Anomalous Hall viscosity at the Weyl-semimetal–insulator transition

Abstract

We show that 3D Lifshitz fermions arising as the critical theory at the Weyl semimetal/insulator transition naturally develop an anomalous Hall viscosity at finite temperature. We discuss how to couple the system to non-relativistic background sources for stress-tensor and momentum currents via a form of Newton-Cartan geometry with torsion and derive the Kubo formulas for the Hall viscosities. While the Lifshitz system that arises most naturally has scaling exponent $z=2$ we also generalize the theory for arbitrary Lifshitz scaling $z$ and show that, in the limit $z \to 0$, it may be given a Chern-Simons interpretation by dimensionally reducing along the anisotropic direction. The Hall viscosities are expressed in terms of zeta functions and their temperature dependence is dictated by the scaling exponent.