Spin-conserved current in altermagnets

Altermagnets have attracted significant interest due to their unique combination of zero net magnetization and novel non-relativistic responses. A key fingerprint of altermagnets is the generation of spin-conserved currents, which cannot be adequately described by conventional group symmetry analysis that treats all symmetry elements equally from a mathematical tensor perspective. Such approaches fail to capture the distinct physical roles played by different symmetries. To address this limitation, we introduce a physical framework that classifies symmetries into site and pair symmetries through coset decomposition. These symmetry classes govern different microscopic mechanisms and macroscopic responses of novel spin currents. By systematically enumerating all symmetry combinations in altermagnets, we comprehensively analyze spin responses including pure spin currents and spin-polarized currents. Moreover, we identify two distinct microscopic pathways corresponding to breaking these symmetry classes separately. Furthermore, this approach provides a unified framework for crystal symmetry-paired spin-momentum locking. This framework governs response phenomenology and guides the identification of materials and perturbations that realize unconventional altermagnetic phenomena.