Interplay of Topology and Electron-Electron Interactions in Rarita-Schwinger-Weyl semimetals

We study, for the first time, the effects of strong short-range electron-electron interactions in generic Rarita-Schwinger-Weyl semimetals hosting spin-$3/2$ electrons with linear dispersion at a fourfold band crossing point. The emergence of this novel quasiparticle, which is absent in high-energy physics, has recently been confirmed experimentally in the solid state. We combine symmetry considerations and a perturbative renormalization group analysis to discern three interacting phases that are prone to emerge in the strongly correlated regime: The chiral topological semimetal breaks a ${\mathbb{Z}}_2$ symmetry and features four Weyl nodes of monopole charge $+1$ located at vertices of a tetrahedron in momentum space. The $s$-wave superconducting state opens a Majorana mass gap for the fermions and is the leading superconducting instability. The Weyl semimetal phase removes the fourfold degeneracy and creates two Weyl nodes with either equal or opposite chirality depending on the anisotropy of the band structure. We find that symmetry breaking occurs at weaker coupling if the total monopole charge remains constant across the transition.

Figure 1

Energy dispersion of the spin-$3/2$ Rarita-Schwinger-Weyl fermion at a fourfold linear band crossing point. We set one of the momentum components of $\mathbf{p}$ to zero and plot the eigenvalues of Eq. (1) versus the remaining two components for $v_2=0$ (left) and $v_2=1/3$ (right). For $v_2=0$ the spectrum is rotation invariant, while for generic values of $v_2$, it features cubic anisotropy.

Figure 2

Renormalization group fixed points. Upper panel: Couplings $g_1$, $g_2$, $g_3$ (solid, dashed, dotted lines) at the fixed points $W$ (black), SC (red), and $V$ (blue). At $W$, we have $g_1=-g_2=g_3$, and so, we only show the first coupling. Lower panel: Susceptibility exponent $\eta$ of the order parameter at the fixed points. At $W$, we have $\eta =3$ for $⟨{\psi }^{†}\mathcal{W}\psi ⟩$ for all $\alpha >0$ (solid black line). This is the dominant divergence for $\alpha >0.7$, while for $\alpha <0.7$ the order parameter $N_i(\alpha )=⟨{\psi }^{†}[V_i+\kappa (\alpha )U_i]\psi ⟩$ with $\kappa (\alpha )\gg 1$ has the largest susceptibility (dashed black line). At SC, the most divergent channel is the $s$-wave superconductor (red). At $V$, $N_i(\alpha )$ has the largest susceptibility exponent (dashed blue line). Since, here, $\kappa (\alpha )$ is generically very small, we can neglect it and only consider the approximate order parameter $⟨{\psi }^{†}{V}_i\psi ⟩$ (solid blue line).