Third-order intrinsic anomalous Hall effect as a transport fingerprint of altermagnets

Abstract: The intrinsic anomalous Hall effect (IAHE) provides a powerful transport fingerprint of quantum magnets, with its linear and second-order responses distinguishing ferromagnets and $\mathcal{P}\mathcal{T}$-symmetric antiferromagnets, respectively. Altermagnets, as an emergent class of quantum magnets, have recently been shown to host a third-order extrinsic anomalous Hall effect, raising a question of whether an \textit{intrinsic} counterpart can serve as a diagnostic of altermagnetic order. Based on spin-group symmetry analysis, we demonstrate that the third-order IAHE is generically allowed in the ten spin Laue groups relevant to altermagnets when spin-orbit coupling (SOC) is taken into account. By combining these symmetry constraints with the anomalous velocity induced by the second-order Berry curvature, we uncover a resonant third-order IAHE arising near the altermagnetic band crossings at generic momenta in both the Lieb-lattice altermagnet and the experimentally realized altermagnet V$_2$Se$_2$O. Notably, we identify the Berry curvature quadrupole, encoded in the second-order Berry curvature and activated by finite SOC, as the microscopic quantum geometric origin of this resonance. Our results establish the third-order IAHE as an intrinsic quantum geometric transport fingerprint of altermagnets and extend the hierarchy of IAHE across collinear quantum magnets.