Spin-momentum locking plays a fundamental role in spintronics and, more broadly, is an important concept in condensed matter physics. In 2D and 3D, spin-momentum locking typically does not allow spin conservation because the spin-1/2 operators of electrons anticommute. Instead, here we study spin-momentum locking terms with conserved, commuting pseudospins built from a combination of spin and orbitals. We find that 2D spin-momentum locking terms with conserved pseudospins generally lead to linearly dispersing modes at low-energy with anomalous charge and pseudospin currents. To cure the anomaly, we show that such anomalous modes can be realized on the surface of a 3D Weyl semimetal (or an associated weak topological insulator) with a nonzero mixed spin-momentum quadrupole moment, which is determined by the momentum location and pseudospin eigenvalues of Weyl points at the Fermi level. Crucially, this mixed quadrupole moment captures a mixed pseudospin-charge bulk response that cancels the anomaly of surface modes, and can generate a giant 3D spin Hall effect, among other phenomena.

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• Spin-orbit coupling
• Topological phases of matter
• Weyl semimetal
• Tight-binding model