Multifractal-enriched mobility edges and emergent quantum phases in Rydberg atomic arrays

TL;DR

This work presents a class of exactly solvable one-dimensional quasiperiodic flat-band lattices that host multifractal-enriched mobility edges and multiple coexisting quantum phases. By leveraging Avila's global theorem, the authors derive analytic Lyapunov exponents in both lattice and dual spaces, yielding explicit mobility-edge conditions and phase diagrams. They further show universality by analyzing cross-stitch and Lieb flat-band lattices and propose a concrete, experimentally accessible realization in Rydberg atom arrays with a spectroscopic protocol to measure real- and dual-space IPRs, enabling direct observation of localized, extended, and multifractal phases in systems of tens of qubits. The combination of exact theoretical results and a practical measurement scheme provides a path to experimentally explore Anderson localization and multifractal physics in synthetic quantum matter.

Abstract

Anderson localization describes disorder-induced phase transitions, distinguishing between localized and extended states. In quasiperiodic systems, a third multifractal state emerges, characterized by unique energy and wave functions. However, the corresponding multifractal-enriched mobility edges and three-state-coexisting quantum phases have yet to be experimentally detected. In this work, we propose exactly-solvable one-dimensional quasiperiodic lattice models that simultaneously host three-state-coexisting quantum phases, with their phase boundaries analytically derived via Avila's global theorem. Furthermore, we propose experimental protocols via Rydberg atom arrays to realize these states. Notably, we demonstrate a spectroscopic technique capable of measuring inverse participation ratios across real-space and dual-space domains, enabling simultaneous characterization of localized, extended, and multifractal quantum phases in systems with up to tens of qubits. Our work opens new avenues for the experimental exploration of Anderson localization and multifractal states in artificial quantum systems.

Figures (19)

• Figure 1: The diamond lattice model represented in (a) lattice space, (b) dual space, (c) and its experimental implementation with Rydberg atomic array.
• Figure 2: Phase diagram of the diamond lattice model. (a) The lattice space IPR $\xi_{R}$ and (b) the dual space IPR $\xi_{K}$ as functions of potential strength $\lambda$ and energy $E$. The phase boundaries, marked by dashed lines, are determined from the critical energies that have been exactly solved. The parameters $N=377$ and $t=1$.
• Figure 3: The expected experimental IPRs. (a,b) The real and dual space IPRs $\xi_{R/K}$ as functions of potential strength $\lambda$ and energy $E/J$ for $N=13$. Double space IPRs at $\lambda=1.5$ vs $E/J$ for (c) N=13 and (d) N=34.
• Figure S1: (a) The IPR $\xi_{R}$ versus $\lambda$, where the black dashed line is the mobility edge (ME) $E_{c}$. (b) The IPR $\xi_{R}$ versus $E$ for $\lambda=1.5$, where the black dashed line is the ME $E_{c}=\pm\frac{4}{3}+1$. (c) Amplitudes of the flat-band $E=-t$ eigenstates with IPR $\xi_{R}= 0.039$ in the first three unit cells at $\lambda = 1.5$. Throughout, $N = 377$ and $t = 1$.
• Figure S2: (a) Four critical points on the energy axis. (b-d) Three possible relative positions of the four critical points.