Entropy production far from equilibrium in a chiral charged plasma in the presence of external electromagnetic fields

We report on the time evolution of a charged strongly coupled $N=4$ SYM plasma with an axial anomaly subjected to strong electromagnetic fields. The evolution of this plasma corresponds to a fully backreacted asymptotically AdS$_5$ solution to the Einstein-Maxwell-Chern-Simons theory. We explore the evolution of the axial current and production of axial charges. As an application we show that after a sufficiently long time both the entropy and the holographic entanglement entropy of a strip-like topology ( both parallel to and transverse to the flow of axial current) grow linearly in time.


Introduction
It is expected that extremely large magnetic fields are generated during the collisions of heavy ions which produce a QGP [1,2] 1 . At high energy chiral symmetry is restored in the QCD Lagrangian leading to the presence of a chiral anomaly. This has led to the proposal of possible anomalous effects which might be seen on an event by event basis during the generation of a QGP, such as the chiral magnetic effect (CME) [3,4]. The CME is due to the asymmetry between the number of particles and antiparticles with right handed and left handed helicity. And it can be shown that when one applies a magnetic field to such a system an electromagnetic current is generated in the direction of the magnetic field [5]. An observable two point correlation sensitive to the CME effect was first proposed in [3,6] 2 . By studying the azimuthally asymmetric distribution of charged hadron production both the STAR collaboration at RHIC and the ALICE collaboration at the LHC have observed the predicted fluctuation [9][10][11]. However the measurement may be obscured by the background with the geometry of the collision responsible for the observation. To correct for this, efforts are currently under way at RHIC with a dedicated isobar (nuclei with the same mass numbers and size but different electric charge) run [12,13]. In condensed matter physics the effect has already been found to exist in 1 Standard lore states that the magnetic fields generated during collisions will not last long enough to produce an observable effect. 2 Another observable was recently proposed in [7] (see also [8]).
Dirac semi-metals [14]. As this still remains inconclusive in heavy ion collisions it therefore motivates further study of thermalizing strongly coupled systems with a chiral anomaly.
A powerful method for obtaining information about strongly coupled systems is via holography. There is a vast amount of literature 3 dedicated to thermalization [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31], often utilizing Vaidya spacetime, and to the study of dynamical holographic systems as analogues for heavy ion collisions [32][33][34][35][36][37][38][39][40][41]. These studies simulate the evolution of SYM plasmas via numerical evolution of bulk Einstein equations and are meant to mimic the conditions of heavy ion collisions. The majority of these studies have been devoted to the collision of gravitational shock waves. However they have not included the time dependent magnetic fields which we know to be present during heavy ion collisions [1,2]. This is in part due to the difficulty of including even static magnetic fields in equilibrium.
Early works with SYM plasma subjected to external magnetic fields were concerned with their thermodynamic properties [42][43][44] with the first example of perturbative studies in [45]. Recently the importance of including magnetic fields in the study of SYM plasma for application to heavy ion collisions was demonstrated by showing the ratio of the transverse to longitudinal pressure (P T /P L ) as a function of the B/T 2 agree between QCD and N = 4 SYM plasma [46]. There have been two studies conducted in which the dynamical evolution of the Einstein equations include a fully back reacted magnetic field [47,48]. However both of these studies do not include a Chern-Simons term in the dual gravitational theory. This term, when included, provides for us an axial anomaly in the dual field theory. Other authors have utilized Vaidya spacetimes to include this term in a linearized analysis [49,50]. In this work we make use of the techniques developed by [32-41, 47, 48, 51] to extend the analysis of [47,48] to include the axial anomaly. This provides for us the simplest such setup in which to study the time-dependent relaxation of a far from equilibrium plasma with a chiral anomaly subjected to electromagnetic fields. It should be stressed that the electromagnetic fields created during a heavy ion collision are dynamically generated (i. e. a local gauge field). Our setup includes an external electric and magnetic field aligned along the x 3 -direction (i. e. a global gauge field). In the presence of a chiral anomaly the aligned electric and magnetic field stimulates the production of axial charges. The increasing axial charge density contributes to the current density along the x 3 -direction in which the chiral charges are accelerated by the electric field leading to Joule heating of the plasma.
As an application of our numerical model we study the growth of entropy during the evolution. Entropy has been repeatedly shown to be a meaningful quantity to compare to experiments (some examples [52][53][54]). An interesting aspect of the CME is that it produces a dissipation-less [55] current and hence does not contribute to thermal entropy production [1]. Despite this lack of thermal or classical entropy production of the current associated with the CME we may expect there is a notion of entropy production due to the anomalous production of axial charges. Our results demonstrate that the production and subsequent acceleration of axial charges by the electric field produces a linear growth in the entropy. In addition we also compute the entanglement entropy in the dual field theory via methods used in [48,56,57]. We also find linear growth of the reduced entropy of strip like subsystems extending in directions both transverse to and parallel to the axial current flow.
Our work is divided as follows. We begin in section 2.1 by introducing our the holographic description of our system. We then discuss the asymptotic analysis and introduce the dual energy-momentum tensor and current for our system. In section 3 we briefly discuss the numerical techniques used to construct solutions to the Einstein equations. In section 4 we display for the first time the energy-momentum tensor of a strongly coupled far from equilibrium charged plasma with chiral anomaly subjected to external electromagnetic fields. We also display for the first time the dynamical evolution of the axial current and axial charge density. Finally we investigate a simple application of our work by investigating the entropy production during the evolution. We compare the evolution of the thermal and entanglement entropy during the evolution with and without the production of axial charges.

Holographic Description
We employ the characteristic formulation of general relativity first formulated in [58,59] and implemented in a myriad of subsequent publications for the study of dynamical systems in asymptotically anti-de-Sitter spacetime (some examples [32-41, 48, 51]). The action is, with G 5 = π 2 L 3 /N 2 c the five dimensional Newton's constant. The cosmoslogical constant Λ is related to the AdS radius L via Λ = −6/L 2 and the Chern-Simons coupling k will be written in a dimensionless form as k = 2γπG 5 . We will soon set 4 , for the remainder of the paper unless otherwise stated. The equations of motion which result from variation of the action are, To pick the ansatz we can consider symmetries of our system. We wish to have aligned electric ( E) and magnetic ( B) fields in order to see the desired effect of the production of chiral charges. We choose to align both these fields along the x 3 -direction. This breaks the O(3)-symmetry to an O(2) in the x 1 − x 2 plane. Additionally we expect the presence of a heat current along x 3 , as a result we break the remaining parity symmetry in x 3 requiring a component of the metric g t3 = g 3t = 0. With these symmetry considerations in mind the simplest ansatz for our desired setup is as follows, with the one form ω = (−A(v, r)dv + F (v, r)dx 3 + 2dr). Our gauge field ansatz in radial gauge is of the form, with a constant magnetic field B. The Maxwell equations reduce to three equations, two of these equations can be used to solve for the bulk electric field E, which appears in the equations of motion for the metric (the prime denotes derivatives in the radial direction P (v, r) = ∂ r P (v, r)).
where here = fṖ is a source term which depends on the included metric components and their radial or dotted derivatives. Utilizing the characteristic derivative and including the final Maxwell equation into the characteristic Einstein equations we find the following form, We choose to not write the full equations here due to their length. Inspecting these equations one finds that eq. (2.9a) is no longer a linear ODE, the first equation of the nested list structure has developed a non-linearity by the inclusion of the Chern-Simons term and now requires two pieces of initial data, the anisotropy profile at the initial time v 0 , B(v 0 , r), and the bulk electric field profile P (v 0 , r). In addition we find the equations forṖ andḂ fail to nest and must be solved simultaneously.
We employ Ryu-Takayanagi's conjecture for the entanglement entropy (EE) [60,61]. The entanglement entropy S A for a subsystem A of a CFT in R 3,1 is defined as, where A(γ A ) is the "area" of a 3 dimensional static minimal surface in AdS 5 with boundary ∂A ⊂ R 3,1 . The area functional of the codimension 2 surface γ A in AdS 5 is, where χ are the embedding coordinates of the surface andg ab = g µν ∂χ µ ∂σ a ∂χ ν ∂σ b is the induced metric on this surface. Following the work of [56] we can specialize eq. (2.10) to the metric given in eq. (2.5) for surfaces bounded by strips in the field theory aligned along the transverse x 1 (x 2 ) and longitudinal (or parallel) x 3 directions, Where V ⊥ = dx 2 dx 3 and V = dx 1 dx 2 are infinite volume contributions with which we measure with respect to. The expressions are essentially identical to those used in the case of colliding gravitational shock waves [57] where we have also suppressed the dependence of the metric components on the time and radial direction and represented dY (σ)/dσ =Ẏ . The areas we compute are divergent quantities which require regularization. To regulate our results for the time evolution we subtract the value for the entanglement entropy of empty AdS spacetime, as proven to be a valid regularization procedure in [57].

Asymptotic Analysis
A near boundary solution is needed to extract field theory information. We seek solutions which asymptotically approach AdS 5 as r → ∞. This is the case if g µν (x, r) → diag(−1, 1, 1, 1) as r → ∞. Schematically this expansion has the following form provided we are in the appropriate coordinate system [47], Expanding the metric components in a power series around r → ∞ one can simultaneously solve the Einstein and Maxwell equations order by order. The near boundary expansion yields the time dependence of the asymptotic coefficients for f 4 and a 4 , Interestingly the near boundary solution entirely determines the behavior of the asymptotic coefficient f 4 dual to the heat current, It's here that we begin to see that a system with only an axial gauge field is partially pathological. There is no dampening in this system. As can be seen from the solution eq. (2.17) the heat current will grow without bound. This is due to the fact that we are constantly pumping axial charges into the system. One can calculate the total charge in the system by looking at the near boundary expansion of the time component of the gauge field A 0 = φ. Typically the charge density is encoded in the coefficient A Clearly eq. (2.18) indeed shows that with a nonzero value for E (the source for P is p 0 +Et) leads to a total charge that grows without bound (the definition of the charge density is a little more subtle in our current choice of Eddington-Finkelstien coordinates, the total charge density with our choice in eq. (2.2) is displayed in eq. (2.26)). Although f 4 can be calculated analytically the coefficient dual to the energy density a 4 cannot. The second term in eq. (2.16) shows two contributions the first from the change in the location of the apparent horizon changing the effective energy the system has. This contribution can be removed by working in a fixed frame ξ = 0. The second is a Joule heating term. One can show that the non-equilibrium contribution to the dual current is J 3 ∼ p 2 (t) (see eq. (2.26)) and hence a 4 ∼ J 3 E 3 .
We follow the same conventions set in [47] for the procedure of holographic renormalization (see [43,47,62,63]). The field theory energy-momentum tensor can be computed by including the proper counter terms to the action and utilizing the near boundary expansion eq. (A.1) to eq. (A.5). The correct (and simplest to utilize) expression for the holographic relation between the bulk gravity and boundary field theory can be found in [47]. This procedure yields the following boundary stress-energy tensor 5 , The renomalization point dependence of the energy-momentum tensor was carefully discussed in [47].
Displaying the results of our calculation requires a choice of µr. A rather un-physical choice is µr = 1/L which we adopt for this work. A more detailed discussion of this choice in this system will be carried out in future work. Please also see [64] for further discussion of these points in the context of generalized global symmetries in holography.
Computing the trace of this energy-momentum tensor gives the expected conformal anomaly, Following [43] we can also extract the following global current using, Given the choice made in eq. (2.2) the one point function of the axial current density is given by 6 , Applying equation eq. (2.25) we find the following form of the dual current one point function, Clearly by inspection of eq. (2.19) to eq. (2.26) the inclusion of the Chern-Simons term has led to new physics in the field theory. As expected the external electric field E contributes to the total energy density of the field theory. In addition due to the anomalous current flow there is now a time dependent heat current f 4 (t). The system is anisotropic with a transverse and longitudinal pressure. The external electric field in the x 3 -direction provides a contribution to the pressure in the x 1 − x 2 plane and x 3 -direction. The source of this pressure contribution can be attributed again to the presence of the Chern-Simons coupling.
Both the x 1 and x 2 component of the current eq. (2.26) are non-zero. These components indicate there is a azimuthally symmetric inflow of axial charges. The x 3 component of the current also contains both an equilibrium ( 1 3 Bγµ(t) see [65]) and a non-equilibrium (8p 2 (t)) contribution.

Numerical Techniques
The numerical solution to the characteristic Einstein equations have been carefully described in many works [32,33,36,37,40,48,51,[66][67][68][69] etc. for particularly nice treatments see [40,67]. In addition the techniques used to compute the entanglement entropy have also been described in detail in [48,56,70] with a particularly nice treatment in [70]. With this in mind we will not describe in depth the methods of construction for these solutions. We will only give a brief statement of the methods used.
Each of our radial differential equations is solved by means of a Chebyshev spectral method (for an introduction see [71]). In order to tame CFL instabilities we employed domain decomposition in the radial grid, typically using 6 sub-domains each with N = 24 grid points (see [40] for a quick explanation). The number of needed grid points is larger then that found in [40] for instance. This is due to the presence of the logarithmic terms which appear due to the electric and magnetic field. These terms ruin the typical "exponential" convergence of a spectral scheme.
In order to step forward in time we employed a standard 4th order Runga-Kutta scheme with a time step of the order dt ≈ 1 4N 2 . Our system contains a thermalizing black brane so we use the residual diffeomorphism symmetry to fix the location of the apparent horizon during the evolution of our system. In our previous work [48] we followed a method provided in [68], calculating an explicit differential equation for ξ. However in this work we have changed this to something similar to what is done in [40], fixing the behavior of the metric function A on the apparent horizon and extracting ∂ t ξ from the near boundary behavior of A via ξ (t) = −1 2 A s (t, z)| z=0 . We will outline our solution algorithm since it differs slightly from previous works. In order to construct solutions we do the following. 3. Fix P (v, r) on the initial time step and solve the nonlinear system for S(v, r) using Frechet differentiation and Newton iteration. The linear solution S Linear serves as an initial guess.

Extract time derivatives
8. On the next time step use the previous solution to eq. (2.9a) as an initial guess for newton iteration of the non-linear system.
9. Repeat steps 4-8 for the duration of the evolution.
In order to begin our time evolution on the initial time step we must repeatedly follow steps 1-4 in order to fix the location of apparent horizon to a numerically convenient location. In our case we fix this location to be at z h = 1.
We utilize a relaxation method to compute solutions to the geodesic equation as done in [48,56] (a basic introduction can be found in [72]). We typically use 350 grid points to approximate the solutions. The method computes the geodesics on a cutoff surface located at z U V = .075. The method takes empty conformal AdS geodesics as an initial guess on the first time step. Once a solution is found it serves as the guess on the next time step.

Isotropization
In figure 1 we display the non-zero components of the energy-momentum tensor along side the non-zero components of the axial current. For this evolution we chose to use the following form of the subtracted functions B s and P s at the initial time step t = 0 7 , displayed here in the ξ = 0 frame with β = 1/10. In figure 1 we fix B = 1/2, γ = 1/2, ρ = 0.429, p 0 = 1/2 and E = 2/5. We begin the evolution with a 4 (v = 0) = −5/4 and f 4 (v = 0) = 5/100. In figure 1 one can see the energy T 00 of the solution continues to grow as time goes on. This is due as stated in section 3 to the acceleration of axial charges in the electric field. One can see the continuous growth of the charge density due to the anomalous production of axial charges. As the total number of charges grows so does the x 3 component of the dual current J 3 = 8p 2 (t) − Bγµ(t)/3 and the heat current at the boundary T 03 = 4f 4 (t). The transverse ( T 11 + T 22 )/2 and longitudinal pressures T 33 oscillate as they undergo the isotropization process. However the continuous growth of the energy can be seen overtaking the isotropization process. It is interesting to note that while the energy density is increasing the transverse pressure at later times stays roughly constant. It is the longitudinal pressure which grows in order to satisfy the trace condition on the energy-momentum tensor. This makes sense considering the continued growth of the x 3 component of the current density. Our work can be compared to previous work [47,48] which demonstrates that the transverse and longitudinal pressures relax to a final anisotropic state due to the presence of the magnetic field.
In figure 2 we display the evolution of the spatial components J i (t) of the dual current. We display this vector field at three different times during the evolution of the plasma. The left image of figure 2 is taken when the system begins its evolution at t = 0. We can see that we have an azimuthally symmetric flow of axial charge directed approximately towards the x 3 -axis. We can see the beginning of a flow of this current in the x 3 -direction with the vectors all pointing slightly down along the x 3 . The middle image of figure 2 displays the current J i (t) at approximately half way through the evolution with t = 1.87445. In this image we can continue to see the current flowing in towards the x 3 -axis. However we also see a more significant change in the orientation of the vector field. At this point in the evolution it is clear the flow is directed along the x 3 -axis. In the right image of figure 2 we are near the end of the simulation window at t = 3.74976. At this point in the evolution the flow within the spatial window displayed is almost entirely directed in along the x 3 -axis. It is interesting to note that if we choose our window to include a larger spatial extent we would see an image similar to the left image of figure 2. Within a spatial range of (x, y) ∈ (−200, 200) × (−200, 200) the current is directed almost entirely along the x 3 -axis at the late times in our evolution. However outside this range the vectors asymptote to an azimuthally symmetric radially inflowing current. The same three time slices are displayed in figure 3 plotted in the x 2 − x 3 at x 1 = 0.  The current is initially directed radially inward toward the x 3axis. Mid: As the total charge increases and is accelerated by the electric field the current flow is closer to being directed entirely along x 3 . Right: Near the end of the simulated window the total charge has increased significantly, near the axis the contribution of the current in the transverse plane is dwarfed by the contribution in the x 3 -direction along the aligned electric and magnetic fields. The current is initially directed radially inward toward the x 3 -axis. Mid: As the total charge increases and is accelerated by the electric field the current flow is closer to being directed entirely along x 3 . Right: Near the end of the simulated window the total charge has increased significantly, near the axis the contribution of the current in the transverse plane is dwarfed by the contribution in the x 3 -direction along the aligned electric and magnetic fields.

Application: Entropy Production
As an application we consider the entropy produced during the process of isotropization.
A notion of the out of equilibrium thermal entropy is given by the area of the apparent horizon. This is not unique, there are many notions of entropy for spacetimes undergoing dynamical processes along with many area increase laws [61,[73][74][75]. Keeping in mind eq. (2.2) the entropy density can be calculated via the spatial scale factor [68], although it should be stated that only near equilibrium can we truly call this quantity the entropy density in the dual theory. In order to put in context the generation of entropy during the production of axial charges we choose to compare our data to the same setup only with the Chern-Simons coupling γ = 0. In figure 4 we compare the results of evolving our system with and without the Chern-Simons coupling. The dashed lines represent the evolution with γ = 0. In the left image of figure 4 we can see that without the Chern-Simons coupling we have a decrease in the growth of the energy density. This is due to a decrease in the current density component J 3 . This decease in J 3 can be seen in the right image of figure 4. Accompanying this curve we also see that we have a fixed charge density throughout the evolution as without the Chern-Simons coupling there is no anomalous production of charges. The difference in the evolution of the energy density leads to changes in the evolution of the pressures while the transverse pressure is roughly the same the longitudinal pressure is decreased. In figure 5 we display both the thermal and entanglement entropy produced during isotropization of the plasma with aligned electric and magnetic fields. We can see in the left image of figure 5 the growth of entropy in the system is a monotonic function of time.
After a sufficiently long time the function approaches a linear growth. Displayed in the figure is a fit to this linear growth with a growth rate of ds/dt = 1.85245. The linear growth of the thermal entropy in the evolution of SYM plasma is not a new phenomenon it was recently seen and discussed in the context of phenomenological insights gained from holographic heavy ion collisions [41] (see their work for more information).
In the right image of figure 5 we display the growth of the entanglement entropy in both the transverse and longitudinal directions. We can see the entanglement growth is also linear in time. The linear growth of the entanglement entropy is a familiar feature of systems undergoing a global quench [76] (see also [15,16] for early examples in holography). We also display a fit to this data with the rate of growth of the entanglement entropy in the transverse and longitudinal directions, It is interesting to note that the entanglement entropy grows at a faster rate in the transverse direction rather then the longitudinal direction. The rate of growth of the entanglement entropy during the linear regime is proportional to the entanglement velocity. This linear regime is what is referred to as the post-local-equilibration regime in [77]. In this regime S(t) = Av E s eq t with A the area of the region and s eq the value of the entropy density of the equilibrium state. It is however unclear what equilibrium state we should compare to. In figure 6 we display the evolution of the entropy and the entanglement entropy with and without the production of chiral charges. We compute this in both the direction parallel and transverse to the aligned electric and magnetic field. In the left image of figure 6 we display the entropy during the production of axial charges as a solid line and without the production of axial charges as a dashed line. We can see in the left image of figure 6 that turning on the Chern-Simons coupling leads to smaller growth rate of the entropy, In the right image of figure 6 we display the entanglement entropy during the production of axial charges as solid blue lines and without the production of axial charges as solid black lines. We provide the linear fits to all of these curves in the plot to help guide the eye towards the late time linear regime. The colors of the dashed fit lines are in correspondence with colors of the solid lines. We can see in both the transverse and longitudinal direction that although the entanglement entropy is larger at earlier times when the Chern-Simons coupling is turned off, it has a smaller growth rate (see table 1), Hence we observe a increased entanglement velocity with a non-zero Chern-Simons coupling. We suspect this increase in the entanglement velocity is related to the azimuthally symmetric inflow of current re-aligning itself to a flow along the x 3 axis and the increasing 6.13563 20.9491 6.00739 21.9211 γ = 1/2 9.508 10.9098 9.2715 11.9024 Table 1: We display the parameters found by fitting the late time evolution of the entanglement entropy to a linear curve of the form S ⊥, = a ⊥, t + b ⊥, . We fit this data for both γ = 0 and γ = 1/2 while holding fixed all other parameters.
number of axial charges. We also suspect the initially larger value of the entanglement entropy without a Chern-Simons coupling is due to an already aligned current flowing along the x 3 axis.

Summary and Discussion
In this work we compute for the first time the dynamical evolution of a charged strongly coupled far from equilibrium plasma with a chiral anomaly subjected to external electromagnetic fields. We have computed this evolution as a numerical solution to the Einstein-Maxwell-Chern-Simons equations for an asymptotically anti-de-Sitter spacetime in 5 dimensions. We have (for the first time in asymptotically AdS 5 spacetimes to the author's knowledge) included the dynamical equations for the gauge field into the characteristic formulation of the Einstein equations and evolved them in time alongside the metric components (see eq. (2.9a) to eq. (2.9h)). We have computed the one point functions of the The inset displays the same information of the evolution of the entanglement entropy for a strip like topology but with embedding coordinates (v(σ), z(σ), x 1 (σ)). In both cases the entangling region had a width of = 0.4. To avoid unnecessary clutter the fit parameters displayed in table 1 are not displayed on the plot. field theory energy-momentum tensor dual to the evolving metric and axial current dual to the evolving bulk gauge field (see eq. (2.19) to eq. (2.22) and eq. (2.26)). We have displayed the axial current density in the simplest dynamical setup possible to capture the evolution of the current generated due to the axial anomaly during the isotropization of a plasma. Our setup was chosen to mimic conditions found in heavy ion collisions. We have found that aligning external electric and magnetic fields in a plasma with an axial anomaly leads to an azimthually symmetric inflow of axial charge towards the x 3 -axis. (see figure 2). This current inflowing from infinity can be considered the source of the generated axial charges which are accelerated along the direction of the electric field (see figure 1). As the system evolves the current aligns itself as a flow along the x 3 beginning along the x 3 itself and moving azimuthally outward along the cylindrical coordinate r c = x 2 + y 2 due to the electric field.
As an application of our solutions we have computed the evolution of the entropy and entanglement entropy. We have found the production and acceleration of axial charges charges by the electric field leads to the linear growth of both entropy and entanglement entropy in the late time (see figure 5). We provided linear fits of these curves which encode a notion of entanglement velocity. Turning off the Chern-Simons coupling leads to an increased rate of growth of the thermal entropy. While the thermal entropy growth increases as we turn off the Chern-Simons coupling the rate of growth of the entanglement entropy decreases (see figure 6 and table 1). We hypothesize these changes are a reflection of the difference in the time evolution of the dual current with and without an axial anomaly.
Looking to the future there are many interesting avenues we can now explore however we will mention just three possible directions: We are interested in finding a simple holographic model in which we can further study the production of axial charges and the CME in an analytic setting. Luckily there have been many works targeting the holographic Schwinger effect [78][79][80][81][82]. In [83] the authors consider extending this calculation for the inclusion of magnetic fields both perpendicular and parallel to the electric fields. Continuing their work to study the holographic entanglement entropy in a theory producing axial charges in this setting would be a logical continuation of this work.
Our discussion of the generation of axial currents naturally leads us to the topic of chiral transport. There have been many works interested in chiral transport phenomena (some excellent examples [49,50,84,85]). The author is current engaged in studying these effects far from equilibrium in anisotropic systems.
In the current work we have static electric and magnetic fields. This is not the case in heavy ion collisions where the electromagnetic fields generated during collisions are highly time dependent [2]. Recent works have tried to address the effect of time dependent electromagnetic fields on heavy ion collisions [86][87][88][89][90]. It would be very interesting to extend our current work to include time dependent electromagnetic fields. The Bianchi identity in the Maxwell sector is no longer trivially satisfied when we include time-dependent magnetic fields. This leads to a significantly more complex evolution. However if we want to provide a meaningful comparison to heavy ion collisions this is a necessary step. It will also be necessary to include both a vector and axial gauge field rather then just the axial gauge field displayed in this work. Furthermore as seen in eq. (2.26) the charges density in the system grow without bound, one possible solution via the Stückelberg mechanism [91,92]. In addition to the time dependence of the fields, the gauge fields should also be dynamic rather then external fields. Recent work has displayed it is possible to include fully dynamic gauge fields in the dual field theory picture [39,64,[93][94][95][96] allowing us to compute, in principle, gauge field correlation functions. 8

Acknowledgments
The author wishes to thank and acknowledge the following individuals. Matthias Kaminski for technical discussions, comments on an early draft of this work and the encouragement to complete this project independently. Thank you Matthias. Larry Yaffe for discussions about this work during the Holographic QCD (HQCD) conference at NORDITA in Stockholm Sweden. The organizers of HQCD for an engaging conference and lively discussions. Dirk Rischke and Goethe University for the opportunity to present previous work and for local accommodations in Frankfurt am Main where part of this current work was completed. Dmitri Kharzeev for a brief discussion about this work and his suggestion that entanglement entropy would be interesting to study in this system. Sašo Grozdanov for helpful comments on a draft of this work. Karl Landsteiner for helpful comments on a draft of this work. This work was partially supported by the U. S. Department of Energy grant DE-SC-0012447.

B Scaling Relations
It is useful to consider independent scalings of field theory directions spanned by x and r given by, These rescalings we used in [47] to demonstrate the independence of the field theory from the AdS radius L without the presence of a Chern-Simons term. Due to the omission of this term is worth our time to verify that with this additional boundary term we again find our field theory to be independent of changes in L. Inspection of line element reveals the scalings eq. (B.1) will produce an overall confor-mal factor of the line element if the metric components transform as, B(x,r) = B(x(x), r(r)) (B.2) S(x,r) = α ψ S(x(x), r(r)) (B.3) A(x,r) = α 2 ψ 2 A(x(x), r(r)) (B.4) F (x,r) = α 2 ψ 2 F (x(x), r(r)).

(B.7)
Finally we must additionally transform the parameters as follows, ρ = α 3 ρ,B = α 2 B,L = ψ −1 L,γ = γ. (B.8) Performing the scaling transformation on the action shows that the action is invariant with respect to these scalings.S = S. (B.9) Clearly our scaling transformation has no effect on the equations of motion. We can therefore independently scale the AdS radius without changing the boundary theory by taking α = 1, ψ = 1 hence justifying our choice of setting L = 1.