Symmetries of Spin-Splitting Induced by Spin-Orbit Coupling in Non-magnetic Crystals

Abstract: Spin-orbit coupling (SOC) leads to splitting of otherwise spin-degenerate bands in noncentrosymmetric materials, even if time-reversal symmetry is present. While this gives rise to well-known phenomena such as the Rashba and Dresselhaus effects, various other terms are allowed based on the point group of the crystal and the electronic Hamiltonian. In this study, we utilize point group representations to illustrate that four different types of SOC terms (Rashba, Dresselhaus, Weyl, and Ising) can emerge in periodic solids. We construct reciprocal space energy expressions for each type of SOC-induced splitting of opposite spin bands, and follow a similar procedure to also obtain minimal tight-binding models that capture all types of spin-splittings for subgroups of the cubic parent group $m\bar{3}m$. Furthermore, we also obtain a complete list of nodal features in the electronic band structure in these systems, distinguishing between crystallographic-symmetry-imposed nodal lines and those imposed by time-reversal-symmetry only. Finally, we conclude by presenting a list of materials that host each type of inversion-breaking SOC effects. Our classification of the spin-splitting symmetries in non-magnetic systems with SOC is the counterpart of the recent classification of spin-splitting symmetries in unconventional magnetic systems without SOC, such as altermagnets and odd-parity magnets. More broadly, our work provides a basis for studying superconductivity and other collective electronic phenomena that are impacted by SOC-induced band splittings in noncentrosymmetric materials.