All-In-All-Out Pyrochlore Iridates as Noncollinear Spin-Orbit Coupled Counterparts of Altermagnets

Abstract

Altermagnets are collinear magnetically ordered states that exhibit momentum-dependent spin splitting in the absence of net magnetization and spin-orbit coupling (SOC). Related spin-splitting patterns, however, can also emerge in noncollinear magnetic systems with large SOC. Here we show, via a microscopic model, that the all-in-all-out (AIAO) state in pyrochlore iridates constitutes a noncollinear counterpart of a $d$-wave altermagnet stabilized by strong SOC. Starting from a microscopic $j_{\mathrm{eff}} = 1/2$ tight-binding model on the pyrochlore lattice, we demonstrate that electronic interactions favor the AIAO phase and analyze its symmetry properties. We show that the AIAO order parameter transforms as an $A_{2g}^{-}$ octupolar magnetic moment, breaking time-reversal symmetry while preserving inversion and zero net magnetization. Using group-theory analysis and mean-field calculations, we demonstrate that this symmetry enforces both a spin-polarized momentum-dependent lifting of band degeneracies that is similar to that of a collinear $d$-wave cubic altermagnet, but also a band splitting at zero-momentum. We show that the latter feature is captured by a low-energy model similar to the Luttinger-Kohn model for cubic semiconductors. Our results identify pyrochlore iridates as a platform for noncollinear counterparts of altermagnetism and provide a general symmetry framework for spin-split phenomena in spin-orbit coupled materials.