Almost half-quantized planar Hall effects in X -wave magnets with X = p , d , f , g , i

The planar Hall effect is a phenomenon that the Hall conductivity emerges perpendicular to the electric field in the presence of an in-plane magnetic field. We investigate the planar Hall effect in two-dimensional metal coupled with higher symmetric X -wave magnets with X = p , d , f , g , i , where those with X = d , g , i are altermagnets. The X -wave magnet is characterized by the number N X of the nodes in the band structure, where N X = 1 , 2 , 3 , 4 , 6 corresponds to X = p , d , f , g , i . Although the system is metallic, provided the Dirac gap is tiny, we demonstrate that the Hall conductivities are almost half quantized and well approximated by the formula σ x y = ± ( e 2 / 2 h ) sgn ( J sin N X Φ ) , where J is the coefficient of the coupling between the X -wave magnet and the electrons and Φ is the direction of the applied magnetic field. Hence, the Hall conductivity is periodic in Φ , and the periodicity is equal to the number N X of the nodes. This property may be used to confirm that the target material is indeed an X -wave magnet. Furthermore, the sign of J may be used as a bit for antiferromagnetic spintronics.