(AHE), in the absence of any measurable magnetization [@nakatsuji; @machida].

With these studies as motivation, we utilize prior studies of the electronic structure of pyrochlore iridates[@ashvin; @ybkim1; @ybkim2] to show that a minimal description for the electronic states in their paramagnetic phase is a storied Hamiltonian from semiconductor physics: the Luttinger model of inverted band gap semiconductors [@luttinger]. This model has gained recent notoriety for its relevance to HgTe, the starting material for some topological insulators [@TI; @qi; @zhang; @zhang2]. While HgTe is a weakly correlated material where band structure alone provides a sufficient description, in the 5d materials the Luttinger Hamiltonian must be supplanted by interactions.

In this paper, we carry out such an analysis, rediscovering and extending a storied analysis by Abrikosov and Beneslavskii of Coulomb forces on the Luttinger problem [@ab1; @ab2]. Using modern renormalization group (RG) techniques, we confirm Abrikosov’s conclusion that long-range Coulomb interactions convert the quadratic band touching into a quantum critical NFL, prove the stability of the state within an $\epsilon$ expansion, and calculate the full set of anomalous dimensions characterizing the state. Consequently, we call the resulting phase a [*Luttinger-Abrikosov-Beneslavskii*]{} (LAB) state. While the LAB phase is stable in the presence of 