Family of self-dual quasicrystals with critical Phases

Abstract

We propose a general framework for constructing self-dual one-dimensional quasiperiodic lattice models with arbitrary-range hoppings and multifractal behaviors. Our framework generates a broad spectrum of one dimensional quasicrystals, ranging from the off-diagonal Aubry-André-Harper models on one end, to those featuring long-range hoppings with varied quasiperiodic modulations on another. Focusing on models with off-diagonal quasiperiodic hoppings with power-law decay, we exploit the fact that, when the self-dual condition is satisfied, the system must be in the critical state with multifractal properties. This enables the engineering of models with competing extended, critical, and localized phases, with richly structured mobility edges separating them. As an outstanding example, we show that a limiting case of our family of self-dual quasicrystals can be implemented using Rydberg-atom arrays. Our work offers a systematic route toward critical phases from self-duality considerations, and would facilitate the experimental simulation of these exotic states.

Figures (3)

• Figure 1: Fractal dimension $\text{FD}$ of $\hat{H}_p(a,4-a)$ for (a) $d=2$, and (b) $d=10$. (c) The FD as a function of the eigenstate index $j/N$, with fixed $a=3$ [the vertical black dashed line in (a)], for $N= 2584$ (red dots) and $N= 46\,368$ (black dots), respectively. (d) The even-odd eigen-level spacings $s_{j}^{e-o}$ for even $j$ (black dots) and $s_{j}^{o-e}$ for odd $j$ (red dots), both as functions of $j/N$ for $N= 28\,657$. The vertical blue dashed lines in (c) and (d) indicate the positions of the mobility edge, separating four distinct regions corresponding to the localized (I, IV), critical (II), and extended (III) phases. We take $N=2584$ in (a)(b), and impose the periodic boundary condition for all calculations. For numerical calculations, we approximate the golden ratio $\tau$ as the ratio of two adjacent $u$-Fibonacci numbers, with $\tau=1597/2584$ here gaoxianlongsupp.
• Figure 2: The FDs for different eigenstates with $d=2$ for (a) $\hat{H}_p(a,+\infty)$ and (b) $\hat{H}_p(+\infty,a)$. Four characteristic regions with step-wise mobility edges in $j/N$ can be identified, labeled by $j/N=P_{1,2,3,4}$. (c) MFDs as functions of $1/\log_{10}N$ for eigenstates in different spectral regions for $a=3$ in (a)(b). Red squares and blue circles correspond to the extended [in (a)] and localized phases [in (b)], respectively. Green triangles and stars indicate the MFDs for the critical states in (a) and (b), respectively. Solid lines denote results of linear fit. (d) The Lyapunov exponent as a function of $a$ and $j/N$. We take $N=2584$ in (a)(b)(d), and impose the periodic boundary condition for all calculations.
• Figure 3: (a) Level scheme of the microwave-dressed Rydberg states of the $n$th atom in the one-dimensional array. Here $2\Delta_n$ is the site-dependent energy splitting between the Rydberg states $|+\rangle_n$ and $|-\rangle_n$, $\Omega_n$ is the Rabi frequency of the coupling between $|+\rangle_n$ and $|-\rangle_n$. (b) Level scheme for the microwave-dressed basis of the $n$th atom. (c)(d) Fractal dimension $\text{FD}$ of different eigenstates with varying interaction (hopping) range $d$ for (c) $H(3,+\infty)$ and (d) $H(+\infty,3)$, respectively. The two mobility edges are located at $j/N = 2-2\tau$ and $2\tau-1$, respectively. For all calculations, the system size is $N=2584$ and we impose the periodic boundary condition.