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Based on ab initio relativistic ${\mathbf k}\cdot{\mathbf p}$ theory, wederive an effective two-band model for surface states of three-dimensionaltopological insulators up to seventh order in $\mathbf{k}$. It provides acomprehensive description of the surface spin structure characterized by anon-orthogonality between momentum and spin. We show that the oscillation ofthe non-orthogonality with the polar angle of $\mathbf{k}$ with a $\pi/3$periodicity can be seen as due to effective six-fold symmetric spin-orbitmagnetic fields with a quintuple and septuple winding of the field vectors persingle rotation of $\mathbf{k}$. Owing to the dominant effect of the classicalRashba field, there remains a single-winding helical spin structure but with aperiodic few-degree deviation from the orthogonal locking between momentum andspin.

Ab initio k.p theory of spin-momentum locking: Application to topological surface states