Second Harmonic Hall Response in Insulators: Inter-band Quantum Geometry and Breakdown of Kleinman's Conjecture

Abstract

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 quasi-static electric fields of low frequencies, the extension of the response to higher frequencies is another promising frontier, which introduces quantum geometry into inter-band transitions. Here, we demonstrate that a specific nonlinear Hall response, namely the second harmonic Hall (SHH) response, can arise from inter-band transitions. We establish the quantum geometric origin of the SHH response and show that inter-band quantum geometry dominates the SHH response when driven near inter-band resonance. Crucially, we find that the inter-band SHH response in insulators exhibits strong frequecy 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.