On Conductivities of Magnetic Quark-Gluon Plasma at Strong Coupling

In the presence of strong magnetic field, the quark gluon plasma is magnetized, leading to anisotropic transport coefficients. In this work, we focus on electric conductivity. We generalize longitudinal and transverse conductivities to finite frequencies. For transverse conductivity, a separation of contribution from fluid velocity is needed. We study the dependence of the conductivities on magnetic field and frequency using holographic magnetic brane model. The longitudinal conductivity scales roughly linearly in magnetic field while the transverse conductivity is rather insensitive to magnetic field. Furthermore, we find the conductivities can be significantly enhanced at large frequency. This can possibly extend lifetime of magnetic field, which is a key component of chiral magnetic effect.


I. INTRODUCTION
Relativistic hydrodynamics modeling of bulk evolution of quark-gluon plasma produced in heavy ion collisions has been remarkably successful over the past decades [1,2].
The model is simple and universal in the sense that it reduces a many-body problem to only a few macroscopic variables: fluid velocity, energy density and baryon number density etc. In order to complete hydrodynamics modeling, microscopic inputs are needed: these include equation of state and transport coefficients such as viscosities and conductivities etc.
Recently, it has been realized that a strong magnetic field can be produced in offcentral heavy ion collisions. The magnetic field plays an important role in the description of anomalous transport phenomena, in particular the chiral magnetic effect [3][4][5]. There have been growing efforts in applying hydrodynamics to study chiral magnetic effect [6][7][8][9]. These studies assume a weak magnetic field such that the system remains isotropic. For strong magnetic field, both pressure and transports become anisotropic. A systematic modification of the current hydrodynamics framework to the so called magnetohydrodynamics (MHD) is needed. This has been carried out by Hernandez and Kovtun (HK) [10], see also dual formulation [11] and early works [12][13][14][15]. The MHD including effect of axial anomaly is constructed in [16]. Evaluation of anisotropic transport coefficients is needed for application of MHD. Viscosities in magnetic quark gluon plasma have been studied in [17,18]. Another interesting transport coefficient is the electric conductivity. In the presence of magnetic field, it splits into longitudinal and transverse conductivities. The longitudinal conductivity has been calculated at weak coupling by lowest Landau approximation in [19,20] and beyond lowest Landau approximation in [21]. At strong coupling, the longitudinal conductivity has been calculated in [22]. The conductivity in 2 + 1 dimensional plasma has been obtained in [23]. In the absence of magnetic field, the isotropic conductivity has also been calculated by lattice simulation [24,25].
The situation of transverse conductivity is quite different. The corresponding Kubo formula for longitudinal and transverse conductivities are derived by HK [10], assuming B field in the y-direction and charge neutrality of plasma [43]: where w 0 = + p is the enthalpy density in equilibrium, σ and σ ⊥ are longitudinal and transverse conductivities respectively. The appearance of σ ⊥ in the denominator may seem odd. Essentially this is due to the interplay between transverse current and fluid velocity.
It holds in the regime ω T and ω B/T . The former is the hydrodynamic limit while the latter requires the B field to be not too small. B/T can be regarded as inverse of time scale for cyclotron motion of plasma particles.
The aim of this work is to calculate both longitudinal and transverse conductivities in holographic magnetic quark-gluon plasma model. The paper is organized as follows: In Section II, we give an intuitive derivation of the Kubo formula for both transverse and longitudinal conductivities. The derivation naturally generalize conductivities in the hydrodynamic limit to finite frequency regime. Section III is devoted to the calculation of conductivities in holographic magnetic brane model. We discuss our results and phenomenological implications in Section IV.

II. KUBO FORMULAS
We can reproduce the transverse Kubo formula in the following intuitive way: let us turn on a weak and slow varying homogeneous electric field E along x 1 -direction. The positive and negative charged particles will move in ±x 2 direction. By the Lorentz force in the B field, both positive and negative particles gain momentum along x 2 . This induces a net flow along x 2 . No net flow is generated along x 1 due to the neutrality of plasma. The net effect of the flow along x 2 will cancel the current along x 1 , again due to Lorentz force. This is the reason why transverse conductivity enters current only at higher order in ω.
We can formulate it more rigorously in the homogeneous limit Here the current j i consists of conducting current and polarization current, with E i + ijk v j B k being effective field experienced by plasma particles and P i being electric polarization vector. The energy flow T 0i contains fluid comoving contribution and medium contribution to Poynting vector, with M i being magnetization vector. The third equation is momentum non-conservation equation due to Lorentz force. When medium in equilibrium has magnetization only, electric polarization is only induced by motion of fluid [26,27]: as is required by Lorentz symmetry. To compare with HK, we note M µ = 2p ,B 2 B µ and (2) reproduces the constitutive equations of HK [10]. (2) is slightly more general in the sense that v i and σ ⊥ can be ω-dependent, thus (2) in fact defines transverse conductivity at finite frequency. Note that the use of fluid velocity at finite frequency is in the same spirit of resummed hydrodynamics [28][29][30][31].
We can then solve for v i : This gives the following current Note that E = ∂ t A = −iω A. We readily obtain the correlator for transverse current: Expanding (6) in ω, we easily obtain: We immediately see (7) gives Kubo formula for σ ⊥ in (1). However it is singular as B → 0 due to non-commutativity of hydrodynamic limit and isotropic limit. (6) can be safely used in both limits. We solve the ω dependent conductivity as The case of longitudinal conductivity is trivial because Lorentz force is not relevant.
The corresponding Kubo formula is given by

A. Magnetic brane background
We use magnetic brane background [32] for the computation of conductivities. The background is a solution to five-dimensional Einstein-Maxwell theory with a negative cosmological constant: Here L is the AdS radius set to unity below, F = dA is the Maxwell field strength, and the second term in the action corresponds to Chern-Simons term. The Chern-Simons term corresponds to axial anomaly. We will not consider the effect of anomaly on conductivities, thus we turn off the Chern-Simons term. The resulting equations of motion (EOM) read The magnetic solution is given by [32] The warping factor U (r) contains a zero at r = r H , which is the location of horizon. This corresponds to a temperature of the plasma T H = U (r H ) 4π . The solution of the background can only be obtained numerically. It is convenient to compactify the radial coordinate by defining r = r H u −1/2 , which puts the horizon at u = 1. The background in terms of u coordinate becomes The EOM read with the derivatives taken with respect to u. The numerical solution is to be obtained by integrating the following horizon solution to the boundary of AdS: We can put v 0 = w 0 = 0 by rescaling of x and y coordinates. We also put u 1 = −2, which sets the unit by fixing the temperature to T = 1 4π . The magnetic field after the rescaling is denoted as b, which is to replace B in (14). The higher order coefficients in (15) can be determined recursively from EOM as: For a particular b we can numerically solve the metric functions. Near boundary, the solution behaves like Thus we need the following rescaling t →t = t, y →ŷ = w(b)y to bring the background to the standard AdS asymptotics. After the rescaling, the full background reads where Below we use tilded symbols for metric functions with standard AdS asymptotics.

B. Transverse and longitudinal conductivities
To calculate transverse conductivity, we consider the following linear perturbation about the background It is convenient to use metric perturbation with mixed indices h After substituting into the equations of motion we obtain the following ordinary differential equations Near the horizon, the solution behave as a The incoming exponent is given by α = −iω, β = α + 1. We will look for solution of the form Near the boundary, the incoming wave solution behaves like In fact this set of incoming solution is determined by only one parameter, which does not match the number of unknown fields. In fact, we can find another constant solution This is a pure gauge solution of the following type Fixing the background gauge field as A M = −Bx 2 δ 1 M , we find the constant solution (24) is Note that the gauge choice of the background is necessary to ensure the vanishing of all other perturbations. Thus the general solution is a linear combination of these two solutions. a x 1 (u) = a inc x 1 (u) + a con x 1 (u), In order to calculate the retarded correlator G R x 1 x 1 , we need to eliminate the contribution to current from response to metric perturbation, thus we should turn off boundary value of metric perturbation. It amounts to setting lim u→0 G tx 2 (u) = 0. This fixes C 2 to Therefore the retarded correlator G R x 1 x 1 reads To calculate the longitudinal conductivity (in y direction), we only have to consider the following perturbation The perturbed field a y (u) satisfies the following EOM a y (u) + a y (u) Ũ (u) Near the horizon, the incoming wave solution behaves as a y ∼ (1 − u) −iω . We look for the solution of the form Near the boundary, the solution behaves like Therefore the retarded correlator G yy reads We will study (28) and (33) in different regimes in the following.

C. Hydrodynamic regime
In hydrodynamic regime we can solve the equation perturbatively in ω, where A, G, D are defined in (31) and (22). Let us study transverse equations first. The coupled EOM of A and G read We first expand the fields h tx 2 and a x 1 and background solutionŨ (u),Ṽ (u),W (u) near horizon using (15). And then we numerically solve (35) by giving an initial condition on the horizon that A 0 (1) = 1. This fixes normalization of the solution but does not affect result of correlators. Note that A 0 (u) has a constant solution A 0 (u) = 1 with this specific initial condition. We further require all higher order functions vanish on the horizon.
The perturbative solution give the following perturbative expansion of transverse retarded Here the functions A (1) i are defined through the following boundary (36) is the expected form of transverse correlator in hydrodynamic regime. The imaginary part starts from ω 3 , whose coefficient can be used to determine transverse conductivity with the corresponding Kubo formula in (1).
The longitudinal equation can be studied similarly. The EOM in terms of D 0 and D 1 are given by Again we numerically solve (38) by giving the initial condition that D(1) = 1. For D 0 (u), we find that D 0 (1) = D 0 (1) = 0, thus it admits a constant solution D 0 (u) = 1. Then we numerically solve D 1 (u). The perturbative solution gives the following perturbative expansion of longitudinal retarded correlator where D (1) 1 is defined through boundary expansion of D 1 Our boundary condition fixes D (0) 1 = 0. We can thus simply identify D 1 with longitudinal conductivity in the hydrodynamic regime based on (1). We show in Fig. 1 the dependence of σ and σ ⊥ on B. We observe nearly linear dependence of σ on B. This is consistent with the picture that all the charge carriers are from the lowest Landau level in large B limit, with the density of charge carriers proportional to B. On the other hand, σ ⊥ tends to a constant at large B. Although we cannot take the limit B → 0 in hydrodynamic regime, we do find at small B, σ and σ ⊥ are numerically consistent with each other. The two limits are also obtained in [33], although in that case, the mixing of perturbation in transverse case was not taken into account. Interestingly, the approach of [33] turns out to give the correct answer in hydrodynamic regime. We show this by membrane paradigm in appendix.

D. Conductivities at arbitrary frequency
Beyond hydrodynamic regime, we should use (8) and (9) as definitions of conductivities at finite frequency. Note that beyond hydrodynamic regime, the conductivities are in general complex. We solve the transverse and longitudinal EOM numerically to obtain complex conductivities. We plot |σ| and Arg[σ] as a function of ω in Fig. 2 and Fig. 3.
|σ| characterizes the magnitude of current induced in magnetic plasma by external electric field. Fig. 2 shows both |σ | and |σ ⊥ | can be significantly larger than their hydrodynamic counterparts at large ω. The large ω limit of |σ | is rather insensitive to B, while for |σ ⊥ |, its large ω limit is non-monotonic in B. We also plot the B-dependence of |σ ⊥ | at large values of ω in Fig. 4. On the other hand, Arg[σ] characterizes the phase difference of current and external electric field. As ω → 0, the conductivities are real meaning that the current is in phase with applied electric field. The large ω limit of Arg In fact, the large ω limit of σ can be obtained analytically by noting that B ω 2 becomes irrelevant. Ignoring the magnetic field, we can use the known result for retarded current-current correlator (adapted to our choice of unit) [34]: which gives us the following asymptotics of conductivity lim ω→∞ σ (ω) = 2i (−iπ − ln(2) + 2 ln(ω)) ω.
|σ | is linear in ω up to logarithmic correction. Arg[σ ] approaches π/2 slowly from below. This is consistent with our numerical results in Fig. 2 and Fig. 3.
The dependence of |σ | with ω can be understood qualitatively as follows: kinetic theory estimate of conductivity is σ ∼ nτ m , with n, m and τ being charge carrier density,

IV. DISCUSSION
We study longitudinal and transverse conductivities at finite magnetic field B and frequency ω. While the former is a straight forward generalization the static (hydrodynamic) limit, the latter involves a careful subtraction of fluid velocity contribution. We arrive at a Kubo formula that is applicable at finite frequencies. It reduces to Kubo formula in the hydrodynamic regime [10].
Using holographic background dual to quark gluon plasma with external magnetic field, we study the B and ω dependence of conductivities. In the hydrodynamic regime, we find the longitudinal conductivity σ scales linearly with B at large B, consistent with lowest Landau level picture. The transverse conductivity is not sensitive to B field in a wide region.
The ω dependence of conductivities is more interesting. We find both conductivities scales nearly linearly in ω at large ω. This could be understood qualitatively as the relaxation time increases with frequency of electric field. The B dependence of the large ω limits of σ and σ ⊥ differ: The former is nearly independent of B, while the latter shows a non-monotonic dependence on B.
The obtained values of conductivities might be relevant for the physics of chiral magnetic effect [3]. The effect of conductivity on lifetime of magnetic field is studied in [35].
It is found that only very large conductivities can extend the lifetime of magnetic field.
In heavy ion collisions experiment, the produced magnetic field [36,37] can be estimated as B/T 2 m 2 π /T 2 0.26, with T = 350MeV. The magnetic field itself might not have significant effect on conductivity from Fig. 1. However, the rapid decaying magnetic field induces rapid changing electric field, which calls for use of conductivities at finite frequency.
At this frequency, the conductivities are enhanced by a factor of 3 from Fig. 2. A lifetime of magnetic field τ 0.2f m would lead to a factor of 10 for the conductivity! A re-evaluation of the effect based on finite frequency conductivities is needed. We leave it for future analysis. In the appendix, we obtain the conductivities using membrane paradigm [38]. While conventional membrane paradigm works for σ , it fails for σ ⊥ due to mixing of current and energy flow. The resolution is that σ ⊥ can also be obtained from correlator of heat current.
The corresponding Kubo formula is given by [10] 1 ω For convenience we revert to the original r coordinate (12).
To study current and energy flow in response to external electric field and metric perturbation, we turn on the following perturbations [39][40][41].
Using (A1) and (1), we obtain σ and σ ⊥ in terms of horizon quantities, We have confirmed that (A8) agrees with our numerical results in the hydrodynamic regime for arbitrary B.