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We compute the effects of electronic interactions on gapless spin-3/2excitations that in a noninteracting system emerge at a bi-quadratic touchingof Kramers degenerate valence and conduction bands, known as Luttingersemimetal. This model can describe the low-energy physics of HgTe, gray-Sn, 227pyrochlore iridates and half-Heuslers. For the sake of concreteness we onlyconsider the short-range components of the Coulomb interaction. By combiningmean-field analysis with a renormalization group (RG) calculation, we constructmultiple cuts of the global phase diagram of interacting spin-3/2 fermions atzero and finite temperature and chemical doping. Such phase diagrams display arich confluence of competing orders, among which rotational symmetry breakingnematic insulators and time reversal symmetry breaking magnetic orders are theprominent excitonic phases. We also show that even repulsive interactions canbe conducive for both s-wave and d-wave pairings. The reconstructed bandstructure inside the ordered phases allows us to organize them according to theenergy (entropy) gain in the following (reverse) order: s-wave pairing, nematicphases, magnetic orders and d-wave pairings, at zero chemical doping. But, thepaired states are always energetically superior over the excitonic ones forfinite doping. The phase diagrams obtained from the RG analysis show that anordered phase with higher energy (entropy) gain is realized at low (high)temperature. In addition, we establish a "selection rule" between theinteraction channels and the resulting ordered phases, suggesting thatrepulsive interactions in the magnetic (nematic) channels are conducive for thenucleation of d-wave (s-wave) pairing. The proposed methodology can shed lighton the global phase diagram of other strongly interacting multi-band systems,such as doped Dirac semimetal, topological insulators and the like.

Interacting spin-3/2 fermions in a Luttinger semimetal: Competing phases and their selection in the global phase diagram