Torsional Landau levels and geometric anomalies in condensed matter Weyl systems

We consider the role of coordinate-dependent Fermi velocities, equivalent to effective tetrad (or vierbein) frame fields characterizing momentum space geometry, and torsional Landau levels (LLs) in condensed matter systems with low-energy Weyl quasiparticles. In contrast to their relativistic counterparts, they arise at finite momenta and with an explicit cutoff to the linear spectrum. Via the universal coupling of tetrads to momentum, they experience geometric chiral and axial anomalies with gravitational character. More precisely, at low energy, the fermions experience background fields corresponding to emergent anisotropic Riemann-Cartan and Newton-Cartan space-times, depending on the form of the low-energy dispersion. On these backgrounds, we show how torsion and the Nieh-Yan (NY) anomaly appear in condensed matter Weyl systems with an ultraviolet (UV) parameter with dimensions of momentum. The torsional NY anomaly arises from the spectral flow of torsional LLs and the linear Weyl description with a cutoff. We carefully review the torsional anomaly and spectral flow for relativistic fermions at zero momentum and contrast this with the spectral flow, nonzero torsional anomaly, and the appearance of the dimensionful UV-cutoff parameter in condensed matter systems at finite momentum. We apply this to chiral transport anomalies sensitive to the emergent tetrads in nonhomogeneous chiral superconductors, superfluids, and Weyl semimetals under elastic strain. This leads to previously overlooked suppression of anomalous density at nodes from momentum space geometry, as compared to (pseudo)gauge fields. We also briefly discuss torsion in anomalous thermal transport for nonrelativistic Weyl fermions, which arises via Luttinger's fictitious gravitational field corresponding to thermal gradients.