{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,11,10]],"date-time":"2026-11-10T21:16:27Z","timestamp":1794345387355,"version":"4.9.0"},"reference-count":30,"publisher":"Springer Science and Business Media LLC","issue":"8","license":[{"start":{"date-parts":[[2020,6,1]],"date-time":"2020-06-01T00:00:00Z","timestamp":1590969600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,6,1]],"date-time":"2020-06-01T00:00:00Z","timestamp":1590969600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Nat. Phys."],"published-print":{"date-parts":[[2020,8]]},"DOI":"10.1038\/s41567-020-0908-7","type":"journal-article","created":{"date-parts":[[2020,6,1]],"date-time":"2020-06-01T12:03:55Z","timestamp":1591013035000},"page":"832-836","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":143,"title":["Emergence of criticality through a cascade of delocalization transitions in quasiperiodic chains"],"prefix":"10.1038","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1113-9185","authenticated-orcid":false,"given":"V.","family":"Goblot","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9062-6001","authenticated-orcid":false,"given":"A.","family":"\u0160trkalj","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"N.","family":"Pernet","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9916-1589","authenticated-orcid":false,"given":"J. L.","family":"Lado","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"C.","family":"Dorow","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1892-9726","authenticated-orcid":false,"given":"A.","family":"Lema\u00eetre","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"L.","family":"Le Gratiet","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"A.","family":"Harouri","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8068-6599","authenticated-orcid":false,"given":"I.","family":"Sagnes","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1746-5245","authenticated-orcid":false,"given":"S.","family":"Ravets","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2508-8315","authenticated-orcid":false,"given":"A.","family":"Amo","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7823-8420","authenticated-orcid":false,"given":"J.","family":"Bloch","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1759-4920","authenticated-orcid":false,"given":"O.","family":"Zilberberg","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2020,6,1]]},"reference":[{"key":"908_CR1","doi-asserted-by":"publisher","first-page":"1492","DOI":"10.1103\/PhysRev.109.1492","volume":"109","author":"PW Anderson","year":"1958","unstructured":"Anderson, P. W. Absence of diffusion in certain random lattices. Phys. Rev. 109, 1492\u20131505 (1958).","journal-title":"Phys. Rev."},{"key":"908_CR2","doi-asserted-by":"crossref","unstructured":"Akkermans, E. & Montambaux, G. Mesoscopic Physics of Electrons and Photons (Cambridge Univ. Press, 2007).","DOI":"10.1017\/CBO9780511618833"},{"key":"908_CR3","unstructured":"Suck, J.-B., Schreiber, M. & H\u00e4ussler, P. Quasicrystals: An Introduction to Structure, Physical Properties and Applications Vol. 55 (Springer, 2013)."},{"key":"908_CR4","doi-asserted-by":"publisher","first-page":"116404","DOI":"10.1103\/PhysRevLett.109.116404","volume":"109","author":"YE Kraus","year":"2012","unstructured":"Kraus, Y. E. & Zilberberg, O. Topological equivalence between the Fibonacci quasicrystal and the Harper model. Phys. Rev. Lett. 109, 116404 (2012).","journal-title":"Phys. Rev. Lett."},{"key":"908_CR5","first-page":"133","volume":"3","author":"S Aubry","year":"1980","unstructured":"Aubry, S. & Andr\u00e9, G. Analyticity breaking and Anderson localization in incommensurate lattices. Ann. Israel Phys. Soc. 3, 133\u2013140 (1980).","journal-title":"Ann. Israel Phys. Soc."},{"key":"908_CR6","doi-asserted-by":"publisher","first-page":"1159","DOI":"10.2307\/121066","volume":"150","author":"SY Jitomirskaya","year":"1999","unstructured":"Jitomirskaya, S. Y. Metal\u2013insulator transition for the almost Mathieu operator. Ann. Math. 150, 1159\u20131175 (1999).","journal-title":"Ann. Math."},{"key":"908_CR7","doi-asserted-by":"publisher","first-page":"1870","DOI":"10.1103\/PhysRevLett.50.1870","volume":"50","author":"M Kohmoto","year":"1983","unstructured":"Kohmoto, M., Kadanoff, L. P. & Tang, C. Localization problem in one dimension: mapping and escape. Phys. Rev. Lett. 50, 1870\u20131872 (1983).","journal-title":"Phys. Rev. Lett."},{"key":"908_CR8","doi-asserted-by":"publisher","first-page":"1873","DOI":"10.1103\/PhysRevLett.50.1873","volume":"50","author":"S Ostlund","year":"1983","unstructured":"Ostlund, S., Pandit, R., Rand, D., Schellnhuber, H. J. & Siggia, E. D. One-dimensional Schr\u00f6dinger equation with an almost periodic potential. Phys. Rev. Lett. 50, 1873\u20131876 (1983).","journal-title":"Phys. Rev. Lett."},{"key":"908_CR9","doi-asserted-by":"publisher","first-page":"287","DOI":"10.1103\/RevModPhys.57.287","volume":"57","author":"PA Lee","year":"1985","unstructured":"Lee, P. A. & Ramakrishnan, T. V. Disordered electronic systems. Rev. Mod. Phys. 57, 287\u2013337 (1985).","journal-title":"Rev. Mod. Phys."},{"key":"908_CR10","doi-asserted-by":"publisher","first-page":"1355","DOI":"10.1103\/RevModPhys.80.1355","volume":"80","author":"F Evers","year":"2008","unstructured":"Evers, F. & Mirlin, A. D. Anderson transitions. Rev. Mod. Phys. 80, 1355\u20131417 (2008).","journal-title":"Rev. Mod. Phys."},{"key":"908_CR11","doi-asserted-by":"publisher","first-page":"197","DOI":"10.1038\/nphoton.2013.30","volume":"7","author":"M Segev","year":"2013","unstructured":"Segev, M., Silberberg, Y. & Christodoulides, D. Anderson localization of light. Nat. Photon. 7, 197\u2013204 (2013).","journal-title":"Nat. Photon."},{"key":"908_CR12","doi-asserted-by":"publisher","first-page":"70","DOI":"10.1088\/1367-2630\/6\/1\/070","volume":"6","author":"C Aulbach","year":"2004","unstructured":"Aulbach, C., Wobst, A., Ingold, G.-L., Hnggi, P. & Varga, I. Phase-space visualization of a metal\u2013insulator transition. New J. Phys. 6, 70 (2004).","journal-title":"New J. Phys."},{"key":"908_CR13","doi-asserted-by":"publisher","first-page":"180401","DOI":"10.1103\/PhysRevLett.115.180401","volume":"115","author":"V Mastropietro","year":"2015","unstructured":"Mastropietro, V. Localization of interacting fermions in the Aubry\u2013Andr\u00e9 model. Phys. Rev. Lett. 115, 180401 (2015).","journal-title":"Phys. Rev. Lett."},{"key":"908_CR14","doi-asserted-by":"publisher","first-page":"842","DOI":"10.1126\/science.aaa7432","volume":"349","author":"M Schreiber","year":"2015","unstructured":"Schreiber, M. et al. Observation of many-body localization of interacting fermions in a quasirandom optical lattice. Science 349, 842\u2013845 (2015).","journal-title":"Science"},{"key":"908_CR15","doi-asserted-by":"publisher","first-page":"460","DOI":"10.1038\/nphys4020","volume":"13","author":"P Bordia","year":"2017","unstructured":"Bordia, P., Lschen, H., Schneider, U., Knap, M. & Bloch, I. Periodically driving a many-body localized quantum system. Nat. Phys. 13, 460\u2013464 (2017).","journal-title":"Nat. Phys."},{"key":"908_CR16","doi-asserted-by":"publisher","first-page":"050","DOI":"10.21468\/SciPostPhys.6.4.050","volume":"6","author":"N Mac\u00e9","year":"2019","unstructured":"Mac\u00e9, N., Laorencie, N. & Alet, F. Many-body localization in a quasiperiodic Fibonacci chain. SciPost Phys. 6, 050 (2019).","journal-title":"SciPost Phys."},{"key":"908_CR17","doi-asserted-by":"publisher","first-page":"085105","DOI":"10.1103\/PhysRevB.100.085105","volume":"100","author":"VK Varma","year":"2019","unstructured":"Varma, V. K. & \u017dnidari\u010d, M. Diffusive transport in a quasiperiodic Fibonacci chain: absence of many-body localization at weak interactions. Phys. Rev. B 100, 085105 (2019).","journal-title":"Phys. Rev. B"},{"key":"908_CR18","doi-asserted-by":"publisher","first-page":"076403","DOI":"10.1103\/PhysRevLett.110.076403","volume":"110","author":"M Verbin","year":"2013","unstructured":"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).","journal-title":"Phys. Rev. Lett."},{"key":"908_CR19","doi-asserted-by":"publisher","first-page":"064201","DOI":"10.1103\/PhysRevB.91.064201","volume":"91","author":"M Verbin","year":"2015","unstructured":"Verbin, M., Zilberberg, O., Lahini, Y., Kraus, Y. E. & Silberberg, Y. Topological pumping over a photonic Fibonacci quasicrystal. Phys. Rev. B 91, 064201 (2015).","journal-title":"Phys. Rev. B"},{"key":"908_CR20","doi-asserted-by":"publisher","first-page":"624","DOI":"10.1038\/nphys3784","volume":"12","author":"YE Kraus","year":"2016","unstructured":"Kraus, Y. E. & Zilberberg, O. Quasiperiodicity and topology transcend dimensions. Nat. Phys. 12, 624\u2013626 (2016).","journal-title":"Nat. Phys."},{"key":"908_CR21","doi-asserted-by":"publisher","first-page":"106402","DOI":"10.1103\/PhysRevLett.109.106402","volume":"109","author":"YE Kraus","year":"2012","unstructured":"Kraus, Y. E., Lahini, Y., Ringel, Z., Verbin, M. & Zilberberg, O. Topological states and adiabatic pumping in quasicrystals. Phys. Rev. Lett. 109, 106402 (2012).","journal-title":"Phys. Rev. Lett."},{"key":"908_CR22","doi-asserted-by":"publisher","first-page":"874","DOI":"10.1088\/0370-1298\/68\/10\/304","volume":"68","author":"PG Harper","year":"1955","unstructured":"Harper, P. G. Single band motion of conduction electrons in a uniform magnetic field. Proc. Phys. Soc. A 68, 874\u2013878 (1955).","journal-title":"Proc. Phys. Soc. A"},{"key":"908_CR23","doi-asserted-by":"publisher","first-page":"2714","DOI":"10.1103\/PhysRevLett.62.2714","volume":"62","author":"H Hiramoto","year":"1989","unstructured":"Hiramoto, H. & Kohmoto, M. New localization in a quasiperiodic system. Phys. Rev. Lett. 62, 2714\u20132717 (1989).","journal-title":"Phys. Rev. Lett."},{"key":"908_CR24","doi-asserted-by":"publisher","first-page":"77","DOI":"10.1088\/0022-3719\/5\/1\/010","volume":"5","author":"DJ Thouless","year":"1972","unstructured":"Thouless, D. J. A relation between the density of states and range of localization for one dimensional random systems. J. Phys. C 5, 77\u201381 (1972).","journal-title":"J. Phys. C"},{"key":"908_CR25","doi-asserted-by":"publisher","first-page":"146404","DOI":"10.1103\/PhysRevLett.112.146404","volume":"112","author":"D Tanese","year":"2014","unstructured":"Tanese, D. et al. Fractal energy spectrum of a polariton gas in a Fibonacci quasiperiodic potential. Phys. Rev. Lett. 112, 146404 (2014).","journal-title":"Phys. Rev. Lett."},{"key":"908_CR26","doi-asserted-by":"publisher","first-page":"161114","DOI":"10.1103\/PhysRevB.95.161114","volume":"95","author":"F Baboux","year":"2017","unstructured":"Baboux, F. et al. Measuring topological invariants from generalized edge states in polaritonic quasicrystals. Phys. Rev. B 95, 161114 (2017).","journal-title":"Phys. Rev. B"},{"key":"908_CR27","doi-asserted-by":"publisher","first-page":"075105","DOI":"10.1103\/PhysRevB.83.075105","volume":"83","author":"J Biddle","year":"2011","unstructured":"Biddle, J., Priour, D. J., Wang, B. & Das Sarma, S. Localization in one-dimensional lattices with nonnearest-neighbor hopping: generalized Anderson and Aubry\u2013Andr\u00e9 models. Phys. Rev. B 83, 075105 (2011).","journal-title":"Phys. Rev. B"},{"key":"908_CR28","doi-asserted-by":"publisher","first-page":"146601","DOI":"10.1103\/PhysRevLett.114.146601","volume":"114","author":"S Ganeshan","year":"2015","unstructured":"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).","journal-title":"Phys. Rev. Lett."},{"key":"908_CR29","doi-asserted-by":"publisher","first-page":"160404","DOI":"10.1103\/PhysRevLett.120.160404","volume":"120","author":"HP L\u00fcschen","year":"2018","unstructured":"L\u00fcschen, H. P. et al. Single-particle mobility edge in a one-dimensional quasiperiodic optical lattice. Phys. Rev. Lett. 120, 160404 (2018).","journal-title":"Phys. Rev. Lett."},{"key":"908_CR30","doi-asserted-by":"publisher","first-page":"895","DOI":"10.1038\/nature07071","volume":"453","author":"G Roati","year":"2008","unstructured":"Roati, G. et al. Anderson localization of a non-interacting Bose\u2013Einstein condensate. Nature 453, 895\u2013898 (2008).","journal-title":"Nature"}],"container-title":["Nature Physics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.nature.com\/articles\/s41567-020-0908-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/www.nature.com\/articles\/s41567-020-0908-7","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/www.nature.com\/articles\/s41567-020-0908-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,5,20]],"date-time":"2023-05-20T18:17:13Z","timestamp":1684606633000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.nature.com\/articles\/s41567-020-0908-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,6,1]]},"references-count":30,"journal-issue":{"issue":"8","published-print":{"date-parts":[[2020,8]]}},"alternative-id":["908"],"URL":"https:\/\/doi.org\/10.1038\/s41567-020-0908-7","relation":{},"ISSN":["1745-2473","1745-2481"],"issn-type":[{"value":"1745-2473","type":"print"},{"value":"1745-2481","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,6,1]]},"assertion":[{"value":"18 November 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 April 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 June 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare no competing interests.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}]}}