Disentangling Anomalous Hall Effect Mechanisms and Extra Symmetry Protection in Altermagnetic Systems

We investigate the evolution of Anomalous Hall Conductivity (AHC) in a coplanar and collinear antiferromagnetic system with varying spin canting angles. A tight-binding model based on three t2g-orbitals in a body-centered tetragonal lattice is constructed, where the inclusion of third-nearest neighbor hopping is demonstrated to be essential for capturing the characteristic energy band splitting of altermagnetic materials. By employing a symmetry analysis based on spin space groups and treating spin-orbit coupling (SOC) as a perturbation, we theoretically distinguish and numerically verify two origins of the transverse transport: the conventional anomalous Hall effect (AHE) induced by net magnetization and the Crystal Hall Effect (CHE) arising from specific crystal symmetries. Our results show that the conductivity components driven by these two mechanisms follow distinct trigonometric dependencies on the canting angle. Crucially, we identify a hidden C110 rotational symmetry that has been previously overlooked in static magnetic group analyses. By expanding the AHC in terms of spin orientation vectors, we demonstrate that this symmetry acts as a bridge connecting distinct magnetic configurations with different canting angles, thereby strictly protecting the equivalence of orthogonal conductivity components in the collinear system.