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 $N_{X}=1,2,3,4,6$ corresponding 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 $\sigma _{xy}=\pm (e^{2}/2h)$ sgn$\left( J\sin N_{X}\Phi \right)$, where $J$ is the coefficient of the coupling between the $X$-wave magnet and the electrons, and $\Phi$ is the direction of the applied magnetic field. Hence, the Hall conductivity is periodic in $\Phi$, 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.