Interacting Spin-3/2 Fermions in three-dimensional Luttinger Metal: Phases, Phase Transitions and Global Phase Diagram

The notion of quasiparticles constitutes the foundation of condensed matter physics. Recently it became evident that Dirac or Weyl materials support linearly dispersing quasi-relativisitc nodal quasiparticles around few isolated points in the Brillouin zone. In this talk, I will focus on a collection of spin-3/2 excitations, which in three dimensions display a bi-quadratic touching of the valence and conduction bands at the $\Gamma$ point. Such peculiar band touching can be realized in the normal state of 227 pyrochlore iridates, half-Heuslers, HgTe and gray tin, for example. Once electron-electron interactions are accounted for, this system can display a rich confluence of a plethora of exotic broken symmetry phases, among which nematic, magnetic, and superconducting (s-wave and topological d-waves) orders are the most prominent ones. Using a controlled renormalization group analysis at finite temperature and chemical doping, I will demonstrate the competition among these phases, as well as the associated quantum critical phenomena. In addition, I will also present various cuts of the global phase diagram of interacting spin-3/2 fermions, and argue that nematic and magnetic interactions are conducive for s-wave and d-wave pairings of Luttinger fermions, respectively.