One-Dimensional Quasicrystals with Power-Law Hopping

One-dimensional quasiperiodic systems with power-law hopping, $1/r^a$, differ from both the standard Aubry-André (AA) model and from power-law systems with uncorrelated disorder. Whereas in the AA model all single-particle states undergo a transition from ergodic to localized at a critical quasidisorder strength, short-range power-law hops with $a>1$ can result in mobility edges. We find that there is no localization for long-range hops with $a\le 1$, in contrast to the case of uncorrelated disorder. Systems with long-range hops rather present ergodic-to-multifractal edges and a phase transition from ergodic to multifractal (extended but nonergodic) states. Both mobility and ergodic-to-multifractal edges may be clearly revealed in experiments on expansion dynamics.

Figure 1

Regimes of 1D quasicrystals with power-law hopping, for $\beta =(\sqrt{5}-1)/2$. For small quasidisorder strength $\mathrm{\Delta }$ all SPS are ergodic (AE) and for large $\mathrm{\Delta }$ (for hopping power $a>1$) all are localized (AL). The $P_s$ regimes are characterized by a fraction ${\beta }^s$ of ergodic SPS, whereas the rest are localized ($a>1$) or multifractal ($a\le 1$). The different behavior for $a>1$ and $a<1$ is indicated in the figure with a slightly different color. The results were obtained for 987 sites, with periodic boundary conditions. Calculations for larger systems do not modify the results [42].

Figure 2

Results for $\beta =(\sqrt{5}-1)/2$. Level spacing $s^{e-o}$ (red) and $s^{o-e}$ (blue) for the AA model (a1-a3), $a=1.5$ (b1,b2), and $a=0.5$ (c1,c2) for different $\mathrm{\Delta }$. In the AA model all SPS are either localized (LOC), multifractal (MF), or ergodic (ERG). In the GAA model, $P_s$ regimes appear, in which the lowest ${\beta }^s$ fraction of SPS is ergodic, whereas the rest is localized ($a>1$) or multifractal ($a<1$). These graphs were obtained from calculations for $L=28657$ sites with periodic boundary conditions. (b3) and (c3) show $D_2$ for the SPS between ${\beta }^{2}L$ and $\beta L$ at $a=1.5$ and 0.5, respectively. For $a=1.5$ (0.5) a blocklike localization (ergodic-to-multifractal) transition occurs when crossing from $P_1$ to $P_2$. The results of panels (b3) and (c3) were obtained from calculations with up to $L=75025$ sites with periodic boundary conditions and then extrapolated to infinite systems. See Ref. [42] for more details about the energy and $\mathrm{\Delta }$ dependence of $D_2$.

Figure 3

Survival probability $F(R=L/2)$ in the long-time limit ($Jt={10}^4$) for $a=0.5$ (blue circles) and $a=3$ (red squares), for open boundary conditions with $L=987$ sites and $\beta =(\sqrt{5}-1)/2$. The intermediate $P_s$ regimes lead to a step-wise dependence on $\mathrm{\Delta }$.

Figure 4

(a) Long-time survival probability $F(R)$ for the AA model with $L=100$, for the AE, AL, and MF cases, assuming an initially localized excitation at $x=0$ in $L=100$ sites. (b) $F(R)$ for the GAA model with open boundary conditions and $L=987$ sites for $a=0.5$, 1, and 3, within the $P_2$ regime.

Figure 5

Fractal dimension $D_2$ for $a=0.5$ (a) and 1.5 (b) as a function of $\mathrm{\Delta }$. Figure (c) shows $D_2$ and $\gamma$ [with $l(t)\propto t^{-\gamma }$] as a function of $a$ for $\mathrm{\Delta }=2$. Dashed lines depict the fitted relations $D_2(a)\approx \frac{1}{3}(1-a{)}^{1/2}$ and $\gamma \simeq D_2/(2-a)$. See Ref. [42] for a detailed discussion on our calculations of $D_2$ and $\gamma$, as well as on the error bars.