Abstract
We decompose the intrinsic second-order nonlinear Hall effect (NLHE) of a generic multiband system into its quantum-geometric contributions within a fully quantum-mechanical, projector-based formalism. By expanding the nonlinear conductivity in powers of the quasiparticle lifetime τ, we recover the established Berry curvature dipole at order τ and clarify discrepancies in previous literature concerning the (interband) quantum metric dipole (or Berry curvature polarizability) contribution at order τ⁰. Crucially, our method reveals an additional contribution at order τ⁰, determined by the {\it intraband} quantum metric dipole (intraQMD), arising from additional virtual interband transitions captured within the fully quantum-mechanical treatment. The intraQMD contribution is generically nonzero in systems with broken time-reversal symmetry and can be distinguished from other geometric contributions by symmetry. Analytical results for low-energy models of topological band crossings, which are hotspots of quantum geometry, demonstrate how band topology influences each contribution. In particular, the intraQMD contribution is especially large in gapped Dirac cones in antiferromagnets. Through a comprehensive symmetry classification of all magnetic space groups, we identify several candidate materials that are expected to exhibit large intrinsic NLHE, including the topological antiferromagnets Yb₃Pt₄, CuMnAs, and CoNb₃S₆, as well as the nodal-plane material MnNb₃S₆.