Chiral kinetic theory in curved spacetime
Many-body systems with chiral fermions exhibit anomalous transport phenomena originated from quantum anomalies. Based on quantum field theory, we derive the kinetic theory for chiral fermions interacting with an external electromagnetic field and a background curved geometry. The resultant framework respects the covariance under the U(1) gauge, local Lorentz, and diffeomorphic transformations. It is particularly useful to study the gravitational or non-inertial effects for chiral systems. As the first application, we study the chiral dynamics in a rotating coordinate and clarify the roles of the Coriolis force and spin-vorticity coupling in generating the chiral vortical effect (CVE). We also show that the CVE is an intrinsic phenomenon of a rotating chiral fluid, and thus independent of observer’s frame.
Introduction.— Quantum anomaly is a prominent concept in the transport phenomena of chiral fermions. One of its most novel consequences is the generation of parity-breaking currents, typified by the chiral magnetic effect (CME) Kharzeev et al. (2008); Fukushima et al. (2008) and chiral vortical effect (CVE) Vilenkin (1979); Erdmenger et al. (2009); Banerjee et al. (2011); Son and Surowka (2009). A crucial feature of these anomalous currents is that they are insensitive to the details of interactions and are thus universal. For this reason, such phenomena have received a lot of attention in a wide context of physics ranging from high-energy nuclear physics Kharzeev et al. (2016); Huang (2016a); Hattori and Huang (2017); Koch et al. (2017) and the astrophysics Charbonneau and Zhitnitsky (2010); Ohnishi and Yamamoto (2014); Masada et al. (2018) to condensed matter physics Miransky and Shovkovy (2015); Armitage et al. (2018); Burkov (2015).
To study the real-time dynamics of the anomalous transport phenomena, the chiral kinetic theory (CKT) is a promising approach which is applicable when the system is dilute and external fields are weak Stephanov and Yin (2012); Son and Yamamoto (2013); Gao et al. (2012); Chen et al. (2013); Huang (2016b); Hidaka et al. (2017); Huang et al. (2018); Mueller and Venugopalan (2017). In CKT, the chiral anomaly is encoded through the Berry curvature Berry (1984), which modifies the Boltzmann equation and the phase space measure. Recently, various aspects of the CKT were investigated, including the Lorentz covariance Chen et al. (2014, 2015); Hidaka et al. (2017); Huang et al. (2018), consistent versus covariant anomalies Gorbar et al. (2017); Carignano et al. (2018), particle collisions Chen et al. (2015); Hidaka et al. (2017, 2018); Hidaka and Yang (2018), etc.
Despite these developments, so far the CKT is restricted to flat spacetime and thus not conventional to explore the anomalous transport phenomena induced by gravitational or non-inertial effects Erdmenger et al. (2009); Banerjee et al. (2011); Landsteiner et al. (2011); Golkar and Son (2015); Golkar and Sethi (2016); Glorioso et al. (2017). Although the classical Boltzmann equation is readily extended to curved spacetime, the formulation of the CKT in curved spacetime is highly nontrivial. The very few attempts so far Dayi et al. (2017); Huang and Sadofyev (2018) considered only a special curved spacetime, that is, the rotating coordinate 111In this Letter, the rotating coordinate will be regarded as a curved spacetime even though its Riemann curvature is zero, while the term “flat spacetime” is specifically referred to as the Minkowski spacetime., and lacked the general covariance. The more rigorous derivation should start with quantum field theory in curved spacetime.
In this Letter, we derive the CKT in an arbitrary curved spacetime and external electromagnetic field, based on the Wigner function that respects the U(1) gauge, diffeomorphism, and local Lorentz covariance Winter (1985); Calzetta et al. (1988); Fonarev (1994). We apply the resultant framework to a rotating coordinate and examine the frame dependence of the CVE, which is so far unclear. We show that, depending on the observer’s frame, the Coriolis force and spin-vorticity coupling (and the side-jump effect) can be responsible to the generation of the CVE, but the total CVE current is always independent of the observer’s frame.
Throughout this Letter, we choose the unit (with the electric charge), but keep explicit; (un)hatted Greek indices denote local flat (curved) spacetime coordinates; is the Minkowski metric; denotes the covariant derivative for the diffeomorphic and local Lorentz transformations; the Levi-Civita symbol is with and .
Phase space and horizontal lift.— In curved spacetime the definition of the phase space is subtle because a global notion of momentum is usually not permitted. This means that for each position on the spacetime manifold , one can introduce a momentum space attached to . The phase space is then the collection of , which constitutes a fiber bundle. One of the natural choices is to define as the tangent or cotangent spaces so that the usual momentum space is reproduced in Minkowski spacetime. In this Letter, we employ the latter. That is, we define the momentum variable as a point in the cotangent space, and the corresponding phase space is the cotangent bundle . Similarly, a point in the tangent space is defined as the position variable canonically conjugate to . The set of builds the tangent bundle .
where the electromagnetic field is introduced to keep the gauge covariance and . We can confirm that the last term in Eq. (1) correctly ensures the diffeomorphism covariance in , as follows. Suppose that is a function in . Under the infinitesimal transformation , the variation of reads with . Hence the last term in Eq. (1) eliminates such a residue . Similarly to Eq. (1), the covariant derivative in is defined by .
The implementation of the horizontal lift brings a great advantage in analysis. That is, we can regard and as “-independent” variables under the parallel transport by , because of
As a result, for arbitrary function on , its lifted image in is represented as the function translated by : . Furthermore, the Fourier transformation from to is expressed as .
Quantum transport in curved spacetime.— With the above preparation, we define the fermionic Wigner function covariantly under the U(1) gauge, diffeomorphic, and local Lorentz transformations, as follows:
In this Letter, we focus on the collisionless fermions, so the spinor field obeys the Dirac equation
where the covariant derivative is with , and being vierbein.
Computing and with the help of the Dirac equation (4), we derive
where represents a commutator. We defined as the total curvature tensor on or . For instance, we have , where the Riemann tensor is with . The transport equation (5) involves the full quantum correction coupled with electromagnetic field and curved background. In Minkowski spacetime, Eq. (5) reproduces the transport equation derived in Ref. Vasak et al. (1987).
In practice, Eq. (5) is a powerful tool for the semiclassical analysis with the systematic expansion in terms of . Let us adopt the power counting scheme with and . Then the Wigner function up to is determined from 222We actually keep the terms, which are necessary to derive Eqs (8)-(10) at .
where is the Ricci tensor. Further we decompose Eq. (6) with the basis of the Clifford algebra: . After a lengthy but straightforward calculation, we obtain
with . The first equation will be the kinetic equation for right-handed Weyl fermions, while the second and third serve as constraints. The equations for are the same, except for a sign change in front of the first term of Eq. (10).
with and being the distribution function. The last term is called the side-jump term Chen et al. (2014); we introduced the spin tensor , where is an arbitrary vector to satisfy and . This vector field accounts for an ambiguity in defining the spin for massless particles Chen et al. (2015). Different ’s correspond to different spin-frames and they are connected via with being a matrix representation of the local Lorentz transformation.
This is the curved-spacetime generalization of the conventional chiral kinetic equation Hidaka et al. (2017); Huang et al. (2018). Several comments are in order. (I) In the classical limit we reproduce the Einstein-Vlasov equation: . (II) The spin connection is unrelated to . Indeed, since is a vector, such a connection can never appear in Eq. (11). (III) The Riemann curvature naively seems to be an correction, as Eqs. (8)-(10) show. However, once coupled with the side-jump term, it emerges even at in Eq. (15). This term represents the so-called spin-curvature force Mathisson (1937); Papapetrou (1951) for chiral fermions. (IV) On the other hand, the curvature does not appear in the delta function, which designates the on-shell condition. However, this would not be the case at . In fact, from the viewpoint of field theory, the dispersion relation (without U(1) gauge field) reads due to Parker and Toms (2009) (see also Ref. Flachi and Fukushima (2018) for a curvature correction to the CVE).
Equilibrium state.— To reveal the physical content of Eq. (15), we consider the equilibrium state. We drop for simplicity. At equilibrium, is generally written as a function of the linear combination of the collisional conserved quantities, i.e., the particle number, the linear momentum, and the angular momentum. Therefore we have with . Note that the orbital angular momentum is involved in the second term. Plugging into Eq. (15) and requiring it to hold for arbitrary , we arrive at the following constraints:
where is an arbitrary scalar function and represents the component perpendicular to . We have three comments about the above equations. (I) Equation (16) is the conformal Killing equation. Choosing a timelike , we define the fluid velocity and temperature via (with ) and , respectively. The physical meaning of is the expansion rate of the fluid: , which follows from Eq. (16). Thus the fluid is kept equilibrium under such an expansion. This is understood as the conformal invariance in the massless Dirac theory. Note that for massive particles must vanish, as the expansion can drive the system out of equilibrium. (II) From Eq. (17), we find that is a constant scalar. We define the chemical potential through . (III) The equilibrium distribution is eventually given by with
The last term expresses the spin-vorticity coupling.
Rotating coordinate.— As the first application, we use our framework to revisit the derivation of CVE by considering a rotating coordinate. Let us choose a constant angular velocity and hereafter set . The corresponding metric tensor reads
with . The nonzero components of the Christoffel symbol are and (with ), which lead to . The metric has an infinite red-shift surface at distance away from the rotating axis. We focus on the spacetime region inside this surface where the metric (19) admits two timelike Killing vectors; and . Note that the former (latter) corresponds to the inertial (rotating) observers 333To be more specific, the rotating (Minkowski) coordinate is considered as the coordinate chart of the rotating (inertial) observer.. The velocities of these two observers are
which are normalized as . From the on-shell condition , we obtain
where denotes the group velocity and the three-momentum is defined as . Similarly, we can obtain and . Thus is identified as the momentum observed by the inertial observer. Note that the second terms in Eqs. (21) and (22) correspond to the rotating energy and velocity shifts, respectively. In the following, we analyze the CKT with several choices of and .
Case 1: Inertial fluid.— First of all, we consider an inertial fluid (i.e., a fluid at rest in flat spacetime) with a rotating observer. We set and . Performing the -integration of Eq. (15), we find (for the particle channel only; antiparticle channel is similar)
with . From the above equation, we identify and , which reproduce the Coriolis and centrifugal force: . From Eq. (14), the particle number current reads
with and being the Berry curvature. Note that due to , at equilibrium all the corrections disappear in Eq. (24), and thus it is just the classical Liouville current: . Also from Eqs. (14) and (18) for , we can check that the same is true for arbitrary . Therefore, the CVE is never induced by an inertial fluid, independently of the observer’s reference frame and the spin-frame choosing vector .
Case 2: Rotating fluid.— In this case we adopt , that is, we consider a fluid at rest in the rotating coordinate. Hereafter let us focus on the small limit to simplify the discussions.
First, we choose , which leads to the kinetic equation and the current as the same forms as Eq. (23) and (24), respectively. However, physical quantities are affected by quantum corrections. When we take and append the antiparticle contribution (for which is replaced with ), the terms in Eq. (24) yield
which is the well-known CVE current. The spin-vorticity coupling term in Eq. (18) are prominent to induce . Becase of this coupling, the first term in Eq. (24) gives of , while the second yields Chen et al. (2014); Huang and Sadofyev (2018).
Second, we employ . It is more convenient to work with a new three-momentum defined as , whose physical meaning will be explained later. After the integration, the kinetic equation reads
with the modified velocity and energy dispersion
The above kinetic equation exhibits an analogy between magnetism and rotation under two types of the correspondence, i.e., in (and so in ), and elsewhere, reflecting the fact that the Landé factor is for spin-1/2 particles. In other words, the spin-vorticity coupling plays a role of the magnetization coupling, and the Coriolis force can be regarded as a fictitious Lorentz force. This suggests that is the momentum observed by the rotating observer. Indeed, Eq. (27) shows that the classical dispersion is linear to . For this reason, Eq. (26) are represented only with quantities in the rotating coordinate. We note that the factor in front of in Eq. (26) represents the quantum modification to the phase space measure Huang and Sadofyev (2018).
From Eq. (14), we compute the particle number current as
which, once substituted with , reproduces Eq. (25) again. Note that the first term does not contribute to the CVE current. In other words, the Coriolis force is responsible for generating the CVE whereas the spin-vorticity coupling is not. This explains why the heuristic replacement works correctly Stephanov and Yin (2012) in computing the CVE current.
Some comments are in order. (I) The above analysis shows that the origin of the CVE can be interpreted differently for different . For the inertial (rotating) spin-frame vector (), the CVE is induced through the spin-vorticity coupling (the Coriolis force). This is a clear demonstration for the nature of spinning massless particles: the total angular momentum is frame-dependently decomposed into the spin and the orbital parts Skagerstam (1992); Chen et al. (2015); Stone et al. (2015). (II) However, in both cases with and , we derive the same CVE current (25). Indeed, the choice of is superficially irrelevant to the CVE, as it is compensated by the side-jump effect. This is confirmed from the fact that for arbitrary , the equilibrium current is derived in the following spin-frame-independent form:
with and . At the same time, we note that Eq. (29) also holds for arbitrary curved spacetime. This explains why the CVE current (25) is the same as that in Minkowski coordinate Chen et al. (2015); Dayi and Kilincarslan (2018); Gao et al. (2018). The CVE is hence intrinsic for rotating fluid, of which the velocity configuration satisfies .
Summary and outlook.— We extended the framework of the chiral kinetic theory (CKT) to curved spacetime, based on quantum field theory. The CKT in curved spacetime is a primary tool for non-equilibrium chiral dynamics under the general-relativistic effect. This enables us to investigate the anomalous transport phenomena in various chiral matter systems with (effective) gravitational field or non-inertial forces, such as supernova or neutron star environment Janka (2012); Yamamoto (2016), rotating/expanding quark-gluon plasma Jiang et al. (2016); Deng and Huang (2016); Ebihara et al. (2017), thermal systems with the temperature gradient Luttinger (1964); Hayata et al. (), and Weyl/Dirac semimetals under strain Cortijo et al. (2015, 2016); Grushin et al. (2016) or torsion Sumiyoshi and Fujimoto (2016).
As an application, we analyzed the CKT in a rotating coordinate, and clarified the frame-dependent interpretation for the chiral vortical effect (CVE). Our calculation showed that although the CVE receives contributions from both the spin-vorticity coupling and Coriolis force depending on the choice of the defining frame of spin, their sum is independent of both the observer’s frame and the spin-frame. In this Letter, we did not discuss about relation between the finite-temperature term in and the gravitational anomaly Landsteiner et al. (2011). It is still left open if such a term is induced by the global anomaly Golkar and Son (2015); Golkar and Sethi (2016); Glorioso et al. (2017). The CKT in curved spacetime is an auspicious candidate to lead to a model-independent answer to this mystery.
Acknowledgements.Acknowledgments.— We thank Tomoya Hayata, Yoshimasa Hidaka, Bei-Lok Hu, Jinfeng Liao, and Qun Wang for useful discussions and valuable comments. This work is supported by the China Postdoctoral Science Foundation under grant No. 2017M621345 (K. M.), and the Young 1000 Talents Program of China, NSFC through Grants Nos. 11535012 and 11675041 (X.-G. H.).
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