Infrared fixed points of higher-spin fermions in topological semimetals

We determine the fate of interacting fermions described by the Hamiltonian $H=\textbf{p}\cdot \textbf{J}$ in three-dimensional topological semimetals with linear band crossing, where $\textbf{p}$ is momentum and $\textbf{J}$ are the spin-$j$ matrices for half-integer pseudospin $j\geq 3/2$. While weak short-range interactions are irrelevant at the crossing point due to the vanishing density of states, weak long-range Coulomb interactions lead to a renormalization of the band structure. Using a self-consistent perturbative renormalization group approach, we show that band crossings of the type $\textbf{p}\cdot \textbf{J}$ are unstable for $j\leq 7/2$. Instead, through an intriguing interplay between cubic crystal symmetry, band topology, and interaction effects, the system is attracted to a variety of infrared fixed points. We also unravel several other properties of higher-spin fermions for general $j$, such as the relation between fermion self-energy and free energy, or the vanishing of the renormalized charge. An $\text{O}(3)$ symmetric fixed point composed of equal chirality Weyl fermions is stable for $j\leq 7/2$ and very likely so for all $j$. We then explore the rich fixed point structure for $j=5/2$ in detail. We find additional attractive fixed points with enhanced $\text{O}(3)$ symmetry that host both emergent Weyl or massless Dirac fermions, and identify a puzzling, infrared stable, anisotropic fixed point without enhanced symmetry in close analogy to the known case of $j=3/2$.