Weakly nonlinear quantum transport: an exactly solvable model
  We have studied the weakly non-linear quantum transport properties of a two-dimensional quantum wire which can be solved exactly. The non-linear transport coefficients have been calculated and interesting physical properties revealed. In particular we found that as the incoming electron energy approaches a resonant point given by energy $E=E_r$, where the transport is characterized by a complete reflection, the second order non-linear conductance changes its sign. This has interesting implications to the current-voltage characteristics. We have also investigated the establishment of the gauge invariance condition. We found that for systems with a finite scattering region, correction terms to the theoretical formalism are needed to preserve the gauge invariance. These corrections were derived analytically for this model. 
Jian Wang, Qingrong Zheng, and Hong Guo