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Description
Altermagnetism has recently emerged as a distinct magnetic phase with strong potential for spintronic applications, characterized by momentum-dependent spin splitting in the absence of net magnetization \cite{PhysRevX.12.040501}. In real materials, the presence of unavoidable spin–orbit coupling (SOC) further enriches this picture. Recent studies \cite{PhysRevB.109.024404} have shown that the presence or absence of an anomalous Hall effect (AHE) distinguishes two types of altermagnets: mixed and pure altermagnets, a classification rooted in the ireducible representations (irreps) of the point group of the underlying crystal. When the altermagnetic order parameter and a Hall vector transform according to the same irreps, they can coexist in superposition, giving rise to a mixed altermagnetic phase with a finite AHE; otherwise, the phase remains purely altermagnetic.
Motivated by this framework, we investigate the evolution from pure to mixed altermagnetic behavior in two paradigmatic layered oxides, Sr$_2$RuO$_4$ and Sr$_2$IrO$_4$. In Sr$_2$RuO$_4$ \cite{jr65-4273}, the surface of RuO$_6$ octahedra has point group $C_{4v}$, a collinear compensated magnetic order with moments polarized along the $z$ axis transforms as a pure altermagnet, which is symmetry-incompatible with a Hall vector and therefore does not generate an AHE. Importantly, additional symmetry lowering—arising from strain, or secondary electronic orders—can transmute the nominally pure $z$-polarized altermagnetic state into a mixed one, thereby enabling a finite AHE.
By comparison, Sr$_2$IrO$_4$ \cite{Wang2026} exhibits a more intricate situation. Depending on the stacking of single or multiple IrO$_6$ layers, the allowed magnetic order parameters transform according to different irrps, permitting both pure and mixed altermagnetic phases.
Through symmetry analysis, we show how structural distortion or other secondary orders govern the transition from pure to mixed altermagnetism, and outline experimental signatures in transport measurements. Our results identify layered ruthenates and iridates as fertile ground for realizing and tuning altermagnetic phenomena, and suggest concrete pathways for engineering altermagnetism in correlated materials and future quantum technologies.