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

TL;DR

This paper analytically investigates the localization properties, critical states, and phase transitions in a generalized Aubry-Andre model with unbounded quasiperiodic potential, revealing exact mobility edges and critical indices.

Contribution

It provides exact expressions for Lyapunov exponents and mobility edges in the unbounded potential case, and identifies distinct critical indices for different parameter regimes.

Findings

Exact Lyapunov exponent and mobility edges derived.

Critical states exhibit larger spatial fluctuations and a scaling exponent of 0.5.

Different critical indices distinguish localized-critical from localized-extended transitions.

Abstract

In this work, we investigate the Anderson localization problems of the generalized Aubry-Andr\'{e} model (Ganeshan-Pixley-Das Sarma's model) with an unbounded quasi-periodic potential where the parameter $|\alpha|\geq1$. 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 $x\simeq0.5$. Furthermore, it is found that the critical indices of localized length $\nu=1$ for bounded ($|\alpha|<1$) case and $\nu=1/2$ for unbounded ($|\alpha|\geq1$) 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 $|\alpha|\geq1$ can be classified by the Lyapunov exponent $\gamma(E)$ and Avila's quantized acceleration $\omega(E)$.