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Response properties that are purely intrinsic to physical systems are ofparamount importance in physics research, as they probe fundamental propertiesof band structures and allow quantitative calculation and comparison withexperiment. For anomalous Hall transport in magnets, an intrinsic effect canappear at the second order to the applied electric field. We show that thisintrinsic second-order anomalous Hall effect is associated with an intrinsicband geometric property -- the dipole moment of Berry-connection polarizability(BCP) in momentum space. The effect has scaling relation and symmetryconstraints that are distinct from the previously studied extrinsiccontributions. Particularly, in antiferromagnets with $\mathcal{PT}$ symmetry,the intrinsic effect dominates. Combined with first-principles calculations, wedemonstrate the first quantitative evaluation of the effect in theantiferromagnet Mn$_{2}$Au. We show that the BCP dipole and the resultingintrinsic second-order conductivity are pronounced around band neardegeneracies. Importantly, the intrinsic response exhibits sensitive dependenceon the N\'{e}el vector orientation with a $2\pi$ periodicity, which offers anew route for electric detection of the magnetic order in$\mathcal{PT}$-invariant antiferromagnets.

Intrinsic Second-Order Anomalous Hall Effect and Its Application in Compensated Antiferromagnets