Fig. 1: Illustration of weak and strong altermagnetic LKSD.

a, Top and middle panels, ab initio band structure of MnTe at k_z = 0 along the Γ–K path for relativistic SOC turned off and on, respectively. The Néel vector is along the crystal y axis (see panels c and d), corresponding to the Γ–M axis (see bottom panel of b). Red and blue colours correspond to opposite z components of spin. Bottom panel, schematics of the Brillouin zone with four spin-degenerate nodal planes in the electronic structure with SOC turned off. b, Same as a at k_z  ≈ 0.35 Å⁻¹ along the \(\overline{{\boldsymbol{\Gamma }}}-\overline{{\bf{M}}}\) path illustrating the strong altermagnetic LKSD. Red and blue colours correspond to opposite y components of spin. Bottom panel highlights the \(\overline{{\boldsymbol{\Gamma }}}-\overline{{\bf{M}}}\) path outside the four nodal planes (the red and blue colours highlight the alternating symmetry of the spin polarization in the plane). c,d, Schematic view of the crystal and magnetic structure of MnTe in the y–z and x–y planes, respectively The red and blue shadings in c and d mark Te octahedra around the Mn sites A and B with opposite spins, which are related by spin rotation combined with sixfold crystal rotation and half-unit cell translation along the z axis.