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

TL;DR

This work shows that in one-dimensional systems with slowly varying potentials or hoppings, the Lyapunov exponent γ(E) can be non analytic even within the localized phase when multiple localization modes coexist. By analyzing models with two localization channels—node localization and extremum localization—the authors reveal that γ(E) becomes non analytic at the energy where the two modes exchange dominance, an effect traced to singularities in the density of states via the Thouless relation. They develop phase diagrams, introduce metrics such as the fractal dimension D2 and a transfer-matrix based γ, and provide numerical evidence for the mechanism across several slowly varying models. A square-well counterexample with a single localization mode shows that the non analytic γ is not guaranteed in all localized phases, highlighting the crucial role of mode competition; the results suggest a broader and more nuanced picture of Anderson localization that may generalize beyond one dimension.

Abstract

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 non-analytic 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 non-analytic 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 non-analytic 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.

Figures (6)

• Figure 1: (a) Two distinct localization modes in the slowly varying potential [see Eq. (\ref{['eq-hami-dia']})]. Symbols represent wave function localized at the nodes and extrema of the potential. (b) Phase diagram of the model. The labels 'Extended', 'NL', 'EL', and 'Empty' indicate extended states, node localized states, extremum localized states, and absence of states, respectively. (c) Schematic Lyapunov exponents for NL and EL, denoted as $\gamma_\text{nl}$ and $\gamma_\text{el}$ respectively.
• Figure 2: (a)-(b) DOS, and (c)-(d) Lyapunov exponent $\gamma$ and its derivative $\partial \gamma$ as functions of energy $E$ for various potential strength $V$. The potential is given by $V_i = V\cos(\pi \alpha i^\nu)$. The dashed lines denote the boundaries $E = \pm(2t-V)$, which separate the extended states from the localized states when $V<2t$, and distinguish between two kinds of localized states when $V>2t$.
• Figure 3: (a) The potential at the localization center $f_\nu(\bar{i})$ versus energy $E$ when $V =3t$. The dashed lines denote the boundaries between two localized modes at $E = \pm t$. (b) Spatial distribution of three representative eigenstates [as marked in Fig. \ref{['fig-fig3']}(a)]. The orange solid line represents the potential.
• Figure 4: (a) Fractal dimension $D_2$ versus eigenenergy $E$ and hopping strength $\lambda$ for the slowly varying model described by Eq. (\ref{['eq-model2']}), with $b = 0.3$. (b)-(c) DOS, and (d)-(e) Lyapunov exponent $\gamma$ and its derivative $\partial \gamma$ versus $E$ for different $V$. The dashed lines in (b) and (d) represent the boundaries $E = -4t/7$ and $E =16t/13$, whereas in (d) and (f), the boundaries are located at $E = -4t/13$ and $E =16t/7$.
• Figure 5: (a) Fractal dimension $D_2$ versus eigenenergy $E$ and hopping strength $\lambda$ for the quasiperiodic slowly varying hopping model described by Eq. (\ref{['eq-gnu']}), with $b = 0.3$. (b) Three representative eigenstates as marked in Fig. \ref{['fig-fig4']}(a). (c)-(d) DOS, and (e)-(f) Lyapunov exponent $\gamma$ and its derivative $\partial \gamma$ versus $E$ for different $\lambda$. The dashed lines in (c) and (e) represent the boundaries $E = 7t/13$ and $E =19t/13$, whereas in (d) and (f), the boundaries are located at $E = -21t/13$ and $E =47t/13$.