Multipole analysis of spin currents in altermagnetic MnTe

Altermagnets, a class of unconventional antiferromagnets where antiparallel spins are connected by combined rotational and translational symmetries, have recently emerged as promising candidates for spintronic applications, as they can efficiently generate spin currents while maintaining vanishing net magnetization. Here, we investigate charge transport and spin currents in $\alpha$-MnTe, a prototypical altermagnet, using symmetry analysis within the multipole framework and fully relativistic first-principles calculations using the Kubo formalism. Our results show that different magnetic configurations with N\'eel vectors $\hat{N}\parallel y$ and $\hat{N}\parallel x$ in MnTe induce distinct order parameters. This distinction gives rise to spin-momentum locking with different parities and magnetic spin Hall effects (magnetic SHEs) with different anisotropies. Strikingly, our calculations show that the combination of intrinsic spin-orbit coupling and altermagnetic spin splitting yields a large magnetic spin Hall angle of up to 16 \% rivaling or exceeding that of heavy metals such as Pt. On the other hand, the anisotropy of the magnetic SHE provides a practical means to identify the type of order parameter. This establishes, through the powerful framework of multipoles, a general approach for studying transport phenomena that extends to a broader class of altermagnets beyond MnTe.