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

In this work, we investigate the Anderson localization problems of the generalized Aubry-André model (Ganeshan-Pixley-Das Sarma's model) with an unbounded quasiperiodic potential where the parameter $\left|\alpha \right|\ge 1$. The Lyapunov exponent $\gamma \left(E\right)$ and the mobility edges $E_c$ are exactly obtained for the unbounded quasiperiodic potential. With the Lyapunov exponent, we find that there exists a critical region in the parameter $\lambda \text{−}E$ 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 the bounded ($\left|\alpha \right|<1$) case and $\nu =1/2$ for the unbounded ($\left|\alpha \right|\ge 1$) 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 $\left|\alpha \right|<1$ and $\left|\alpha \right|\ge 1$ can be classified by the Lyapunov exponent $\gamma \left(E\right)$ and Avila's quantized acceleration $\omega \left(E\right)$.

Figure 1

Lyapunov exponents for the bounded case ($\alpha =1/2$) and potential strength $\lambda /t=1,3,5$. The discrete points are the numerical results for eigenenergies. The solid lines are given by Eq. (26). The mobility edges for $\lambda /t=1$ are indicated by black arrows. Near mobility edges of the localized-extended transition (e.g., $E_c=0$ and $-8t$ for $\lambda /t=1$), the Lyapunov exponent $\gamma \left(E\right)\propto |E-E_c|$ approaches zero (as $E\to E_c$). The critical index of the localized length $\nu =1$.

Figure 2

Lyapunov exponents for the unbounded case ($\alpha =2$) and potential strength $\lambda /t=1,3,5$. The discrete points are the numerical results for eigenenergies. The solid lines are given by Eq. (26). The mobility edges for $\lambda /t=1$ are indicated by black arrows. Near the localized-critical transition (e.g., $E_c=t$ and $-3t$ for $\lambda /t=1$), the Lyapunov exponent $\gamma \left(E\right)\propto |E-E_c{|}^{1/2}$ (as $E\to E_c$), and the critical index of the localized length $\nu =1/2$.

Figure 3

Phase diagram in $(\lambda ,E)$ plane for the bounded case ($\alpha =1/2$). When $\alpha =1/2$, there exist localized-extended transitions. The blue solid lines are the phase boundaries (mobility edges $E_c$), which are given by Eq. (28). Standard deviations are represented with different colors.

Figure 4

Phase diagram in $(\lambda ,E)$ plane for the unbounded case ($\alpha =2$). When $\alpha =2$, there exist localized-critical transitions. The blue solid lines are the phase boundaries (mobility edges $E_c$), which are given by Eq. (33). Standard deviations are represented with different colors. Within the critical region, there are large fluctuations in standard deviations.

Figure 5

Several typical wave functions for extended, localized, and critical states.

Figure 6

Standard deviations of localized states and extended states for parameters $\alpha =1/2$ and $\lambda /t=1$. The eigenenergy $E_n$ increases gradually as eigenstate index $n$ runs from 1 to 1000.

Figure 7

Standard deviations of localized states and extended states for parameters $\alpha =2$ and $\lambda /t=1$. The eigenenergy $E_n$ increases gradually as eigenstate index $n$ runs from 1 to 1000.

Figure 8

The inverse participation ratio ${\text{IPR}}_n$ of all the eigenstates for system size $N=500,1000,2000$ and $N=4000$.

Figure 9

The scaling law of $\overline{\text{IPR}}$. We calculate the average participation ratio $\overline{\text{IPR}}$ for the eigenstates in some typical energy intervals. The energy intervals for localized states, critical states, and extended states are $-3.5<E/t<-3.2, -0.5<E/t<-0.3$, and $-2.1<E/t<-1.5$, respectively. The system sizes for the discrete points are $N=500,1000,2000$ and $N=4000$, respectively. It is found that for localized states, the scaling exponent $x\simeq 0$. While for extended states, the scaling exponent $x\simeq 1$. For critical states, the scaling exponent $x\simeq 0.5$.

Figure 10

Lyapunov exponents and Avila's accelerations for localized states and extended states. (a) Lyapunov exponents for $\alpha =1/2$ and $\lambda /t=1$. (b) Avila's accelerations for $\alpha =1/2$. The mobility edges $E_c=0$ are indicated by black arrows.

Figure 11

Lyapunov exponents and Avila's accelerations for localized states and extended states. (a) Lyapunov exponents for $\alpha =2$ and $\lambda /t=1$. (b) Avila's accelerations for $\alpha =2$. The mobility edges $E_c=-3t$ and $t$ are indicated by black arrows.