Quantum geometric map of magnetotransport

We propose a quantum geometric map for the magnetononlinear Hall effect (MNHE), the planar Hall effect (PHE), and the ordinary Hall effect (OHE). These magnetotransport phenomena originate from the bilinear charge current of Bloch electrons in electromagnetic fields, incorporating both spin Zeeman coupling and orbital minimal coupling to the applied magnetic field. Benchmarked against Onsager reciprocity, we demonstrate that the spin- and orbital-induced MNHEs are governed by the time-reversal-even Zeeman quantum metric dipole and conventional quantum metric quadrupole, respectively; the spin- and orbital-induced PHEs are dominated by the time-reversal-odd Zeeman Berry curvature dipole and conventional Berry curvature quadrupole, respectively. We further show that the OHE contains an interband contribution that is related to the quantum metric quadrupole, contrary to conventional wisdom. Navigated by this map, we study the previously unexplored spin-induced PHE in the surface Dirac cone of topological insulators, where we uncover a step-like PHE. Our work offers a unified quantum geometric framework for understanding magnetotransport experiments.