will be interesting to understand how their
We introduced a general numerical method to compute the holographic retarded
Greens functions when many fields are coupled. This method, used to compute three
conductivities in this paper, can be applied also to other models and problems such
as [19, 20, 22, 24, 25]. It would be interesting to extend our analysis to dyonic black
holes and holographic superconductors [49]. It would be also interesting to study the
models based on other free massless form fields introduced in [36], which may be used to
engineer certain desired properties of condensed matter systems.
Acknowledgments
We would like to thank Yunkyu Bang, Richard Davison, Aristomenis Donos, Jerome
Gauntlett, Xian-Hui Ge, Blaise Gouteraux, Takaaki Ishii, Yan Liu, Ya-Wen Sun, and
Marika Taylor for valuable discussions and correspondence. 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
(NRF2014R1A1A1003220). The work of SS and YS was supported by Mid-career Researcher
Program through the National Research Foundation of Korea (NRF) grant No.
NRF2013R1A2A2A05004846. 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), and at CERN
(Numerical holography, Dec. 2014), where part of this work was done.
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