  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.
