Localization transitions and mobility edges in coupled Aubry-André chains

We study the localization transitions for coupled one-dimensional lattices with quasiperiodic potential. In addition to the localized and extended phases, there is an intermediate mixed phase that can be easily explained decoupling the system so as to deal with effective uncoupled Aubry-André chains with different transition points. We clarify, therefore, the origin of such an intermediate phase, finding the conditions for getting a uniquely defined mobility edge for such coupled systems. Finally, we consider many coupled chains with an energy shift that compose an extension of the Aubry-André model in two dimensions. We study the localization behavior in this case comparing the results with those obtained for a truly aperiodic two-dimensional (2D) Aubry-André model, with quasiperiodic potentials in any directions, and the 2D Anderson model.

Figure 1

IPR for two different eigenvalues of the Aubry-André Hamiltonian with nearest-neighbor hopping, with $L=1000$ sites, and $\lambda$ in units of $t_1$.

Figure 2

Two coupled chains: the red line describes the nearest-neighbor intrachain hopping $t_1$, the blue line denotes the transverse hopping $t_0^d$, and the green line denotes the nearest-neighbor interchain hopping $t_1^d$.

Figure 3

IPR, in 10-base logarithmic scale, for two coupled Aubry-André chains with $L=200$ sites each and with nearest-neighbor hopping $t_1=1,t_1^d=0.2$, and $t_0^d=0.5$. The two vertical lines are given by Eqs. (18) (red right line) and (19) (blue left line). (Below) Energy spectra, $E^{+}$ and $E^{-}$, of the two effective decoupled chains at $\lambda ={\lambda }_c^{-}=1.6$.

Figure 4

Two coupled chains: the red line describes the nearest-neighbor hopping $t_1$, the pink line the second-nearest-neighbor hopping $t_2$, the blue line the transverse hopping $t_0^d$, the green line the nearest-neighbor interchain hopping $t_1^d$, and the yellow line the second-nearest-neighbor interchain hopping $t_2^d$.

Figure 5

IPR, in 10-base logarithmic scale, for two coupled Aubry-André chains $\left(L=200\right)$ with second-nearest-neighbor hopping: $t_1=1,t_1^d=0.1,t_2=0.1,t_2^d=0.05$, and $t_0^d=0.5$. The lines are given by Eqs. (26) (blue line) and (27) (red line) in clockwise order. (Below) Energy spectra, $E^{+}$ and $E^{-}$, of the two effective decoupled chains along $\lambda ={\lambda }_c^{+}$ (upper plot) and $\lambda ={\lambda }_c^{-}$ (lower plot). The yellow vertical line indicates the crossing energy point $E^{*}$.

Figure 6

IPR, in 10-base logarithmic scale, for two coupled Aubry-André chains $\left(L=200\right)$ with second-nearest-neighbor hopping: $t_1=1,t_1^d=0.1,t_2=0.1,t_2^d=0.01$, and $t_0^d=0.5$. The lines are described by Eqs. (26) (blue left line) and (27) (red right line). (Below) Energy spectra, $E^{+}$ and $E^{-}$, of the two effective decoupled chains along $\lambda ={\lambda }_c^{-}$.

Figure 7

IPR, in 10-base logarithmic scale, for two coupled Aubry-André chains $\left(L=200\right)$ with second-nearest-neighbor hopping: $t_1=1,t_1^d=0.1,t_2=0.1,t_2^d=0.01$, and $t_0^d=1$. Since Eqs. (31) and (32) are fulfilled, so that the critical line can be written in terms of only the ratio $t_1/t_2$, the mobility edge, described by the blue straight line, is given by Eq. (3). (Below) Energy spectra, $E^{+}$ and $E^{-}$, of the two effective decoupled chains along $\lambda ={\lambda }_c^{-}={\lambda }_c^{+}$.

Figure 8

The averaged IPR and NPR, $〈I_P〉$ and $〈N_P〉$, got from the spectrum reported in Fig. 3 (nearest-neighbor case) and in Fig. 6 (next-nearest-neighbor case).

Figure 9

Many coupled chains: the red line describes the nearest-neighbor intrachain hopping $t_1$, the blue line the transverse interchain hopping $t_0^d$, and the green line the nearest-neighbor interchain hopping $t_1^d$.

Figure 10

IPR, in 10-base logarithmic scale, for $S=5$ coupled Aubry-André chains $\left(L=200\right)$ with nearest-neighbor hopping $t_1=1,t_1^d=0.2$, and $t_0^d=0.5$. The vertical lines are described by Eq. (43), with $k=1,\cdots ,S$, from right (red line) to left (blue line).

Figure 11

Critical amplitudes of the effective uncoupled Aubry-André chains, Eq. (43), for different numbers $S$ of originally coupled chains, for $t_1=1$ and $t_1^d=0.2$.

Figure 12

Many coupled chains: the red line describes the nearest-neighbor intrachain hopping $t_1$, the blue line the transverse interchain hopping $t_0^d$, the green line the nearest-neighbor interchain hopping $t_1^d$, the pink line the next-nearest-neighbor hopping $t_2$, and the yellow line the interchain hopping $t_2^d$.

Figure 13

IPR, in 10-base logarithmic scale, for $S=3$ and 5 coupled Aubry-Andé chains $\left(L=200\right)$ with next-nearest-neighbor hopping parameters $t_1=1,t_2=t_1^d=0.1,t_2^d=0.05$, and $t_0^d=0.5$. The straight lines are given by Eq. (44), with $k=1,\cdots ,S$, from red to blue lines in a counterclockwise order.

Figure 14

IPR, in 10-base logarithmic scale, for $S=3$ coupled Aubry-Andé chains $\left(L=200\right)$ with next-nearest-neighbor hopping parameters $t_1=1,t_1^d=t_2=0.1,t_2^d=0.01,t_0^d=1$. Since Eqs. (31) and (32) are fulfilled, so that the critical line can be written in terms of only the ratio $t_1/t_2$, the mobility edge, described by the blue straight line, is given by Eq. (3).

Figure 15

IPR, in 10-base log-scale, for two coupled Aubry-André chains $(S=2$ and $L=200)$, with single-site shifted energies, $\epsilon (i+1,1)=\epsilon (i,2)$, and with nearest-neighbor hopping $t_1=t_0^d=1$.

Figure 16

IPR, in 10-base log-scale, for many coupled Aubry-André models, Eqs. (52) and (53), with $L=S=50$, and nearest-neighbor hopping $t_1=t_0^d=1$.

Figure 17

IPR, in 10-base log-scale, for the 2D Aubry-André, with energies given by Eq. (54), with $p=\sqrt{2}$, and where $L=S=50$, and nearest-neighbor hopping $t_1=t_d=1$.

Figure 18

Average inverse participation ratio, over all the eigenstates, in 10-base log-scale, for the generalized 2D Aubry-André model obtained by uniformly shifting the chains, Eq. (53) (red curve), for the same Aubry-André model but where the coupled chains are randomly shifted (violet dashed line), for the 2D Anderson model (blue curve), and for the truly aperiodic 2D Aubry-André model, Eq. (54) (green line). All these models are defined on a square lattice with size $L=S=30$ and nearest-neighbor hopping $t_1=t_0^d=1$. Inset: Inverse participation ratio, in 10-base log scale, for the 2D Anderson model with $S=L=30$ and $t_1=t_0^d=1$, as function of the energy levels and $\lambda$.