Altermagnet skyrmions: Macroscopic realization of d-wave symmetry

Altermagnetic (ATM) skyrmions embody a macroscopic realization of microscopic d-wave symmetry, as revealed by their unique spin textures in materials. Through lattice rotational symmetry and spin ordering, ATMs show exchange-driven spin splitting with directional dependence, imprinting d-wave anisotropy—modulated by the exchange difference J₁ − J₂—onto skyrmion configurations. Our analysis, combining simulations and theoretical modeling, uncovers a 90^◦ interlayer orientation between elliptical skyrmion pairs, a geometric hallmark validated by C₄ symmetry and anisotropic Berry curvature. We propose this angle as a rotational invariant, underscoring ATM skyrmions’ potential as topological probes for spintronics and beyond.