Delocalization and scaling properties of low-dimensional quasiperiodic systems

In this paper, we explore the localization transition and the scaling properties of both quasi-one-dimensional and two-dimensional quasiperiodic systems, which are constituted from coupling several Aubry-André (AA) chains along the transverse direction, in the presence of next-nearest-neighbor (NNN) hopping. The localization length, two-terminal conductance, and participation ratio are calculated within the tight-binding Hamiltonian. Our results reveal that a metal-insulator transition could be driven in these systems not only by changing the NNN hopping integral but also by the dimensionality effects. These results are general and hold by coupling distinct AA chains with various model parameters. Furthermore, we show from finite-size scaling that the transport properties of the two-dimensional quasiperiodic system can be described by a single parameter and the scaling function can reach the value 1, contrary to the scaling theory of localization of disordered systems. The underlying physical mechanism is discussed.

Figure 1

Schematic view of the QP system (white region with open symbols) connected by two leads (gray region with filled circles). The black, blue, and red lines denote the intrachain, interchain, and NNN hopping integrals, respectively. Here, the width of the system is $S=3$.

Figure 2

Energy-dependent localization length $\xi$ of the two-leg ladder for (a) $t_d=0$ (solid line), (b) $t_d=0.3t$, and (c) $t_d=0.6t$ with $L={10}^5$. The insets show the corresponding conductance $G$ and the dotted line in (a) denotes the DOS. The above results are performed for a single QP sample. (d) Length-dependent averaged conductance $\langle G\rangle$ for typical values of $E$ with $t_d=0.6t$, which is obtained from ${10}^4$ QP samples. Other parameters are $S=2$, $\lambda =5t$, $W_1=t$, and $W_2=4t$.

Figure 3

Energy-dependent $G$ of the two-leg ladder by coupling two distinct AA chains with $t_d=0$ (solid line) and $t_d=0.6t$ (dotted line). (a) $W_1=2.2t$, $W_2=3t$, and $\alpha =1/\left(10\pi \right)$. (b) $W_1=t$, $W_2=4t$, and $\alpha =(1+\sqrt{5})/2$. Other parameters are $S=2$, $L={10}^5$, and $\lambda =t$. The results are calculated from a single QP sample.

Figure 4

(a) Eigenvalue spectrum of the two-leg ladder by varying the irrational number $\alpha$ within $(0,1)$. (b) 2D plot of $P_r$ vs the eigenvalue number $k$ [see the horizontal coordinate $E_k$ in (a)] and $\alpha$. Larger $k$ refers to higher $E_k$. Here, $L=400$ and other parameters are the same as those in Fig. 2. All of these results are performed for a single QP sample and are similar when other QP samples are considered or by coupling two identical AA chains with $W_1=W_2$.

Figure 5

$G$ vs $E$ of several quasi-1D QP systems with different width $S$. (a) $W_j=2.4t$ for all $j$'s. (b) $W_j=t$ when $j$ is odd and $W_j=4t$ when $j$ is even. Other parameters are $L={10}^5$, $\lambda =t$, and $t_d=0.2t$. The results are performed for a single QP sample.

Figure 6

Scaling properties of $\langle lng\rangle$ for the 2D QP system with $S=L$ and all $W_j=W$. (a) $\langle lng\rangle$ vs $E$ for different $L$ and (b) $\langle lng\rangle$ vs $L$ for several values of $E$ with $W=2.4t$ and $t_d=0.2t$. (c) $\langle lng\rangle$ vs $t_d$ for distinct $L$, (d) $\langle lng\rangle$ vs $L$ for various $t_d$ which is very close to the critical value $t_d^c$ (see text), and (e) $\langle lng\rangle$ vs $L$ for several $t_d$ which is distant from $t_d^c$, by fixing $W=3t$ and $E=1.8t$. (d) The squares, circles, down triangles, diamonds, and left triangles denote $t_d=0.2508t$, $0.251t$, $0.252t$, $0.253t$, and $0.255t$, respectively. (e) The circles and down triangles represent, respectively, $t_d=0.3t$ and $0.5t$ with their vertical axis on the left side, while the squares correspond to $t_d=0.24t$ with its vertical axis on the right side (as indicated by the arrow). (b),(e) The solid lines are the linear fitting curves $\langle lng\rangle \propto {\beta }_{0}lnL$ with ${\beta }_0$ being about 1. The results are averaged over ${10}^4$ QP samples.

Figure 7

Scaling function $\beta$ vs $\langle lng\rangle$ for the 2D QP system. $\beta$ is calculated from a wide range of $W$, $t_d$, and $E$. Here, the length is taken as $L>{10}^2$ to capture the QP character of the system, the step of the length is chosen as $dlnL=0.2$, and ${10}^4$ QP samples are considered. The oblique line is the fitting curve of $\beta \propto 1.02\langle lng\rangle$ for $\langle lng\rangle <0$ and the horizontal line is $\beta =1$. The top inset displays the MIT phase diagram in the $W$-$t_d$ space (black symbols), which is extracted from Fig. 6, and the solid line denotes the curve $W=2(t+2t_d)$. The bottom inset shows $\langle lng\rangle$ as a function of the sample number $N_S$ with $E=1.8t$, $L={10}^2$, and other parameters being identical to those in Fig. 6.

Figure 8

Statistical properties of the Thouless conductance for the 2D QP system. (a) Distribution of $g$ and (b) energy-dependent standard deviation ${\sigma }_g=\sqrt{\langle g^2\rangle -{\langle g\rangle }^2}$. The results are obtained from ${10}^5$ QP samples and other parameters are identical to those in Fig. 6.