Solution of electric-field-driven tight-binding lattice coupled to fermion reservoirs

We study electrons in a tight-binding lattice driven by a dc electric field with their energy dissipated through on-site fermionic thermostats. Due to the translational invariance in the transport direction, the problem can be block diagonalized. We solve this time-dependent quadratic problem and demonstrate that the problem has a well-defined steady state. The steady-state occupation number shows that the Fermi surface shifts at small fields by the drift velocity, in agreement with the Boltzmann transport theory, but it then deviates significantly at high fields due to strong nonlinear effect. Despite the lack of momentum scattering, the conductivity takes the same form as the semiclassical Ohmic expression from the relaxation-time approximation.