Different shapes of spin textures as a journey through the Brillouin zone

Crystallographic point group symmetry (CPGS) such as polar and nonpolar crystal classes have long been known to classify compounds that spin-orbit-induced spin splitting. While taking a journey through the Brillouin zone (BZ) from one $k$-point another for fixed CPGS, it is expected wave vector (WPGS) can change, consequently, qualitative change in texture of polarization occur [the expectation value operator ${\stackrel{P\vec}{S}}_{n{k}_{0}}$ Bloch state $u(n,k)$ ${k}_{0}$]. However, nature (ST) generally unsuspected. In this paper, we determine full classification linear-in-$k$ ST patterns based on polarity chirality reflected WPGS at ${k}_{0}$. The spin-polarization controlling bound be parallel rotation axis perpendicular mirror planes, hence, operation types WPGSs impose restriction ST. For instance, always $k$ chiral since they contain only rotational symmetries. Some consequences operations include observation are unexpected according crystal. example, usually established spin-momentum locking effect (spin vector) requires inversion breaking by an asymmetric electric potential. find even without dipoles or external fields. We use determined relation between design principle select with multiple STs near band edges different valleys. Based high-throughput calculations 1481 compounds, 37 previously fabricated materials edges. well predicted platform potential application spin-valleytronics control accessing