On thermal Nieh-Yan anomaly in Weyl superfluids

We discuss the possibility of the torsional Nieh-Yan anomaly of the type $\mu (ej^\mu_5) =\gamma T^2({\cal T}^a \wedge {\cal T}_a)$ in Weyl superfluids, where $T$ is temperature and ${\cal T}^a$ is the effective or emergent torsion from the superfluid order parameter. As distinct from the dimensionful ultraviolet (UV) parameter $\Lambda^2$ in the conventional torsional Nieh-Yan anomaly, the parameter $\gamma$ is dimensionless in canonical units. This suggests that such dimensionless parameter may be fundamental, being determined by the geometry, topology and number of chiral quantum fields in the system. By comparing this to a Weyl superfluid with low-temperature corrections, $T\ll\Delta_0$, we show that such a term does exist in the hydrodynamics of a chiral $p$-wave superfluid, such as $^3$He-A, or a chiral superconductor. We also discuss and show how other $T^2$ terms of similar form and of the same order in gradients, coming from e.g. Fermi-liquid corrections and the chiral chemical potential, can also be expressed in terms of dimensionless fundamental parameters with emergent low-energy relativistic fields. Lastly, we discuss our results in comparison to relativistic Weyl fermions and the connection of the torsional gravitational anomalies to thermal transport in Weyl systems.