In this seminar, we argue that a class of strongly spin-orbit coupled materials, including some pyrochlore iridates, may be described by a minimal model consisting of the Luttinger Hamiltonian supplemented by Coulomb interactions.

It contains two-fold degenerate conduction and valence bands touching quadratically at the zone center. Using modern renormalization group methods, we show that interactions induce a quantum critical non-Fermi liquid phase (Luttinger Abrikosov Beneslavskii phase), stable provided time-reversal and cubic symmetries are maintained. We determine the universal power-law exponents describing various observables, which include conductivity, specific heat, non-linear susceptibility and magnetic Gruneisen number. Furthermore, we determine the phase diagram in the presence of cubic and/or time-reversal symmetry breaking perturbations, which includes topological insulator and Weyl semimetal phases. Many of these phases possess an extraordinarily large anomalous Hall effect.

If time permits, we discuss quantum phase transitions nearby the non-Fermi liquid phase,  induced by either time-reversal breaking or cubic-symmetry breaking. We find novel universality class for the quantum phase transitions in three spatial dimensions, where fluctuation effects are believed to be weak.