Beyond spin-1/2: Multipolar spin-orbit coupling in noncentrosymmetric crystals with time-reversal symmetry

We develop a symmetry-adapted multipolar $\mathbf{k}\cdot \mathbf{p}$ theory close to the bulk $\mathrm{\Gamma }$ point for time-reversal-symmetric, noncentrosymmetric $C_{3v}$ crystals in the strong atomic spin-orbit-coupling ($jj$-coupling) limit. Using a $j\in \{1\mathrm{/}2,3\mathrm{/}2,5\mathrm{/}2\}$ multiplet basis appropriate for heavy-element $\mathit{\text{p}}$- and $\mathit{\text{d}}$-bands, we systematically construct all symmetry-allowed spin-orbit coupling terms up to fifth order in momentum and generalize the usual spin texture to a total-angular-momentum texture. For $j>1\mathrm{/}2$, multipolar spin-orbit coupling qualitatively reshapes Fermi surfaces and makes the topology of Bloch states band dependent. This leads to anisotropic high-$j$ textures that go beyond a single Rashba helix. We classify these textures by their total-angular-momentum vorticity $W_n$ for every energy band and identify distinct $\mathrm{\mid }{W}_n\mathrm{\mid }=1,2,5$ phases. We show that their crossovers generate enhanced and nonmonotonic current-induced spin-polarization responses, namely the Edelstein effect, upon tuning the chemical potential. Our results provide a symmetry-based framework for analyzing and predicting multipolar spin-orbit coupling, total-angular-momentum textures, and spintronic responses in heavy-element materials without an inversion center.