Localization of wave functions in the disordered models 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 a smooth
function of eigenenergy in the same phase, thus its non-smoothness can serve as
strong evidence to determine the phase transition from the extended phase to
the localized phase. Cependant, logically, there is no fundamental reason that
prohibits this Lyapunov exponent from being non-smooth in the localized phase.
Dans ce travail, we show that if the localization centers are inhomogeneous in the
whole chain and if the system possesses (at least) two different localization
modes, the Lyapunov exponent can become non-smooth in the localized phase at
the boundaries between the different localization modes. We demonstrate these
results using several slowly varying models and show that the singularities of
density of states are essential to these non-smoothness, according to the
Thouless formula. These results can be generalized to higher-dimensional
models, suggesting the possible delicate structures in the localized phase,
which can revise our understanding of localization hence greatly advance our
comprehension of Anderson localization.

Cet article explore les excursions dans le temps et leurs implications.

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