Fig. 2: Simplified Feynman diagrams for the nonlinear Hall conductivity.

The simplified triangular (a–c) and two-photon (d–f) diagrams after the redefinition of the vertex functions. The gray shadow represents the modified diagonal velocity that has appeared in Fig. 1, while the orange, green, and blue shadows represent the newly defined modified off-diagonal, side-jump, and skew-scattering velocities respectively. \({{{\Sigma }}}_{{{{{{{{\boldsymbol{k}}}}}}}}{{{{{{{\boldsymbol{k}}}}}}}}^{\prime} {{{{{{{\boldsymbol{k}}}}}}}}^{\prime\prime} }\) is the indecomposable self-energy. The symbols ↻ × 3 and ↕ × 2 are the same as those in Fig. 1, and vertex × 2 in b means that for the shown diagram one can label either \({v}_{a}^{+-}\) or \({v}_{b}^{-+}\) as the modified off-diagonal vertex. Detailed definition of each quantity within the non-crossing approximation and all the simplified diagrams can be found in Supplementary Note 6.