Nonlinear transverse current on $\mathcal{C}_{3v}$-symmetric honeycomb lattice

At nonlinear orders in the electric field, the vanishing of the Hall conductivity does not prevent the nonlinear component of the current from being transverse for selected field directions. We study electrons on $\mathcal{C}_{3v}$-symmetric honeycomb lattice for which the Hall conductivity vanishes at first and second order. Nevertheless, the second-order current component is transverse for fields perpendicular to the three mirror lines. The $\mathcal{C}_{3v}$ symmetry constrains the first-order and second-order conductivity tensors to have only one independent component each, which we calculate using the quantum kinetic equation. In linearly\hyp{}polarized oscillating field, the current has a zero\hyp{}frequency component switching sign upon $\pi/2$-rotation of the polarization angle.