Torsional Landau levels and geometric anomalies in condensed matter Weyl systems

Abstract

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.

Geometrizing non-relativistic bilinear deformations

“…It was first obtained as a geometrization of Newtonian spacetime by Cartan [23,24], but has seen a revival in recent years. There is extensive literature on the subject and its applications to a wide selection of areas such as generalized holography [26,[32][33][34][35][36][37][38][39][40][41][42], condensed matter theory [29,[43][44][45][46][47], hydrodynamics [48][49][50][51][52][53][54], string theory [55][56][57][58][59][60][61][62][63] and more [64][65][66][67][68]. We shall consider the 2-dimensional gravity relevant to our aims.…”

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

“…In a more recent development, there has been renewed interest in the field theoretic and hydrodynamic anomalies with non-zero torsion (and relatedly, so-called pseudo gauge fields) in systems with independent frame fields and connection, in contrast to the metric. Several authors have considered non-zero torsional transport coefficients, given some assumptions beyond standard relativistic field theory and/or finite chemical potential and temperature [59][60][61][62][63][64][65][66][67][68][69][70][71][72][73][74][75][76][77][78] . The independent gravitational anomaly term from torsion [79][80][81] , the Nieh-Yan anomaly term [82][83][84][85][86][87] , is special in that it's 2nd order in the gradients, in contrast to the 4th mixed R 2 gravitational anomaly from topological anomaly polynomials, and appears with a dimensionful scale parameter that is, apparently, nonuniversal and unquantized.…”

Topological polarization, dual invariants, and surface flat bands in crystalline insulators

“…[20,68,69]. More generally, the connection and evolution of the gapped crystalline responses and invariants with elasticity tetrads to unquantized anomalies in adjacent semimetal phases remains an interesting problem [70][71][72][73][74] as well as topological elasticity [35,36,44].…”