Fractality in nonequilibrium steady states of quasiperiodic systems

We investigate the nonequilibrium response of quasiperiodic systems to boundary driving. In particular, we focus on the Aubry-André-Harper model at its metal-insulator transition and the diagonal Fibonacci model. We find that opening the system at the boundaries provides a viable experimental technique to probe its underlying fractality, which is reflected in the fractal spatial dependence of simple observables (such as magnetization) in the nonequilibrium steady state. We also find that the dynamics in the nonequilibrium steady state depends on the length of the chain chosen: generic length chains harbour qualitatively slower transport (different scaling exponent) than Fibonacci length chains, which is in turn slower than in the closed system. We conjecture that such fractal nonequilibrium steady states should arise in generic driven critical systems that have fractal properties.

Figure 1

NESS magnetization profile in the Aubry-André-Harper model at criticality for Fibonacci chain lengths $L$. Each profile was obtained by averaging over ${10}^3\varphi$ values. As one zooms in, finer and finer details are revealed with peaks located at fractions of the inverse golden ratio after rescaling along the $x$-axis [10]. Fractal dimension is $D_{\mathrm{f}}\approx 1.11$ [10]. The curves lie within the standard deviations of each other.

Figure 2

Magnetization profile in the AAH model at criticality for generic non-Fibonacci chain length $L=8192$. Fractal dimension is $D_{\mathrm{f}}\approx 1.60$ [10], larger than in the case with Fibonacci lengths shown in Fig. 1. Note that strong oscillations are not noise; in the inset, where we show individual points, the error bars (after averaging over 1000 random phases) are of the same size as points.

Figure 3

NESS current at criticality. Average current is shown, except in the single-shot cases (open symbols). Top panel: The AAH model where the scaling is subdiffusive, and depending on the length, Fibonacci vs. generic vs. single shot symmetric quasidisorder, the scaling exponent $\gamma$ is different. The insets show resonances in the NESS current at the Fibonacci length (red circles); few of the many “satellite” resonances at $F_n\pm F_m$ are also highlighted (this is equivalent to subdividing the intervals as in Fig. 1), which are not noise but each of which have a sequence and scaling associated with them [10]. The system with the symmetric quasiperiodic potential is almost diffusive, as in the closed system dynamics. Although we present results for $\mathrm{\Gamma }=1.0$, we have checked that a weaker coupling $\mathrm{\Gamma }=0.1$ shows similar qualitative behavior. Bottom panel: The Fibonacci model with $h=0.5$ showing similar qualitative dependence on system length and quasidisorder realization, is superdiffusive in all three cases. In both panels $L=4096$ deviates from the generic scaling because it is quite close to a secondary resonance; i.e., $4096\approx F_{19}-F_{11}=4181-89$.

Figure 4

Root mean-squared displacement in wave-packet spreading with Hamiltonian dynamics in critical AAH Hamiltonian with free boundaries, $L=8001$. For symmetric realization of the quasidisorder (blue full line), a nice fit (dashed black line, indistinguishable from data) to subdiffusive spreading is observed, whereas upon phase averaging (red dots, $\approx 500$ samples) normal diffusion is restored.

Figure 5

Distribution of the real parts $x$ of complex eigenvalues ${\lambda }_j$ of $A$. For Fibonacci lengths (data not shown) the power is also $\approx 1.40$.

Figure 6

Two-point correlations in the NESS for the critical Aubry-André-Harper model with $L=610$ (Fibonacci length). We averaged over 1000 random phases $\varphi$. Minima in the correlations (vertical and horizontal “blue” lines occur for Fibonacci numbers $j$, and a self similarity is visually present here too as with the magnetization.

Figure 7

Fractal dependence of standard deviation of magnetization $\langle {\sigma }_i^z\rangle$ in the AAH model on the rescaled spatial index. Fibonacci length $L=4181$ was chosen.

Figure 8

Nonprimary resonances and off-criticality. Left panel: Scaling of phase-averaged NESS current for $m=2$ Fibonacci length sequence [Eq. (A9)] in the critical Aubry-André-Harper model. A nice power-law subdiffusive scaling with exponent 1.29 (very close, if not equal within statistical errors, to $m=1$ shown in main text) is obtained. This vindicates the point that the “satellite” resonances displayed in Fig. 3 of the main text are real in that they follow a scaling depending on the particular length sequence chosen and that they are not noise. Right panel: Scaling of NESS current slightly away from criticality. Immediately to the left of the closed system critical point $h_c=2.0$, we find a saturation of the current setting in as the system size is increased, signaling ballistic transport of spin in the metallic phase. Whereas immediately to the right of $h_c$ we find an exponential decay of the current, characteristic of localization and absence of any transport. All plots correspond to $\mathrm{\Gamma }=0.1$.

Figure 9

Stability of secondary resonances. Scaled NESS currents as a function of deviation from the secondary Fibonacci sequence. The collapse at $L-F_n^{\left(2\right)}=0$ is another representation of the power-law scaling observed in left panel of Fig. 8, with the constant width of the central peak for various scalings $F_n^{\left(2\right)}=322,521,843,1364$ indicative of its stability with respect to fluctuations around these lengths. At $m=1$ Fibonacci lengths, further peaks are observed indicated by dashed lines.

Figure 10

Magnetization profile in the Fibonacci model at $h=0.5$ for Fibonacci chain lengths. Like in the Aubry-André-Harper model's result in Fig. 1 of the main text, finer and finer details are revealed with peaks located at fractions of the inverse golden ratio. Fractal dimension is $D_{\mathrm{f}}\approx 1.09$; $600–1000$ disorder samples were used.

Figure 11

Magnetization profile in the Fibonacci model at $h=0.5$ and Fibonacci chain lengths $L=1597$. Left panel: Profiles $m$ for two different realizations of the disorder (no averaging). Right panel: Profile $M=\overline{m}$ obtained upon averaging over 1100 disorder samples. The error bars on the estimation of $M$ are smaller than the thickness of the line. The light-blue background is defined by the standard deviation of $m$ from its average value $M$, in which profiles are more likely to lie from one realization of the disorder to another.

Figure 12

Magnetization profile in the Fibonacci model at $h=0.5$ for non-Fibonacci chain lengths. Here the rescaling of the $x$ axis was done by $F_p$, which is the smallest Fibonacci number larger than $L_p$. We find that there is a universality among the magnetization profiles for lengths related by $\tilde{L}=L/g^n$, for integer $n$. 1000 ($L=2000$)–2000 ($L=1236$) disorder samples were used.

Figure 13

Root mean-squared displacement in wave-packet spreading with Hamiltonian dynamics in Fibonacci Hamiltonian with free boundaries, $L=8001$. A superdiffusive scaling is observed [results have been sample-averaged as explained in the text, but results are the same even for single realization, i.e., the sequence given by Eq. (A10)]. The exponent is consistent with that observed in Ref. [5] but not with that obtained while employing scaling relation $\beta =1/(1+\gamma )$ [25] (see Fig. 3 of the main text for $\gamma$ results), as also observed in the Aubry-André-Harper model.

Figure 14

Box counting dimension $D_{\mathrm{f}}$ of NESS magnetization profiles in Aubry-André-Harper model obtained as linear fit to logarithm of number $N_b$ of boxes required to cover the data to the logarithm of the box size $\delta$. Left panel: ballistic phase with $h=1.5$; for $L=F_m$ (red closed symbols) there is no fractality, which emerges when $L\ne F_m$ (blue open symbols). Inset shows the magnetization profiles for Fibonacci (red lines) and non-Fibonacci (blue lines) lengths used to compute the $D_{\mathrm{f}}$ in the main panel. Right panel: critical point with $h=2.0$; there is fractality both for $L=F_n$ and for $L\ne F_m$, with the latter having a larger fractal dimension.