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 $\Gamma$ point for time-reversal-symmetric, noncentrosymmetric $C_{3v}$ crystals in the strong atomic spin-orbit-coupling ($jj$-coupling) limit. Using a $j\in\{1/2,3/2,5/2\}$ multiplet basis appropriate for heavy-element \textit{p}- and \textit{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 $j>1/2$, 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 $|W_{n}|=1,2,5$ 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.