Emergent spacetime and gravitational Nieh-Yan anomaly in chiral Weyl superfluids and superconductors*

Momentum transport in topological Weyl superfluids (and superconductors) is anomalous in the presence of textures and superflow. Using the semi-classical approximation and gradient expansion, the gauge and Galilean symmetries of the system induce via the superfluid hydrodynamics an emergent "Riemann-Cartan" spacetime with torsion and curvature for the low-energy Weyl quasiparticles. Furthermore, the background exhibits local conservation laws corresponding to emergent quasi-relativistic Lorentz symmetry. We show that the momentum anomaly can be given a consistent interpretation as the chiral Nieh-Yan gravitational anomaly experienced by Weyl fermions on the emergent curved spacetime. The coefficient of this anomaly term is not universally quantized and seems to be determined by the underlying non-relativistic Fermi-liquid and Galilean symmetries. Finally, the connection and extension of the emergent spacetime to finite temperature corrections and thermal effects will be discussed.