Non-collinear Altermagnetic Phases in the Mott Insulator NiS$_2$

Altermagnets (A$\ell$Ms) constitute a novel family of magnetic materials characterized by the absence of net magnetization and the presence of spin-polarized band structures. Whereas A$\ell$M phases were initially proposed in collinear structures, the recently discovered noncollinear chiral A$\ell$Ms stand out for their distinct hedgehog spin texture and multifunctionality in spintronics. In this work, we deepen the characterization of these systems by constructing a Landau theory for noncollinear achiral A$\ell$Ms. Furthermore, we demonstrate that the achiral symmetry of the crystal is reflected in the spin texture in reciprocal space, which presents only spatial-even multipoles. These multipoles, distinguished from those in collinear A$\ell$Ms via the high-order secondary order parameters, can couple to many phenomena such as the spin Hall effect and piezomagnetic effect. To exemplify our theory, we study the noncollinear achiral magnet NiS$_2$ within the framework of altermagnetism, showcasing both spin Hall and piezomagnetic effects in a prototypical correlated Mott insulator that provides an ideal platform to explore the interplay between strong electronic correlations, crystal symmetry, and altermagnetic spin textures. Interestingly, altermagnetism emerges in two magnetic ordered phases of NiS$_2$ upon lowering the temperature. The non-collinearity strengthens the robustness of A$\ell$M order, as the anti-ferromagnetism induced by the strong correlations will not impose effective time-reversal symmetry as in the collinear case. Our results suggest non-collinear achiral A$\ell$Ms as a promising platform for spintronics applications due to the potential to achieve various spin textures with different magnetic orders.