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Emergence of criticality through a cascade of delocalization transitions in quasiperiodic chains


Conduction through materials crucially depends on how ordered the materials are. Periodically ordered systems exhibit extended Bloch waves that generate metallic bands, whereas disorder is known to limit conduction and localize the motion of particles in a medium1,2. In this context, quasiperiodic systems, which are neither periodic nor disordered, demonstrate exotic conduction properties, self-similar wavefunctions and critical phenomena3. Here, we explore the localization properties of waves in a novel family of quasiperiodic chains obtained when continuously interpolating between two paradigmatic limits4: the Aubry–André model5,6, famous for its metal-to-insulator transition, and the Fibonacci chain7,8, known for its critical nature. We discover that the Aubry–André model evolves into criticality through a cascade of band-selective localization/delocalization transitions that iteratively shape the self-similar critical wavefunctions of the Fibonacci chain. Using experiments on cavity-polariton devices, we observe the first transition and reveal the microscopic origin of the cascade. Our findings offer (1) a unique new insight into understanding the criticality of quasiperiodic chains, (2) a controllable knob by which to engineer band-selective pass filters and (3) a versatile experimental platform with which to further study the interplay of many-body interactions and dissipation in a wide range of quasiperiodic models.

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Fig. 1: IAAF modulation and theoretical localization phase diagram.
Fig. 2: Continuum IAAF model and its experimental implementation.
Fig. 3: Experimental localization phase diagram.
Fig. 4: Spatial evolution with β of the lowest-energy eigenstate.

Data availability

Source data are available for this paper. All other data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.


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We thank Y.E. Kraus and Y. Lahini for fruitful discussions. A.Š. and O.Z. acknowledge financial support from the Swiss National Science Foundation through grant no. PP00P2 163818. J.L.L. acknowledges financial support from the ETH Fellowship programme. This work was supported by ERC grant Honeypol, H2020-FETFLAG project PhoQus (820392), QUANTERA project Interpol (ANR-QUAN-0003-05), the French National Research Agency project Quantum Fluids of Light (ANR-16-CE30-0021), the Paris Ile-de-France Région in the framework of DIM SIRTEQ, the French government through the Programme Investissement d’Avenir (I-SITE ULNE/ANR-16-IDEX-0004 ULNE) managed by the Agence Nationale de la Recherche, the French RENATECH network, Labex NanoSaclay (ICQOQS, grant no. ANR-10-LABX-0035), Labex CEMPI (ANR-11-LABX-0007), the CPER Photonics for Society P4S and the Métropole Européenne de Lille (MEL) via the project TFlight.

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Authors and Affiliations



A.Š. and J.L.L. performed the tight-binding theoretical work. V.G. and N.P. developed the continuum model simulations. V.G. and C.D. designed the samples. A.L., L.L.G., A.H. and I.S. fabricated the samples. V.G., N.P. and C.D. performed the experiments. V.G., A.Š., N.P., J.L.L., S.R., A.A., J.B. and O.Z. contributed to the data analysis (simulations and experiments), scientific discussions and the writing of the manuscript. J.B. and O.Z. supervised the work.

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Correspondence to J. Bloch or O. Zilberberg.

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Supplementary information

Supplementary Information

Supplementary Figs. 1–9, discussion and the captions of Videos 1 and 2.

Supplementary Video 1

This video shows the mechanism behind the first localization–delocalization transition together with the explanation of the relocalization on two sites.

Supplementary Video 2

This video shows the localization on four sites at higher beta-s.

Source data

Source Data Fig. 2

Experimental data for Fig. 2f,g.

Source Data Fig. 3

Experimental data for Fig. 3a–e.

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Goblot, V., Štrkalj, A., Pernet, N. et al. Emergence of criticality through a cascade of delocalization transitions in quasiperiodic chains. Nat. Phys. 16, 832–836 (2020).

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