MPP-2007-42
Isospin diffusion in thermal AdS/CFT with flavor
Johanna Erdmenger, Matthias Kaminski, Felix Rust∗
Max-Planck-Institut für Physik (Werner-Heisenberg-Institut),
Föhringer Ring 6, 80805 München, Germany
We study the gauge/gravity dual of a finite temperature field theory at finite isospin chem-
ical potential by considering a probe of two coincident D7-branes embedded in the AdS-
Schwarzschild black hole background. The isospin chemical potential is obtained by giving
a vev to the time component of the non-Abelian gauge field on the brane. The fluctuations
of the non-Abelian gauge field on the brane are dual to the SU(2) flavor current in the field
theory. For the embedding corresponding to vanishing quark mass, we calculate all Green
functions corresponding to the components of the flavor current correlator. We discuss the
physical properties of these Green functions, which go beyond linear response theory. In par-
ticular, we show that the isospin chemical potential leads to a frequency-dependent isospin
diffusion coefficient.
PACS numbers: 11.25.Tq, 11.25.Wx, 12.38.Mh, 11.10.Wx
Contents
I. Introduction 2
II. Hydrodynamics and AdS/CFT 4
III. Supergravity background and action 5
A. Finite temperature background and brane configuration 5
B. Introducing a non-Abelian chemical potential 6
C. Dirac-Born-Infeld action 7
D. Equations of motion 8
1. Equations for Aa1- and A
2-components 8
2. Equations for Aa0- and A
3-components 9
3. Solutions 9
IV. Isospin diffusion and correlation functions 12
A. Current correlators 12
1. Green functions: Calculation 13
2. Green functions: Results 14
B. Isospin diffusion 17
V. Conclusion 19
∗Electronic address: jke@mppmu.mpg.de, kaminski@mppmu.mpg.de, rust@mppmu.mpg.de
http://arxiv.org/abs/0704.1290v2
mailto:jke@mppmu.mpg.de, kaminski@mppmu.mpg.de, rust@mppmu.mpg.de
Acknowledgments 19
A. Notation 20
B. Solutions to equations of motion 20
1. Solutions for Xα, X̃α and A
2. Solutions for X ′0, X̃
0 and A
3. Solutions for X ′3, X̃
3 and A
4. Comparison of numerical and analytical results 23
C. Abelian Correlators 23
D. Correlation functions 24
E. Thermal spectral functions 24
References 25
I. INTRODUCTION
Over the past years, there have been a number of lines of investigation for describing QCD-
like theories with gravity duals. In this way, considerable progress towards a gauge/gravity dual
description of phenomenologically relevant models has been made. One of these lines of investi-
gation is the gravity dual description of the quark-gluon plasma obtained by applying AdS/CFT
to relativistic hydrodynamics [1, 2, 3, 4, 5, 6, 7]. The central result of this approach is the cal-
culation of the shear viscosity from AdS/CFT. More recently, an R charge chemical potential has
been introduced by considering gravity backgrounds with R charged black holes [8, 9, 10], and
also the heat conductivity has been calculated by considering the R current correlators in these
backgrounds [8].
A further approach to generalizing the AdS/CFT correspondence to more realistic field theories
is the addition of flavor to gravity duals via the addition of probe branes [11, 12, 13, 14, 15]. This
allows in particular for the calculation of meson masses.
These two approaches have been combined in order to study the flavor contribution to finite
temperature field theories from the gravity dual perspective. This began with [13] where the
embedding of a D7-brane probe into the AdS-Schwarzschild black hole background was studied
and a novel phase transition was found, which occurs when the D7 probe reaches the black hole
horizon. This transition was shown to be of first order in [16] (see [14] for a similar transition
in the D4/D6 system), and studied in further detail in [17, 18]. Related phase transitions appear
in [19, 20, 21]. Mesons in gravity duals of finite temperature field theories have been studied in
[22, 23, 24, 25].
Recently, in view of adding flavor to the quark-gluon plasma, the flavor contribution to the
shear viscosity has been calculated in [26, 27], where it was found that ηfund ∝ λNcNfT 3.
For a thermodynamical approach in the grand canonical ensemble, the inclusion of a chemical
potential and a finite number density is essential. In [28], an isospin chemical potential was in-
troduced by considering two coincident D7 probes, and by giving a vev to the time component of
the SU(2) gauge field on this probe. This was shown to give rise to a thermodynamical instability
comparable to Bose-Einstein condensation, compatible with the field-theoretical results of [29].
For the gauge/gravity dual analysis, a potential generated by an SU(2) instanton on the D7 probe
in the gravity background was used [30, 31, 32, 33].
A baryon chemical potential µB is obtained by turning on the diagonal U(1) ⊂ U(Nf ) gauge
field on the D7-brane probe [34]. Contributions to the D7-brane action arise from the deriva-
tive of this U(1) gauge field with respect to the radial direction. The effects of this potential on
the first-order phase transition described above have been studied in [34], where regions of ther-
modynamical instability have been found in the (T, µB) phase diagram. – For the D4/D8/D̄8
Sakai-Sugimoto model [15, 35], the phase transitions in presence of a baryon number chemical
potential, as well as physical processes such as photoemission and vector meson screening, have
been studied in [36, 37, 38].
A related approach has been used to calculate the rate of energy loss of a heavy quark moving
through a supersymmetric Yang-Mills plasma at large coupling [39]. In this approach the heavy
quark is given by a classical string attached to the D7-brane probe. – A first study of flavors in
thermal AdS/CFT beyond the quenched approximation, i.e. with Nf ∼ Nc, was performed in [40].
Here we study finite-temperature field theories with finite isospin chemical potential by consid-
ering two coincident D7-brane probes in the Lorentzian signature AdS-Schwarzschild black hole
background. As in [28], we introduce an isospin chemical potential by defining
, (1.1)
for the time component of the SU(2) gauge field on the two coincident D7-branes. This constant
chemical potential is a solution to the D7-brane equations of motion and is present even for the
D7-brane embedding corresponding to massless quarks. We consider small µ, such that the Bose-
Einstein instability mentioned above, which is of order O(µ2), does not affect our discussion here.
For simplicity we consider only the D7 probe embedding for which the quark mass vanishes,
m = 0. This embedding is constant and terminates at the horizon. We establish the SU(2) non-
Abelian action for a probe of two coincident D7-branes. We obtain the equations of motion for
fluctuations about the background given by (1.1). These are dual to the SU(2) flavor current Jµa.
We find an ansatz for decoupling the equations of motion for the different Lorentz and flavor com-
ponents, and solve them by adapting the method developed in [2, 3]. This involves Fourier trans-
forming to momentum space, and using a power expansion ansatz for the equations of motion. We
discuss the approximation necessary for an analytical solution, which amounts to considering fre-
quencies with ω < µ < T . With this approach we obtain the complete current-current correlator.
The key point is that the constant chemical potential effectively replaces a time derivative in the ac-
tion and in the equations of motion. In the Fourier transformed picture, this leads to a square-root
dependence of physical observables on the frequency,
ω. This non-linear behavior goes beyond
linear response theory. We discuss the physical properties of the Green functions contributing to
the current-current correlator. In particular, for small frequencies we find a frequency-dependent
diffusion coefficient D(ω) ∝ 1
ω/µ. Whereas frequency-dependent diffusion has – to our
knowledge – not yet been discussed in the context of the quark-gluon plasma, it is well-known in
the theory of quantum liquids. For instance, for small frequencies the square-root behavior we find
agrees qualitatively with the results of [41, 42] for liquid para-hydrogen. Generally, frequency-
dependent diffusion leads to a non-exponential decay of time-dependent fluctuations, as discussed
for a classical fluid in [43].
Physically, the isospin chemical potential corresponds to the energy necessary to inverting the
isospin of a given particle. Within nuclear physics, such a chemical potential is of relevance
for the description neutron stars. Moreover, isospin diffusion has been measured in heavy ion
reactions [44, 45]. – For two-flavor QCD, effects of a finite isospin chemical potential have been
discussed for instance in [46, 47, 48]. The phase diagrams discussed there are beyond the scope
of the present paper. We expect to return to similar diagrams in the gauge/gravity dual context in
the future.
This paper is organized as follows. In section 2 we summarize the AdS/CFT hydrodynamics
approach to calculating Green functions, which we use in the subsequent. Moreover we comment
on frequency-dependent diffusion within hydrodynamics. In section 3 we establish the D7 probe
action in presence of the isospin chemical potential, derive the corresponding equations of motion
and solve them. In section 4 we obtain the associated Green functions in the hydrodynamical
approximation. We discuss their pole structure and obtain the frequency-dependent diffusion co-
efficient. We conclude in section 5 with an interpretation of our results. An explanation of our
notation as well as a series of calculations are relegated to a number of appendices.
II. HYDRODYNAMICS AND ADS/CFT
Thermal Green functions have proven to be a useful tool for analyzing the structure of hydro-
dynamic theories and for calculating hydrodynamic quantities such as transport coefficients. For
instance, given a retarded current correlation function G(~k)µν in Minkowski space, the spectral
function can be written in terms of its imaginary part,
χµν(~k) = −2 ImGµν(~k) . (2.1)
For the gravity dual approach, this is discussed for instance in [7, 49].
In this paper we use the gauge/gravity dual prescription of [3] for calculating Green functions
in Minkowski spacetime. For further reference, we outline this prescription in the subsequent. It is
based on the AdS/CFT-correspondence relating supergravity fields A in a black hole background
to operators J in the dual gauge theory. The black hole background is asymptotically Anti-de
Sitter space and places the dual field theory at finite temperature. This temperature corresponds
to the Hawking temperature of the black hole, or more generally speaking, of the black branes.
Starting out from a classical supergravity action Scl for the gauge field A, according to [3] we
extract the function B(u) (containing metric factors and the metric determinant) in front of the
kinetic term (∂uA)
Scl =
dud4xB(u) (∂uA)
2 + . . . (2.2)
Then we perform a Fourier transformation and solve the linearized equations of motion in momen-
tum space. This is a second order differential equation, so we have to fix two boundary conditions.
The first one at the boundary of AdS at u = 0 can be written as
A(u,~k) = f(u,~k)Abdy(~k) , (2.3)
where Abdy(~k) is the value of the supergravity field at the boundary of AdS depending only on
the four flat boundary coordinates. Thus by definition we have f(u,~k)
= 1. For the other
boundary, located at the horizon u = 1, we impose the incoming wave condition. This requires
that any Fourier mode A(~k) with timelike ~k can travel into the black hole, but is not allowed to
cross the horizon in the opposite direction. For spacelike ~k, the components of A have to be regular
at the horizon. Then the retarded thermal Green function is given by
G(ω, q) = −2B(u) f(u,−~k) ∂u f(u,~k)
. (2.4)
The thermal correlators obtained in this way display hydrodynamic properties, such as poles
located at complex frequencies. Generically, for the R current component correlation functions
calculated from supergravity, there are retarded contributions of the form
G(ω, q) ∝
iω −Dq2
. (2.5)
This may be identified with the the Green function for the hydrodynamic diffusion equation
∂0 J0(t,x) = D∇2 J0(t,x) , (2.6)
with J0 the time component of a diffusive current. D is the diffusion constant. In Fourier space
this equation reads
iωJ0(ω,k) = Dk
2J0(ω,k) . (2.7)
In position space, this corresponds to an exponential decay of J0 with time.
For the non-Abelian case with an isospin chemical potential, in sections III and IV we will
obtain retarded Green functions of the form
G(ω, q) ∝ 1
iω −D(ω)q2
. (2.8)
Retarded Green functions of this type have been discussed for instance in [43]. (2.8) corresponds
to frequency-dependent diffusion with coefficient D(ω), such that (2.7) becomes
iωJ0(ω,k) = D(ω)k
2J0(ω,k) . (2.9)
In our case, J0 is the averaged isospin at a given point in the liquid.
This is a non-linear behavior which goes beyond linear response theory. In particular, when
Fourier-transforming back to position space, we have to use the convolution for the product D · J0
and obtain
∂0J0(t,x) +∇2
ds J0(s,x)D(t− s) = 0 (2.10)
for the redarded Green function. This implies together with the continuity equation ∂0 J0 +∇·J =
0, with J the three-vector current associated to J0, that
J = −∇(D ∗ J0) , (2.11)
where ∗ denotes the convolution. This replaces the linear response theory constitutive equation
J = −D∇J0. Note that for D(t− s) = Dδ(t− s) with D constant, (2.10) reduces again to (2.6).
III. SUPERGRAVITY BACKGROUND AND ACTION
A. Finite temperature background and brane configuration
We consider an asymptotically AdS5 × S5 spacetime as the near horizon limit of a stack of Nc
coincident D3-branes. More precisely, as in [2], our background is an AdS black hole, which is
the geometry dual to a field theory at finite temperature. The Minkowski signature background is
ds2 =
−f(u) dx20 + dx21 + dx22 + dx23
4u2f(u)
du2 +R2dΩ25,
0 ≤ u ≤ 1, xi ∈ R, C0123 =
(3.1)
with the metric dΩ25 of the unit 5-sphere, and the function f(u), AdS radius R and temperature
parameter b given in terms of the string coupling gs, temperature T , inverse string tension α
′ and
number of colors Nc by
f(u) = 1− u2, R4 = 4πgsNcα′2, b = πT. (3.2)
The geometry is asymptotically AdS5 × S5 with the boundary of the AdS part located at u = 0.
At the black hole horizon the radial coordinate u has the value u = 1.
Into this ten-dimensional spacetime we embed Nf = 2 coinciding D7-branes, hosting flavor
gauge fields Aµ. The embedding we choose extends the D7-branes in all directions of AdS space
and wraps an S3 on the S5. In this work we restrict ourselves to the most straightforward case,
that is the embedding of the branes through the origin along the AdS radial coordinate u. This
corresponds to massless quarks in the dual field theory. On the brane, the metric in this case simply
reduces to
ds2 =
−f(u) dx20 + dx21 + dx22 + dx23
4u2f(u)
du2 +R2dΩ23,
0 ≤ u ≤ 1, xi ∈ R.
(3.3)
Due the choice of our gauge field in the next subsection, the remaining three-sphere in this metric
will not play a prominent role.
The table below gives an overview of the indices we use to refer to certain directions and
subspaces.
AdS5 S
coord. names x0 x1 x2 x3 u –
µ, ν. . .
indices i, j. . . u
B. Introducing a non-Abelian chemical potential
A gravity dual description of a chemical potential amounts to a non-dynamical time component
of the gauge field Aµ in the action for the D7-brane probe embedded into the background given
above. There are essentially two different ways to realize a non-vanishing contribution from a
chemical potential to the field strength tensor F = 2∂[µAν] + f
abcAbµA
ν . The first is to consider
a u-dependent baryon chemical potential for a single brane probe. The second, which we pursue
here, is to consider a constant isospin chemical potential. This requires a non-Abelian probe brane
action and thus a probe of at least two coincident D7-branes, as suggested in [28]. Here the time
component of the gauge field is taken to be
A0 = A
a, (3.4)
where we sum over indices which occur twice in a term and denote the gauge group generators by
ta. The brane configuration described above corresponds to an SU(Nf) gauge group with Nf = 2
on the brane, which corresponds to a global SU(Nf ) in the dual field theory. For Nf = 2, the
generators of the gauge group on the brane are given by ta = σ
, with Pauli matrices σa. We will
see that (3.4) indeed produces non-trivial new contributions to the action.
Using the standard background field method of quantum field theory, we will consider the
chemical potential as a fixed background and let the gauge fields fluctuate. We single out a par-
ticular direction in flavor space by taking A30 = µ as the only non-vanishing component of the
background field. From now on we use the symbol Aaν to refer to gauge field fluctuations around
the fixed background,
Aaν → µδν0δa3 + Aaν . (3.5)
We pick the gauge in which Au ≡ 0 and assume that Aµ ≡ 0 for µ = 5, 6, 7. Due to the
symmetries of the background, we effectively examine gauge field fluctuations Aµ living in the
five-dimensional subspace on the brane spanned by the coordinates x0, x1, x2, x3 and by the radial
AdS coordinate u. The magnitude of all components of A and the background chemical potential
µ are considered to be small. This allows us to simplify certain expressions by dropping terms of
higher order in A and in the chemical potential µ.
C. Dirac-Born-Infeld action
The action describing the dynamics of the flavor gauge fields in the setup of this work is the
Dirac-Born-Infeld action. There are no contributions from the Chern-Simons action, which would
require non-zero gauge field components in all of the 4,5,6,7-directions. As mentioned, we con-
sider the D7 probe embedding whose asymptotic value at the boundary is chosen such that it cor-
responds to vanishing quark mass, m = 0. The metric on the brane is then given by (3.3). Since
we are interested in two-point correlators only, it is sufficient to consider the action to second order
in α′,
SD7 = −T7
(2πα′)2
2π2R3 Tr
uh=1∫
du d4x
−g gµµ′ gνν′ F aµν F aµ′ν′ , (3.6)
where we use the following definitions for the D7-brane tension T7 and the trace over the repre-
sentation matrices ta,
T7 = (2π)
7g−1s (α
′)−4 , (3.7)
tr(ta tb) = Tr δ
ab . (3.8)
In our case we have Tr = 1/2. The overall factor 2π
2R3 comes from the integration over the
5, 6, 7-directions, which are the directions along the S3.
Evaluating the DBI action given in (3.6) with the substitution rule (3.5), we arrive at
SD7 = − T7
(2πα′)2
2π2R3 Tr
uh=1∫
du d4x
−ggµµ′gνν′
4∂[µA
ν] ∂[µ′A
ν′] − 8δ0νδ0ν′fabc∂[0Aaµ] Abµ′ µc
(3.9)
where we use the short-hand notation µc = µδ3c and neglect terms of higher than linear order in
µ, and higher than quadratic order in A since both are small in our approach.
Up to the sum over flavor indices a, the first term in the bracket in (3.9) is reminiscent of the
Abelian super-Maxwell action in five dimensions, considered already for the R charge current
correlators in [2]. The new second term in our action arises from the non-Abelian nature of the
gauge group, giving terms proportional to the gauge group’s structure constants fabc in the field
strength tensor F aµν = 2∂[µA
+ fabcAbµA
D. Equations of motion
We proceed by calculating the retarded Green functions for the action (3.9), following the
prescription of [3] as outlined in section 2 above. According to this prescription, as a first step we
consider the equations of motion obtained from the action (3.9), which are given by
0 = 2∂µ
−g gµµ′gνν′ ∂[µ′Aaν′]
+ fabc
−gg00gνν′ µc
0 − 2∂0Abν′
+ δ0 ν∂µ
−g g00gµµ′Abµ′µc
(3.10)
It is useful to work in momentum space from now on. We therefore expand the bulk gauge fields
in Fourier modes in the xi directions,
Aµ(u, ~x) =
(2π)4
e−iωx0+ik·xAµ(u,~k). (3.11)
As we work in the gauge where Au = 0, we only have to take care of the components Ai with
i = 0, 1, 2, 3.
For the sake of simplicity, we choose the momentum of the fluctuations to be along the x3
direction, so their momentum four-vector is ~k = (ω, 0, 0, q). With this choice we have specified to
gauge fields which only depend on the radial coordinate u, the time coordinate x0 and the spatial
x3 direction.
1. Equations for Aa1- and A
2-components
Choosing the free Lorentz index in the equations of motion (3.10) to be ν = α = 1, 2 gives two
identical differential equations for A1 and A2,
0 = Aaα
2 − fq2
Aaα + 2i
Acα, (3.12)
where we indicated the derivative with respect to u with a prime and have introduced the dimen-
sionless quantities
, q =
, m =
. (3.13)
We now make use of the structure constants of SU(2), which are fabc = εabc, where εabc is the
totally antisymmetric epsilon symbol with ε123 = 1. Writing out (3.12) for the three different
choices of a = 1, 2, 3 results in
0 = A1α
2 − fq2
A1α − 2i
A2α , (3.14)
0 = A2α
2 − fq2
A2α + 2i
A1α , (3.15)
0 = A3α
2 − fq2
A3α . (3.16)
The first two of these equations are coupled, the third one is the same equation that was solved in
the Abelian Super-Maxwell case [2].
2. Equations for Aa0- and A
3-components
The remaining choices for the free Lorentz index ν = 0, 3, u in (3.10) result in three equations
which are not independent. The choices ν = 0 and ν = u give
0 = Aa0
Aa0 −
Aa3 − i
Ac3 , (3.17)
0 = wAa0
′ + qfAa3
′ + ifabc
′ . (3.18)
Solving (3.18) for Aa0
′, differentiating it once with respect to u and using (3.17) results in equation
(3.10) for ν = 3,
0 = Aa3
Aa3 +
Aa0 + i
Ac0 + 2i
Ac3 . (3.19)
We will make use of the equations (3.17) and (3.18) which look more concise. These equations of
motion for Aa0 and A
3 are coupled in Lorentz and flavor indices. To decouple them with respect
to the Lorentz structure, we solve (3.18) for Aa3
′ and insert the result into the differentiated version
of (3.17). This gives
0 = Aa0
′′′ +
(uf)′
2 − fq2
′ + 2i
′. (3.20)
The equations for a = 1, 2 are still coupled with respect to their gauge structure. The case
a = 3 was solved in [2]. We will solve (3.20) for Aa0
′ and can obtain Aa3
′ from (3.18). Note that it
is sufficient for our purpose to obtain solutions for the derivatives of the fields. These contribute to
(2.4), while the functions A = f(u,~k)Abdy(~k) themselves simply contribute a factor of f(u,−~k)
which is one at the boundary.
3. Solutions
Generally, we follow the methods developed in [2], since our differential equations are very
similar to the ones considered there. Additionally, we need to respect the flavor structure of the
gauge fields. The equations for flavor index a = 3 resemble the ones analyzed in [2]. However,
those for a = 1, 2 involve extra terms, which couple these equations. In our case the equations are
coupled not only via their Lorentz indices, but also with respect to the flavor indices. We already
decoupled the Lorentz structure in the previous section. As shown below, the equations of motion
which involve different gauge components will decouple if we consider the variables
Xi = A
i + iA
X̃i = A
i − iA2i .
(3.21)
Here the A1i , A
i are the generally complex gauge field components in momentum space. Note
that up to SU(2) transformations, the combinations (3.21) are the only ones which decouple the
equations of motion for a = 1, 2. These combinations are reminiscent of the non-Abelian SU(2)
gauge field in position space,
Ai = A
A3i A
i − iA2i
A1i + iA
i −A3i
. (3.22)
The equations of motion for the flavor index a = 3 were solved in [2]. To solve the equations
of motion for the fields Aai with a = 1, 2, we rewrite them in terms of Xi and X̃i. Applying the
transformation (3.21) to the equations of motion (3.14) and (3.15) and the a = 1, 2 versions of
(3.20) and (3.18) leads to
0 = X ′′α +
X ′α +
2 − fq2 ∓ 2mw
Xα, α = 1, 2, (3.23)
0 = X ′′′0 +
(uf)′
X ′′0 +
2 − fq2 ∓ 2mw
X ′0, (3.24)
0 = (w∓m)X ′0 + qfX ′3, (3.25)
where the upper signs correspond to X and the lower ones to X̃.
We see that some coefficients of these functions are divergent at the horizon u = 1. Such
differential equations with singular coefficients are generically solved by an ansatz
Xi = (1− u)β F (u), X̃i = (1− u)
eβ F̃ (u), (3.26)
with regular functions F (u) and F̃ (u). To cancel the singular behaviour of the coefficients, we
have to find the adequate β and β̃, the so-called indices, given by equations known as the indicial
equations for β and β̃. We eventually get for all Xi and X̃i
β = ±iw
1− 2m
, β̃ = ±iw
. (3.27)
Note that these exponents differ from those of the Abelian Super-maxwell theory [2] by a de-
pendence on
w in the limit of small frequencies (w < m). In the limit of vanishing chemical po-
tential m → 0, the indices given in [2] are reproduced from (3.27). In order to solve (3.23), (3.24)
and (3.25), we wish to introduce a series expansion ansatz in the momentum variables w and q. In
fact, the physical motivation behind this expansion is that we aim for thermodynamical quantities
which are known from statistical mechanics in the hydrodynamic limit of small four-momentum
~k. So the standard choice would be
F (u) = F0 +wF1 + q
2G1 + . . . . (3.28)
On the other hand, we realize that our indices will appear linearly (and quadratically) in the differ-
ential equations’ coefficients after inserting (3.26) into (3.23), (3.24) and (3.25). The square root
in β and β̃ mixes different orders of w. In order to sort coefficients in our series ansatz, we assume
w < m and keep only the leading w contributions to β and β̃, such that
β ≈ ∓
, β̃ ≈ ±i
. (3.29)
This introduces an additional order O(w1/2), which we include in our ansatz (3.28) yielding
F (u) = F0 +w
1/2F1/2 +wF1 + q
2G1 + . . . , (3.30)
and analogously for the tilded quantities. If we had not included O(w1/2) the resulting system
would be overdetermined. The results we obtain by using the approximations (3.29) and (3.30)
have been checked against the numerical solution for exact β with exact F (u). These approxima-
tions are useful for fluctuations with q,w < 1 (see subsection B 4 in Appendix B). Note that by
dropping the 1 in (3.27) we also drop the Abelian limit.
Consider the indices (3.27) for positive frequency first. In order to meet the incoming wave
boundary condition introduced in section II, we restrict the solution β̃ to the negative sign only.
For the approximate β̃ in (3.29) we therefore choose the lower (negative) sign. This exponent
describes a mode that travels into the horizon of the black hole. In case of β we demand the mode
to decay towards the horizon, choosing the lower (positive) sign in (3.29) consistently. Note that
for negative frequencies ω < 0 the indices β and β̃ exchange their roles.
Using (3.29) in (3.26) and inserting the ansatz into the equations of motion, we find equations
for each order in q2 and w separately. After solving the equations of motion for the coefficient
functions F0, F1/2, F1 and G1, we eventually can assemble the solutions to the equations of motion
for X as defined in (3.21),
X(u) = (1− u)β F (u) = (1− u)β
wF1/2 +wF1 + q
2G1 + . . .
. (3.31)
and a corresponding formula for X̃(u) from the ansatz (3.26).
Illustrating the method, we now write down the equations of motion order by order for the
function Xα. To do so, we use (3.31) with (3.29) in (3.23) with the upper sign for Xα. Then we
examine the result order by order in w and q2,
O(const) : 0 = F ′′0 +
F ′0 , (3.32)
w) : 0 = F ′′1/2 +
F ′1/2 −
F ′0 −
F0 , (3.33)
O(w) : 0 = F ′′1 +
F ′1 −
F ′1/2 −
F1/2 −m
4− u(1 + u)2
2uf 2
F0 , (3.34)
O(q2) : 0 = G′′1 +
G′1 −
F0 . (3.35)
At this point we observe that the differential equations we have to solve for each order are
shifted with respect to the solutions found in [2]. The contributions of order wn in [2] now show
up in order wn/2. Their solutions will exhibit factors of order µn/2.
Solving the system (3.32) to (3.35) of coupled differential equations is straightforward in the
way that they can be reduced to several uncoupled first order ordinary differential equations in
the following way. Note that there obviously is a constant solution F0 = C for the first equation.
Inserting it into (3.33) and (3.35) leaves us with ordinary differential equations for F ′1/2 and G
respectively. Using the solutions of F0 and F1/2 in (3.34) gives one more such equation for F
To fix the boundary values of the solutions just mentioned, we demand the value of F (uH = 1)
to be given by the constant F0 and therefore choose the other component functions’ solutions
such that limu→1 F1/2 = 0, and the same for F1 and G1. The remaining integration constant C is
determined by taking the boundary limit u → 0 of the explicit solution (3.31), making use of the
second boundary condition
X(u) = Xbdy, (3.36)
see appendix B. Eventually, we end up with all the ingredients needed to construct the gauge
field’s fluctuations X(u) as in (3.31).
We solve the equations (3.23) with lower sign for X̃α and (3.24) for X
0 and its tilded partner in
exactly the same way as just outlined, only some coefficients of these differential equations differ.
The solution for X ′3 is then obtained from (3.25).
All solutions are given explicitly in Appendix B together with all other information needed to
construct the functions Xα, X̃α, X
0, X̃
3 and X̃
IV. ISOSPIN DIFFUSION AND CORRELATION FUNCTIONS
A. Current correlators
In this section we obtain the momentum space correlation functions for the gauge field compo-
nent combinations X and X̃ defined in equation (3.21). Recall that the imaginary part of the
retarded correlators essentially gives the thermal spectral functions (see also section II). The
following discussion of the correlators’ properties is therefore equivalent to a discussion of the
corresponding spectral functions.
First note that the on-shell action gets new contributions from the non-Abelian structure,
SD7 = − T7
(2πα′)2
2π2R3 Tr (4.1)
(2π)4
√−gguugjj′ Aaj
′(~q)Aaj′(−~q)
− 4iq fabcµc
−gg00g33Aa[3Ab0]
where j, j′ = 0 , 1 , 2 , 3 and the index u denotes the radial AdS-direction. Up to the sum over
flavor indices, the first term in the bracket is similar to the Abelian Super-Maxwell action of [2].
The second term is a new contribution depending on the isospin chemical potential. It is a contact
term which we will neglect. The correlation functions however get a structure that is different
from the Abelian case. This is due to the appearance of the chemical potential in the equations of
motion and their solutions. Writing (4.1) as a function of X and X̃ results in
SD7 = − T7
(2πα′)2
2π2R3 Tr
(2π)4
−g guugjj′
′X̃j′ + X̃
+ A3j
]∣∣∣∣
(4.2)
− 4qµ
−gg00g33
X[0X̃3] + X̃[3X0]
In order to find the current correlators, we apply the method outlined in section II to (4.2),
with the solutions for the fields given in appendix B. As an example, we derive the correlators
G0e0 = 〈J0(~q)J̃0(−~q) 〉 and Ge00 = 〈J̃0(~q) J0(−~q)〉 of the flavor current time components J0 and J̃0,
coupling to the bulk fields X0 and X̃0, respectively. Correlation functions of all other components
are derived analogously. For the notation see appendix A.
1. Green functions: Calculation
First, we extract the factor B(u) of (2.2),
B(u) = −T7
(2πα′)2
2π2R3Tr
−g guu g00 . (4.3)
The second step, finding the solutions to the mode equations of motion, has already been per-
formed in section III D 3. In the example at hand we need the solutions X0 and X̃0. From (3.31)
and from appendix B we obtain
′ =− (1− u)
0 +wqX̃
2mw+wm ln 2 + q2
1−w1/2
π2 + 3 ln2 2 + 3 ln2(1 + u) + 6 ln 2 ln
1 + u
(4.4)
+12Li2(1− u) + 12Li2(−u)− 12Li2
+ q2 ln
′ = (1− u)−i
0 +wqX
2mw+wm ln 2− q2
1 +w1/2 i
π2 + 3 ln2 2 + 3 ln2(1 + u) + 6 ln 2 ln
1 + u
(4.5)
+12Li2(1− u) + 12Li2(−u)− 12Li2
+ q2 ln
Note that we need the derivatives to apply (2.4).
Now we perform the third step and insert (4.3), (4.4) and (4.5) into (2.4). Our solutions X0
and X̃0 replace the solution f(u,~k) and f(u,−~k) in (2.4). The resulting expression is evaluated
at ub = 0, which comes from the lower limit of the u-integral in the on-shell action (4.2). At small
u = ǫ → 0, (4.4) and (4.5) give
0 +wqX̃
2mw+wm ln 2 + q2
− lim
0 +wqX̃
ln ǫ , (4.6)
0 +wqX
2mw+wm ln 2− q2
+ lim
0 +wqX
ln ǫ . (4.7)
In the next to leading order of (4.6) and (4.7) there appear singularities, just like in the Abelian
Super-Maxwell calculation [2, equation (5.15)]. However, in the hydrodynamic limit, we consider
only the finite leading order.
2. Green functions: Results
Putting everything together, for the two Green functions for the field components X0, X̃0 given
in (3.21) by
X0 = A
0 + iA
0, X̃0 = A
0 − iA20,
we obtain
G0e0 =
2πT q2
2mw− q2 −wm ln2
, (4.8)
Ge00 =
2πT q2
2mw− q2 +wm ln2
. (4.9)
These are the Green functions for the time components in Minkowski space, perpendicular to the
chemical potential in flavor space. All Green functions are obtained considering hydrodynamic
approximations in O(w1/2,w, q2), neglecting mixed and higher orders O(w3/2,w1/2q2, q4).
The prefactor in (4.8), (4.9) is obtained using T7 as in (3.7), and carefully inserting all metric
factors, together with the standard AdS/CFT relation R4 = 4πgsNcα
′2. As in other settings with
flavor [26], we concordantly get an overall factor of Nc, and not N
c , for all correlators. Contrary
to those approaches, we do not get a factor of Nf when summing over the different flavors. This is
due to the fact that in our setup, the individual flavors yield distinct contributions. Most striking is
the non-trivial dependence on the (dimensionless) chemical potential m in both correlators. Note
also the distinct structures in the denominators. The first one (4.8) has an explicit relative factor
of i between the terms in the denominator. In the second correlator (4.9) there is no explicit factor
of i. The correlator (4.8) has a complex pole structure for ω > 0, but is entirely real for ω < 0.
On the other hand, (4.9) is real for ω > 0 but develops a diffusion structure for ω < 0. So the
correlators G0e0 and Ge00 essentially exchange their roles as ω changes sign (see also Fig. 1). We
find a similar behavior for all correlators G
and Gejl with j, l = 0, 1, 2, 3. This behavior is a
consequence of the insertion of O(w1/2) in the hydrodynamic expansion (3.30).
We assume m to be small enough in order to neglect the denominator term of order O(wm) ≪
wm, q2). Moreover, using the definitions of w, q and m from (3.13) we may write (4.8)
and (4.9) as
G0e0 = −
ω + q2D(ω)
, (4.10)
Ge00 =
iω − q2D(ω)
, (4.11)
where the frequency-dependent diffusion coefficient D(ω) is given by
D(ω) =
. (4.12)
We observe that this coefficient also depends on the inverse square root of the chemical potential µ.
Its physical interpretation is discussed below in section IV B.
In the same way we derive the other correlation functions
G3e3 =−
ω3/2 (ω − µ)
Q̃(ω, q)
, Ge33 =
ω3/2 (ω + µ)
Q(ω, q)
, (4.13)
G0e3 =−
ω q(ω − µ)
Q̃(ω, q)
, Ge03 =
ω q(ω + µ)
Q(ω, q)
, (4.14)
G3e0 =−
ω3/2 q
Q̃(ω, q)
, Ge30 =
ω3/2 q
Q(ω, q)
. (4.15)
with the short-hand notation
Q(ω, q) = iω − q2D(ω), Q̃(ω, q) = ω + q2D(ω) . (4.16)
Note that most of these functions are proportional to powers of q and therefore vanish in the limit
of vanishing spatial momentum q → 0. Only the 33-combinations from (4.13) survive this limit.
In contrast to the Abelian Super-Maxwell correlators [2] given in appendix C, it stands out that our
results (4.10), (4.11) and (4.13) and (4.15) have a new zero at ω = µ or −µ. Nevertheless bear in
mind that we took the limit ω < µ in order to obtain our solutions. Therefore the apparent zeros at
±µ lie outside of the range considered. Compared to the Abelian case there is an additional factor
ω. The dependence on temperature remains linear.
In the remaining X-correlators we do not find any pole structure to order
ω, subtracting an
O(q2) contribution as in [2],
G1e1 =G2e2 =
µω , (4.17)
Ge11 =Ge22 = −
µω . (4.18)
As seen from (4.17), Gαeα (with α = 1, 2) are real for negative ω and imaginary for positive ω. The
opposite is true for Geαα, as is obvious from the relative factor of i.
The correlators of components, pointing along the isospin potential in flavor space (a = 3), are
found to be
iω −D0q2
, GA3
= GA3
iω −D0q2
, (4.19)
= GA3
NcT iω
, GA3
iω −D0q2
, (4.20)
with the diffusion constant D0 = 1/(2πT ) . Note that these latter correlators have the same
structure but differ by a factor 4/Nc from those found in the Abelian super-Maxwell case [2] (see
also (C1) and (C2)). In particular the correlators in equation (4.19) do not depend on the chemical
potential.
To analyze the novel structures appearing in the other correlators, we explore their real and
imaginary parts as well as the interrelations among them,
ReG0e0(ω ≥ 0) = ReGe00(ω < 0) = −
2µ |ω|+ q2/(2πT )
, (4.21)
ReG0e0(ω < 0) = ReGe00(ω ≥ 0) = −
2µ |ω|+ q4/(2πT )2
, (4.22)
ImG0e0(ω < 0) = −ImGe00(ω ≥ 0) =
2µ |ω|
2µ |ω|+ q4/(2πT )2
, (4.23)
ImG0e0(ω ≥ 0) = 0, ImGe00(ω < 0) = 0. (4.24)
Now we see why, as discussed below (4.11), G0e0 and Ge00 exchange their roles when crossing the
origin at ω = 0. This is due to the fact that the real parts of all G
and Gejl are mirror images
of each other by reflection about the vertical axis at ω = 0. In contrast, the imaginary parts are
inverted into each other at the origin. Figure 1 shows the real and imaginary parts of correlators
G0e0 and Ge00. The different curves correspond to distinct values of the chemical potential µ. The
real part shows a deformed resonance behavior. The imaginary part has a deformed interference
shape with vanishing value for negative frequencies. All curves are continuous and finite at ω = 0.
However due to the square root dependence, they are not differentiable at the origin. Parts of the
correlator which are real for positive ω are shifted into the imaginary part by the change of sign
when crossing ω = 0, and vice versa.
To obtain physically meaningful correlators, we follow a procedure which generalizes the
Abelian approach of [6]. In the Abelian case, gauge-invariant components of the field strength ten-
sor, such as Eα = ωAα, are considered as physical variables. This procedure cannot be transferred
directly to the non-Abelian case. Instead, we consider the non-local part of the gauge invariant
trF 2 which contributes to the on-shell action (4.1). In this action, the contribution involving the
non-Abelian structure constant – as well as µ – is a local contact term. The non-local contribution
however generates the Green function combination
. (4.25)
We take this sum as our physical Green function. This choice is supported further by the fact
that it may be written in terms of the linear combinations (3.21) which decouple the equations of
motion. For example, for the time component, written in the variables X0, X̃0 given by (3.21),
the combination (4.25) reads (compare to (4.2))
G0e0 +Ge00 +GA30A30 . (4.26)
The contribution from GA3
is of order O(µ0), while the combination for the first two flavor
directions, G0e0 +Ge00, is of order O(µ).
We proceed by discussing the physical behavior of the Green function combinations introduced
above. GA3
is plotted in Fig. 2 on the right. Its frequency dependence is of the same form as in
the Abelian correlator obtained in [2], as can be seen from (C1). Since we are interested in effects
of order O(µ), we drop the third flavor direction a = 3 from the sum (4.26) in the following. It
is reassuring to observe that the flavor directions a = 1, 2 which are orthogonal to the chemical
potential, combine to give a correlator spectrum qualitatively similar to the one found in [2] for
PSfrag replacements
−0.05−0.1 0 0.05 0.1
PSfrag replacements
−0.05−0.1
0 0.05 0.1
FIG. 1: Real (left plot) and imaginary part (right plot) of the correlator G
as a function of frequency ω/T
at different chemical potential values µ/T = 0.5 (solid line), µ/T = 0.3 (short-dashed line) and µ/T =
0.1 (long-dashed line). The corresponding plots for the correlator G
would look like the mirror image of
the ones given. The real part would be reflected about the vertical axis at ω = 0, the imaginary part would
be reflected about the origin. All dimensionful quantities are given in units of temperature. The numerical
values used for the parameters are q/T = 0.1, Nc = 100.
the Abelian Super-Maxwell action (see Fig. 2). However, we discover intriguing new effects such
as the highly increased steepness of the curves near the origin due to the square root dependence
and a kink at the origin.
We observe a narrowing of the inverse resonance peak compared to the form found for the
Abelian Super-Maxwell action (and also compared to the form of our GA3
, as is seen from
comparing the left with the right plot in Fig. 2). At the origin the real and imaginary part are
finite and continuous, but they are not continuously differentiable. However, the imaginary part
of GA3
has finite steepness at the origin. The real part though has vanishing derivative at ω = 0.
Note that the imaginary part of flavor directions a = 1, 2 on the left plot in Fig. 2 never drops
below the real part. In the third flavor direction, as well as in the Abelian solution, such a drop
occurs on the positive ω-axis.
The correlators G3e3, Ge33, G0e3 and Ge03 have the same interrelations between their respective
real and imaginary parts as G0e0 and Ge00. Nevertheless, their dependence on the frequency and
momentum is different, as can be seen from (4.13) to (4.15). A list of the 33-direction Green
functions split into real and imaginary parts can be found in appendix D.
Thermal spectral functions in different directions are compared graphically in appendix E.
B. Isospin diffusion
The attenuated poles in hydrodynamic correlation functions have specific meanings (for ex-
emplary discussions of this in the AdS/CFT setup see e.g. [7], [50]). In our case we observe an
attenuated pole in the sum G0e0 + Ge00 at ω = 0. As can be seen from the plots in Fig. 2, our pole
lies at Reω = 0. This structure appears in hydrodynamics as the signature of a diffusion pole
located at purely imaginary ω. Its location on the imaginary ω-axis is given by the zeros of the
denominators of our correlators as (neglecting O(ω , q4 ))
ω = −i q
. (4.27)
PSfrag replacements
0.5 1 1.5−0.5−1−1.5
PSfrag replacements
0.010.005−0.005−0.01
FIG. 2: In the left plot the sum of both correlators in 00-directions is split into its imaginary (dashed
line) and real (solid line) part and plotted against frequency. For comparison the right plot shows the
corresponding real and imaginary parts for the GA3
. It is qualitatively similar to the Abelian correlator in
i = 0 Lorentz direction computed from the Super-Maxwell action in [2]. Note the different frequency scales
in the two plots. The curves in a = 1, 2-directions are much narrower due to their square root dependence
on ω. Furthermore they have a much larger maximum amplitude. All dimensionful quantities are given in
units of temperature. The numerical values used for the parameters are, as in Fig. 1, q/T = 0.1, Nc = 100
and only in the left plot µ/T = 0.2.
Squaring both sides of (4.27) we see that this effect is of order O(q4). On the other hand, looking
for poles in the correlator involving the third flavor direction GA3
, we obtain dominant contri-
butions of order O(q2) and O(µ0) (neglecting O(ω2 , q2 ))
ω = −i
. (4.28)
This diffusion pole is reminiscent of the result of the Abelian result of [2] given in appendix (C).
As discussed in section IV A, we consider gauge-invariant combinations G0e0 + Ge00 + GA30A30 . In
order to inspect the non-Abelian effects of order O(µ) showing up in the first two correlators in
this sum, we again drop the third flavor direction which is of order O(µ0).
Motivated by the diffusion pole behavior of our correlators in flavor-directions a = 1, 2 corre-
sponding to the combinations X, X̃ (see (4.27)), we wish to regain the structure of the diffusion
equation given in (2.9), which in our coordinates (k = (ω , 0 , 0 , q )) reads
i ω J0 = D(ω) q
2 J0 . (4.29)
Our goal is to rewrite (4.27) such that a term O(ω) and one term in order O(q2) appears. Further-
more there should be a relative factor of −i between these two terms. The obvious manipulation
to meet these requirements is to multiply (4.27) by
ω in order to get
ω = −i q2
. (4.30)
Comparing the gravity result (4.30) with the hydrodynamic equation (4.29), we obtain the
frequency-dependent diffusion coefficient
D(ω) =
. (4.31)
Our argument is thus summarized as follows: Given the isospin chemical potential as in (1.1),
(3.5), J0 from (2.9) is the isospin charge density in (4.29). According to (4.29), the coeffi-
cient (4.31) describes the diffusive response of the quark-gluon plasma to a gradient in the isospin
charge distribution. For this reason we interprete D(ω) as the isospin diffusion coefficient.
Near the pole, the strongly-coupled plasma behaves analogously to a diffractive medium with
anomalous dispersion in optics. In the presence of the isospin chemical potential, the propaga-
tion of non-Abelian gauge fields in the black hole background depends on the square root of the
frequency. In the dual gauge theory, this corresponds to a non-exponential decay of isospin fluc-
tuations with time.
The square root dependence of our diffusion coefficient is valid for small frequencies. As
long as ω/T < 1/4, the square root is larger than its argument and at ω/T = 1/4, the differ-
ence to a linear dependence on frequency is maximal. Therefore in the regime of small frequen-
cies ω/T < 1/4, which is accessible to our approximation, diffusion of modes close to 1/4 is
enhanced compared to modes with frequencies close to zero.
V. CONCLUSION
In this paper we have considered a relatively simple gauge/gravity dual model for a finite tem-
perature field theory, consisting of an isospin chemical potential µ obtained from a time component
vev for the SU(2) gauge field on two coincident brane probes. We have considered the D7-brane
embedding corresponding to vanishing quark mass, for which µ is a constant, independent in
particular of the radial holographic coordinate. The main result of this paper is that this model,
despite its simplicity, leads to a hydrodynamical behavior of the dual field theory which goes be-
yond linear response theory. We find in particular a frequency-dependent diffusion coefficient
with a non-analytical behavior. Frequency-dependent diffusion is a well-known phenomenon in
condensed matter physics. Here it originates simply from the fact that due to the non-Abelian
structure of the gauge field on the brane probe, the chemical potential replaces a time derivative in
the action and in the equations of motion from which the Green functions are obtained.
Of course the calculation presented has some limitations as far as the approximations made are
concerned. This applies in particular to the approximation (3.29) of the so-called indices in the
ansatz for solving the equations of motion. Here we have dropped the constant present under the
square root and used an expression proportional to the square root of the frequency. This allows
for a closed solution without having to use numerics. However using this approximation we have
dropped the Abelian limit. This leads ultimately to the square root dependence of the diffusion
coefficient on the frequency. This dependence is unphysical for ω → 0, since the diffusion coeffi-
cient is expected to be non-zero for zero frequency. We expect physical behavior to be restored if
the Abelian limit is included in the calculation. To avoid the approximation described, this requires
a numerical approach. A suitable solution method for all momenta and all frequencies has been
presented in [7] and in [49]. We are going to study the application of this method to the model
presented here in the future.
Acknowledgments
We are grateful to R. Apreda, G. Policastro, C. Sieg, A. Starinets and L. Yaffe for useful dis-
cussions and correspondence.
APPENDIX A: NOTATION
The five-dimensional AdS Schwarzschild black hole space in which we work is endowed with
a metric of signature (−,+,+,+,+), as given explicitly in (3.3). We make use of the Einstein
notation to indicate sums over Lorentz indices, and additionally simply sum over non-Lorentz
indices, such as gauge group indices, whenever they occur twice in a term.
To distinguish between vectors in different dimensions of the AdS space, we use bold symbols
like q for vectors in the the three spatial dimensions which do not live along the radial AdS
coordinate. Four-vectors which do not have components along the radial AdS coordinates are
denoted by symbols with an arrow on top, as ~q.
The Green functions G = 〈JĴ〉 considered give correlations between currents J and Ĵ . These
currents couple to fields A and Â respectively. In our notation we use symbols such as GAa
denote correlators of currents coupling to fields Aak and A
l , with flavor indices a, b and Lorentz
indices k, l = 0, 1, 2, 3. For the gauge field combinations Xk and X̃l given in (3.21), we obtain
Green functions G
denoting correlators of the corresponding currents.
APPENDIX B: SOLUTIONS TO EQUATIONS OF MOTION
Here we explicitly write down the component functions used to construct the solutions to the
equations of motion for the gauge field fluctuations up to order w and q2. The functions themselves
are then composed as in (3.31).
The solutions for the components with flavor index a = 3 where obtained in [2].
1. Solutions for Xα, X̃α and A3α
The function Xα(u) solves (3.23) with the upper sign and is constructed as in (3.31) from the
following component functions,
+O(ω), (B1)
F0 = C, (B2)
F1/2 = −C
1 + u
, (B3)
F1 = −C
π2 − 9 ln2 2 + 3 ln(1− u) (ln 16− 4 ln(1 + u))
+ 3 ln(1 + u) (ln(4(1 + u))− 4 ln u) (B4)
Li2(1− u) + Li2(−u) + Li2
1 + u
+ lnu ln(1 + u) + Li2(1− u) + Li2(−u)
, (B5)
where the constant C can be expressed it in terms of the field’s boundary value Xbdy =
limu→0X(u, k),
C = Xbdy
ln 2 +mw
ln2 2
2 +O(w3/2, q4)
. (B6)
The solutions of the equations of motion (3.23) with lower sign for the functions X̃α(u) are
given by
β̃ = −i
+O(ω), (B7)
F̃0 = C̃, (B8)
F̃1/2 = iC̃
1 + u
, (B9)
F̃1 = C̃
π2 − 9 ln2 2 + 3 ln(1− u) (ln 16− 4 ln(1 + u))
+ 3 ln(1 + u) (ln(4(1 + u))− 4 lnu) (B10)
Li2(1− u) + Li2(−u) + Li2
1 + u
G̃1 =
+ ln u ln(1 + u) + Li2(1− u) + Li2(−u)
, (B11)
with C̃ given by
C̃ = X̃bdy
ln 2−mw
ln2 2
2 +O(w3/2, q4)
, (B12)
so that limu→0 X̃(u, k) = X̃
The solution for A3α solves (3.16) up to order w and q
2 with boundary value (A3α)
. It is
A3α =
8 (A3α)
(1− u)− iw2
8− 4iw ln 2 + π2q2
1 + i
1 + u
+ ln u ln(1 + u) + Li2(1− u) + Li2(−u)
(B13)
2. Solutions for X ′0, X̃
0 and A
Here we state the solutions to (3.20). This formula describes three equations, differing in
the choice of a = 1, 2, 3. The cases a = 1, 2 give coupled equations which are decoupled by
transformation from A1,20 to X0 and X̃0. The choice a = 3 gives a single equation.
The function X ′0 is solution to (3.24) with upper sign. We specify the component functions as
+O(ω), (B14)
F0 = C, (B15)
F1/2 = −C
1 + u
, (B16)
F1 = −C
π2 + 3 ln2 2 + 3 ln2(1 + u) + 6 ln 2 ln
1 + u
(B17)
Li2(1− u) + Li2(−u)− Li2
, (B18)
G1 = C ln
1 + u
, (B19)
where the constant C can be expressed in terms of the field’s boundary value Xbdy =
limu→0X(u, k),
C = −
0 +wqX
2mw+mw ln 2 + q2
. (B20)
To get the function X̃ ′0, we solve (3.24) with the lower sign and obtain
β̃ = −i
+O(ω), (B21)
F̃0 = C̃, (B22)
F̃1/2 = iC̃
1 + u
, (B23)
F̃1 = C̃
π2 + 3 ln2 2 + 3 ln2(1 + u) + 6 ln 2 ln
1 + u
(B24)
Li2(1− u) + Li2(−u)− Li2
, (B25)
G̃1 = C̃ ln
1 + u
, (B26)
where the constant C̃ can be expressed it in terms of the field’s boundary value X̃bdy =
limu→0 X̃(u, k),
0 +wqX̃
2mw+mw ln 2− q2
. (B27)
The solution for (3.20) with a = 3 is the function A30
, given by
= (1− u)−
0 + wqA
iw− q2
1 + u
+ q2 ln
1 + u
. (B28)
PSfrag replacements
µ = 0.4 T
ω = q = 0.2 T
0.2 0.4 0.6 0.8 1
0.195
0.200
0.205
0.210
PSfrag replacements
µ = 0.4 T
ω = q = 0.2 T
µ = 0.4 T
ω = q = 0.2 T
Re F (u)/T
0.2 0.4 0.6 0.8 1
0.095
0.100
0.105
0.110
FIG. 3: These plots show the real and imaginary part of the function F (u) which is part of Xα = (1 −
u)βF (u). The solid line depicts the analytical approximation, obtained in this paper. As a check we solved
the equations of motion for F (u) numerically. They are drawn as dashed lines. In this example we used
T = 1. The numerical solution was chosen to agree with the analytical one at the horizon and boundary.
3. Solutions for X ′3, X̃
3 and A
We give the derivatives of X3 and X̃3 as
X ′3 = −
X ′0 (B29)
X̃ ′3 = −
X̃ ′0. (B30)
The solution for A33
. (B31)
4. Comparison of numerical and analytical results
As an example, in Fig. 3 we show the numerical and analytical solutions for the function F (u)
in Xα = (1− u)βF (u). Here we compare the numerical result for F (u) obtained from the ansatz
(3.26) with (3.27) in (3.23) with the analytically obtained approximation given above in (B1) to
(B6).
APPENDIX C: ABELIAN CORRELATORS
For convenient reference we quote here the correlation functions of the Abelian super-Maxwell
theory found in [2]. The authors start from a 5-dimensional supergravity action and not from a
Dirac-Born-Infeld action as we do. Therefore there is generally a difference by a factor Nc/4.
Note also that here all Nf flavors contribute equally. In our notation
Gab11 = G
22 = −
iN2c Tω δ
+ · · · , Gab00 =
N2c Tq
2 δab
16π(iω −Dq2)
+ · · · , (C1)
Gab03 = G
30 = −
N2c Tωq δ
16π(iω −Dq2)
+ · · · , Gab33 =
N2c Tω
2 δab
16π(iω −Dq2)
+ · · · , (C2)
where D = 1/(2πT ) .
APPENDIX D: CORRELATION FUNCTIONS
In this appendix we list the real and imaginary parts of the flavor currents in the first two
flavor-directions a = 1, 2 and in the third Lorentz-direction coupling to the supergravity-fields X3
and X̃3 (as defined in (3.21)).
Re{G3e3(ω ≥ 0)} = Re{Ge33(ω < 0)} = −
2 (ω2 + µ |ω|)
16π2 [2µ |ω|+ q4/(2πT )2]
, (D1)
Im{G3e3(ω ≥ 0)} = −Im{Ge33(ω < 0)} = −
2µ |ω| (ω2 + µ |ω|)
8π [2µ |ω|+ q4/(2πT )2]
, (D2)
Re{G3e3(ω < 0)} = Re{Ge33(ω ≥ 0)} = −
NcT (ω
2 − µ |ω|)
2µ |ω|+ q2/(2πT )
] , (D3)
Im{G3e3(ω < 0)} = 0, Im{Ge33(ω ≥ 0)} = 0. (D4)
APPENDIX E: THERMAL SPECTRAL FUNCTIONS
We include here a comparision of the sizes of spectral functions in distinct flavor- and Lorentz-
directions (see also (2.1) in section II).PSfrag replacements
0.5 1 1.5 2
PSfrag replacements
0.5 1 1.5 2
FIG. 4: Here the thermal spectral functions in distinct Lorentz- and flavor-directions are plotted against
frequency ω/T in units of temperature. In the left plot the chemical potential was chosen to be µ/T = 0.7,
in the right one µ/T = 0.2. Flavor-directions a = 1, 2 are summed and displayed as one curve. The
frequency-dependence of 00- (solid red line) and 03-Lorentz-directions (short-dashed blue line) is shown.
By the dotted line we denote the spectral curve in 11- or 22-directions. This curve was scaled by a factor 100
in order to make it visible in these plots. The third flavor-direction is only plotted for the spectral function in
Lorentz-directions 00 (long-dashed green curve). We do not show the 33-direction spectral function which
has a square root dependence and is comparable in size with the 11-direction.
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	Contents
	Introduction
	Hydrodynamics and AdS/CFT
	Supergravity background and action
	Finite temperature background and brane configuration
	Introducing a non-Abelian chemical potential
	Dirac-Born-Infeld action
	Equations of motion
	Equations for A1a- and A2a-components
	Equations for A0a- and A3a-components
	Solutions
	Isospin diffusion and correlation functions
	Current correlators
	Green functions: Calculation
	Green functions: Results
	Isospin diffusion
	Conclusion
	Acknowledgments
	Notation
	Solutions to equations of motion
	Solutions for X, X"0365X and A3
	Solutions for X0', X"0365X0' and A30'
	Solutions for X3', X"0365X3' and A33'
	Comparison of numerical and analytical results
	Abelian Correlators
	Correlation functions
	Thermal spectral functions
	References

  We study the gauge/gravity dual of a finite temperature field theory at
finite isospin chemical potential by considering a probe of two coincident
D7-branes embedded in the AdS-Schwarzschild black hole background. The isospin
chemical potential is obtained by giving a vev to the time component of the
non-Abelian gauge field on the brane. The fluctuations of the non-Abelian gauge
field on the brane are dual to the SU(2) flavor current in the field theory.
For the embedding corresponding to vanishing quark mass, we calculate all Green
functions corresponding to the components of the flavor current correlator. We
discuss the physical properties of these Green functions, which go beyond
linear response theory. In particular, we show that the isospin chemical
potential leads to a frequency-dependent isospin diffusion coefficient.

Introduction 2
II. Hydrodynamics and AdS/CFT 4
III. Supergravity background and action 5
A. Finite temperature background and brane configuration 5
B. Introducing a non-Abelian chemical potential 6
C. Dirac-Born-Infeld action 7
D. Equations of motion 8
1. Equations for Aa1- and A
2-components 8
2. Equations for Aa0- and A
3-components 9
3. Solutions 9
IV. Isospin diffusion and correlation functions 12
A. Current correlators 12
1. Green functions: Calculation 13
2. Green functions: Results 14
B. Isospin diffusion 17
V. Conclusion 19
∗Electronic address: jke@mppmu.mpg.de, kaminski@mppmu.mpg.de, rust@mppmu.mpg.de
http://arxiv.org/abs/0704.1290v2
mailto:jke@mppmu.mpg.de, kaminski@mppmu.mpg.de, rust@mppmu.mpg.de
Acknowledgments 19
A. Notation 20
B. Solutions to equations of motion 20
1. Solutions for Xα, X̃α and A
2. Solutions for X ′0, X̃
0 and A
3. Solutions for X ′3, X̃
3 and A
4. Comparison of numerical and analytical results 23
C. Abelian Correlators 23
D. Correlation functions 24
E. Thermal spectral functions 24
References 25
I. INTRODUCTION
Over the past years, there have been a number of lines of investigation for describing QCD-
like theories with gravity duals. In this way, considerable progress towards a gauge/gravity dual
description of phenomenologically relevant models has been made. One of these lines of investi-
gation is the gravity dual description of the quark-gluon plasma obtained by applying AdS/CFT
to relativistic hydrodynamics [1, 2, 3, 4, 5, 6, 7]. The central result of this approach is the cal-
culation of the shear viscosity from AdS/CFT. More recently, an R charge chemical potential has
been introduced by considering gravity backgrounds with R charged black holes [8, 9, 10], and
also the heat conductivity has been calculated by considering the R current correlators in these
backgrounds [8].
A further approach to generalizing the AdS/CFT correspondence to more realistic field theories
is the addition of flavor to gravity duals via the addition of probe branes [11, 12, 13, 14, 15]. This
allows in particular for the calculation of meson masses.
These two approaches have been combined in order to study the flavor contribution to finite
temperature field theories from the gravity dual perspective. This began with [13] where the
embedding of a D7-brane probe into the AdS-Schwarzschild black hole background was studied
and a novel phase transition was found, which occurs when the D7 probe reaches the black hole
horizon. This transition was shown to be of first order in [16] (see [14] for a similar transition
in the D4/D6 system), and studied in further detail in [17, 18]. Related phase transitions appear
in [19, 20, 21]. Mesons in gravity duals of finite temperature field theories have been studied in
[22, 23, 24, 25].
Recently, in view of adding flavor to the quark-gluon plasma, the flavor contribution to the
shear viscosity has been calculated in [26, 27], where it was found that ηfund ∝ λNcNfT 3.
For a thermodynamical approach in the grand canonical ensemble, the inclusion of a chemical
potential and a finite number density is essential. In [28], an isospin chemical potential was in-
troduced by considering two coincident D7 probes, and by giving a vev to the time component of
the SU(2) gauge field on this probe. This was shown to give rise to a thermodynamical instability
comparable to Bose-Einstein condensation, compatible with the field-theoretical results of [29].
For the gauge/gravity dual analysis, a potential generated by an SU(2) instanton on the D7 probe
in the gravity background was used [30, 31, 32, 33].
A baryon chemical potential µB is obtained by turning on the diagonal U(1) ⊂ U(Nf ) gauge
field on the D7-brane probe [34]. Contributions to the D7-brane action arise from the deriva-
tive of this U(1) gauge field with respect to the radial direction. The effects of this potential on
the first-order phase transition described above have been studied in [34], where regions of ther-
modynamical instability have been found in the (T, µB) phase diagram. – For the D4/D8/D̄8
Sakai-Sugimoto model [15, 35], the phase transitions in presence of a baryon number chemical
potential, as well as physical processes such as photoemission and vector meson screening, have
been studied in [36, 37, 38].
A related approach has been used to calculate the rate of energy loss of a heavy quark moving
through a supersymmetric Yang-Mills plasma at large coupling [39]. In this approach the heavy
quark is given by a classical string attached to the D7-brane probe. – A first study of flavors in
thermal AdS/CFT beyond the quenched approximation, i.e. with Nf ∼ Nc, was performed in [40].
Here we study finite-temperature field theories with finite isospin chemical potential by consid-
ering two coincident D7-brane probes in the Lorentzian signature AdS-Schwarzschild black hole
background. As in [28], we introduce an isospin chemical potential by defining
, (1.1)
for the time component of the SU(2) gauge field on the two coincident D7-branes. This constant
chemical potential is a solution to the D7-brane equations of motion and is present even for the
D7-brane embedding corresponding to massless quarks. We consider small µ, such that the Bose-
Einstein instability mentioned above, which is of order O(µ2), does not affect our discussion here.
For simplicity we consider only the D7 probe embedding for which the quark mass vanishes,
m = 0. This embedding is constant and terminates at the horizon. We establish the SU(2) non-
Abelian action for a probe of two coincident D7-branes. We obtain the equations of motion for
fluctuations about the background given by (1.1). These are dual to the SU(2) flavor current Jµa.
We find an ansatz for decoupling the equations of motion for the different Lorentz and flavor com-
ponents, and solve them by adapting the method developed in [2, 3]. This involves Fourier trans-
forming to momentum space, and using a power expansion ansatz for the equations of motion. We
discuss the approximation necessary for an analytical solution, which amounts to considering fre-
quencies with ω < µ < T . With this approach we obtain the complete current-current correlator.
The key point is that the constant chemical potential effectively replaces a time derivative in the ac-
tion and in the equations of motion. In the Fourier transformed picture, this leads to a square-root
dependence of physical observables on the frequency,
ω. This non-linear behavior goes beyond
linear response theory. We discuss the physical properties of the Green functions contributing to
the current-current correlator. In particular, for small frequencies we find a frequency-dependent
diffusion coefficient D(ω) ∝ 1
ω/µ. Whereas frequency-dependent diffusion has – to our
knowledge – not yet been discussed in the context of the quark-gluon plasma, it is well-known in
the theory of quantum liquids. For instance, for small frequencies the square-root behavior we find
agrees qualitatively with the results of [41, 42] for liquid para-hydrogen. Generally, frequency-
dependent diffusion leads to a non-exponential decay of time-dependent fluctuations, as discussed
for a classical fluid in [43].
Physically, the isospin chemical potential corresponds to the energy necessary to inverting the
isospin of a given particle. Within nuclear physics, such a chemical potential is of relevance
for the description neutron stars. Moreover, isospin diffusion has been measured in heavy ion
reactions [44, 45]. – For two-flavor QCD, effects of a finite isospin chemical potential have been
discussed for instance in [46, 47, 48]. The phase diagrams discussed there are beyond the scope
of the present paper. We expect to return to similar diagrams in the gauge/gravity dual context in
the future.
This paper is organized as follows. In section 2 we summarize the AdS/CFT hydrodynamics
approach to calculating Green functions, which we use in the subsequent. Moreover we comment
on frequency-dependent diffusion within hydrodynamics. In section 3 we establish the D7 probe
action in presence of the isospin chemical potential, derive the corresponding equations of motion
and solve them. In section 4 we obtain the associated Green functions in the hydrodynamical
approximation. We discuss their pole structure and obtain the frequency-dependent diffusion co-
efficient. We conclude in section 5 with an interpretation of our results. An explanation of our
notation as well as a series of calculations are relegated to a number of appendices.
II. HYDRODYNAMICS AND ADS/CFT
Thermal Green functions have proven to be a useful tool for analyzing the structure of hydro-
dynamic theories and for calculating hydrodynamic quantities such as transport coefficients. For
instance, given a retarded current correlation function G(~k)µν in Minkowski space, the spectral
function can be written in terms of its imaginary part,
χµν(~k) = −2 ImGµν(~k) . (2.1)
For the gravity dual approach, this is discussed for instance in [7, 49].
In this paper we use the gauge/gravity dual prescription of [3] for calculating Green functions
in Minkowski spacetime. For further reference, we outline this prescription in the subsequent. It is
based on the AdS/CFT-correspondence relating supergravity fields A in a black hole background
to operators J in the dual gauge theory. The black hole background is asymptotically Anti-de
Sitter space and places the dual field theory at finite temperature. This temperature corresponds
to the Hawking temperature of the black hole, or more generally speaking, of the black branes.
Starting out from a classical supergravity action Scl for the gauge field A, according to [3] we
extract the function B(u) (containing metric factors and the metric determinant) in front of the
kinetic term (∂uA)
Scl =
dud4xB(u) (∂uA)
2 + . . . (2.2)
Then we perform a Fourier transformation and solve the linearized equations of motion in momen-
tum space. This is a second order differential equation, so we have to fix two boundary conditions.
The first one at the boundary of AdS at u = 0 can be written as
A(u,~k) = f(u,~k)Abdy(~k) , (2.3)
where Abdy(~k) is the value of the supergravity field at the boundary of AdS depending only on
the four flat boundary coordinates. Thus by definition we have f(u,~k)
= 1. For the other
boundary, located at the horizon u = 1, we impose the incoming wave condition. This requires
that any Fourier mode A(~k) with timelike ~k can travel into the black hole, but is not allowed to
cross the horizon in the opposite direction. For spacelike ~k, the components of A have to be regular
at the horizon. Then the retarded thermal Green function is given by
G(ω, q) = −2B(u) f(u,−~k) ∂u f(u,~k)
. (2.4)
The thermal correlators obtained in this way display hydrodynamic properties, such as poles
located at complex frequencies. Generically, for the R current component correlation functions
calculated from supergravity, there are retarded contributions of the form
G(ω, q) ∝
iω −Dq2
. (2.5)
This may be identified with the the Green function for the hydrodynamic diffusion equation
∂0 J0(t,x) = D∇2 J0(t,x) , (2.6)
with J0 the time component of a diffusive current. D is the diffusion constant. In Fourier space
this equation reads
iωJ0(ω,k) = Dk
2J0(ω,k) . (2.7)
In position space, this corresponds to an exponential decay of J0 with time.
For the non-Abelian case with an isospin chemical potential, in sections III and IV we will
obtain retarded Green functions of the form
G(ω, q) ∝ 1
iω −D(ω)q2
. (2.8)
Retarded Green functions of this type have been discussed for instance in [43]. (2.8) corresponds
to frequency-dependent diffusion with coefficient D(ω), such that (2.7) becomes
iωJ0(ω,k) = D(ω)k
2J0(ω,k) . (2.9)
In our case, J0 is the averaged isospin at a given point in the liquid.
This is a non-linear behavior which goes beyond linear response theory. In particular, when
Fourier-transforming back to position space, we have to use the convolution for the product D · J0
and obtain
∂0J0(t,x) +∇2
ds J0(s,x)D(t− s) = 0 (2.10)
for the redarded Green function. This implies together with the continuity equation ∂0 J0 +∇·J =
0, with J the three-vector current associated to J0, that
J = −∇(D ∗ J0) , (2.11)
where ∗ denotes the convolution. This replaces the linear response theory constitutive equation
J = −D∇J0. Note that for D(t− s) = Dδ(t− s) with D constant, (2.10) reduces again to (2.6).
III. SUPERGRAVITY BACKGROUND AND ACTION
A. Finite temperature background and brane configuration
We consider an asymptotically AdS5 × S5 spacetime as the near horizon limit of a stack of Nc
coincident D3-branes. More precisely, as in [2], our background is an AdS black hole, which is
the geometry dual to a field theory at finite temperature. The Minkowski signature background is
ds2 =
−f(u) dx20 + dx21 + dx22 + dx23
4u2f(u)
du2 +R2dΩ25,
0 ≤ u ≤ 1, xi ∈ R, C0123 =
(3.1)
with the metric dΩ25 of the unit 5-sphere, and the function f(u), AdS radius R and temperature
parameter b given in terms of the string coupling gs, temperature T , inverse string tension α
′ and
number of colors Nc by
f(u) = 1− u2, R4 = 4πgsNcα′2, b = πT. (3.2)
The geometry is asymptotically AdS5 × S5 with the boundary of the AdS part located at u = 0.
At the black hole horizon the radial coordinate u has the value u = 1.
Into this ten-dimensional spacetime we embed Nf = 2 coinciding D7-branes, hosting flavor
gauge fields Aµ. The embedding we choose extends the D7-branes in all directions of AdS space
and wraps an S3 on the S5. In this work we restrict ourselves to the most straightforward case,
that is the embedding of the branes through the origin along the AdS radial coordinate u. This
corresponds to massless quarks in the dual field theory. On the brane, the metric in this case simply
reduces to
ds2 =
−f(u) dx20 + dx21 + dx22 + dx23
4u2f(u)
du2 +R2dΩ23,
0 ≤ u ≤ 1, xi ∈ R.
(3.3)
Due the choice of our gauge field in the next subsection, the remaining three-sphere in this metric
will not play a prominent role.
The table below gives an overview of the indices we use to refer to certain directions and
subspaces.
AdS5 S
coord. names x0 x1 x2 x3 u –
µ, ν. . .
indices i, j. . . u
B. Introducing a non-Abelian chemical potential
A gravity dual description of a chemical potential amounts to a non-dynamical time component
of the gauge field Aµ in the action for the D7-brane probe embedded into the background given
above. There are essentially two different ways to realize a non-vanishing contribution from a
chemical potential to the field strength tensor F = 2∂[µAν] + f
abcAbµA
ν . The first is to consider
a u-dependent baryon chemical potential for a single brane probe. The second, which we pursue
here, is to consider a constant isospin chemical potential. This requires a non-Abelian probe brane
action and thus a probe of at least two coincident D7-branes, as suggested in [28]. Here the time
component of the gauge field is taken to be
A0 = A
a, (3.4)
where we sum over indices which occur twice in a term and denote the gauge group generators by
ta. The brane configuration described above corresponds to an SU(Nf) gauge group with Nf = 2
on the brane, which corresponds to a global SU(Nf ) in the dual field theory. For Nf = 2, the
generators of the gauge group on the brane are given by ta = σ
, with Pauli matrices σa. We will
see that (3.4) indeed produces non-trivial new contributions to the action.
Using the standard background field method of quantum field theory, we will consider the
chemical potential as a fixed background and let the gauge fields fluctuate. We single out a par-
ticular direction in flavor space by taking A30 = µ as the only non-vanishing component of the
background field. From now on we use the symbol Aaν to refer to gauge field fluctuations around
the fixed background,
Aaν → µδν0δa3 + Aaν . (3.5)
We pick the gauge in which Au ≡ 0 and assume that Aµ ≡ 0 for µ = 5, 6, 7. Due to the
symmetries of the background, we effectively examine gauge field fluctuations Aµ living in the
five-dimensional subspace on the brane spanned by the coordinates x0, x1, x2, x3 and by the radial
AdS coordinate u. The magnitude of all components of A and the background chemical potential
µ are considered to be small. This allows us to simplify certain expressions by dropping terms of
higher order in A and in the chemical potential µ.
C. Dirac-Born-Infeld action
The action describing the dynamics of the flavor gauge fields in the setup of this work is the
Dirac-Born-Infeld action. There are no contributions from the Chern-Simons action, which would
require non-zero gauge field components in all of the 4,5,6,7-directions. As mentioned, we con-
sider the D7 probe embedding whose asymptotic value at the boundary is chosen such that it cor-
responds to vanishing quark mass, m = 0. The metric on the brane is then given by (3.3). Since
we are interested in two-point correlators only, it is sufficient to consider the action to second order
in α′,
SD7 = −T7
(2πα′)2
2π2R3 Tr
uh=1∫
du d4x
−g gµµ′ gνν′ F aµν F aµ′ν′ , (3.6)
where we use the following definitions for the D7-brane tension T7 and the trace over the repre-
sentation matrices ta,
T7 = (2π)
7g−1s (α
′)−4 , (3.7)
tr(ta tb) = Tr δ
ab . (3.8)
In our case we have Tr = 1/2. The overall factor 2π
2R3 comes from the integration over the
5, 6, 7-directions, which are the directions along the S3.
Evaluating the DBI action given in (3.6) with the substitution rule (3.5), we arrive at
SD7 = − T7
(2πα′)2
2π2R3 Tr
uh=1∫
du d4x
−ggµµ′gνν′
4∂[µA
ν] ∂[µ′A
ν′] − 8δ0νδ0ν′fabc∂[0Aaµ] Abµ′ µc
(3.9)
where we use the short-hand notation µc = µδ3c and neglect terms of higher than linear order in
µ, and higher than quadratic order in A since both are small in our approach.
Up to the sum over flavor indices a, the first term in the bracket in (3.9) is reminiscent of the
Abelian super-Maxwell action in five dimensions, considered already for the R charge current
correlators in [2]. The new second term in our action arises from the non-Abelian nature of the
gauge group, giving terms proportional to the gauge group’s structure constants fabc in the field
strength tensor F aµν = 2∂[µA
+ fabcAbµA
D. Equations of motion
We proceed by calculating the retarded Green functions for the action (3.9), following the
prescription of [3] as outlined in section 2 above. According to this prescription, as a first step we
consider the equations of motion obtained from the action (3.9), which are given by
0 = 2∂µ
−g gµµ′gνν′ ∂[µ′Aaν′]
+ fabc
−gg00gνν′ µc
0 − 2∂0Abν′
+ δ0 ν∂µ
−g g00gµµ′Abµ′µc
(3.10)
It is useful to work in momentum space from now on. We therefore expand the bulk gauge fields
in Fourier modes in the xi directions,
Aµ(u, ~x) =
(2π)4
e−iωx0+ik·xAµ(u,~k). (3.11)
As we work in the gauge where Au = 0, we only have to take care of the components Ai with
i = 0, 1, 2, 3.
For the sake of simplicity, we choose the momentum of the fluctuations to be along the x3
direction, so their momentum four-vector is ~k = (ω, 0, 0, q). With this choice we have specified to
gauge fields which only depend on the radial coordinate u, the time coordinate x0 and the spatial
x3 direction.
1. Equations for Aa1- and A
2-components
Choosing the free Lorentz index in the equations of motion (3.10) to be ν = α = 1, 2 gives two
identical differential equations for A1 and A2,
0 = Aaα
2 − fq2
Aaα + 2i
Acα, (3.12)
where we indicated the derivative with respect to u with a prime and have introduced the dimen-
sionless quantities
, q =
, m =
. (3.13)
We now make use of the structure constants of SU(2), which are fabc = εabc, where εabc is the
totally antisymmetric epsilon symbol with ε123 = 1. Writing out (3.12) for the three different
choices of a = 1, 2, 3 results in
0 = A1α
2 − fq2
A1α − 2i
A2α , (3.14)
0 = A2α
2 − fq2
A2α + 2i
A1α , (3.15)
0 = A3α
2 − fq2
A3α . (3.16)
The first two of these equations are coupled, the third one is the same equation that was solved in
the Abelian Super-Maxwell case [2].
2. Equations for Aa0- and A
3-components
The remaining choices for the free Lorentz index ν = 0, 3, u in (3.10) result in three equations
which are not independent. The choices ν = 0 and ν = u give
0 = Aa0
Aa0 −
Aa3 − i
Ac3 , (3.17)
0 = wAa0
′ + qfAa3
′ + ifabc
′ . (3.18)
Solving (3.18) for Aa0
′, differentiating it once with respect to u and using (3.17) results in equation
(3.10) for ν = 3,
0 = Aa3
Aa3 +
Aa0 + i
Ac0 + 2i
Ac3 . (3.19)
We will make use of the equations (3.17) and (3.18) which look more concise. These equations of
motion for Aa0 and A
3 are coupled in Lorentz and flavor indices. To decouple them with respect
to the Lorentz structure, we solve (3.18) for Aa3
′ and insert the result into the differentiated version
of (3.17). This gives
0 = Aa0
′′′ +
(uf)′
2 − fq2
′ + 2i
′. (3.20)
The equations for a = 1, 2 are still coupled with respect to their gauge structure. The case
a = 3 was solved in [2]. We will solve (3.20) for Aa0
′ and can obtain Aa3
′ from (3.18). Note that it
is sufficient for our purpose to obtain solutions for the derivatives of the fields. These contribute to
(2.4), while the functions A = f(u,~k)Abdy(~k) themselves simply contribute a factor of f(u,−~k)
which is one at the boundary.
3. Solutions
Generally, we follow the methods developed in [2], since our differential equations are very
similar to the ones considered there. Additionally, we need to respect the flavor structure of the
gauge fields. The equations for flavor index a = 3 resemble the ones analyzed in [2]. However,
those for a = 1, 2 involve extra terms, which couple these equations. In our case the equations are
coupled not only via their Lorentz indices, but also with respect to the flavor indices. We already
decoupled the Lorentz structure in the previous section. As shown below, the equations of motion
which involve different gauge components will decouple if we consider the variables
Xi = A
i + iA
X̃i = A
i − iA2i .
(3.21)
Here the A1i , A
i are the generally complex gauge field components in momentum space. Note
that up to SU(2) transformations, the combinations (3.21) are the only ones which decouple the
equations of motion for a = 1, 2. These combinations are reminiscent of the non-Abelian SU(2)
gauge field in position space,
Ai = A
A3i A
i − iA2i
A1i + iA
i −A3i
. (3.22)
The equations of motion for the flavor index a = 3 were solved in [2]. To solve the equations
of motion for the fields Aai with a = 1, 2, we rewrite them in terms of Xi and X̃i. Applying the
transformation (3.21) to the equations of motion (3.14) and (3.15) and the a = 1, 2 versions of
(3.20) and (3.18) leads to
0 = X ′′α +
X ′α +
2 − fq2 ∓ 2mw
Xα, α = 1, 2, (3.23)
0 = X ′′′0 +
(uf)′
X ′′0 +
2 − fq2 ∓ 2mw
X ′0, (3.24)
0 = (w∓m)X ′0 + qfX ′3, (3.25)
where the upper signs correspond to X and the lower ones to X̃.
We see that some coefficients of these functions are divergent at the horizon u = 1. Such
differential equations with singular coefficients are generically solved by an ansatz
Xi = (1− u)β F (u), X̃i = (1− u)
eβ F̃ (u), (3.26)
with regular functions F (u) and F̃ (u). To cancel the singular behaviour of the coefficients, we
have to find the adequate β and β̃, the so-called indices, given by equations known as the indicial
equations for β and β̃. We eventually get for all Xi and X̃i
β = ±iw
1− 2m
, β̃ = ±iw
. (3.27)
Note that these exponents differ from those of the Abelian Super-maxwell theory [2] by a de-
pendence on
w in the limit of small frequencies (w < m). In the limit of vanishing chemical po-
tential m → 0, the indices given in [2] are reproduced from (3.27). In order to solve (3.23), (3.24)
and (3.25), we wish to introduce a series expansion ansatz in the momentum variables w and q. In
fact, the physical motivation behind this expansion is that we aim for thermodynamical quantities
which are known from statistical mechanics in the hydrodynamic limit of small four-momentum
~k. So the standard choice would be
F (u) = F0 +wF1 + q
2G1 + . . . . (3.28)
On the other hand, we realize that our indices will appear linearly (and quadratically) in the differ-
ential equations’ coefficients after inserting (3.26) into (3.23), (3.24) and (3.25). The square root
in β and β̃ mixes different orders of w. In order to sort coefficients in our series ansatz, we assume
w < m and keep only the leading w contributions to β and β̃, such that
β ≈ ∓
, β̃ ≈ ±i
. (3.29)
This introduces an additional order O(w1/2), which we include in our ansatz (3.28) yielding
F (u) = F0 +w
1/2F1/2 +wF1 + q
2G1 + . . . , (3.30)
and analogously for the tilded quantities. If we had not included O(w1/2) the resulting system
would be overdetermined. The results we obtain by using the approximations (3.29) and (3.30)
have been checked against the numerical solution for exact β with exact F (u). These approxima-
tions are useful for fluctuations with q,w < 1 (see subsection B 4 in Appendix B). Note that by
dropping the 1 in (3.27) we also drop the Abelian limit.
Consider the indices (3.27) for positive frequency first. In order to meet the incoming wave
boundary condition introduced in section II, we restrict the solution β̃ to the negative sign only.
For the approximate β̃ in (3.29) we therefore choose the lower (negative) sign. This exponent
describes a mode that travels into the horizon of the black hole. In case of β we demand the mode
to decay towards the horizon, choosing the lower (positive) sign in (3.29) consistently. Note that
for negative frequencies ω < 0 the indices β and β̃ exchange their roles.
Using (3.29) in (3.26) and inserting the ansatz into the equations of motion, we find equations
for each order in q2 and w separately. After solving the equations of motion for the coefficient
functions F0, F1/2, F1 and G1, we eventually can assemble the solutions to the equations of motion
for X as defined in (3.21),
X(u) = (1− u)β F (u) = (1− u)β
wF1/2 +wF1 + q
2G1 + . . .
. (3.31)
and a corresponding formula for X̃(u) from the ansatz (3.26).
Illustrating the method, we now write down the equations of motion order by order for the
function Xα. To do so, we use (3.31) with (3.29) in (3.23) with the upper sign for Xα. Then we
examine the result order by order in w and q2,
O(const) : 0 = F ′′0 +
F ′0 , (3.32)
w) : 0 = F ′′1/2 +
F ′1/2 −
F ′0 −
F0 , (3.33)
O(w) : 0 = F ′′1 +
F ′1 −
F ′1/2 −
F1/2 −m
4− u(1 + u)2
2uf 2
F0 , (3.34)
O(q2) : 0 = G′′1 +
G′1 −
F0 . (3.35)
At this point we observe that the differential equations we have to solve for each order are
shifted with respect to the solutions found in [2]. The contributions of order wn in [2] now show
up in order wn/2. Their solutions will exhibit factors of order µn/2.
Solving the system (3.32) to (3.35) of coupled differential equations is straightforward in the
way that they can be reduced to several uncoupled first order ordinary differential equations in
the following way. Note that there obviously is a constant solution F0 = C for the first equation.
Inserting it into (3.33) and (3.35) leaves us with ordinary differential equations for F ′1/2 and G
respectively. Using the solutions of F0 and F1/2 in (3.34) gives one more such equation for F
To fix the boundary values of the solutions just mentioned, we demand the value of F (uH = 1)
to be given by the constant F0 and therefore choose the other component functions’ solutions
such that limu→1 F1/2 = 0, and the same for F1 and G1. The remaining integration constant C is
determined by taking the boundary limit u → 0 of the explicit solution (3.31), making use of the
second boundary condition
X(u) = Xbdy, (3.36)
see appendix B. Eventually, we end up with all the ingredients needed to construct the gauge
field’s fluctuations X(u) as in (3.31).
We solve the equations (3.23) with lower sign for X̃α and (3.24) for X
0 and its tilded partner in
exactly the same way as just outlined, only some coefficients of these differential equations differ.
The solution for X ′3 is then obtained from (3.25).
All solutions are given explicitly in Appendix B together with all other information needed to
construct the functions Xα, X̃α, X
0, X̃
3 and X̃
IV. ISOSPIN DIFFUSION AND CORRELATION FUNCTIONS
A. Current correlators
In this section we obtain the momentum space correlation functions for the gauge field compo-
nent combinations X and X̃ defined in equation (3.21). Recall that the imaginary part of the
retarded correlators essentially gives the thermal spectral functions (see also section II). The
following discussion of the correlators’ properties is therefore equivalent to a discussion of the
corresponding spectral functions.
First note that the on-shell action gets new contributions from the non-Abelian structure,
SD7 = − T7
(2πα′)2
2π2R3 Tr (4.1)
(2π)4
√−gguugjj′ Aaj
′(~q)Aaj′(−~q)
− 4iq fabcµc
−gg00g33Aa[3Ab0]
where j, j′ = 0 , 1 , 2 , 3 and the index u denotes the radial AdS-direction. Up to the sum over
flavor indices, the first term in the bracket is similar to the Abelian Super-Maxwell action of [2].
The second term is a new contribution depending on the isospin chemical potential. It is a contact
term which we will neglect. The correlation functions however get a structure that is different
from the Abelian case. This is due to the appearance of the chemical potential in the equations of
motion and their solutions. Writing (4.1) as a function of X and X̃ results in
SD7 = − T7
(2πα′)2
2π2R3 Tr
(2π)4
−g guugjj′
′X̃j′ + X̃
+ A3j
]∣∣∣∣
(4.2)
− 4qµ
−gg00g33
X[0X̃3] + X̃[3X0]
In order to find the current correlators, we apply the method outlined in section II to (4.2),
with the solutions for the fields given in appendix B. As an example, we derive the correlators
G0e0 = 〈J0(~q)J̃0(−~q) 〉 and Ge00 = 〈J̃0(~q) J0(−~q)〉 of the flavor current time components J0 and J̃0,
coupling to the bulk fields X0 and X̃0, respectively. Correlation functions of all other components
are derived analogously. For the notation see appendix A.
1. Green functions: Calculation
First, we extract the factor B(u) of (2.2),
B(u) = −T7
(2πα′)2
2π2R3Tr
−g guu g00 . (4.3)
The second step, finding the solutions to the mode equations of motion, has already been per-
formed in section III D 3. In the example at hand we need the solutions X0 and X̃0. From (3.31)
and from appendix B we obtain
′ =− (1− u)
0 +wqX̃
2mw+wm ln 2 + q2
1−w1/2
π2 + 3 ln2 2 + 3 ln2(1 + u) + 6 ln 2 ln
1 + u
(4.4)
+12Li2(1− u) + 12Li2(−u)− 12Li2
+ q2 ln
′ = (1− u)−i
0 +wqX
2mw+wm ln 2− q2
1 +w1/2 i
π2 + 3 ln2 2 + 3 ln2(1 + u) + 6 ln 2 ln
1 + u
(4.5)
+12Li2(1− u) + 12Li2(−u)− 12Li2
+ q2 ln
Note that we need the derivatives to apply (2.4).
Now we perform the third step and insert (4.3), (4.4) and (4.5) into (2.4). Our solutions X0
and X̃0 replace the solution f(u,~k) and f(u,−~k) in (2.4). The resulting expression is evaluated
at ub = 0, which comes from the lower limit of the u-integral in the on-shell action (4.2). At small
u = ǫ → 0, (4.4) and (4.5) give
0 +wqX̃
2mw+wm ln 2 + q2
− lim
0 +wqX̃
ln ǫ , (4.6)
0 +wqX
2mw+wm ln 2− q2
+ lim
0 +wqX
ln ǫ . (4.7)
In the next to leading order of (4.6) and (4.7) there appear singularities, just like in the Abelian
Super-Maxwell calculation [2, equation (5.15)]. However, in the hydrodynamic limit, we consider
only the finite leading order.
2. Green functions: Results
Putting everything together, for the two Green functions for the field components X0, X̃0 given
in (3.21) by
X0 = A
0 + iA
0, X̃0 = A
0 − iA20,
we obtain
G0e0 =
2πT q2
2mw− q2 −wm ln2
, (4.8)
Ge00 =
2πT q2
2mw− q2 +wm ln2
. (4.9)
These are the Green functions for the time components in Minkowski space, perpendicular to the
chemical potential in flavor space. All Green functions are obtained considering hydrodynamic
approximations in O(w1/2,w, q2), neglecting mixed and higher orders O(w3/2,w1/2q2, q4).
The prefactor in (4.8), (4.9) is obtained using T7 as in (3.7), and carefully inserting all metric
factors, together with the standard AdS/CFT relation R4 = 4πgsNcα
′2. As in other settings with
flavor [26], we concordantly get an overall factor of Nc, and not N
c , for all correlators. Contrary
to those approaches, we do not get a factor of Nf when summing over the different flavors. This is
due to the fact that in our setup, the individual flavors yield distinct contributions. Most striking is
the non-trivial dependence on the (dimensionless) chemical potential m in both correlators. Note
also the distinct structures in the denominators. The first one (4.8) has an explicit relative factor
of i between the terms in the denominator. In the second correlator (4.9) there is no explicit factor
of i. The correlator (4.8) has a complex pole structure for ω > 0, but is entirely real for ω < 0.
On the other hand, (4.9) is real for ω > 0 but develops a diffusion structure for ω < 0. So the
correlators G0e0 and Ge00 essentially exchange their roles as ω changes sign (see also Fig. 1). We
find a similar behavior for all correlators G
and Gejl with j, l = 0, 1, 2, 3. This behavior is a
consequence of the insertion of O(w1/2) in the hydrodynamic expansion (3.30).
We assume m to be small enough in order to neglect the denominator term of order O(wm) ≪
wm, q2). Moreover, using the definitions of w, q and m from (3.13) we may write (4.8)
and (4.9) as
G0e0 = −
ω + q2D(ω)
, (4.10)
Ge00 =
iω − q2D(ω)
, (4.11)
where the frequency-dependent diffusion coefficient D(ω) is given by
D(ω) =
. (4.12)
We observe that this coefficient also depends on the inverse square root of the chemical potential µ.
Its physical interpretation is discussed below in section IV B.
In the same way we derive the other correlation functions
G3e3 =−
ω3/2 (ω − µ)
Q̃(ω, q)
, Ge33 =
ω3/2 (ω + µ)
Q(ω, q)
, (4.13)
G0e3 =−
ω q(ω − µ)
Q̃(ω, q)
, Ge03 =
ω q(ω + µ)
Q(ω, q)
, (4.14)
G3e0 =−
ω3/2 q
Q̃(ω, q)
, Ge30 =
ω3/2 q
Q(ω, q)
. (4.15)
with the short-hand notation
Q(ω, q) = iω − q2D(ω), Q̃(ω, q) = ω + q2D(ω) . (4.16)
Note that most of these functions are proportional to powers of q and therefore vanish in the limit
of vanishing spatial momentum q → 0. Only the 33-combinations from (4.13) survive this limit.
In contrast to the Abelian Super-Maxwell correlators [2] given in appendix C, it stands out that our
results (4.10), (4.11) and (4.13) and (4.15) have a new zero at ω = µ or −µ. Nevertheless bear in
mind that we took the limit ω < µ in order to obtain our solutions. Therefore the apparent zeros at
±µ lie outside of the range considered. Compared to the Abelian case there is an additional factor
ω. The dependence on temperature remains linear.
In the remaining X-correlators we do not find any pole structure to order
ω, subtracting an
O(q2) contribution as in [2],
G1e1 =G2e2 =
µω , (4.17)
Ge11 =Ge22 = −
µω . (4.18)
As seen from (4.17), Gαeα (with α = 1, 2) are real for negative ω and imaginary for positive ω. The
opposite is true for Geαα, as is obvious from the relative factor of i.
The correlators of components, pointing along the isospin potential in flavor space (a = 3), are
found to be
iω −D0q2
, GA3
= GA3
iω −D0q2
, (4.19)
= GA3
NcT iω
, GA3
iω −D0q2
, (4.20)
with the diffusion constant D0 = 1/(2πT ) . Note that these latter correlators have the same
structure but differ by a factor 4/Nc from those found in the Abelian super-Maxwell case [2] (see
also (C1) and (C2)). In particular the correlators in equation (4.19) do not depend on the chemical
potential.
To analyze the novel structures appearing in the other correlators, we explore their real and
imaginary parts as well as the interrelations among them,
ReG0e0(ω ≥ 0) = ReGe00(ω < 0) = −
2µ |ω|+ q2/(2πT )
, (4.21)
ReG0e0(ω < 0) = ReGe00(ω ≥ 0) = −
2µ |ω|+ q4/(2πT )2
, (4.22)
ImG0e0(ω < 0) = −ImGe00(ω ≥ 0) =
2µ |ω|
2µ |ω|+ q4/(2πT )2
, (4.23)
ImG0e0(ω ≥ 0) = 0, ImGe00(ω < 0) = 0. (4.24)
Now we see why, as discussed below (4.11), G0e0 and Ge00 exchange their roles when crossing the
origin at ω = 0. This is due to the fact that the real parts of all G
and Gejl are mirror images
of each other by reflection about the vertical axis at ω = 0. In contrast, the imaginary parts are
inverted into each other at the origin. Figure 1 shows the real and imaginary parts of correlators
G0e0 and Ge00. The different curves correspond to distinct values of the chemical potential µ. The
real part shows a deformed resonance behavior. The imaginary part has a deformed interference
shape with vanishing value for negative frequencies. All curves are continuous and finite at ω = 0.
However due to the square root dependence, they are not differentiable at the origin. Parts of the
correlator which are real for positive ω are shifted into the imaginary part by the change of sign
when crossing ω = 0, and vice versa.
To obtain physically meaningful correlators, we follow a procedure which generalizes the
Abelian approach of [6]. In the Abelian case, gauge-invariant components of the field strength ten-
sor, such as Eα = ωAα, are considered as physical variables. This procedure cannot be transferred
directly to the non-Abelian case. Instead, we consider the non-local part of the gauge invariant
trF 2 which contributes to the on-shell action (4.1). In this action, the contribution involving the
non-Abelian structure constant – as well as µ – is a local contact term. The non-local contribution
however generates the Green function combination
. (4.25)
We take this sum as our physical Green function. This choice is supported further by the fact
that it may be written in terms of the linear combinations (3.21) which decouple the equations of
motion. For example, for the time component, written in the variables X0, X̃0 given by (3.21),
the combination (4.25) reads (compare to (4.2))
G0e0 +Ge00 +GA30A30 . (4.26)
The contribution from GA3
is of order O(µ0), while the combination for the first two flavor
directions, G0e0 +Ge00, is of order O(µ).
We proceed by discussing the physical behavior of the Green function combinations introduced
above. GA3
is plotted in Fig. 2 on the right. Its frequency dependence is of the same form as in
the Abelian correlator obtained in [2], as can be seen from (C1). Since we are interested in effects
of order O(µ), we drop the third flavor direction a = 3 from the sum (4.26) in the following. It
is reassuring to observe that the flavor directions a = 1, 2 which are orthogonal to the chemical
potential, combine to give a correlator spectrum qualitatively similar to the one found in [2] for
PSfrag replacements
−0.05−0.1 0 0.05 0.1
PSfrag replacements
−0.05−0.1
0 0.05 0.1
FIG. 1: Real (left plot) and imaginary part (right plot) of the correlator G
as a function of frequency ω/T
at different chemical potential values µ/T = 0.5 (solid line), µ/T = 0.3 (short-dashed line) and µ/T =
0.1 (long-dashed line). The corresponding plots for the correlator G
would look like the mirror image of
the ones given. The real part would be reflected about the vertical axis at ω = 0, the imaginary part would
be reflected about the origin. All dimensionful quantities are given in units of temperature. The numerical
values used for the parameters are q/T = 0.1, Nc = 100.
the Abelian Super-Maxwell action (see Fig. 2). However, we discover intriguing new effects such
as the highly increased steepness of the curves near the origin due to the square root dependence
and a kink at the origin.
We observe a narrowing of the inverse resonance peak compared to the form found for the
Abelian Super-Maxwell action (and also compared to the form of our GA3
, as is seen from
comparing the left with the right plot in Fig. 2). At the origin the real and imaginary part are
finite and continuous, but they are not continuously differentiable. However, the imaginary part
of GA3
has finite steepness at the origin. The real part though has vanishing derivative at ω = 0.
Note that the imaginary part of flavor directions a = 1, 2 on the left plot in Fig. 2 never drops
below the real part. In the third flavor direction, as well as in the Abelian solution, such a drop
occurs on the positive ω-axis.
The correlators G3e3, Ge33, G0e3 and Ge03 have the same interrelations between their respective
real and imaginary parts as G0e0 and Ge00. Nevertheless, their dependence on the frequency and
momentum is different, as can be seen from (4.13) to (4.15). A list of the 33-direction Green
functions split into real and imaginary parts can be found in appendix D.
Thermal spectral functions in different directions are compared graphically in appendix E.
B. Isospin diffusion
The attenuated poles in hydrodynamic correlation functions have specific meanings (for ex-
emplary discussions of this in the AdS/CFT setup see e.g. [7], [50]). In our case we observe an
attenuated pole in the sum G0e0 + Ge00 at ω = 0. As can be seen from the plots in Fig. 2, our pole
lies at Reω = 0. This structure appears in hydrodynamics as the signature of a diffusion pole
located at purely imaginary ω. Its location on the imaginary ω-axis is given by the zeros of the
denominators of our correlators as (neglecting O(ω , q4 ))
ω = −i q
. (4.27)
PSfrag replacements
0.5 1 1.5−0.5−1−1.5
PSfrag replacements
0.010.005−0.005−0.01
FIG. 2: In the left plot the sum of both correlators in 00-directions is split into its imaginary (dashed
line) and real (solid line) part and plotted against frequency. For comparison the right plot shows the
corresponding real and imaginary parts for the GA3
. It is qualitatively similar to the Abelian correlator in
i = 0 Lorentz direction computed from the Super-Maxwell action in [2]. Note the different frequency scales
in the two plots. The curves in a = 1, 2-directions are much narrower due to their square root dependence
on ω. Furthermore they have a much larger maximum amplitude. All dimensionful quantities are given in
units of temperature. The numerical values used for the parameters are, as in Fig. 1, q/T = 0.1, Nc = 100
and only in the left plot µ/T = 0.2.
Squaring both sides of (4.27) we see that this effect is of order O(q4). On the other hand, looking
for poles in the correlator involving the third flavor direction GA3
, we obtain dominant contri-
butions of order O(q2) and O(µ0) (neglecting O(ω2 , q2 ))
ω = −i
. (4.28)
This diffusion pole is reminiscent of the result of the Abelian result of [2] given in appendix (C).
As discussed in section IV A, we consider gauge-invariant combinations G0e0 + Ge00 + GA30A30 . In
order to inspect the non-Abelian effects of order O(µ) showing up in the first two correlators in
this sum, we again drop the third flavor direction which is of order O(µ0).
Motivated by the diffusion pole behavior of our correlators in flavor-directions a = 1, 2 corre-
sponding to the combinations X, X̃ (see (4.27)), we wish to regain the structure of the diffusion
equation given in (2.9), which in our coordinates (k = (ω , 0 , 0 , q )) reads
i ω J0 = D(ω) q
2 J0 . (4.29)
Our goal is to rewrite (4.27) such that a term O(ω) and one term in order O(q2) appears. Further-
more there should be a relative factor of −i between these two terms. The obvious manipulation
to meet these requirements is to multiply (4.27) by
ω in order to get
ω = −i q2
. (4.30)
Comparing the gravity result (4.30) with the hydrodynamic equation (4.29), we obtain the
frequency-dependent diffusion coefficient
D(ω) =
. (4.31)
Our argument is thus summarized as follows: Given the isospin chemical potential as in (1.1),
(3.5), J0 from (2.9) is the isospin charge density in (4.29). According to (4.29), the coeffi-
cient (4.31) describes the diffusive response of the quark-gluon plasma to a gradient in the isospin
charge distribution. For this reason we interprete D(ω) as the isospin diffusion coefficient.
Near the pole, the strongly-coupled plasma behaves analogously to a diffractive medium with
anomalous dispersion in optics. In the presence of the isospin chemical potential, the propaga-
tion of non-Abelian gauge fields in the black hole background depends on the square root of the
frequency. In the dual gauge theory, this corresponds to a non-exponential decay of isospin fluc-
tuations with time.
The square root dependence of our diffusion coefficient is valid for small frequencies. As
long as ω/T < 1/4, the square root is larger than its argument and at ω/T = 1/4, the differ-
ence to a linear dependence on frequency is maximal. Therefore in the regime of small frequen-
cies ω/T < 1/4, which is accessible to our approximation, diffusion of modes close to 1/4 is
enhanced compared to modes with frequencies close to zero.
V. CONCLUSION
In this paper we have considered a relatively simple gauge/gravity dual model for a finite tem-
perature field theory, consisting of an isospin chemical potential µ obtained from a time component
vev for the SU(2) gauge field on two coincident brane probes. We have considered the D7-brane
embedding corresponding to vanishing quark mass, for which µ is a constant, independent in
particular of the radial holographic coordinate. The main result of this paper is that this model,
despite its simplicity, leads to a hydrodynamical behavior of the dual field theory which goes be-
yond linear response theory. We find in particular a frequency-dependent diffusion coefficient
with a non-analytical behavior. Frequency-dependent diffusion is a well-known phenomenon in
condensed matter physics. Here it originates simply from the fact that due to the non-Abelian
structure of the gauge field on the brane probe, the chemical potential replaces a time derivative in
the action and in the equations of motion from which the Green functions are obtained.
Of course the calculation presented has some limitations as far as the approximations made are
concerned. This applies in particular to the approximation (3.29) of the so-called indices in the
ansatz for solving the equations of motion. Here we have dropped the constant present under the
square root and used an expression proportional to the square root of the frequency. This allows
for a closed solution without having to use numerics. However using this approximation we have
dropped the Abelian limit. This leads ultimately to the square root dependence of the diffusion
coefficient on the frequency. This dependence is unphysical for ω → 0, since the diffusion coeffi-
cient is expected to be non-zero for zero frequency. We expect physical behavior to be restored if
the Abelian limit is included in the calculation. To avoid the approximation described, this requires
a numerical approach. A suitable solution method for all momenta and all frequencies has been
presented in [7] and in [49]. We are going to study the application of this method to the model
presented here in the future.
Acknowledgments
We are grateful to R. Apreda, G. Policastro, C. Sieg, A. Starinets and L. Yaffe for useful dis-
cussions and correspondence.
APPENDIX A: NOTATION
The five-dimensional AdS Schwarzschild black hole space in which we work is endowed with
a metric of signature (−,+,+,+,+), as given explicitly in (3.3). We make use of the Einstein
notation to indicate sums over Lorentz indices, and additionally simply sum over non-Lorentz
indices, such as gauge group indices, whenever they occur twice in a term.
To distinguish between vectors in different dimensions of the AdS space, we use bold symbols
like q for vectors in the the three spatial dimensions which do not live along the radial AdS
coordinate. Four-vectors which do not have components along the radial AdS coordinates are
denoted by symbols with an arrow on top, as ~q.
The Green functions G = 〈JĴ〉 considered give correlations between currents J and Ĵ . These
currents couple to fields A and Â respectively. In our notation we use symbols such as GAa
denote correlators of currents coupling to fields Aak and A
l , with flavor indices a, b and Lorentz
indices k, l = 0, 1, 2, 3. For the gauge field combinations Xk and X̃l given in (3.21), we obtain
Green functions G
denoting correlators of the corresponding currents.
APPENDIX B: SOLUTIONS TO EQUATIONS OF MOTION
Here we explicitly write down the component functions used to construct the solutions to the
equations of motion for the gauge field fluctuations up to order w and q2. The functions themselves
are then composed as in (3.31).
The solutions for the components with flavor index a = 3 where obtained in [2].
1. Solutions for Xα, X̃α and A3α
The function Xα(u) solves (3.23) with the upper sign and is constructed as in (3.31) from the
following component functions,
+O(ω), (B1)
F0 = C, (B2)
F1/2 = −C
1 + u
, (B3)
F1 = −C
π2 − 9 ln2 2 + 3 ln(1− u) (ln 16− 4 ln(1 + u))
+ 3 ln(1 + u) (ln(4(1 + u))− 4 ln u) (B4)
Li2(1− u) + Li2(−u) + Li2
1 + u
+ lnu ln(1 + u) + Li2(1− u) + Li2(−u)
, (B5)
where the constant C can be expressed it in terms of the field’s boundary value Xbdy =
limu→0X(u, k),
C = Xbdy
ln 2 +mw
ln2 2
2 +O(w3/2, q4)
. (B6)
The solutions of the equations of motion (3.23) with lower sign for the functions X̃α(u) are
given by
β̃ = −i
+O(ω), (B7)
F̃0 = C̃, (B8)
F̃1/2 = iC̃
1 + u
, (B9)
F̃1 = C̃
π2 − 9 ln2 2 + 3 ln(1− u) (ln 16− 4 ln(1 + u))
+ 3 ln(1 + u) (ln(4(1 + u))− 4 lnu) (B10)
Li2(1− u) + Li2(−u) + Li2
1 + u
G̃1 =
+ ln u ln(1 + u) + Li2(1− u) + Li2(−u)
, (B11)
with C̃ given by
C̃ = X̃bdy
ln 2−mw
ln2 2
2 +O(w3/2, q4)
, (B12)
so that limu→0 X̃(u, k) = X̃
The solution for A3α solves (3.16) up to order w and q
2 with boundary value (A3α)
. It is
A3α =
8 (A3α)
(1− u)− iw2
8− 4iw ln 2 + π2q2
1 + i
1 + u
+ ln u ln(1 + u) + Li2(1− u) + Li2(−u)
(B13)
2. Solutions for X ′0, X̃
0 and A
Here we state the solutions to (3.20). This formula describes three equations, differing in
the choice of a = 1, 2, 3. The cases a = 1, 2 give coupled equations which are decoupled by
transformation from A1,20 to X0 and X̃0. The choice a = 3 gives a single equation.
The function X ′0 is solution to (3.24) with upper sign. We specify the component functions as
+O(ω), (B14)
F0 = C, (B15)
F1/2 = −C
1 + u
, (B16)
F1 = −C
π2 + 3 ln2 2 + 3 ln2(1 + u) + 6 ln 2 ln
1 + u
(B17)
Li2(1− u) + Li2(−u)− Li2
, (B18)
G1 = C ln
1 + u
, (B19)
where the constant C can be expressed in terms of the field’s boundary value Xbdy =
limu→0X(u, k),
C = −
0 +wqX
2mw+mw ln 2 + q2
. (B20)
To get the function X̃ ′0, we solve (3.24) with the lower sign and obtain
β̃ = −i
+O(ω), (B21)
F̃0 = C̃, (B22)
F̃1/2 = iC̃
1 + u
, (B23)
F̃1 = C̃
π2 + 3 ln2 2 + 3 ln2(1 + u) + 6 ln 2 ln
1 + u
(B24)
Li2(1− u) + Li2(−u)− Li2
, (B25)
G̃1 = C̃ ln
1 + u
, (B26)
where the constant C̃ can be expressed it in terms of the field’s boundary value X̃bdy =
limu→0 X̃(u, k),
0 +wqX̃
2mw+mw ln 2− q2
. (B27)
The solution for (3.20) with a = 3 is the function A30
, given by
= (1− u)−
0 + wqA
iw− q2
1 + u
+ q2 ln
1 + u
. (B28)
PSfrag replacements
µ = 0.4 T
ω = q = 0.2 T
0.2 0.4 0.6 0.8 1
0.195
0.200
0.205
0.210
PSfrag replacements
µ = 0.4 T
ω = q = 0.2 T
µ = 0.4 T
ω = q = 0.2 T
Re F (u)/T
0.2 0.4 0.6 0.8 1
0.095
0.100
0.105
0.110
FIG. 3: These plots show the real and imaginary part of the function F (u) which is part of Xα = (1 −
u)βF (u). The solid line depicts the analytical approximation, obtained in this paper. As a check we solved
the equations of motion for F (u) numerically. They are drawn as dashed lines. In this example we used
T = 1. The numerical solution was chosen to agree with the analytical one at the horizon and boundary.
3. Solutions for X ′3, X̃
3 and A
We give the derivatives of X3 and X̃3 as
X ′3 = −
X ′0 (B29)
X̃ ′3 = −
X̃ ′0. (B30)
The solution for A33
. (B31)
4. Comparison of numerical and analytical results
As an example, in Fig. 3 we show the numerical and analytical solutions for the function F (u)
in Xα = (1− u)βF (u). Here we compare the numerical result for F (u) obtained from the ansatz
(3.26) with (3.27) in (3.23) with the analytically obtained approximation given above in (B1) to
(B6).
APPENDIX C: ABELIAN CORRELATORS
For convenient reference we quote here the correlation functions of the Abelian super-Maxwell
theory found in [2]. The authors start from a 5-dimensional supergravity action and not from a
Dirac-Born-Infeld action as we do. Therefore there is generally a difference by a factor Nc/4.
Note also that here all Nf flavors contribute equally. In our notation
Gab11 = G
22 = −
iN2c Tω δ
+ · · · , Gab00 =
N2c Tq
2 δab
16π(iω −Dq2)
+ · · · , (C1)
Gab03 = G
30 = −
N2c Tωq δ
16π(iω −Dq2)
+ · · · , Gab33 =
N2c Tω
2 δab
16π(iω −Dq2)
+ · · · , (C2)
where D = 1/(2πT ) .
APPENDIX D: CORRELATION FUNCTIONS
In this appendix we list the real and imaginary parts of the flavor currents in the first two
flavor-directions a = 1, 2 and in the third Lorentz-direction coupling to the supergravity-fields X3
and X̃3 (as defined in (3.21)).
Re{G3e3(ω ≥ 0)} = Re{Ge33(ω < 0)} = −
2 (ω2 + µ |ω|)
16π2 [2µ |ω|+ q4/(2πT )2]
, (D1)
Im{G3e3(ω ≥ 0)} = −Im{Ge33(ω < 0)} = −
2µ |ω| (ω2 + µ |ω|)
8π [2µ |ω|+ q4/(2πT )2]
, (D2)
Re{G3e3(ω < 0)} = Re{Ge33(ω ≥ 0)} = −
NcT (ω
2 − µ |ω|)
2µ |ω|+ q2/(2πT )
] , (D3)
Im{G3e3(ω < 0)} = 0, Im{Ge33(ω ≥ 0)} = 0. (D4)
APPENDIX E: THERMAL SPECTRAL FUNCTIONS
We include here a comparision of the sizes of spectral functions in distinct flavor- and Lorentz-
directions (see also (2.1) in section II).PSfrag replacements
0.5 1 1.5 2
PSfrag replacements
0.5 1 1.5 2
FIG. 4: Here the thermal spectral functions in distinct Lorentz- and flavor-directions are plotted against
frequency ω/T in units of temperature. In the left plot the chemical potential was chosen to be µ/T = 0.7,
in the right one µ/T = 0.2. Flavor-directions a = 1, 2 are summed and displayed as one curve. The
frequency-dependence of 00- (solid red line) and 03-Lorentz-directions (short-dashed blue line) is shown.
By the dotted line we denote the spectral curve in 11- or 22-directions. This curve was scaled by a factor 100
in order to make it visible in these plots. The third flavor-direction is only plotted for the spectral function in
Lorentz-directions 00 (long-dashed green curve). We do not show the 33-direction spectral function which
has a square root dependence and is comparable in size with the 11-direction.
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	Contents
	Introduction
	Hydrodynamics and AdS/CFT
	Supergravity background and action
	Finite temperature background and brane configuration
	Introducing a non-Abelian chemical potential
	Dirac-Born-Infeld action
	Equations of motion
	Equations for A1a- and A2a-components
	Equations for A0a- and A3a-components
	Solutions
	Isospin diffusion and correlation functions
	Current correlators
	Green functions: Calculation
	Green functions: Results
	Isospin diffusion
	Conclusion
	Acknowledgments
	Notation
	Solutions to equations of motion
	Solutions for X, X"0365X and A3
	Solutions for X0', X"0365X0' and A30'
	Solutions for X3', X"0365X3' and A33'
	Comparison of numerical and analytical results
	Abelian Correlators
	Correlation functions
	Thermal spectral functions
	References
