Quantum Geometric Origin of the Intrinsic Nonlinear Hall Effect

We analyze the quantum geometric contribution to the intrinsic second-order nonlinear Hall effect (NLHE) for a general multiband Hamiltonian. The nonlinear conductivity, obtained in Green's function formalism, is decomposed into its quantum geometric constituents using a projector-based approach. In addition to the previously identified Berry curvature and interband quantum metric dipoles, we obtain a third term of quantum geometric origin, given by the momentum derivative of the $intraband$ quantum metric. This contribution, which we term the intraband quantum metric dipole, provides substantial corrections to the NLHE in topological magnets and becomes the dominant geometric term in topological antiferromagnets with gapped Dirac cones. Considering generalized 2D and 3D Weyl/Dirac Hamiltonians, describing a large class of topological band crossings with sizable quantum geometry, we derive analytical expressions of the NLHE, thereby revealing the individual contributions of the three quantum geometric terms. Combined with an exhaustive symmetry classification of all magnetic space groups, this analysis leads to the identification of several candidate materials expected to exhibit large intrinsic NLHE, including the antiferromagnets $\text{Yb}_3\text{Pt}_4$, $\text{CuMnAs}$, and $\text{CoNb}_3\text{S}_6$, as well as the nodal-plane material $\text{MnNb}_3\text{S}_6$. Finally, our projector-based approach yields a compact expression for the NLHE in terms of momentum derivatives of the Bloch Hamiltonian matrix alone, enabling efficient numerical evaluation of each contribution in the aforementioned materials.