Nonlinear Hall effect induced by internal Coulomb interaction and phase relaxation process in a four-terminal system with time-reversal symmetry

We numerically investigate nonlinear Hall transport properties in a four-terminal system with time-reversal symmetry and broken inversion symmetry. Within the nonequilibrium Green's function formalism, the second-order nonlinear conductances are derived, where the internal Coulomb potential in response to external voltages is explicitly included to guarantee the gauge invariance. For the system with a single mirror symmetry $\mathcal{M}_{x}$, nonlinear Hall properties are only observable in the $y$ direction and contributed solely from the second-order nonlinear effect. In addition to the intrinsic nonlinear Hall effect originated from nonzero Berry curvature dipole, it is found that the internal Coulomb potential has the same symmetry of the four-terminal system, which gives rise to an extra nonlinear Hall response. Furthermore, the phase relaxation mechanism modeled by virtual probes leads to additional dephasing-induced nonlinear Hall effect.