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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.


  1. 1.

    Anderson, P. W. Absence of diffusion in certain random lattices. Phys. Rev. 109, 1492–1505 (1958).

    ADS  Article  Google Scholar 

  2. 2.

    Akkermans, E. & Montambaux, G. Mesoscopic Physics of Electrons and Photons (Cambridge Univ. Press, 2007).

  3. 3.

    Suck, J.-B., Schreiber, M. & Häussler, P. Quasicrystals: An Introduction to Structure, Physical Properties and Applications Vol. 55 (Springer, 2013).

  4. 4.

    Kraus, Y. E. & Zilberberg, O. Topological equivalence between the Fibonacci quasicrystal and the Harper model. Phys. Rev. Lett. 109, 116404 (2012).

    ADS  Article  Google Scholar 

  5. 5.

    Aubry, S. & André, G. Analyticity breaking and Anderson localization in incommensurate lattices. Ann. Israel Phys. Soc. 3, 133–140 (1980).

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Jitomirskaya, S. Y. Metal–insulator transition for the almost Mathieu operator. Ann. Math. 150, 1159–1175 (1999).

    MathSciNet  Article  Google Scholar 

  7. 7.

    Kohmoto, M., Kadanoff, L. P. & Tang, C. Localization problem in one dimension: mapping and escape. Phys. Rev. Lett. 50, 1870–1872 (1983).

    ADS  MathSciNet  Article  Google Scholar 

  8. 8.

    Ostlund, S., Pandit, R., Rand, D., Schellnhuber, H. J. & Siggia, E. D. One-dimensional Schrödinger equation with an almost periodic potential. Phys. Rev. Lett. 50, 1873–1876 (1983).

    ADS  MathSciNet  Article  Google Scholar 

  9. 9.

    Lee, P. A. & Ramakrishnan, T. V. Disordered electronic systems. Rev. Mod. Phys. 57, 287–337 (1985).

    ADS  Article  Google Scholar 

  10. 10.

    Evers, F. & Mirlin, A. D. Anderson transitions. Rev. Mod. Phys. 80, 1355–1417 (2008).

    ADS  Article  Google Scholar 

  11. 11.

    Segev, M., Silberberg, Y. & Christodoulides, D. Anderson localization of light. Nat. Photon. 7, 197–204 (2013).

    ADS  Article  Google Scholar 

  12. 12.

    Aulbach, C., Wobst, A., Ingold, G.-L., Hnggi, P. & Varga, I. Phase-space visualization of a metal–insulator transition. New J. Phys. 6, 70 (2004).

    ADS  Article  Google Scholar 

  13. 13.

    Mastropietro, V. Localization of interacting fermions in the Aubry–André model. Phys. Rev. Lett. 115, 180401 (2015).

    ADS  Article  Google Scholar 

  14. 14.

    Schreiber, M. et al. Observation of many-body localization of interacting fermions in a quasirandom optical lattice. Science 349, 842–845 (2015).

    ADS  MathSciNet  Article  Google Scholar 

  15. 15.

    Bordia, P., Lschen, H., Schneider, U., Knap, M. & Bloch, I. Periodically driving a many-body localized quantum system. Nat. Phys. 13, 460–464 (2017).

    Article  Google Scholar 

  16. 16.

    Macé, N., Laorencie, N. & Alet, F. Many-body localization in a quasiperiodic Fibonacci chain. SciPost Phys. 6, 050 (2019).

    ADS  Article  Google Scholar 

  17. 17.

    Varma, V. K. & Žnidarič, M. Diffusive transport in a quasiperiodic Fibonacci chain: absence of many-body localization at weak interactions. Phys. Rev. B 100, 085105 (2019).

    ADS  Article  Google Scholar 

  18. 18.

    Verbin, M., Zilberberg, O., Kraus, Y. E., Lahini, Y. & Silberberg, Y. Observation of topological phase transitions in photonic quasicrystals. Phys. Rev. Lett. 110, 076403 (2013).

    ADS  Article  Google Scholar 

  19. 19.

    Verbin, M., Zilberberg, O., Lahini, Y., Kraus, Y. E. & Silberberg, Y. Topological pumping over a photonic Fibonacci quasicrystal. Phys. Rev. B 91, 064201 (2015).

    ADS  Article  Google Scholar 

  20. 20.

    Kraus, Y. E. & Zilberberg, O. Quasiperiodicity and topology transcend dimensions. Nat. Phys. 12, 624–626 (2016).

    Article  Google Scholar 

  21. 21.

    Kraus, Y. E., Lahini, Y., Ringel, Z., Verbin, M. & Zilberberg, O. Topological states and adiabatic pumping in quasicrystals. Phys. Rev. Lett. 109, 106402 (2012).

    ADS  Article  Google Scholar 

  22. 22.

    Harper, P. G. Single band motion of conduction electrons in a uniform magnetic field. Proc. Phys. Soc. A 68, 874–878 (1955).

    ADS  Article  Google Scholar 

  23. 23.

    Hiramoto, H. & Kohmoto, M. New localization in a quasiperiodic system. Phys. Rev. Lett. 62, 2714–2717 (1989).

    ADS  MathSciNet  Article  Google Scholar 

  24. 24.

    Thouless, D. J. A relation between the density of states and range of localization for one dimensional random systems. J. Phys. C 5, 77–81 (1972).

    ADS  Article  Google Scholar 

  25. 25.

    Tanese, D. et al. Fractal energy spectrum of a polariton gas in a Fibonacci quasiperiodic potential. Phys. Rev. Lett. 112, 146404 (2014).

    ADS  Article  Google Scholar 

  26. 26.

    Baboux, F. et al. Measuring topological invariants from generalized edge states in polaritonic quasicrystals. Phys. Rev. B 95, 161114 (2017).

    ADS  Article  Google Scholar 

  27. 27.

    Biddle, J., Priour, D. J., Wang, B. & Das Sarma, S. Localization in one-dimensional lattices with nonnearest-neighbor hopping: generalized Anderson and Aubry–André models. Phys. Rev. B 83, 075105 (2011).

    ADS  Article  Google Scholar 

  28. 28.

    Ganeshan, S., Pixley, J. H. & Das Sarma, S. Nearest neighbor tight binding models with an exact mobility edge in one dimension. Phys. Rev. Lett. 114, 146601 (2015).

    ADS  Article  Google Scholar 

  29. 29.

    Lüschen, H. P. et al. Single-particle mobility edge in a one-dimensional quasiperiodic optical lattice. Phys. Rev. Lett. 120, 160404 (2018).

    ADS  Article  Google Scholar 

  30. 30.

    Roati, G. et al. Anderson localization of a non-interacting Bose–Einstein condensate. Nature 453, 895–898 (2008).

    ADS  Article  Google Scholar 

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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.

Author information




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.

Corresponding authors

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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