Nonlinear Landauer formula: Nonlinear response theory of disordered and topological materials

The Landauer formula provides a general scattering formulation of electrical conduction. Despite its utility, it has been mainly applied to the linear-response regime, and a scattering theory of nonlinear response has yet to be fully developed. Here, we extend the Landauer formula to the nonlinear-response regime. We show that while the linear conductance is directly related to the transmission probability, the nonlinear conductance is given by its derivatives with respect to energy. This sensitivity to the energy derivatives is shown to produce unique nonlinear transport phenomena of mesoscopic systems including disordered and topological materials. By way of illustration, we investigate nonlinear conductance of disordered chains and identify their universal behavior according to symmetry. In particular, we find large singular nonlinear conductance for zero modes, including Majorana zero modes in topological superconductors. We also show the critical behavior of nonlinear response around the mobility edges due to the Anderson transitions. Moreover, we study nonlinear response of graphene as a prime example of topological materials featuring quantum anomaly. Furthermore, considering the geometry of electronic wave functions, we develop a scattering theory of the nonlinear Hall effect. We establish a new connection between the nonlinear Hall response and the nonequilibrium quantum fluctuations. We also discuss the influence of disorder and Anderson localization on the nonlinear Hall effect. Our work opens a new avenue in quantum physics beyond the linear-response regime.

Automated summary

The Landauer formula has been extended to the nonlinear-response regime, revealing that the nonlinear conductance is given by derivatives with respect to energy. This sensitivity to energy derivatives produces unique nonlinear transport phenomena in mesoscopic systems, such as disordered and topological materials. The study also explores the nonlinear conductance of disordered chains and the critical behavior of nonlinear response near mobility edges, as well as the nonlinear response of graphene and the scattering theory of the nonlinear Hall effect.