Divergent density of states and nonanalytic Lyapunov exponent in one-dimensional slowly varying systems

Localization of wave functions in disordered systems can be characterized by the Lyapunov exponent, which is zero in the extended phase and nonzero in the localized phase. Previous studies have shown that this exponent is an analytic function of eigenenergy in a given phase; thus, its nonanalytic behavior has been commonly used to determine the boundaries between the extended and localized phases. In this work, we show that if the localization centers are inhomogeneous across the whole chain and the system possesses (at least) two different localization modes, the Lyapunov exponent can become nonanalytic in the localized phase at the boundaries between the different localization modes. We establish this central result by using several one-dimensional slowly varying models, and reveal that the nonanalytic feature in the Lyapunov exponent is inherently tied to the singularities in the density of states through the Thouless formula. The possible existence of delicate structures in the localized phase effectively broadens our understanding of Anderson localization.