Symmetry Condition and Mechanism for the Anomalous Hall Effect in Odd-Parity Magnets

Abstract

The anomalous Hall effect (AHE) in unconventional magnets has attracted considerable attention. A giant AHE has recently been observed in the candidate $p$-wave magnet Gd$_3$(Ru$_{1-δ}$Rh$_δ$)$_4$Al$_{12}$ [Nature 646, 837(2025)], while its theoretical interpretation invokes a spontaneous magnetization to break time-reversal symmetry ($\mathcal{T}$), although the magnetic order itself already breaks $\mathcal{T}$. Consequently, the microscopic origin of the AHE in odd-parity magnets (OPMs) remains an open question. Here we establish the symmetry condition and microscopic mechanism for the AHE in OPMs. Since the OPM order is classified by the spin group whereas the AHE is governed by the magnetic group, the AHE is allowed only when the magnetic point group is ferromagnetic; conversely, it is forbidden whenever the effective time reversal $\mathcal{T}τ$ (with $τ$ a fractional translation), which is essential to the realization of the OPM, is preserved. Microscopically, OPMs generically host a hidden Zeeman field (HZF) rooted in this intrinsic $\mathcal{T}$ breaking, and its interplay with spin-orbit coupling (SOC) generates the AHE without any external field. We demonstrate this picture with models on the square, honeycomb, and kagome lattices. In particular, a square-lattice helimagnet with sublattice-resolved SOC maps exactly onto an anisotropic Rashba--Zeeman model, whose Zeeman term is nothing but the HZF and directly yields the AHE. Our results establish the symmetry condition for the AHE in OPMs and identify the HZF--SOC interplay as a general route to realize it.