Dynamics of Open Tight-binding Model

In this paper, we investigate the open tight-binding model with $N$ sites coupled to two reservoirs on its edges with the nonequilibrium Green function method to understand effects of open boundaries. As a result, we obtain an analytical expression of the electron density in the steady state and in the transient regime for $N \rightarrow \infty$ and at absolute zero temperature. From the expression of the electron density in the steady state, we show that the open boundaries do not affect the qualitative behavior of the electron density and the phase transition which has been observed for the isolated tight-binding model also exists in our open case. Using the expression of the time-dependent electron density, we show that a two-step decay appears in the relaxation process purely caused by open boundaries. In addition, we show that the dependence of speed of convergence on boundary coupling strength qualitatively changes at a certain value of the boundary parameters. This is a result of special eigenvalues of a non-hermitian matrix whose imaginary parts do not vanish even in the limit $N \rightarrow \infty$.