Spontaneous anomalous Hall effect in two-dimensional altermagnets

The anomalous Hall effect (AHE) is an efficient tool for detecting the N\'eel vector in collinear compensated magnets with spin-split bands, known as altermagnets (AMs). Here, we establish design principles for obtaining nonzero anomalous Hall conductivity in the recently proposed two-dimensional (2D) AMs using spin and magnetic group symmetry analysis. We show that only two of the seven nontrivial spin layer groups exhibit an unconventional in-plane AHE in which the N\'eel vector lies within the plane of the Hall current. Through first-principles simulations on bilayers of ${\mathrm{MnPSe}}_{3}$ and MnSe, we demonstrate the validity of our group theoretic framework for obtaining AHE with $d$- and $i$-wave altermagnetic orders, depending on the stacking of the bilayers. We find that the spin group symmetry is successful in determining the linear and cubic dependence of anomalous Hall conductivity in N\'eel vector space, although AHE is a relativistic effect. This work shows that the AHE in 2D AMs can probe the altermagnetic order and N\'eel vector reversal, thereby facilitating the miniaturization of altermagnetic spintronics.