Quenching the CME via the gravitational anomaly and holography

In the presence of a gravitational contribution to the chiral anomaly, the chiral magnetic effect induces an energy current proportional to the square of the temperature in equilibrium. In holography the thermal state corresponds to a black hole. We numerically study holographic quenches in which a planar shell of scalar matter falls into a black hole and rises its temperature. During the process the momentum density (energy current) is conserved. The energy current has two components, a non-dissipative one induced by the anomaly and a dissipative flow component. The dissipative component can be measured via the drag it asserts on an additional auxiliary color charge. Our results indicate strong suppression very far from equilibrium.

In the presence of a gravitational contribution to the chiral anomaly, the chiral magnetic effect induces an energy current proportional to the square of the temperature in equilibrium. In holography the thermal state corresponds to a black hole. We numerically study holographic quenches in which a planar shell of scalar matter falls into a black hole and rises its temperature. During the process the momentum density (energy current) is conserved. The energy current has two components, a non-dissipative one induced by the anomaly and a dissipative flow component. The dissipative component can be measured via the drag it asserts on an additional auxiliary color charge. Our results indicate strong suppression very far from equilibrium.
Anomaly induced transport phenomena have been in the focus of much research in recent years [1,2]. The prime example is the so called chiral magnetic effect (CME) [3]. The CME has a component due to the gravitational contribution to the chiral anomaly. In the presence of a magnetic field B and at temperature T chiral fermions build up an energy current [4,5] J ε = 32π 2 T 2 λ B . (1) In the background of gauge and gravitational fields chiral fermions have the anomaly In holography the gravitational anomaly is implemented via a mixed Chern-Simons term. Space-time is curved in the additional holographic direction and this is what generates (1) from the mixed Chern-Simons term [6]. Perturbative nonrenormalization has been shown in [7,8]. The relation of (1) with the gravitational contribution to the chiral anomaly has also been derived in a model independent manner combining hydrodynamic and geometric arguments in [9,10]. An additional constraint on effective actions consistent with (1) stems from considerations based on global gravitational anomalies [11,12]. Transport signatures induced by (1) have recently been reported in the Weyl semimetal NbP [13]. So far anomaly induced transport has mostly been studied in a near equilibrium setup in which local versions of temperature and chemical potentials can be defined. An application of anomaly induced transport is the quark gluon plasma created in heavy ion collisions [14]. There the magnetic field is extremely strong. It is also very short lived and might have already decayed before local thermal equilibrium is reached [15]. One therefore needs a better understanding of how anomalies induce transport far out of equilibrium. Holography is an extremely efficient tool to study both, anomalous transport phenomena and out of equilibrium dynamics of strongly coupled quantum systems. Previous studies of anomalous transport in holographic quenches focused on the * guillermo.milans@csic.es, karl.landsteiner@uam.es, esperanza.lopez@uam.es pure U(1) 3 anomaly [16,17]. This motivates us to study anomalous transport induced by the gravitational anomaly in a holographic quantum quench.
To develop intuition we first consider anomalous hydrodynamics [18][19][20] in a spatially homogeneous magnetic field. The anomalous transport effect we want to monitor is the generation of an energy current in the magnetic field (1). We start with an initial state at temperature T 0 and heat the system up to a final temperature T . In a relativistic theory the energy current J ε i = T 0i is the same as the momentum density P i = T i0 . Momentum is a conserved quantity and can not increase if it is not injected into the system. On the other hand the anomaly induced energy current does increase and this increase must be balanced by a collective flow not included in (1). The increase in the the non-dissipative anomalous energy current 1 will be exactly counterbalanced by a dissipative contribution at any given moment. If we introduce a uniform but very light density of impurities carrying some additional charge drag will generate a convective current proportional to the density of impurities [32]. This current measures how much energy current is generated via (1). We thus need to consider a system with two U(1) charges. The first one has the mixed gravitational anomaly (2) and carries the magnetic flux. The second one is a anomaly free auxiliary U(1) charge that serves to monitor the build up of the current (1) as the temperature changes.
We now consider the hydrodynamics of the system. Since all spatial gradients vanish the constitutive relations are The specific form of the anomalous transport coefficients σ B , σ B and σ B,X depend on the hydrodynamic frame choice [22]. In Landau framê Note that despite J X µ having no anomaly it does have a nontrivial chiral magnetic transport coefficient in Landau frame.

arXiv:1709.08384v1 [hep-th] 25 Sep 2017
This makes the effect due to dragging manifest. We assume that the system is neutral with respect to the anomalous charge, i.e. µ = ρ = 0. As initial condition we take J = J X = 0 and the energy current to be given by (1). Thus the energy current at any given moment is where we work in the linear response regime, i.e. we assume λ B to be a small perturbation compared to the the energy density and pressure such that u µ ≈ (1, v). Solving for the velocity v and using the constitutive relation for the current J X we find that it is given by This equation determines the current build up due to drag at any given moment if the system undergoes a slow near equilibrium time evolution such that an instantaneous temperature T can be defined. It is independent of the choice of Landau frame. For a generic conformal field theory p = KT 4 and ε = 3p, and we obtain with j X = If a system undergoes near equilibrium evolution j X must lie for any given moment on that curve. Deviation from (11) will be a benchmark for far from equilibrium behavior.
We will now consider a holographic model that allows to implement the physics described in a non-equilibrium setting. The action of our model is In addition to gravity and a scalar field φ , it involves two gauge fields A M and X M with field strengths F = dA and F X = dX. Only gauge transformations of A are anomalous. The Chern-Simons term is the holographic implementation of the mixed chiral-gravitational anomaly. We do not include possible pure gauge Chern-Simons terms since we want to isolate the effects of the gravitational anomaly. We set from now on 2κ 2 = 1. The dual field theory will be brought out of equilibrium by abruptly varying a coupling. These type of processes are known as quantum quenches [23]. Holographically, the leading mode of bulk fields at the AdS boundary are interpreted as a couplings for the dual field theory [24,25,30]. We will perform a quench on the coupling associated to the scalar φ . For simplicity we have chosen φ to be massless and neutral under the gauge fields A M and X M . We consider the following simple boundary profile for its leading mode Since (12) is invariant under global shifts of φ , the Hamiltonians before and after the quench will be equivalent. The energy density induced by the quench is a function of both its amplitude η and time span τ.
In order to study the transport properties, it is enough to solve the equation of motion to first order both in the Chern-Simons coupling and the magnetic field. Besides, we will work in the decoupling limit of large charge q for the anomaly free gauge field. Equivalently we can work at q = 1 and treat X µ perturbatively to first order. In that case its dynamics does not backreact onto the other sectors of the theory. With these simplifications, the equations of motion reduce to where At zero order in a λ -expansion the gauge fields do not backreact on gravity and the scalar, and thus can be ignored when solving their leading dynamics. We parameterize the zeroorder solution of the metric as Time reparameterizations are used to fix δ (t, 0) = 0, such that the Minkowski metric is reproduced at the AdS boundary. Field theories at thermal equilibrium are dual to black hole backgrounds. Our initial geometry will be a Schwarzschild black hole, for which f = 1−π 4 T 4 0 z 4 and δ = 0. Around t = 0 the quench takes place, creating an extra energy density on the boundary theory and equivalently, a matter shell that enters from AdS boundary into bulk. The shell subsequently undergoes gravitational collapse and is absorbed by the original black hole. The resulting black hole represents the final equilibrium state with T >T 0 . Similar holographic setups have been extensively studied in the last years [26][27][28][29].
The gauge fields A m and X M satisfy the same equation at zero order in λ . However the different roles they should fulfill select different leading solutions. We want A M to induce a constant background magnetic field on the boundary theory and no charge density. Choosing the magnetic field to point in the x 3 direction, this is achieved by the simple solution F 12 =B with B constant throughout the bulk. Contrary we wish X M to induce a non-vanishing charge density and no boundary field strength, which leads to The expectation value of its associated current is given by The integration constant ρ X is the desired charge density.
At linear order in λ , the magnetic field induces a nondiagonal component in the metric where c is an integration constant. In the initial and final state the bulk scalar field stress tensor vanishes. We fix the integration constant by demanding that the initial state reproduces (1), which implies c = 8π 2 T 2 0 . In our conventions the energy current can be read off the asymptotic metric expansion as g 03 = 1 4 T 03 z 2 + · · · [31]. Neither the scalar field and nor anomalous gauge field receive a correction at order λ . However the off-diagonal component of the metric backreacts on X M and generates an entry parallel to the magnetic field. It is governed by the equation Although X 3 is sourced by the Chern-Simons term even in the initial state, it leads to a vanishing J 3 X . It is the subsequent evolution what generates a boundary current parallel to the magnetic field. We also note that due to the fact that J 3 X is treated in the decoupling limit it reacts immediately to the drag.
We focus first on fast quenches. These are processes whose time span is small in units of the inverse final temperature, 2τT < 1. Fig.1a shows the result of two such processes which the same initial temperature T 0 and time span τ but different final temperatures. A comparison with the benchmark curve (11), plotted in the lower inset of Fig.1a, rules out a possible near equilibrium description. In hydrodynamic process the current j X attains the maximum when the temperature reaches T m /T 0 = √ 2, at which it takes the value 1/4. For higher temperatures the equilibrium current decreases, reflecting the large inertia of a hot medium. One of the processes in Fig.1a has been tuned to reach T m , while the other generates a higher temperature T /T 0 =3. In both cases j X exhibits a maximum before stabilizing. This contradicts (11), which would predict a monotonic growth for the process whose final temperature is T m , and shows that the evolution is far from equilibrium. Moreover the current at the maximum is larger than 1/4 in the first quench (blue) and smaller in the second (red), which again disagrees with (11). When the currents of both processes are normalized to one at the final equilibrium state, their profiles in the rescaled time tT are very similar. The current overshoots around 4% its final value before attaining it, as can be seen in the upper inset of Fig.1a. The time span of a fast quench also has very small impact on the evolution of the current. This is illustrated in the inset of Fig.1b, which shows three processes with the same final temperature T /T 0 = 2.5 and different time span.
It is interesting to compare the evolution of j X with that of another important observable, the energy density. The energy density builds up during the quench and attains its final value as soon as the quench ends. In terms of the holographic model the energy pumping into the system happens while the time derivative of the scalar at the AdS boundary is nonvanishing. The profile (13) is within 3% of its final value at t/τ ≈ 1.75. We observe in Fig.1 that instead j X equilibrates at tT ≈ 1.2. Hence the energy density and the current generated in fast quenches have independent equilibration timescales, τ and 1/T respectively. This provides an alternative and simple way to discard a near equilibrium evolution, since no single effective temperature can describe both observables. At the geometrical level the different equilibration timescales have further implications. While the energy density only depends on the bulk total mass, the anomaly free current must be sensitive to the interior geometry close to the emerging horizon. Consistently with this, j X turns out to mainly build up as equilibrium is approached. The body of Fig.1b describes a very fast quench with τT 0 =0.007. The purple curve gives the time profile of the scalar field at the asymptotic boundary (13). The time interval when the quench occurs has been highlighted in blue. Notably, the current is practically zero in this example even sometime after the quench has finished. This clearly shows that the anomaly free current does not react to the initial far from equilibrium state, but to the onset of the equilibrium. A central conclusion of our study is that anomalous transport properties related to the gravitational anomaly are very much linked to the system being at or close to thermal equilibrium.
We wish to now study the transition from fast to slower quenches. Fig.2a shows the last stages in the evolution of the current for several processes with the same initial temperature and time span. For the sake of comparison we normalize the final current to one, and take the span of the quench instead of the final temperature as time unit. Processes with 2τT < 0.5 behave as explained above. When 2τT ≈ 0.65 the maximum before equilibration starts to weaken out, and it practically disappears for 2τT ≈ 1. Processes with 2τT 1 exhibit a monotonic growth of the current to its equilibrium value. We will refer to them as intermediate quenches. We have highlighted the time span of the quench, up to the moment when φ 0 is within 3% of its final value. The transition between fast and intermediate quenches happens when time scale set by the final temperature approaches that of the quench. It is important to stress that, in general, intermediate quenches do not admit either a near equilibrium description. Indeed when the final temperature is higher than T m , the absence of a transient max-imum signals out of equilibrium dynamics. For the parameters associated to Fig.2a we have 2τT m = 0.5, such that processes with final temperature below T m behave as fast quenches. 2ΤT   Finally we consider slow quenches. First we consider process that are slow with respect to the final temperature and can be expected have a sizable final period of near equilibrium evolution. The hydrodynamic current described by the benchmark curve (11) decreases with rising temperature when T > T m . In this case, a final period of near-equilibrium evolution implies that j X must exhibit a transient maximum. In Fig.2b we investigate quenches with 2τT >2 for the same initial conditions as in Fig.2a. The red curve coincides in both plots for the sake of comparison. We observe that a maximum reappears with distinctive characteristic from that exhibited by fast quenches. Contrary to them the maximum of the quotient j X / j Xeq increases with the temperature, and can be attained even before the quench is half way through. It is natural that the larger τT the earlier the system enters the near-equilibrium regime, whose onset is qualitatively signaled by the maximum of the current.
In the previous processes τT 0 = 0.175, such that they are slow with respect to the final temperature but fast with respect to the initial one. This constrains when the near equilibrium regime sets in and hence whether the current can reach 1/4, the maximum of the equilibrium curve (11). Keeping the same τT 0 , Fig.3a shows evolutions with very high final temperature. Since we are interested in the maximal value of the current, j X has not been rescaled as we did in Fig.2. Instead of approaching 1/4, the maximum turns out to slowly decreases with increasing final temperature. This shows that the near equilibrium regime in processes with τT 0 = 0.175 can only be associated with temperatures well above T m . In order for the complete evolution to be hydrodynamic, the quench needs to be also slow with respect to the initial temperature. We plot in Fig.3b processes with fixed T /T 0 = 2.6 and growing time span. The maximum of the current tends now to 1/4 as expected. We have checked that, consistently, the current builds up in a monotonic way for these slow processes whose final temperature is equal or smaller than T m .
We have studied holographic quantum quenches of the CME induced by the gravitational anomaly. A rich phenomenology arises depending on the time scale of the quench.  Figure 3. Left: Evolution of j X in processes with τT 0 = 0.175 and very high final temperature. Right: Processes with fixed initial and final temperatures T /T 0 = 2.6 and growing τ.
If the quench is fast with respect to the initial and final temperatures, the evolution is far from equilibrium until the final exponential approach to stabilization. The current builds up late in the time evolution and slightly overshoots before it achieves its equilibrium value. In equilibrium the anomalous conductivity is proportional to the square of the temperature. The main motivation of this work was to analyze what activates the anomalous conductivity out of equilibrium, where there is no notion of temperature. It could have been governed by energy density, which in equilibrium is also measured by the temperature.
In this case the current should have reacted as soon as energy is injected into the system. Our result on fast quenches shows that this is not the case. Rather the system has to evolve closer to equilibrium to build up the anomalous current.
Intermediate quenches leave the far from equilibrium stage while they are reaching the final state, resulting in a monotonic growth of the current. Processes which are slow with respect to the final temperature but fast with respect to the initial one, have a finite period of near-equilibrium evolution. This extends to the complete evolution for large τT 0 .
We set c = 8π 2 T 2 0 such that our initial current is given by (1). Furthermore we note that for this value of c, g 03 vanishes at the horizon at the initial temperature T 0 .
Equation (22) for the anomaly free gauge field reduces to with d another integration constant. Requiring X 3 to be regular at the horizon imposes d =0 in the initial state, which turns out to yield a vanishing J 3 X . Regularity of X 3 is preserved by the time evolution. Therefore in the final state with T > T 0 we find d = 8λ Bπ −2 (T 2 0 − T 2 )/T 4 . This leads to the current To recover equation (10), note that the energy density can be read off the expansion of the metric when put in Fefferman-Graham coordinates as ε = 3π 4 T 4 and that ε = 3p.