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Relativistic gravitational anomalies lead to anomalous transport coefficientsthat can be activated at finite temperature in condensed matter systems withgapless fermions. The chiral vortical effect (CVE) is an anomalous chiralcurrent along a rotation axis, expressed in terms of a gravimagnetic field anda gravitational anomaly. Another one, the chiral torsional effect (CTE), arisesfrom hydrodynamically independent frame fields and connection. We discuss therelation of CVE, CTE and gravitational anomalies for relativistic fermions fromthe perspective of torsion and the Nieh-Yan anomaly. The DC transportcoefficients of the two anomalies are found to be closely related dependingwhether or not torsion is non-zero in the hydrodynamics. The relativisticanomaly from torsion is well defined if instead of an ultraviolet divergentterm, the chemical potential or temperature enter. The anomaly term is 2ndorder in gradients and contributes in linear response for CTE, implying alsothe same for CVE. As an example, we consider chiral $p+ip$ Weylsuperfluids/conductors. At low-energies, the system is effectively relativisticalong a special anisotropy axis. The hydrodynamics is governed by twovelocities, normal velocity $\boldsymbol{v}_n$ and superfluid velocity$\boldsymbol{v}_s$. The two anomalies follow from the normal component rotationand the dependence of the momentum density on the superfluid velocity (orderparameter). In the CVE the chiral current is produced by solid body rotation ofthe normal component with (angular) velocity $\boldsymbol{v}_n=\boldsymbol\Omega \times \boldsymbol{r}$. In the CTE, a chiral current isproduced the vorticity $\nabla \times \boldsymbol{v}_s$, which in thelow-energy effective theory is torsion. In equilibrium, $\langle\langle \nabla\times \boldsymbol{v_s}\rangle\rangle= 2\boldsymbol\Omega$ on average and thetwo anomaly currents cancel.

physical review. d/physical review. d.

Anomalous chiral transport with vorticity and torsion: Cancellation of  two mixed gravitational anomaly currents in rotating chiral $p+ip$ Weyl  condensates