  We argue that a class of strongly spin-orbit coupled materials, including
some pyrochlore iridates and the inverted band gap semiconductor HgTe, may be
described by a minimal model consisting of the Luttinger Hamiltonian
supplemented by Coulomb interactions, a problem studied by Abrikosov and
collaborators. It contains two-fold degenerate conduction and valence bands
touching quadratically at the zone center. Using modern renormalization group
methods, we update and extend Abrikosov's classic work and show that
interactions induce a quantum critical non-Fermi liquid phase, stable provided
time-reversal and cubic symmetries are maintained. We determine the universal
power-law exponents describing various observables in this "Luttinger Abrikosov
Beneslavskii" state, 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 semi-metal phases.
Many of these phases possess an extraordinarily large anomalous Hall effect,
with the Hall conductivity scaling sub-linearly with magnetization.
