Isospin diffusion in thermal AdS/CFT with flavor

TL;DR

This paper investigates the effects of finite isospin chemical potential on a strongly coupled gauge theory using holography, revealing a frequency-dependent isospin diffusion coefficient through detailed Green function analysis.

Contribution

It introduces a holographic model with non-Abelian gauge fields on D7-branes to study isospin diffusion at finite chemical potential, extending previous linear response analyses.

Findings

Frequency-dependent isospin diffusion coefficient

Green functions for flavor currents calculated at zero quark mass

Insights into non-linear response properties of the system

Abstract

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.

Figures

Equations

$$A_0=\left(\begin{array}{cc}\mu & 0\\ 0& -\mu \end{array}\right),$$

$${\chi }_{\mu \nu }(\vec{k})=-2\mathrm{Im}{G}_{\mu \nu }(\vec{k}).$$

$$S_{\mathrm{cl}}=\int \mathrm{d}u{\mathrm{d}}^{4}xB(u)({\partial }_{u}A{)}^2+\dots$$

$$A(u,\vec{k})=f(u,\vec{k})A^{\text{bdy}}(\vec{k}),$$

$$G(\omega ,\mathbf{q})={-2B(u)f(u,-\vec{k}){\partial }_{u}f(u,\vec{k})|}_{u\to 0}.$$

$$G(\omega ,\mathbf{q})\propto \frac{1}{i\omega -D{\mathbf{q}}^2}.$$

$${\partial }_0{J}_0(t,\mathbf{x})=D{\nabla }^2{J}_0(t,\mathbf{x}),$$

$$i\omega J_0(\omega ,\mathbf{k})=D{\mathbf{k}}^2{J}_0(\omega ,\mathbf{k}).$$

$$G(\omega ,\mathbf{q})\propto \frac{1}{i\omega -D(\omega ){\mathbf{q}}^2}.$$

$$i\omega J_0(\omega ,\mathbf{k})=D(\omega ){\mathbf{k}}^2{J}_0(\omega ,\mathbf{k}).$$

$${\partial }_0{J}_0(t,\mathbf{x})+{\nabla }^2\underset{-\infty }{\overset{t}{\int }}\mathrm{d}s{J}_0(s,\mathbf{x})D(t-s)=0$$

$$\mathbf{J}=-\nabla (D\ast J_0),$$

$$\begin{array}{c}\mathrm{d}{s}^2=\frac{b^2{R}^2}{u}\left(-f(u)\mathrm{d}{x}_0^2+\mathrm{d}{x}_1^2+\mathrm{d}{x}_2^2+\mathrm{d}{x}_3^2\right)+\frac{R^2}{4u^{2}f(u)}\mathrm{d}{u}^2+R^2\mathrm{d}{\Omega }_5^2,\\ 0\le u\le 1,x_i\in \mathbb{R},C_{0123}=\frac{b^4{R}^4}{u^2},\end{array}$$

$$f(u)=1-u^2,R^4=4\pi g_s{N}_c{{\alpha }^{\prime }}^2,b=\pi T.$$

$$\begin{array}{c}\mathrm{d}{s}^2=\frac{b^2{R}^2}{u}\left(-f(u)\mathrm{d}{x}_0^2+\mathrm{d}{x}_1^2+\mathrm{d}{x}_2^2+\mathrm{d}{x}_3^2\right)+\frac{R^2}{4u^{2}f(u)}\mathrm{d}{u}^2+R^2\mathrm{d}{\Omega }_3^2,\\ 0\le u\le 1,x_i\in \mathbb{R}.\end{array}$$

$$A_0^{}=A_0^a{t}^a,$$

$$A_{\nu }^a\to \mu {\delta }_{\nu 0}{\delta }^{a3}+A_{\nu }^a.$$

$$S_{\text{D7}}=-T_7\frac{(2\pi {\alpha }^{\prime }{)}^2}{2}2{\pi }^2{R}^3{T}_r\underset{u_b=0}{\overset{u_h=1}{\int }}\mathrm{d}u{\mathrm{d}}^{4}x\sqrt{-g}{g}^{\mu {\mu }^{\prime }}{g}^{\nu {\nu }^{\prime }}{F}_{\mu \nu }^a{F}_{{\mu }^{\prime }{\nu }^{\prime }}^a,$$

$$T_7$$

$$\mathrm{tr}(t^a{t}^b)$$

$$S_{\mathrm{D}7}=$$

$$\times \underset{u_b=0}{\overset{u_h=1}{\int }}\mathrm{d}u{\mathrm{d}}^{4}x\sqrt{-g}{g}^{\mu {\mu }^{\prime }}{g}^{\nu {\nu }^{\prime }}\left(4{\partial }_{[\mu }^{}{A}_{\nu ]}^a{\partial }_{[{\mu }^{\prime }}^{}{A}_{{\nu }^{\prime }]}^a-8{\delta }_{0\nu }{\delta }_{0{\nu }^{\prime }}{f}^{abc}{\partial }_{[0}^{}{A}_{\mu ]}^a{A}_{{\mu }^{\prime }}^b{\mu }^c\right),$$

$$0=$$

$$+f^{abc}\left[\sqrt{-g}{g}^{00}{g}^{\nu {\nu }^{\prime }}{\mu }^c\left({\partial }_{{\nu }^{\prime }}^{}{A}_0^b-2{\partial }_0^{}{A}_{{\nu }^{\prime }}^b\right)+{\delta }^{0\nu }{\partial }_{\mu }^{}\left(\sqrt{-g}{g}^{00}{g}^{\mu {\mu }^{\prime }}{A}_{{\mu }^{\prime }}^b{\mu }^c\right)\right].$$

$$A_{\mu }(u,\vec{x})=\int \frac{{\mathrm{d}}^{4}k}{(2\pi {)}^4}{e}^{-i\omega x_0+i\mathbf{k}\cdot \mathbf{x}}{A}_{\mu }(u,\vec{k}).$$

$$0={A_{\alpha }^a}^{\prime \prime }+\frac{f^{\prime }}{f}{A_{\alpha }^a}^{\prime }+\frac{{\mathfrak{w}}^2-f{\mathfrak{q}}^2}{u{f}^2}{A}_{\alpha }^a+2i\frac{\mathfrak{w}}{u{f}^2}{f}^{abc}\frac{{\mu }^b}{2\pi T}{A}_{\alpha }^c,$$

$$\mathfrak{w}=\frac{\omega }{2\pi T},\mathfrak{q}=\frac{q}{2\pi T},\mathfrak{m}=\frac{\mu }{2\pi T}.$$

$$0$$

$$0={A_3^a}^{\prime \prime }+\frac{f^{\prime }}{f}{A_3^a}^{\prime }+\frac{{\mathfrak{w}}^2}{u{f}^2}{A}_3^a+\frac{\mathfrak{qw}}{u{f}^2}{A}_0^a+i\frac{\mathfrak{q}}{u{f}^2}{f}^{abc}\frac{{\mu }^b}{2\pi T}{A}_0^c+2i\frac{\mathfrak{w}}{u{f}^2}{f}^{abc}\frac{{\mu }^b}{2\pi T}{A}_3^c.$$

Full text

1. I Introduction

2. II Hydrodynamics and AdS/CFT

3. III Supergravity background and action

4. III.1 Finite temperature background and brane configuration

5. III.2 Introducing a non-Abelian chemical potential

6. III.3 Dirac-Born-Infeld action

7. III.4 Equations of motion

8. III.4.1 Equations for $A_{1}^{a}$- and $A_{2}^{a}$-components

9. III.4.2 Equations for $A_{0}^{a}$- and $A_{3}^{a}$-components

10. III.4.3 Solutions

11. IV Isospin diffusion and correlation functions

12. IV.1 Current correlators

13. IV.1.1 Green functions: Calculation

14. IV.1.2 Green functions: Results

15. IV.2 Isospin diffusion

16. V Conclusion

17. A Notation

18. B Solutions to equations of motion

19. B.1 Solutions for $X_{\alpha}$, $\widetilde{X}_{\alpha}$ and $A^{3}_{\alpha}$

20. B.2 Solutions for $X_{0}^{\prime}$, $\widetilde{X}_{0}^{\prime}$ and ${A^{3}_{0}}^{\prime}$

21. B.3 Solutions for $X_{3}^{\prime}$, $\widetilde{X}_{3}^{\prime}$ and ${A^{3}_{3}}^{\prime}$

22. B.4 Comparison of numerical and analytical results

23. C Abelian Correlators

24. D Correlation functions

25. E Thermal spectral functions

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 investigation is the gravity dual description of the quark-gluon plasma obtained by applying AdS/CFT to relativistic hydrodynamics Policastro:2001yc ; Policastro:2002se ; Son:2002sd ; Kovtun:2003wp ; Kovtun:2004de ; Kovtun:2005ev ; Kovtun:2006pf . The central result of this approach is the calculation 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 Son:2006em ; Mas:2006dy ; Maeda:2006by , and also the heat conductivity has been calculated by considering the R current correlators in these backgrounds Son:2006em .

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 Karch:2002sh ; Kruczenski:2003be ; Babington:2003vm ; Kruczenski:2003uq ; Sakai:2004cn . 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 Babington:2003vm 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 Kirsch:2004km (see Kruczenski:2003uq for a similar transition in the D4/D6 system), and studied in further detail in Mateos:2006nu ; Albash:2006ew . Related phase transitions appear in Karch:2006bv ; Kajantie:2006hv ; Schnitzer:2006xz . Mesons in gravity duals of finite temperature field theories have been studied in Ghoroku:2005kg ; Ghoroku:2006cc ; Peeters:2006iu ; Hoyos:2006gb .

Recently, in view of adding flavor to the quark-gluon plasma, the flavor contribution to the shear viscosity has been calculated in Mateos:2006yd ; Mateos:2007vn , where it was found that $\eta_{\rm fund}\propto\lambda N_{c}N_{f}T^{3}$.

For a thermodynamical approach in the grand canonical ensemble, the inclusion of a chemical potential and a finite number density is essential. In Apreda:2005yz , an isospin chemical potential was introduced 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 Harnik:2003ke . For the gauge/gravity dual analysis, a potential generated by an $SU(2)$ instanton on the D7 probe in the gravity background was used Guralnik:2004ve ; Erdmenger:2005bj ; Apreda:2005hj ; Arean:2007nh .

A baryon chemical potential $\mu_{B}$ is obtained by turning on the diagonal $U(1)\subset U(N_{f})$ gauge field on the D7-brane probe Kobayashi:2006sb . Contributions to the D7-brane action arise from the derivative 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 Kobayashi:2006sb , where regions of thermodynamical instability have been found in the $(T,\mu_{B})$ phase diagram. – For the $D4/D8/\bar{D8}$ Sakai-Sugimoto model Sakai:2004cn ; Sakai:2005yt , 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 Kim:2006gp ; Horigome:2006xu ; Parnachev:2006ev .

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 Herzog:2006gh . 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 $N_{f}\sim N_{c}$, was performed in Bertoldi:2007sf .

Here we study finite-temperature field theories with finite isospin chemical potential by considering two coincident D7-brane probes in the Lorentzian signature AdS-Schwarzschild black hole background. As in Apreda:2005yz , we introduce an isospin chemical potential by defining

[TABLE]

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 $\mu$, such that the Bose-Einstein instability mentioned above, which is of order ${\cal{O}}(\mu^{2})$, 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). These are dual to the $SU(2)$ flavor current $J^{\mu a}$. We find an ansatz for decoupling the equations of motion for the different Lorentz and flavor components, and solve them by adapting the method developed in Policastro:2002se ; Son:2002sd . This involves Fourier transforming 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 frequencies with $\omega<\mu<T$. 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 action and in the equations of motion. In the Fourier transformed picture, this leads to a square-root dependence of physical observables on the frequency, $\sqrt{\omega}$. 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 $D(\omega)\propto\frac{1}{T}\sqrt{\omega/\mu}$. 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 PhysRevLett ; Rabani 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 Bhattacharjee:1980tf .

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 Liu:2006xs ; Tsang:2003td . – For two-flavor QCD, effects of a finite isospin chemical potential have been discussed for instance in Son:2000xc ; Splittorff:2000mm ; Toublan:2003tt . 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 coefficient. 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 hydrodynamic theories and for calculating hydrodynamic quantities such as transport coefficients. For instance, given a retarded current correlation function $G(\vec{k})_{\mu\nu}$ in Minkowski space, the spectral function can be written in terms of its imaginary part,

[TABLE]

For the gravity dual approach, this is discussed for instance in Kovtun:2006pf ; Teaney:2006nc .

In this paper we use the gauge/gravity dual prescription of Son:2002sd 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 $S_{\rm cl}$ for the gauge field $A$, according to Son:2002sd we extract the function $B(u)$ (containing metric factors and the metric determinant) in front of the kinetic term $(\partial_{u}A)^{2}$,

[TABLE]

Then we perform a Fourier transformation and solve the linearized equations of motion in momentum 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

[TABLE]

where $A^{\text{bdy}}(\vec{k})$ 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 $\left.f(u,\vec{k})\right|_{u\to 0}=1$. For the other boundary, located at the horizon $u=1$, we impose the incoming wave condition. This requires that any Fourier mode $A(\vec{k})$ with timelike $\vec{k}$ can travel into the black hole, but is not allowed to cross the horizon in the opposite direction. For spacelike $\vec{k}$, the components of $A$ have to be regular at the horizon. Then the retarded thermal Green function is given by

[TABLE]

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

[TABLE]

This may be identified with the the Green function for the hydrodynamic diffusion equation

[TABLE]

with $J_{0}$ the time component of a diffusive current. $D$ is the diffusion constant. In Fourier space this equation reads

[TABLE]

In position space, this corresponds to an exponential decay of $J_{0}$ 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

[TABLE]

Retarded Green functions of this type have been discussed for instance in Bhattacharjee:1980tf . (9) corresponds to frequency-dependent diffusion with coefficient $D(\omega)$, such that (8) becomes

[TABLE]

In our case, $J_{0}$ 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 $D\cdot J_{0}$ and obtain

[TABLE]

for the redarded Green function. This implies together with the continuity equation $\partial_{0}\,J_{0}\,+\bm{\nabla}\cdot\bm{J}\,=\,0$, with $\bm{J}$ the three-vector current associated to $J_{0}$, that

[TABLE]

where $*$ denotes the convolution. This replaces the linear response theory constitutive equation ${\bm{J}}=-D{\bm{\nabla}}J_{0}$. Note that for $D(t-s)=D\delta(t-s)$ with $D$ constant, (11) reduces again to (7).

III Supergravity background and action

III.1 Finite temperature background and brane configuration

We consider an asymptotically $AdS_{5}\times S^{5}$ spacetime as the near horizon limit of a stack of $N_{c}$ coincident D3-branes. More precisely, as in Policastro:2002se , our background is an $AdS$ black hole, which is the geometry dual to a field theory at finite temperature. The Minkowski signature background is

[TABLE]

with the metric $\mathrm{d}\Omega_{5}^{2}$ of the unit $5$-sphere, and the function $f(u)$, $AdS$ radius $R$ and temperature parameter $b$ given in terms of the string coupling $g_{s}$, temperature $T$, inverse string tension $\alpha^{\prime}$ and number of colors $N_{c}$ by

[TABLE]

The geometry is asymptotically $AdS_{5}\times S^{5}$ 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 $N_{f}=2$ coinciding D7-branes, hosting flavor gauge fields $A_{\mu}$. The embedding we choose extends the D7-branes in all directions of $AdS$ space and wraps an $S^{3}$ on the $S^{5}$. 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

[TABLE]

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.

III.2 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 $A_{\mu}$ 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 $F=2\partial_{[\mu}A_{\nu]}+f^{abc}A^{b}_{\mu}A^{c}_{\nu}$. 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 Apreda:2005yz . Here the time component of the gauge field is taken to be

[TABLE]

where we sum over indices which occur twice in a term and denote the gauge group generators by $t^{a}$. The brane configuration described above corresponds to an $SU(N_{f})$ gauge group with $N_{f}=2$ on the brane, which corresponds to a global $SU(N_{f})$ in the dual field theory. For $N_{f}=2$, the generators of the gauge group on the brane are given by $t^{a}=\frac{\sigma^{a}}{2}$, with Pauli matrices $\sigma^{a}$. We will see that (16) 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 particular direction in flavor space by taking $A^{3}_{0}=\mu$ as the only non-vanishing component of the background field. From now on we use the symbol $A^{a}_{\nu}$ to refer to gauge field fluctuations around the fixed background,

[TABLE]

We pick the gauge in which $A_{u}\equiv 0$ and assume that $A_{\mu}\equiv 0$ for $\mu=5,6,7$. Due to the symmetries of the background, we effectively examine gauge field fluctuations $A_{\mu}$ living in the five-dimensional subspace on the brane spanned by the coordinates $x_{0}$, $x_{1}$, $x_{2}$, $x_{3}$ and by the radial AdS coordinate $u$. The magnitude of all components of $A$ and the background chemical potential $\mu$ 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 $\mu$.

III.3 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 consider the D7 probe embedding whose asymptotic value at the boundary is chosen such that it corresponds to vanishing quark mass, $m=0$. The metric on the brane is then given by (15). Since we are interested in two-point correlators only, it is sufficient to consider the action to second order in $\alpha^{\prime}$,

[TABLE]

where we use the following definitions for the D7-brane tension $T_{7}$ and the trace over the representation matrices $t^{a}$,

[TABLE]

In our case we have $T_{r}=1/2$. The overall factor $2\pi^{2}R^{3}$ comes from the integration over the $5,6,7$-directions, which are the directions along the $S^{3}$.

Evaluating the DBI action given in (18) with the substitution rule (17), we arrive at

[TABLE]

where we use the short-hand notation $\mu^{c}=\mu\delta^{3c}$ and neglect terms of higher than linear order in $\mu$, 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 (21) is reminiscent of the Abelian super-Maxwell action in five dimensions, considered already for the R charge current correlators in Policastro:2002se . 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 $f^{abc}$ in the field strength tensor $F^{a}_{\mu\nu}=2\partial_{[\mu}^{\vphantom{a}}A^{a}_{\nu]}+f^{abc}A^{b}_{\mu}\,A^{c}_{\nu}$.

III.4 Equations of motion

We proceed by calculating the retarded Green functions for the action (21), following the prescription of Son:2002sd as outlined in section 2 above. According to this prescription, as a first step we consider the equations of motion obtained from the action (21), which are given by

[TABLE]

It is useful to work in momentum space from now on. We therefore expand the bulk gauge fields in Fourier modes in the $x^{i}$ directions,

[TABLE]

As we work in the gauge where $A_{u}=0$, we only have to take care of the components $A_{i}$ with $i=0,1,2,3$.

For the sake of simplicity, we choose the momentum of the fluctuations to be along the $x_{3}$ direction, so their momentum four-vector is $\vec{k}=(\omega,0,0,q)$. With this choice we have specified to gauge fields which only depend on the radial coordinate $u$, the time coordinate $x_{0}$ and the spatial $x_{3}$ direction.

III.4.1 Equations for $A_{1}^{a}$- and $A_{2}^{a}$-components

Choosing the free Lorentz index in the equations of motion (22) to be $\nu=\alpha=1,2$ gives two identical differential equations for $A_{1}$ and $A_{2}$,

[TABLE]

where we indicated the derivative with respect to $u$ with a prime and have introduced the dimensionless quantities

[TABLE]

We now make use of the structure constants of $SU(2)$, which are $f^{abc}=\varepsilon^{abc}$, where $\varepsilon^{abc}$ is the totally antisymmetric epsilon symbol with $\varepsilon^{123}=1$. Writing out (24) for the three different choices of $a=1,2,3$ results in

[TABLE]

The first two of these equations are coupled, the third one is the same equation that was solved in the Abelian Super-Maxwell case Policastro:2002se .

III.4.2 Equations for $A_{0}^{a}$- and $A_{3}^{a}$-components

The remaining choices for the free Lorentz index $\nu=0,3,u$ in (22) result in three equations which are not independent. The choices $\nu=0$ and $\nu=u$ give

[TABLE]

Solving (30) for ${A_{0}^{a}}^{\prime}$, differentiating it once with respect to $u$ and using (29) results in equation (22) for $\nu=3$,

[TABLE]

We will make use of the equations (29) and (30) which look more concise. These equations of motion for $A_{0}^{a}$ and $A_{3}^{a}$ are coupled in Lorentz and flavor indices. To decouple them with respect to the Lorentz structure, we solve (30) for ${A_{3}^{a}}^{\prime}$ and insert the result into the differentiated version of (29). This gives

[TABLE]

The equations for $a=1,2$ are still coupled with respect to their gauge structure. The case $a=3$ was solved in Policastro:2002se . We will solve (32) for ${A^{a}_{0}}^{\prime}$ and can obtain ${A^{a}_{3}}^{\prime}$ from (30). Note that it is sufficient for our purpose to obtain solutions for the derivatives of the fields. These contribute to (5), while the functions $A=f(u,\vec{k})A^{\text{bdy}}(\vec{k})$ themselves simply contribute a factor of $f(u,-\vec{k})$ which is one at the boundary.

III.4.3 Solutions

Generally, we follow the methods developed in Policastro:2002se , 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 Policastro:2002se . 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

[TABLE]

Here the $A_{i}^{1}$, $A_{i}^{2}$ are the generally complex gauge field components in momentum space. Note that up to $SU(2)$ transformations, the combinations (33) 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,

[TABLE]

The equations of motion for the flavor index $a=3$ were solved in Policastro:2002se . To solve the equations of motion for the fields $A^{a}_{i}$ with $a=1,2$, we rewrite them in terms of $X_{i}$ and $\widetilde{X}_{i}$. Applying the transformation (33) to the equations of motion (26) and (27) and the $a=1,2$ versions of (32) and (30) leads to

[TABLE]

where the upper signs correspond to $X$ and the lower ones to $\widetilde{X}$.

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

[TABLE]

with regular functions $F(u)$ and $\widetilde{F}(u)$. To cancel the singular behaviour of the coefficients, we have to find the adequate $\beta$ and $\widetilde{\beta}$, the so-called indices, given by equations known as the indicial equations for $\beta$ and $\widetilde{\beta}$. We eventually get for all $X_{i}$ and $\widetilde{X}_{i}$

[TABLE]

Note that these exponents differ from those of the Abelian Super-maxwell theory Policastro:2002se by a dependence on $\sqrt{\mathfrak{w}}$ in the limit of small frequencies ($\mathfrak{w}<\mathfrak{m}$). In the limit of vanishing chemical potential $\mathfrak{m}\to 0$, the indices given in Policastro:2002se are reproduced from (39). In order to solve (35), (36) and (37), we wish to introduce a series expansion ansatz in the momentum variables $\mathfrak{w}$ and $\mathfrak{q}$. 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 $\vec{k}$. So the standard choice would be

[TABLE]

On the other hand, we realize that our indices will appear linearly (and quadratically) in the differential equations’ coefficients after inserting (38) into (35), (36) and (37). The square root in $\beta$ and $\widetilde{\beta}$ mixes different orders of $\mathfrak{w}$. In order to sort coefficients in our series ansatz, we assume $\mathfrak{w}<\mathfrak{m}$ and keep only the leading $\mathfrak{w}$ contributions to $\beta$ and $\widetilde{\beta}$, such that

[TABLE]

This introduces an additional order $\mathcal{O}(\mathfrak{w}^{1/2})$, which we include in our ansatz (40) yielding

[TABLE]

and analogously for the tilded quantities. If we had not included $\mathcal{O}(\mathfrak{w}^{1/2})$ the resulting system would be overdetermined. The results we obtain by using the approximations (41) and (42) have been checked against the numerical solution for exact $\beta$ with exact $F(u)$. These approximations are useful for fluctuations with $\mathfrak{q},\mathfrak{w}<1$ (see subsection B.4 in Appendix B). Note that by dropping the $1$ in (39) we also drop the Abelian limit.

Consider the indices (39) for positive frequency first. In order to meet the incoming wave boundary condition introduced in section II, we restrict the solution $\widetilde{\beta}$ to the negative sign only. For the approximate $\widetilde{\beta}$ in (41) we therefore choose the lower (negative) sign. This exponent describes a mode that travels into the horizon of the black hole. In case of $\beta$ we demand the mode to decay towards the horizon, choosing the lower (positive) sign in (41) consistently. Note that for negative frequencies $\omega<0$ the indices $\beta$ and $\widetilde{\beta}$ exchange their roles.

Using (41) in (38) and inserting the ansatz into the equations of motion, we find equations for each order in $\mathfrak{q}^{2}$ and $\mathfrak{w}$ separately. After solving the equations of motion for the coefficient functions $F_{0}$, $F_{1/2}$, $F_{1}$ and $G_{1}$, we eventually can assemble the solutions to the equations of motion for $X$ as defined in (33),

[TABLE]

and a corresponding formula for $\widetilde{X}(u)$ from the ansatz (38).

Illustrating the method, we now write down the equations of motion order by order for the function $X_{\alpha}$. To do so, we use (43) with (41) in (35) with the upper sign for $X_{\alpha}$. Then we examine the result order by order in $\mathfrak{w}$ and $\mathfrak{q}^{2}$,

[TABLE]

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 Policastro:2002se . The contributions of order $\mathfrak{w}^{n}$ in Policastro:2002se now show up in order $\mathfrak{w}^{n/2}$. Their solutions will exhibit factors of order $\mu^{n/2}$.

Solving the system (44) to (47) 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 $F_{0}=C$ for the first equation. Inserting it into (45) and (47) leaves us with ordinary differential equations for $F_{1/2}^{\prime}$ and $G_{1}^{\prime}$ respectively. Using the solutions of $F_{0}$ and $F_{1/2}$ in (46) gives one more such equation for $F_{1}^{\prime}$.

To fix the boundary values of the solutions just mentioned, we demand the value of $F(u_{H}=1)$ to be given by the constant $F_{0}$ and therefore choose the other component functions’ solutions such that $\lim_{u\to 1}F_{1/2}=0$, and the same for $F_{1}$ and $G_{1}$. The remaining integration constant $C$ is determined by taking the boundary limit $u\to 0$ of the explicit solution (43), making use of the second boundary condition

[TABLE]

see appendix B. Eventually, we end up with all the ingredients needed to construct the gauge field’s fluctuations $X(u)$ as in (43).

We solve the equations (35) with lower sign for $\widetilde{X}_{\alpha}$ and (36) for $X_{0}^{\prime}$ and its tilded partner in exactly the same way as just outlined, only some coefficients of these differential equations differ. The solution for $X_{3}^{\prime}$ is then obtained from (37).

All solutions are given explicitly in Appendix B together with all other information needed to construct the functions $X_{\alpha}$, $\widetilde{X}_{\alpha}$, $X_{0}^{\prime}$, $\widetilde{X}_{0}^{\prime}$, $X_{3}^{\prime}$ and $\widetilde{X}_{3}^{\prime}$.

IV Isospin diffusion and correlation functions

IV.1 Current correlators

In this section we obtain the momentum space correlation functions for the gauge field component combinations $X$ and $\widetilde{X}$ defined in equation (33). 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,

[TABLE]

where $j,\,j^{\prime}\,=\,0\,,1\,,2\,,3$ 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 Policastro:2002se . 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 (49) as a function of $X$ and $\widetilde{X}$ results in

[TABLE]

In order to find the current correlators, we apply the method outlined in section II to (IV.1), with the solutions for the fields given in appendix B. As an example, we derive the correlators $G_{0\widetilde{0}}=\langle J_{0}(\vec{q})\widetilde{J}_{0}(-\vec{q})\,\rangle$ and $G_{\widetilde{0}0}=\langle\widetilde{J}_{0}(\vec{q})\,{J_{0}}(-\vec{q})\rangle$ of the flavor current time components $J_{0}$ and $\widetilde{J}_{0}$, coupling to the bulk fields $X_{0}$ and $\widetilde{X}_{0}$, respectively. Correlation functions of all other components are derived analogously. For the notation see appendix A.

IV.1.1 Green functions: Calculation

First, we extract the factor $B(u)$ of (3),

[TABLE]

The second step, finding the solutions to the mode equations of motion, has already been performed in section III.4.3. In the example at hand we need the solutions $X_{0}$ and $\widetilde{X}_{0}$. From (43) and from appendix B we obtain

[TABLE]

Note that we need the derivatives to apply (5).

Now we perform the third step and insert (51), (IV.1.1) and (IV.1.1) into (5). Our solutions $X_{0}$ and $\widetilde{X}_{0}$ replace the solution $f(u,\vec{k})$ and $f(u,-\vec{k})$ in (5). The resulting expression is evaluated at $u_{b}=0$, which comes from the lower limit of the $u$-integral in the on-shell action (IV.1). At small $u=\epsilon\to 0$, (IV.1.1) and (IV.1.1) give

[TABLE]

In the next to leading order of (54) and (55) there appear singularities, just like in the Abelian Super-Maxwell calculation (Policastro:2002se, , equation (5.15)). However, in the hydrodynamic limit, we consider only the finite leading order.

IV.1.2 Green functions: Results

Putting everything together, for the two Green functions for the field components $X_{0}$, $\widetilde{X}_{0}$ given in (33) by

[TABLE]

we obtain

[TABLE]

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 $\mathcal{O}(\mathfrak{w}^{1/2},\mathfrak{w},\mathfrak{q}^{2})$, neglecting mixed and higher orders $\mathcal{O}(\mathfrak{w}^{3/2},\mathfrak{w}^{1/2}\mathfrak{q}^{2},\mathfrak{q}^{4})$.

The prefactor in (56), (57) is obtained using $T_{7}$ as in (19), and carefully inserting all metric factors, together with the standard AdS/CFT relation $R^{4}=4\pi g_{s}N_{c}\alpha^{\prime}{}^{2}$. As in other settings with flavor Mateos:2006yd , we concordantly get an overall factor of $N_{c}$, and not $N_{c}^{2}$, for all correlators. Contrary to those approaches, we do not get a factor of $N_{f}$ 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 $\mathfrak{m}$ in both correlators. Note also the distinct structures in the denominators. The first one (56) has an explicit relative factor of $i$ between the terms in the denominator. In the second correlator (57) there is no explicit factor of $i$. The correlator (56) has a complex pole structure for $\omega>0$, but is entirely real for $\omega<0$. On the other hand, (57) is real for $\omega>0$ but develops a diffusion structure for $\omega<0$. So the correlators $G_{0\widetilde{0}}$ and $G_{\widetilde{0}0}$ essentially exchange their roles as $\omega$ changes sign (see also Fig. 1). We find a similar behavior for all correlators $G_{j\widetilde{l}}$ and $G_{\widetilde{j}l}$ with $j,l\,=\,0,\,1,\,2,\,3$. This behavior is a consequence of the insertion of $\mathcal{O}(\mathfrak{w}^{1/2})$ in the hydrodynamic expansion (42).

We assume $\mathfrak{m}$ to be small enough in order to neglect the denominator term of order $\mathcal{O}(\mathfrak{w}\mathfrak{m})\ll\mathcal{O}(\sqrt{\mathfrak{w}\mathfrak{m}},\mathfrak{q}^{2})$. Moreover, using the definitions of $\mathfrak{w},\,\mathfrak{q}$ and $\mathfrak{m}$ from (25) we may write (56) and (57) as

[TABLE]

where the frequency-dependent diffusion coefficient $D(\omega)$ is given by

[TABLE]

We observe that this coefficient also depends on the inverse square root of the chemical potential $\mu$. Its physical interpretation is discussed below in section IV.2.

In the same way we derive the other correlation functions

[TABLE]

with the short-hand notation

[TABLE]

Note that most of these functions are proportional to powers of $q$ and therefore vanish in the limit of vanishing spatial momentum $q\to 0$. Only the $33$-combinations from (61) survive this limit. In contrast to the Abelian Super-Maxwell correlators Policastro:2002se given in appendix C, it stands out that our results (58), (59) and (61) and (63) have a new zero at $\omega=\,\mu$ or $-\mu$. Nevertheless bear in mind that we took the limit $\omega<\mu$ in order to obtain our solutions. Therefore the apparent zeros at $\pm\mu$ lie outside of the range considered. Compared to the Abelian case there is an additional factor of $\sqrt{\omega}$. The dependence on temperature remains linear.

In the remaining $X$-correlators we do not find any pole structure to order $\sqrt{\omega}$, subtracting an $\mathcal{O}(q^{2})$ contribution as in Policastro:2002se ,

[TABLE]

As seen from (65), $G_{\alpha\widetilde{\alpha}}$ (with $\alpha=1,2$) are real for negative $\omega$ and imaginary for positive $\omega$. The opposite is true for $G_{\widetilde{\alpha}\alpha}$, 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

[TABLE]

with the diffusion constant $D_{0}=1/(2\pi T)$ . Note that these latter correlators have the same structure but differ by a factor $4/N_{c}$ from those found in the Abelian super-Maxwell case Policastro:2002se  (see also (111) and (112)). In particular the correlators in equation (67) 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,

[TABLE]

and

[TABLE]

Now we see why, as discussed below (59), $G_{0\widetilde{0}}$ and $G_{\widetilde{0}0}$ exchange their roles when crossing the origin at $\omega=0$. This is due to the fact that the real parts of all $G_{j\widetilde{l}}$ and $G_{\widetilde{j}l}$ are mirror images of each other by reflection about the vertical axis at $\omega=0$. In contrast, the imaginary parts are inverted into each other at the origin. Figure 1 shows the real and imaginary parts of correlators $G_{0\widetilde{0}}$ and $G_{\widetilde{0}0}$. The different curves correspond to distinct values of the chemical potential $\mu$. 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 $\omega=0$. However due to the square root dependence, they are not differentiable at the origin. Parts of the correlator which are real for positive $\omega$ are shifted into the imaginary part by the change of sign when crossing $\omega=0$, and vice versa.

To obtain physically meaningful correlators, we follow a procedure which generalizes the Abelian approach of Kovtun:2005ev . In the Abelian case, gauge-invariant components of the field strength tensor, such as $E_{\alpha}=\omega A_{\alpha}$, 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 ${\rm tr}F^{2}$ which contributes to the on-shell action (49). In this action, the contribution involving the non-Abelian structure constant – as well as $\mu$ – is a local contact term. The non-local contribution however generates the Green function combination

[TABLE]

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 (33) which decouple the equations of motion. For example, for the time component, written in the variables  $X_{0},\,\widetilde{X}_{0}$ given by (33), the combination (73) reads (compare to (IV.1))

[TABLE]

The contribution from $G_{A^{3}_{0}A^{3}_{0}}$ is of order $\mathcal{O}(\mu^{0})$, while the combination for the first two flavor directions, $G_{0\widetilde{0}}+G_{\widetilde{0}0}$, is of order $\mathcal{O}(\mu)$.

We proceed by discussing the physical behavior of the Green function combinations introduced above. $G_{A^{3}_{0}A^{3}_{0}}$ is plotted in Fig. 2 on the right. Its frequency dependence is of the same form as in the Abelian correlator obtained in Policastro:2002se , as can be seen from (111). Since we are interested in effects of order $\mathcal{O}(\mu)$, we drop the third flavor direction $a=3$ from the sum (74) 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 Policastro:2002se for 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 $G_{A_{0}^{3}A_{0}^{3}}$, 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 $G_{A_{0}^{3}A_{0}^{3}}$ has finite steepness at the origin. The real part though has vanishing derivative at $\omega=0$. 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 $\omega$-axis.

The correlators $G_{3\widetilde{3}}$, $G_{\widetilde{3}3}$, $G_{0\widetilde{3}}$ and $G_{\widetilde{0}3}$ have the same interrelations between their respective real and imaginary parts as $G_{0\widetilde{0}}$ and $G_{\widetilde{0}0}$. Nevertheless, their dependence on the frequency and momentum is different, as can be seen from (61) to (63). 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.

IV.2 Isospin diffusion

The attenuated poles in hydrodynamic correlation functions have specific meanings (for exemplary discussions of this in the AdS/CFT setup see e.g. Kovtun:2006pf , Policastro:2002tn ). In our case we observe an attenuated pole in the sum $G_{0\widetilde{0}}+G_{\widetilde{0}0}$ at $\omega=0$. As can be seen from the plots in Fig. 2, our pole lies at $\mathrm{Re}\,\omega=0$. This structure appears in hydrodynamics as the signature of a diffusion pole located at purely imaginary $\omega$. Its location on the imaginary $\omega$-axis is given by the zeros of the denominators of our correlators as (neglecting $\mathcal{O}(\omega\,,q^{4}\,)$)

[TABLE]

Squaring both sides of (75) we see that this effect is of order $\mathcal{O}(q^{4})$. On the other hand, looking for poles in the correlator involving the third flavor direction $G_{A_{0}^{3}A_{0}^{3}}$, we obtain dominant contributions of order $\mathcal{O}(q^{2})$ and $\mathcal{O}(\mu^{0})$ (neglecting $\mathcal{O}(\omega^{2}\,,q^{2}\,)$)

[TABLE]

This diffusion pole is reminiscent of the result of the Abelian result of Policastro:2002se given in appendix (C). As discussed in section IV.1, we consider gauge-invariant combinations $G_{0\widetilde{0}}+G_{\widetilde{0}0}+G_{A_{0}^{3}A_{0}^{3}}$. In order to inspect the non-Abelian effects of order $\mathcal{O}(\mu)$ showing up in the first two correlators in this sum, we again drop the third flavor direction which is of order $\mathcal{O}(\mu^{0})$.

Motivated by the diffusion pole behavior of our correlators in flavor-directions $a=1,\,2$ corresponding to the combinations $X,\,\widetilde{X}$ (see (75)), we wish to regain the structure of the diffusion equation given in (10), which in our coordinates ($k\,=\,(\omega\,,0\,,0\,,q\,)$) reads

[TABLE]

Our goal is to rewrite (75) such that a term $\mathcal{O}(\omega)$ and one term in order $\mathcal{O}(q^{2})$ appears. Furthermore there should be a relative factor of $-i$ between these two terms. The obvious manipulation to meet these requirements is to multiply (75) by $\sqrt{\omega}$ in order to get

[TABLE]

Comparing the gravity result (78) with the hydrodynamic equation (77), we obtain the frequency-dependent diffusion coefficient

[TABLE]

Our argument is thus summarized as follows: Given the isospin chemical potential as in (1), (17), $J_{0}$ from (10) is the isospin charge density in (77). According to (77), the coefficient (79) describes the diffusive response of the quark-gluon plasma to a gradient in the isospin charge distribution. For this reason we interprete $D(\omega)$ 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 propagation 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 fluctuations with time.

The square root dependence of our diffusion coefficient is valid for small frequencies. As long as $\omega/T<1/4$, the square root is larger than its argument and at $\omega/T=1/4$, the difference to a linear dependence on frequency is maximal. Therefore in the regime of small frequencies $\omega/T<1/4$, 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 temperature field theory, consisting of an isospin chemical potential $\mu$ 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 $\mu$ 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 beyond 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 (41) 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 $\omega\rightarrow 0$, since the diffusion coefficient 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 Kovtun:2006pf and in Teaney:2006nc . We are going to study the application of this method to the model presented here in the future.

Acknowledgements.

We are grateful to R. Apreda, G. Policastro, C. Sieg, A. Starinets and L. Yaffe for useful discussions 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 (15). 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 $\bm{q}$ 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 $\vec{q}$.

The Green functions $G=\langle J\hat{J}\rangle$ considered give correlations between currents $J$ and $\hat{J}$. These currents couple to fields $A$ and $\hat{A}$ respectively. In our notation we use symbols such as $G_{A^{a}_{k}A^{b}_{l}}$ to denote correlators of currents coupling to fields $A^{a}_{k}$ and $A^{b}_{l}$, with flavor indices $a,b$ and Lorentz indices $k,l=0,1,2,3$. For the gauge field combinations $X_{k}$ and $\widetilde{X}_{l}$ given in (33), we obtain Green functions $G_{k\widetilde{l}}$ 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 $\mathfrak{w}$ and $\mathfrak{q}^{2}$. The functions themselves are then composed as in (43).

The solutions for the components with flavor index $a=3$ where obtained in Policastro:2002se .

B.1 Solutions for $X_{\alpha}$, $\widetilde{X}_{\alpha}$ and $A^{3}_{\alpha}$

The function $X_{\alpha}(u)$ solves (35) with the upper sign and is constructed as in (43) from the following component functions,

[TABLE]

The solutions of the equations of motion (35) with lower sign for the functions $\tilde{X}_{\alpha}(u)$ are given by

[TABLE]

so that $\lim_{u\to 0}\tilde{X}(u,k)=\tilde{X}^{\text{bdy}}$.

The solution for $A^{3}_{\alpha}$ solves (28) up to order $\mathfrak{w}$ and $\mathfrak{q}^{2}$ with boundary value $\left(A^{3}_{\alpha}\right)^{\text{bdy}}$. It is

[TABLE]

B.2 Solutions for $X_{0}^{\prime}$, $\widetilde{X}_{0}^{\prime}$ and ${A^{3}_{0}}^{\prime}$

Here we state the solutions to (32). 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 $A^{1,2}_{0}$ to $X_{0}$ and $\widetilde{X}_{0}$. The choice $a=3$ gives a single equation.

The function $X_{0}^{\prime}$ is solution to (36) with upper sign. We specify the component functions as

[TABLE]

To get the function $\widetilde{X}_{0}^{\prime}$, we solve (36) with the lower sign and obtain

[TABLE]

The solution for (32) with $a=3$ is the function ${A^{3}_{0}}^{\prime}$, given by

[TABLE]

B.3 Solutions for $X_{3}^{\prime}$, $\widetilde{X}_{3}^{\prime}$ and ${A^{3}_{3}}^{\prime}$

We give the derivatives of $X_{3}$ and $\widetilde{X}_{3}$ as

[TABLE]

The solution for ${A^{3}_{3}}^{\prime}$ is

[TABLE]

B.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 $X_{\alpha}=(1-u)^{\beta}F(u)$. Here we compare the numerical result for $F(u)$ obtained from the ansatz (38) with (39) in (35) with the analytically obtained approximation given above in (80) to (85).

Appendix C Abelian Correlators

For convenient reference we quote here the correlation functions of the Abelian super-Maxwell theory found in Policastro:2002se . 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 $N_{c}/4$. Note also that here all $N_{f}$ flavors contribute equally. In our notation

[TABLE]

where $D=1/(2\pi T)$ .

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 $X_{3}$ and $\widetilde{X}_{3}$ (as defined in (33)).

[TABLE]

and

[TABLE]

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) in section II).