Torsion and anomalies in the warped limit of Lifschitz theories

Abstract

We describe how fermionic Lifschitz theories naturally develop non-dissipative response to torsion once the anisotropic scaling exponent is made arbitrarily small. In this limit the system acquires an enhanced (Carrollian) boost symmetry. We show, both through the explicit computation of the path integral Jacobian and through the solution of the Wess-Zumino consistency conditions, that the translation symmetry in the anisotropic direction becomes anomalous. This turns out to be a mixed anomaly between boosts and translations. In a Newton-Cartan formulation of the space-time geometry such anomaly is sourced by torsion. We use these results to give an effective field theory description of the anomalous transport coefficients, which were originally computed through Kubo formulas in [1].

Torsional Landau levels and geometric anomalies in condensed matter Weyl systems

“…In the presence of elastic deformations corresponding to torsion, i.e., phonons, the anomalous chiral symmetry corresponding to translations is manifested as the nonconservation of (lattice) node momentum flux k W e j μ 5 between the Weyl fermions and the background phonons [38,47], as found in superfluid 3 He-A for the (p + ip)-wave paired Fermi liquid [3]. See also [16,76,78,109].…”

“…Aspects of gravitational anomalies in condensed matter have been considered in, e.g., Refs. [38,44,58,59,63,[73][74][75][76][77][78][79][80] including Weyl and Dirac fermions in superfluids, superconductors, and semimetals. The dimensional hierarchy and descent relations of the torsional anomaly were recently analyzed in Ref.…”

“…These considerations are beyond the LL spectral flow argument and we emphasize that the current (38) should be derived from a corresponding Wess-Zumino term, generalized for torsional space-times [18,70,76,79,[100][101][102]. See especially [76], where the consistent and covariant anomalies are discussed in an anisotropic Lifshitz model, closely related to Eq. (B13).…”

“…This leads to the cutoff of the order of (57) direction with uniaxial strain only, with the constraint 2 rel 1, equivalent to q z k F . It is possible that higher orders of k z could be incorporated, following [76]. Here, the same model with different parameters also applies for the Dirac semimetal Na 3 Bi (cf.…”

Quantum Hall effective action for the anisotropic Dirac semimetal

“…In principle, this method can be used to systematically construct the effective theory for such systems as well. This program has already been initiated to describe anomaly-induced transport [49,63].…”