Second order chiral kinetic theory under gravity and antiparallel charge-energy flow

We derive the chiral kinetic theory under the presence of a gravitational Riemann curvature. We reveal that on top of the conventional frame choosing vector, higher order quantum correction to the chiral kinetic theory brings an additional ambiguity to specify the distribution function. Based on this framework, we derive new types of fermionic transport, that is, the charge current and energy-momentum tensor induced by the gravitational Riemann curvature. Such novel phenomena arise not only under genuine gravity but also in a (pseudo-)relativistic fluid with inhomogeneous vorticity or temperature. It is especially found that the charge and energy currents are antiparallelly induced by an inhomogeneous fluid vorticity (more generally, by the Ricci tensor ${R_0}^i$), as a consequence of the spin-curvature coupling. We also briefly discuss possible applications to Weyl/Dirac semimetals and heavy-ion collision experiments.