Anomalous Hall viscosity at the Weyl-semimetal–insulator transition

We show that three-dimensional 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 nonrelativistic 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 $z=\frac{1}{2}$, 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 Dirichlet eta functions and their temperature dependence is dictated by the scaling exponent.

Figure 1

Phases of the two-band Hamiltonian (3). The figure shows the dispersion relation as function of $p_{\perp }$ and $p_3$. The left figure is the insulating phase. The right figure is deep in the Weyl-semimetal phase and the middle figure shows the critical point in-between the two.