Nonlinear transport theory at the order of quantum metric

Quantum metric, a probe to spacetime of the Hilbert space, has been found measurable in the nonlinear electronic transport thus has attracted tremendous interest. However, without comparing with mechanisms tied to disorder, it is still unclear whether the quantum metric dominates in recent experiments. We exhaust all possible mechanisms of nonlinear transport under the same symmetry that the quantum metric emerges, by finding five more disorder-related mechanisms and their nonlinear conductivity formulas. With the help of the formulas, we derive the scaling law to identify distinct mechanisms in the recent experiments [Gao, et al., Science 381, 181 (2023); Wang, et al., Nature 621, 487 (2023)], determine nonzero nonlinear conductivity elements for all 122 magnetic point groups to guide future experiments, and calculate the nonlinear conductivity to explain the sophisticated features in the experiments of the even-layered MnBi$_2$Ti$_4$ thin films. Our theory will facilitate future experiments and applications of nonlinear transport.