Lyapunov exponent, mobility edges, and critical region in the generalized Aubry-Andre model with an unbounded quasiperiodic potential

Abstract

In this work, we investigate the Anderson localization problems of the generalized Aubry-André model (Ganeshan-Pixley-Das Sarma's model) with an unbounded quasi-periodic potential where the parameter $|\alpha |\ge 1$. The Lyapunov exponent $\gamma (E)$ and the mobility edges $E_c$ are exactly obtained for the unbounded quasi-periodic potential. With the Lyapunov exponent, we find that there exists a critical region in the parameter $\lambda -E$ plane. The critical region consists of critical states. In comparison with localized and extended states, the fluctuation of spatial extensions of the critical states is much larger. The numerical results show that the scaling exponent of inverse participation ratio (IPR) of critical states $x\simeq 0.5$. Furthermore, it is found that the critical indices of localized length $\nu =1$ for bounded ($|\alpha |<1$) case and $\nu =1/2$ for unbounded ($|\alpha |\ge 1$) case. The above distinct critical indices can be used to distinguish the localized-extended from localized-critical transitions. At the end, we show that the systems with different $E$ for both cases of $|\alpha |<1$ and $|\alpha |\ge 1$ can be classified by the Lyapunov exponent $\gamma (E)$ and Avila's quantized acceleration $\omega (E)$.

References in corpus (6)

• Nearest neighbor tight binding models with an exact mobility edge in one dimension
• Flat Bands Under Correlated Perturbations
• Metal-insulator transition in an aperiodic ladder network: an exact result
• Self-consistent theory of mobility edges in quasiperiodic chains
• Localization and mobility edges in the off-diagonal quasiperiodic model with slowly varying potentials
• Coexistence of zero Lyapunov exponent and positive Lyapunov exponent for new quasi-periodic Schr$\ddot{o}$dinger operator

Cited by in corpus (2)

• Emergence of multifractality through cascade-like transitions in a mosaic interpolating Aubry-André-Fibonacci chain
• Numerical Investigation of Localization in Two-Dimensional Quasiperiodic Mosaic Lattice