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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Anderson, P. W. Absence of diffusion in certain random lattices. Phys. Rev. 109, 1492–1505 (1958).
Akkermans, E. & Montambaux, G. Mesoscopic Physics of Electrons and Photons (Cambridge Univ. Press, 2007).
Suck, J.-B., Schreiber, M. & Häussler, P. Quasicrystals: An Introduction to Structure, Physical Properties and Applications Vol. 55 (Springer, 2013).
Kraus, Y. E. & Zilberberg, O. Topological equivalence between the Fibonacci quasicrystal and the Harper model. Phys. Rev. Lett. 109, 116404 (2012).
Aubry, S. & André, G. Analyticity breaking and Anderson localization in incommensurate lattices. Ann. Israel Phys. Soc. 3, 133–140 (1980).
Jitomirskaya, S. Y. Metal–insulator transition for the almost Mathieu operator. Ann. Math. 150, 1159–1175 (1999).
Kohmoto, M., Kadanoff, L. P. & Tang, C. Localization problem in one dimension: mapping and escape. Phys. Rev. Lett. 50, 1870–1872 (1983).
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).
Lee, P. A. & Ramakrishnan, T. V. Disordered electronic systems. Rev. Mod. Phys. 57, 287–337 (1985).
Evers, F. & Mirlin, A. D. Anderson transitions. Rev. Mod. Phys. 80, 1355–1417 (2008).
Segev, M., Silberberg, Y. & Christodoulides, D. Anderson localization of light. Nat. Photon. 7, 197–204 (2013).
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).
Mastropietro, V. Localization of interacting fermions in the Aubry–André model. Phys. Rev. Lett. 115, 180401 (2015).
Schreiber, M. et al. Observation of many-body localization of interacting fermions in a quasirandom optical lattice. Science 349, 842–845 (2015).
Bordia, P., Lschen, H., Schneider, U., Knap, M. & Bloch, I. Periodically driving a many-body localized quantum system. Nat. Phys. 13, 460–464 (2017).
Macé, N., Laorencie, N. & Alet, F. Many-body localization in a quasiperiodic Fibonacci chain. SciPost Phys. 6, 050 (2019).
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).
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).
Verbin, M., Zilberberg, O., Lahini, Y., Kraus, Y. E. & Silberberg, Y. Topological pumping over a photonic Fibonacci quasicrystal. Phys. Rev. B 91, 064201 (2015).
Kraus, Y. E. & Zilberberg, O. Quasiperiodicity and topology transcend dimensions. Nat. Phys. 12, 624–626 (2016).
Kraus, Y. E., Lahini, Y., Ringel, Z., Verbin, M. & Zilberberg, O. Topological states and adiabatic pumping in quasicrystals. Phys. Rev. Lett. 109, 106402 (2012).
Harper, P. G. Single band motion of conduction electrons in a uniform magnetic field. Proc. Phys. Soc. A 68, 874–878 (1955).
Hiramoto, H. & Kohmoto, M. New localization in a quasiperiodic system. Phys. Rev. Lett. 62, 2714–2717 (1989).
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).
Tanese, D. et al. Fractal energy spectrum of a polariton gas in a Fibonacci quasiperiodic potential. Phys. Rev. Lett. 112, 146404 (2014).
Baboux, F. et al. Measuring topological invariants from generalized edge states in polaritonic quasicrystals. Phys. Rev. B 95, 161114 (2017).
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).
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).
Lüschen, H. P. et al. Single-particle mobility edge in a one-dimensional quasiperiodic optical lattice. Phys. Rev. Lett. 120, 160404 (2018).
Roati, G. et al. Anderson localization of a non-interacting Bose–Einstein condensate. Nature 453, 895–898 (2008).
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.
The authors declare no competing interests.
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Supplementary Figs. 1–9, discussion and the captions of Videos 1 and 2.
This video shows the mechanism behind the first localization–delocalization transition together with the explanation of the relocalization on two sites.
This video shows the localization on four sites at higher beta-s.
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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). https://doi.org/10.1038/s41567-020-0908-7
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