Emergent Spacetime and Gravitational Nieh-Yan Anomaly in Chiral p plus ip Weyl Superfluids and Superconductors

Momentum transport is anomalous in chiral p + ip superfluids and superconductors in the presence of textures and superflow. Using the gradient expansion of the semiclassical approximation, we show how gauge and Galilean symmetries induce an emergent curved spacetime with torsion and curvature for the quasirelativistic low-energy Majorana-Weyl quasiparticles. We explicitly show the emergence of the spin connection and curvature, in addition to torsion, using the superfluid hydrodynamics. The background constitutes an emergent quasirelativistic Riemann-Cartan spacetime for the Weyl quasiparticles which satisfy the conservation laws associated with local Lorentz symmetry restricted to the plane of uniaxial anisotropy of the superfluid (or superconductor). Moreover, we show that the anomalous Galilean momentum conservation is a consequence of the gravitational Nieh-Yan (NY) chiral anomaly the Weyl fermions experience on the background geometry. Notably, the NY anomaly coefficient features a nonuniversal ultraviolet cutoff scale., with canonical dimensions of momentum. Comparison of the anomaly equation and the hydrodynamic equations suggests that the value of the cutoff parameter. is determined by the normal state Fermi liquid and nonrelativistic uniaxial symmetry of the p-wave superfluid or superconductor.