The nonlinear Hall effect has recently garnered significant attention as a powerful probe of Fermi surface quantum geometry in metals. While current studies mainly focus on the nonlinear Hall response driven by quasistatic electric fields of low frequencies, the extension of the response to higher frequencies is another promising frontier, which introduces quantum geometry into interband transitions. Here, we demonstrate that a specific nonlinear Hall response, namely, the second harmonic Hall (SHH) response, can arise from interband transitions. We establish the quantum geometric origin of the SHH response and show that interband quantum geometry dominates the SHH response when driven near interband resonance. Crucially, we find that the interband SHH response in insulators exhibits strong frequency dispersion, manifesting the breakdown of Kleinman's conjecture in nonlinear optics. This connects the SHH response to the breakdown of Kleinman's conjecture and reveals that frequency-dispersive insulators generally allow the SHH response. Furthermore, we predict a giant SHH susceptibility in gated strained bilayer graphene and propose that one can apply the polarization resolved second harmonic microscopy to detect the SHH response there.

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• Anomalous Hall effect
• Quantum transport
• Topological materials