General method for calculating transport properties of disordered mesoscopic systems based on the nonequilibrium Green's function formalism

Disorder scattering plays important roles in quantum transport as well as various Hall effects, including the second-order nonlinear Hall effect induced by Berry curvature dipole. Calculation of disorder-averaged transport properties usually requires substantial computational resources, especially for higher-order effects. Existing methods are either limited by approximation conditions or constrained by numerical stability, making it difficult to conveniently obtain average physical quantities over a wide range of disorder strength. In this work, we develop a general method for noninteracting system to obtain analytical expressions of disorder averages in finite orders of disorder strength. This method utilizes the Dyson equation to expand physical quantities expressed in terms of the Green's functions into series of disorder-averaged matrices, and the only approximation involved is the truncation of the Dyson equation. Therefore, this method not only avoids the brute force calculation of disorder samples, but also widely applies to different model systems, types of disorder, and the number of Green's functions in the expressions. We demonstrate the applicability of this general method by calculating averages of the linear conductance of a two-terminal system, the spin Hall conductance and the second-order nonlinear conductance of four-terminal Hall setups. It is found that truncation at the fourth order of disorder strength provides a reasonable accuracy and a convenient Pad\'{e} treatment effectively extends its applicable range. Numerical results also confirms disorder enhancement of the second-order nonlinear Hall current in four-terminal systems. Moreover, more accurate predictions for a broader range of disorder strength can be achieved by including higher-order terms in a similar manner.