Chiral Kinetic Theory

• Chiral kinetic theory is the framework for massless Weyl fermions, integrating chirality, Berry curvature, and anomaly effects into phase-space dynamics.
• It extends semiclassical and covariant formulations by incorporating side jumps, Berry corrections, and Bardeen–Zumino currents to ensure gauge-consistent transport.
• The theory underpins phenomena like the chiral magnetic and vortical effects in Weyl materials and plasmas, guiding both experimental and theoretical research.

Chiral kinetic theory is the transport theory of massless or Weyl fermions in which chirality, Berry curvature, and anomaly-induced response are incorporated directly into phase-space dynamics. In its semiclassical form, it modifies the one-particle action, equations of motion, and phase-space measure by a momentum-space Berry monopole; in covariant formulations, it is derived from Wigner functions or world-line representations and accommodates side jumps, collisions, background geometry, and medium self-energies. Across these formulations, the theory is used to describe the chiral magnetic and vortical effects, anomalous continuity equations, collective plasma modes, strong-field magnetized response, Weyl-material transport, and neutrino transport (Stephanov et al., 2012, Hidaka et al., 2018).

1. Semiclassical phase-space structure

The basic semiclassical construction begins from the Weyl Hamiltonian

$H=\boldsymbol{\sigma}\cdot \mathbf p,$

whose helicity branches are separated away from the level crossing at $\mathbf p=0$. Projecting onto a single helicity branch produces a Berry connection $\mathbf a_{\mathbf p}$ in momentum space and the semiclassical action

$$I=\int \left(\mathbf p\cdot \dot{\mathbf x}+\mathbf A\cdot \dot{\mathbf x}-\Phi-|\mathbf p|-\mathbf a_{\mathbf p}\cdot \dot{\mathbf p}\right)dt.$$

Its Berry curvature is the monopole field

$$\mathbf b=\nabla_{\mathbf p}\times \mathbf a_{\mathbf p}=\frac{\hat{\mathbf p}}{2|\mathbf p|^2},$$

with singular divergence $$\nabla_{\mathbf p}\cdot \mathbf b=2\pi\delta^3(\mathbf p)$$. The resulting equations of motion are

$$\sqrt G\,\dot{\mathbf x}=\hat{\mathbf p}+\mathbf E\times \mathbf b+\mathbf B(\hat{\mathbf p}\cdot \mathbf b),\qquad \sqrt G\,\dot{\mathbf p}=\mathbf E+\hat{\mathbf p}\times \mathbf B+\mathbf b(\mathbf E\cdot \mathbf B),$$

with $G=(1+\mathbf b\cdot \mathbf B)^2$, so the invariant measure is $\sqrt G\, d^3x\,d^3p/(2\pi)^3$. In this formulation, the chiral magnetic and chiral vortical effects arise from the same Berry-monopole structure, and the anomaly appears as a source at the nonclassical point $\mathbf p=0$ in the phase-space continuity equation (Stephanov et al., 2012).

This semiclassical framework was extended to relativistic plasmas by emphasizing that finite-temperature consistency requires both fermions and antifermions together with a Berry-corrected dispersion relation,

$\mathbf p=0$0

Within that kinetic theory, the axial anomaly is recovered without thermal renormalization,

$\mathbf p=0$1

and linear response reproduces the parity-even and parity-odd parts of the anomalous Abelian HTL/HDL polarization tensor, including the chiral-magnetic contribution proportional to $\mathbf p=0$2 (Manuel et al., 2013).

2. Covariant and consistent currents

A central structural issue in chiral kinetic theory is the distinction between covariant fermionic currents and conserved consistent gauge currents. In Weyl materials, the low-energy quasiparticles near the two nodes couple to

$\mathbf p=0$3

so standard semiclassical CKT yields, besides the axial anomaly, a mixed nonconservation law for the electric current in simultaneous electromagnetic and pseudoelectromagnetic backgrounds. The resolution is to add the Bardeen–Zumino or Chern–Simons correction

$\mathbf p=0$4

and define the consistent current

$\mathbf p=0$5

which satisfies $\mathbf p=0$6. In Weyl materials this is not merely formal: the axial field $\mathbf p=0$7 is physically realized by Weyl-node separation and strain. The same correction cancels the equilibrium chiral magnetic current when $\mathbf p=0$8 and reproduces the anomalous Hall contribution $\mathbf p=0$9 (Gorbar et al., 2016).

The same consistent-current logic persists beyond linear field order. A second-order consistent CKT for two Weyl fermions includes a first-order field correction to the Berry curvature, a second-order correction to the quasiparticle dispersion, and the same Bardeen–Zumino completion,

$\mathbf a_{\mathbf p}$0

so that the electric current remains locally conserved in the presence of pseudoelectromagnetic fields. This extension is required to capture effects quadratic in $\mathbf a_{\mathbf p}$1, including the $\mathbf a_{\mathbf p}$2 and $\mathbf a_{\mathbf p}$3 corrections to longitudinal plasmon gaps (Gorbar et al., 2017).

3. Lorentz covariance, side jumps, and collisions

Once Lorentz covariance is imposed at order $\mathbf a_{\mathbf p}$4, chiral kinetic theory ceases to be an ordinary scalar Boltzmann theory. In the covariant collisional formulation for massless chiral particles, the spin tensor is

$\mathbf a_{\mathbf p}$5

with $\mathbf a_{\mathbf p}$6 a timelike frame vector. Changing $\mathbf a_{\mathbf p}$7 shifts the worldline by a finite side jump,

$\mathbf a_{\mathbf p}$8

so collisions cannot remain pointlike in every frame if angular momentum is to be conserved. The particle current therefore acquires a collision-induced jump term,

$\mathbf a_{\mathbf p}$9

and the kinetic equation takes the manifestly covariant form $$I=\int \left(\mathbf p\cdot \dot{\mathbf x}+\mathbf A\cdot \dot{\mathbf x}-\Phi-|\mathbf p|-\mathbf a_{\mathbf p}\cdot \dot{\mathbf p}\right)dt.$$0. This framework yields a conserved symmetric stress tensor, a fermionic $$I=\int \left(\mathbf p\cdot \dot{\mathbf x}+\mathbf A\cdot \dot{\mathbf x}-\Phi-|\mathbf p|-\mathbf a_{\mathbf p}\cdot \dot{\mathbf p}\right)dt.$$1-theorem,

$$I=\int \left(\mathbf p\cdot \dot{\mathbf x}+\mathbf A\cdot \dot{\mathbf x}-\Phi-|\mathbf p|-\mathbf a_{\mathbf p}\cdot \dot{\mathbf p}\right)dt.$$2

a rotating equilibrium distribution

$$I=\int \left(\mathbf p\cdot \dot{\mathbf x}+\mathbf A\cdot \dot{\mathbf x}-\Phi-|\mathbf p|-\mathbf a_{\mathbf p}\cdot \dot{\mathbf p}\right)dt.$$3

and the standard chiral-vortical coefficients, including

$$I=\int \left(\mathbf p\cdot \dot{\mathbf x}+\mathbf A\cdot \dot{\mathbf x}-\Phi-|\mathbf p|-\mathbf a_{\mathbf p}\cdot \dot{\mathbf p}\right)dt.$$4

for a Weyl fermion (Chen et al., 2015).

A complementary Wigner-function derivation makes the same side-jump structure explicit in quantum-field-theoretic language. For right-handed fermions, the lesser Wigner function contains the spin tensor

$$I=\int \left(\mathbf p\cdot \dot{\mathbf x}+\mathbf A\cdot \dot{\mathbf x}-\Phi-|\mathbf p|-\mathbf a_{\mathbf p}\cdot \dot{\mathbf p}\right)dt.$$5

and the distribution function is non-scalar under frame changes. The kinetic equation carries a modified shell condition,

$$I=\int \left(\mathbf p\cdot \dot{\mathbf x}+\mathbf A\cdot \dot{\mathbf x}-\Phi-|\mathbf p|-\mathbf a_{\mathbf p}\cdot \dot{\mathbf p}\right)dt.$$6

with a collision term derived from self-energies. In relaxation-time approximation, this formulation yields dissipative $$I=\int \left(\mathbf p\cdot \dot{\mathbf x}+\mathbf A\cdot \dot{\mathbf x}-\Phi-|\mathbf p|-\mathbf a_{\mathbf p}\cdot \dot{\mathbf p}\right)dt.$$7 transport, including anomalous Hall currents driven by $$I=\int \left(\mathbf p\cdot \dot{\mathbf x}+\mathbf A\cdot \dot{\mathbf x}-\Phi-|\mathbf p|-\mathbf a_{\mathbf p}\cdot \dot{\mathbf p}\right)dt.$$8-field gradients and viscous corrections to CME and CVE that are proportional to $$I=\int \left(\mathbf p\cdot \dot{\mathbf x}+\mathbf A\cdot \dot{\mathbf x}-\Phi-|\mathbf p|-\mathbf a_{\mathbf p}\cdot \dot{\mathbf p}\right)dt.$$9 (Hidaka et al., 2018).

4. Geometry, rotation, torsion, and strong magnetic fields

A fully covariant geometric formulation places chiral kinetic theory on the cotangent bundle $$\mathbf b=\nabla_{\mathbf p}\times \mathbf a_{\mathbf p}=\frac{\hat{\mathbf p}}{2|\mathbf p|^2},$$0 of curved spacetime. In that framework, the relevant derivative is the horizontal lift

$$\mathbf b=\nabla_{\mathbf p}\times \mathbf a_{\mathbf p}=\frac{\hat{\mathbf p}}{2|\mathbf p|^2},$$1

and the kinetic equation is derived directly from the Dirac equation in background metric $$\mathbf b=\nabla_{\mathbf p}\times \mathbf a_{\mathbf p}=\frac{\hat{\mathbf p}}{2|\mathbf p|^2},$$2 and gauge field $$\mathbf b=\nabla_{\mathbf p}\times \mathbf a_{\mathbf p}=\frac{\hat{\mathbf p}}{2|\mathbf p|^2},$$3 through the Wigner function. The right-handed current satisfies, at $$\mathbf b=\nabla_{\mathbf p}\times \mathbf a_{\mathbf p}=\frac{\hat{\mathbf p}}{2|\mathbf p|^2},$$4,

$$\mathbf b=\nabla_{\mathbf p}\times \mathbf a_{\mathbf p}=\frac{\hat{\mathbf p}}{2|\mathbf p|^2},$$5

and the final curved-space CKT includes a spin-curvature force through

$$\mathbf b=\nabla_{\mathbf p}\times \mathbf a_{\mathbf p}=\frac{\hat{\mathbf p}}{2|\mathbf p|^2},$$6

Applied to rotating coordinates, this reproduces

$$\mathbf b=\nabla_{\mathbf p}\times \mathbf a_{\mathbf p}=\frac{\hat{\mathbf p}}{2|\mathbf p|^2},$$7

and shows that the CVE is an intrinsic property of a rotating chiral fluid, independent of observer frame (Liu et al., 2018).

A distinct rotating-frame construction modifies the underlying quantum kinetic equation itself by replacing the usual derivative with

$$\mathbf b=\nabla_{\mathbf p}\times \mathbf a_{\mathbf p}=\frac{\hat{\mathbf p}}{2|\mathbf p|^2},$$8

After reduction to three dimensions, this produces a unique rotating-frame CKT whose force law contains the Coriolis term

$$\mathbf b=\nabla_{\mathbf p}\times \mathbf a_{\mathbf p}=\frac{\hat{\mathbf p}}{2|\mathbf p|^2},$$9

while still reproducing the anomalous continuity equation and the CME/CVE currents (Dayi et al., 2018).

When the Weyl Hamiltonian depends on both momentum and position through deformation, momentum-space Berry curvature alone is insufficient. For smooth torsional deformations, the natural object is the full phase-space Berry curvature

$$\nabla_{\mathbf p}\cdot \mathbf b=2\pi\delta^3(\mathbf p)$$0

with modified measure

$$\nabla_{\mathbf p}\cdot \mathbf b=2\pi\delta^3(\mathbf p)$$1

The anomaly is governed by the monopole charge function $$\nabla_{\mathbf p}\cdot \mathbf b=2\pi\delta^3(\mathbf p)$$2, and for a single Weyl cone the resulting continuity equation becomes

$$\nabla_{\mathbf p}\cdot \mathbf b=2\pi\delta^3(\mathbf p)$$3

In a twisted and time-compressed Weyl crystal, this produces a torsion-generated chiral chemical potential which, in an external magnetic field, yields a torsion-induced CME current (Gao et al., 2020).

Two other generalizations alter the kinematic basis rather than the background geometry. In arbitrary even spacetime dimensions, the anomaly and the equilibrium anomalous currents are encoded by the Chern character of a nonabelian Berry connection over the Fermi surface; the master current takes the form

$$\nabla_{\mathbf p}\cdot \mathbf b=2\pi\delta^3(\mathbf p)$$4

which reduces in $$\nabla_{\mathbf p}\cdot \mathbf b=2\pi\delta^3(\mathbf p)$$5 dimensions to the familiar CME/CVE structures (Dwivedi et al., 2016). In a strong background magnetic field, weak-field $$\nabla_{\mathbf p}\cdot \mathbf b=2\pi\delta^3(\mathbf p)$$6-expansion becomes inadequate, and CKT can instead be reorganized around exact Landau levels. In that formulation, the lowest Landau level carries the anomaly and the chiral magnetic current, while the transverse conductivity in the LLL and relaxation-time approximation approaches

$$\nabla_{\mathbf p}\cdot \mathbf b=2\pi\delta^3(\mathbf p)$$7

in the large-$$\nabla_{\mathbf p}\cdot \mathbf b=2\pi\delta^3(\mathbf p)$$8 limit (Lin et al., 2019).

5. Collective modes, correlators, and effective actions

In Weyl materials with a constant magnetic field $$\nabla_{\mathbf p}\cdot \mathbf b=2\pi\delta^3(\mathbf p)$$9, a constant strain-induced pseudomagnetic field $$\sqrt G\,\dot{\mathbf x}=\hat{\mathbf p}+\mathbf E\times \mathbf b+\mathbf B(\hat{\mathbf p}\cdot \mathbf b),\qquad \sqrt G\,\dot{\mathbf p}=\mathbf E+\hat{\mathbf p}\times \mathbf B+\mathbf b(\mathbf E\cdot \mathbf B),$$0, and dynamical electromagnetic perturbations, consistent CKT gives a polarization vector

$$\sqrt G\,\dot{\mathbf x}=\hat{\mathbf p}+\mathbf E\times \mathbf b+\mathbf B(\hat{\mathbf p}\cdot \mathbf b),\qquad \sqrt G\,\dot{\mathbf p}=\mathbf E+\hat{\mathbf p}\times \mathbf B+\mathbf b(\mathbf E\cdot \mathbf B),$$1

where the $$\sqrt G\,\dot{\mathbf x}=\hat{\mathbf p}+\mathbf E\times \mathbf b+\mathbf B(\hat{\mathbf p}\cdot \mathbf b),\qquad \sqrt G\,\dot{\mathbf p}=\mathbf E+\hat{\mathbf p}\times \mathbf B+\mathbf b(\mathbf E\cdot \mathbf B),$$2 term is the Bardeen–Zumino anomalous Hall contribution. At $$\sqrt G\,\dot{\mathbf x}=\hat{\mathbf p}+\mathbf E\times \mathbf b+\mathbf B(\hat{\mathbf p}\cdot \mathbf b),\qquad \sqrt G\,\dot{\mathbf p}=\mathbf E+\hat{\mathbf p}\times \mathbf B+\mathbf b(\mathbf E\cdot \mathbf B),$$3, the longitudinal mode remains at the Langmuir frequency,

$$\sqrt G\,\dot{\mathbf x}=\hat{\mathbf p}+\mathbf E\times \mathbf b+\mathbf B(\hat{\mathbf p}\cdot \mathbf b),\qquad \sqrt G\,\dot{\mathbf p}=\mathbf E+\hat{\mathbf p}\times \mathbf B+\mathbf b(\mathbf E\cdot \mathbf B),$$4

while the transverse modes split,

$$\sqrt G\,\dot{\mathbf x}=\hat{\mathbf p}+\mathbf E\times \mathbf b+\mathbf B(\hat{\mathbf p}\cdot \mathbf b),\qquad \sqrt G\,\dot{\mathbf p}=\mathbf E+\hat{\mathbf p}\times \mathbf B+\mathbf b(\mathbf E\cdot \mathbf B),$$5

These modes are chiral plasmons rather than ordinary plasmons: the longitudinal excitation carries not only electric-current oscillations but also an oscillating chiral current density, with a topological contribution identified as a dynamical chiral electric separation effect. The splitting depends on the chiral shift $$\sqrt G\,\dot{\mathbf x}=\hat{\mathbf p}+\mathbf E\times \mathbf b+\mathbf B(\hat{\mathbf p}\cdot \mathbf b),\qquad \sqrt G\,\dot{\mathbf p}=\mathbf E+\hat{\mathbf p}\times \mathbf B+\mathbf b(\mathbf E\cdot \mathbf B),$$6, so plasma spectroscopy probes Weyl-node separation (Gorbar et al., 2016).

At second order in electromagnetic and pseudoelectromagnetic fields, the same program shows that the would-be chiral magnetic wave and chiral pseudomagnetic wave are not truly gapless once Maxwell dynamics is treated self-consistently. The longitudinal dispersion acquires a nonzero gap

$$\sqrt G\,\dot{\mathbf x}=\hat{\mathbf p}+\mathbf E\times \mathbf b+\mathbf B(\hat{\mathbf p}\cdot \mathbf b),\qquad \sqrt G\,\dot{\mathbf p}=\mathbf E+\hat{\mathbf p}\times \mathbf B+\mathbf b(\mathbf E\cdot \mathbf B),$$7

with corrections quadratic in $$\sqrt G\,\dot{\mathbf x}=\hat{\mathbf p}+\mathbf E\times \mathbf b+\mathbf B(\hat{\mathbf p}\cdot \mathbf b),\qquad \sqrt G\,\dot{\mathbf p}=\mathbf E+\hat{\mathbf p}\times \mathbf B+\mathbf b(\mathbf E\cdot \mathbf B),$$8 and $$\sqrt G\,\dot{\mathbf x}=\hat{\mathbf p}+\mathbf E\times \mathbf b+\mathbf B(\hat{\mathbf p}\cdot \mathbf b),\qquad \sqrt G\,\dot{\mathbf p}=\mathbf E+\hat{\mathbf p}\times \mathbf B+\mathbf b(\mathbf E\cdot \mathbf B),$$9, and the pseudomagnetic mode retains terms linear in $G=(1+\mathbf b\cdot \mathbf B)^2$0. The theory explicitly shows that no self-consistent longitudinal solution with $G=(1+\mathbf b\cdot \mathbf B)^2$1 exists, so the collective excitation is a chiral magnetic or chiral pseudomagnetic plasmon whose oscillations include both electric and chiral currents (Gorbar et al., 2017).

In strongly magnetized chiral plasma, lowest-Landau-level CKT yields linear-response correlators and a nonlocal effective action for perturbative vector and axial gauge fields. The charge densities satisfy

$G=(1+\mathbf b\cdot \mathbf B)^2$2

and the longitudinal correlators have poles at

$G=(1+\mathbf b\cdot \mathbf B)^2$3

the collisionless chiral magnetic wave. The kinetic theory computes covariant currents, and the consistent current is obtained by the explicit Bardeen–Zumino shift

$G=(1+\mathbf b\cdot \mathbf B)^2$4

With this correction, the correlators satisfy derivative symmetry and admit an effective action as generating functional, consistent with Onsager reciprocity (Yang, 2021).

6. Effective-theory interpretations and medium-induced transport

Effective-theory derivations clarify that the kinetic equation is not unique term by term when different quasiparticle variables are used. In high-density effective theory, the low-energy field describes a dressed excitation near the Fermi surface, with momentum decomposed as

$G=(1+\mathbf b\cdot \mathbf B)^2$5

The resulting chiral kinetic equation differs from the direct field-theory equation at $G=(1+\mathbf b\cdot \mathbf B)^2$6, but the difference is traced to reparametrization ambiguity and the use of different degrees of freedom. The explicit map

$G=(1+\mathbf b\cdot \mathbf B)^2$7

shows that the EFT distribution $G=(1+\mathbf b\cdot \mathbf B)^2$8 and the field-theory distribution $G=(1+\mathbf b\cdot \mathbf B)^2$9 generate equivalent dynamics even though their transport equations are not identical term by term (Lin et al., 2019).

World-line constructions sharpen the distinction between Berry phase and anomaly. In that approach, the real part of the fermion determinant is represented by a world-line action of spinning, colored Grassmann point particles, with equations of motion that generalize the Bargmann–Michel–Telegdi and Wong equations. Berry’s phase arises only after taking a nonrelativistic adiabatic limit of the real part. By contrast, the axial anomaly comes from the imaginary part of the determinant,

$\sqrt G\, d^3x\,d^3p/(2\pi)^3$0

so its topological origin is distinct from the Berry phase in this formulation (Mueller et al., 2017).

A more recent Wigner-function theory incorporates self-energy corrections directly into the kinetic operator, on-shell condition, and spin tensor. The relevant shifted momentum is

$\sqrt G\, d^3x\,d^3p/(2\pi)^3$1

and the Wigner function contains

$\sqrt G\, d^3x\,d^3p/(2\pi)^3$2

In core-collapse supernova matter, where the neutrino self-energy is generated by thermal electrons and nucleons, this produces medium-induced neutrino currents along magnetic fields for anisotropic neutrino distributions and a neutrino spin Hall effect driven by density gradients,

$\sqrt G\, d^3x\,d^3p/(2\pi)^3$3

This suggests that, in interacting media, self-energies play the same structural role that external fields play in simpler versions of CKT: they reshape the quasiparticle shell and generate new Berry-type transport channels (Yamamoto et al., 2023).