{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,19]],"date-time":"2026-05-19T01:07:11Z","timestamp":1779152831373,"version":"3.51.4"},"reference-count":60,"publisher":"Springer Science and Business Media LLC","issue":"12","license":[{"start":{"date-parts":[[2024,10,11]],"date-time":"2024-10-11T00:00:00Z","timestamp":1728604800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,10,11]],"date-time":"2024-10-11T00:00:00Z","timestamp":1728604800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Nat. Phys."],"published-print":{"date-parts":[[2024,12]]},"DOI":"10.1038\/s41567-024-02662-2","type":"journal-article","created":{"date-parts":[[2024,10,11]],"date-time":"2024-10-11T06:02:04Z","timestamp":1728626524000},"page":"1933-1940","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["Incommensurability enabled quasi-fractal order in 1D narrow-band moir\u00e9 systems"],"prefix":"10.1038","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4786-539X","authenticated-orcid":false,"given":"Miguel","family":"Gon\u00e7alves","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8566-0718","authenticated-orcid":false,"given":"Bruno","family":"Amorim","sequence":"additional","affiliation":[]},{"given":"Flavio","family":"Riche","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0993-3734","authenticated-orcid":false,"given":"Eduardo V.","family":"Castro","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7630-2054","authenticated-orcid":false,"given":"Pedro","family":"Ribeiro","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,10,11]]},"reference":[{"key":"2662_CR1","unstructured":"Aubry, S. & Andr\u00e9, G. Analyticity breaking and Anderson localization in incommensurate lattices. Ann. Isr. Phys. Soc. 3, 18 (1980)."},{"key":"2662_CR2","doi-asserted-by":"crossref","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-Einstein condensate. Nature 453, 895\u2013898 (2008).","journal-title":"Nature"},{"key":"2662_CR3","doi-asserted-by":"crossref","unstructured":"Lahini, Y et al. Observation of a localization transition in quasiperiodic photonic lattices. Phys. Rev. Lett. 103, 013901 (2009).","DOI":"10.1103\/PhysRevLett.103.013901"},{"key":"2662_CR4","doi-asserted-by":"crossref","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":"2662_CR5","doi-asserted-by":"crossref","unstructured":"L\u00fcschen, H. P. et al. Single-particle mobility edge in a one-dimensional quasiperiodic optical lattice. Phys. Rev. Lett. 120, 160404 (2018).","DOI":"10.1103\/PhysRevLett.120.160404"},{"key":"2662_CR6","doi-asserted-by":"crossref","first-page":"207604","DOI":"10.1103\/PhysRevLett.120.207604","volume":"120","author":"JH Pixley","year":"2018","unstructured":"Pixley, J. H., Wilson, J. H., Huse, D. A. & Gopalakrishnan, S. Weyl semimetal to metal phase transitions driven by quasiperiodic potentials. Phys. Rev. Lett. 120, 207604 (2018).","journal-title":"Phys. Rev. Lett."},{"key":"2662_CR7","doi-asserted-by":"crossref","first-page":"245401","DOI":"10.1103\/PhysRevB.99.245401","volume":"99","author":"MoonJip Park","year":"2019","unstructured":"Park, MoonJip, Kim, HeeSeung & Lee, S. Emergent localization in dodecagonal bilayer quasicrystals. Phys. Rev. B 99, 245401 (2019).","journal-title":"Phys. Rev. B"},{"key":"2662_CR8","doi-asserted-by":"crossref","first-page":"144202","DOI":"10.1103\/PhysRevB.100.144202","volume":"100","author":"B Huang","year":"2019","unstructured":"Huang, B. & Liu, W. V. Moir\u00e9 localization in two-dimensional quasiperiodic systems. Phys. Rev. B 100, 144202 (2019).","journal-title":"Phys. Rev. B"},{"key":"2662_CR9","volume":"5","author":"Y Fu","year":"2020","unstructured":"Fu, Y., K\u00f6nig, E. J., Wilson, J. H., Chou, Yang-Zhi & Pixley, J. H. Magic-angle semimetals. npj Quantum Mater. 5, 71 (2020).","journal-title":"npj Quantum Mater."},{"key":"2662_CR10","doi-asserted-by":"crossref","first-page":"011001","DOI":"10.1088\/2053-1583\/ac3259","volume":"9","author":"M Gon\u00e7alves","year":"2021","unstructured":"Gon\u00e7alves, M. et al. Incommensurability-induced sub-ballistic narrow-band-states in twisted bilayer graphene. 2D Mater. 9, 011001 (2021).","journal-title":"2D Mater."},{"key":"2662_CR11","first-page":"041047","volume":"7","author":"P Bordia","year":"2017","unstructured":"Bordia, P. et al. Probing slow relaxation and many-body localization in two-dimensional quasiperiodic systems. Phys. Rev. X 7, 041047 (2017).","journal-title":"Phys. Rev. X"},{"key":"2662_CR12","doi-asserted-by":"crossref","first-page":"070405","DOI":"10.1103\/PhysRevLett.123.070405","volume":"123","author":"H Yao","year":"2019","unstructured":"Yao, H., Khoudli, H., Bresque, L. \u00e9a & Sanchez-Palencia, L. Critical behavior and fractality in shallow one-dimensional quasiperiodic potentials. Phys. Rev. Lett. 123, 070405 (2019).","journal-title":"Phys. Rev. Lett."},{"key":"2662_CR13","doi-asserted-by":"crossref","first-page":"110401","DOI":"10.1103\/PhysRevLett.126.110401","volume":"126","author":"R Gautier","year":"2021","unstructured":"Gautier, R., Yao, H. & Sanchez-Palencia, L. Strongly interacting bosons in a two-dimensional quasicrystal lattice. Phys. Rev. Lett. 126, 110401 (2021).","journal-title":"Phys. Rev. Lett."},{"key":"2662_CR14","doi-asserted-by":"crossref","first-page":"054204","DOI":"10.1103\/PhysRevB.106.054204","volume":"106","author":"DS Borgnia","year":"2022","unstructured":"Borgnia, D. S., Vishwanath, A. & Slager, R.-J. Rational approximations of quasiperiodicity via projected Green\u2019s functions. Phys. Rev. B 106, 054204 (2022).","journal-title":"Phys. Rev. B"},{"key":"2662_CR15","doi-asserted-by":"crossref","first-page":"014108","DOI":"10.1103\/PhysRevB.91.014108","volume":"91","author":"F Liu","year":"2015","unstructured":"Liu, F., Ghosh, S. & Chong, Y. D. Localization and adiabatic pumping in a generalized Aubry-Andr\u00e9-Harper model. Phys. Rev. B 91, 014108 (2015).","journal-title":"Phys. Rev. B"},{"key":"2662_CR16","doi-asserted-by":"crossref","unstructured":"\u010cade\u017d, T., Mondaini, R. & Sacramento, P. D. Edge and bulk localization of Floquet topological superconductors. Phys. Rev. B 99, 014301 (2019).","DOI":"10.1103\/PhysRevB.99.014301"},{"key":"2662_CR17","doi-asserted-by":"crossref","first-page":"073204","DOI":"10.1103\/PhysRevLett.125.073204","volume":"125","author":"Y Wang","year":"2020","unstructured":"Wang, Y., Zhang, L., Niu, S., Yu, D. & Liu, X.-J. Realization and detection of nonergodic critical phases in an optical Raman lattice. Phys. Rev. Lett. 125, 073204 (2020).","journal-title":"Phys. Rev. Lett."},{"key":"2662_CR18","doi-asserted-by":"crossref","first-page":"27","DOI":"10.21468\/SciPostPhys.12.1.027","volume":"12","author":"T Liu","year":"2022","unstructured":"Liu, T., Xia, X., Longhi, S. & Sanchez-Palencia, L. Anomalous mobility edges in one-dimensional quasiperiodic models. SciPost Phys. 12, 27 (2022).","journal-title":"SciPost Phys."},{"key":"2662_CR19","doi-asserted-by":"crossref","first-page":"L100201","DOI":"10.1103\/PhysRevB.108.L100201","volume":"108","author":"M Gon\u00e7alves","year":"2023","unstructured":"Gon\u00e7alves, M., Amorim, B., Castro, E. V. & Ribeiro, P. Renormalization group theory of one-dimensional quasiperiodic lattice models with commensurate approximants. Phys. Rev. B 108, L100201 (2023).","journal-title":"Phys. Rev. B"},{"key":"2662_CR20","doi-asserted-by":"crossref","first-page":"186303","DOI":"10.1103\/PhysRevLett.131.186303","volume":"131","author":"M Gon\u00e7alves","year":"2023","unstructured":"Gon\u00e7alves, M., Amorim, B., Castro, E. V. & Ribeiro, P. Critical phase dualities in 1D exactly solvable quasiperiodic models. Phys. Rev. Lett. 131, 186303 (2023).","journal-title":"Phys. Rev. Lett."},{"key":"2662_CR21","doi-asserted-by":"crossref","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":"2662_CR22","doi-asserted-by":"crossref","first-page":"1143","DOI":"10.1364\/OME.416552","volume":"11","author":"O Zilberberg","year":"2021","unstructured":"Zilberberg, O. Topology in quasicrystals. Opt. Mater. Express 11, 1143\u20131157 (2021).","journal-title":"Opt. Mater. Express"},{"key":"2662_CR23","doi-asserted-by":"crossref","first-page":"033009","DOI":"10.1103\/PhysRevResearch.1.033009","volume":"1","author":"JL Lado","year":"2019","unstructured":"Lado, J. L. & Zilberberg, O. Topological spin excitations in Harper-Heisenberg spin chains. Phys. Rev. Res. 1, 033009 (2019).","journal-title":"Phys. Rev. Res."},{"key":"2662_CR24","doi-asserted-by":"crossref","first-page":"63404","DOI":"10.1103\/PhysRevA.75.063404","volume":"75","author":"DJ Boers","year":"2007","unstructured":"Boers, D. J., Goedeke, B., Hinrichs, D. & Holthaus, M. Mobility edges in bichromatic optical lattices. Phys. Rev. A 75, 63404 (2007).","journal-title":"Phys. Rev. A"},{"key":"2662_CR25","doi-asserted-by":"crossref","first-page":"060401","DOI":"10.1103\/PhysRevLett.125.060401","volume":"125","author":"H Yao","year":"2020","unstructured":"Yao, H., Giamarchi, T. & Sanchez-Palencia, L. Lieb-Liniger bosons in a shallow quasiperiodic potential: Bose glass phase and fractal Mott lobes. Phys. Rev. Lett. 125, 060401 (2020).","journal-title":"Phys. Rev. Lett."},{"key":"2662_CR26","doi-asserted-by":"crossref","first-page":"040603","DOI":"10.1103\/PhysRevLett.126.040603","volume":"126","author":"FA An","year":"2021","unstructured":"An, F. A. et al. Interactions and mobility edges: observing the generalized Aubry-Andr\u00e9 model. Phys. Rev. Lett. 126, 040603 (2021).","journal-title":"Phys. Rev. Lett."},{"key":"2662_CR27","doi-asserted-by":"crossref","first-page":"170403","DOI":"10.1103\/PhysRevLett.122.170403","volume":"122","author":"T Kohlert","year":"2019","unstructured":"Kohlert, T. et al. Observation of many-body localization in a one-dimensional system with a single-particle mobility edge. Phys. Rev. Lett. 122, 170403 (2019).","journal-title":"Phys. Rev. Lett."},{"key":"2662_CR28","doi-asserted-by":"crossref","first-page":"2001170","DOI":"10.1002\/adom.202001170","volume":"8","author":"AD Sinelnik","year":"2020","unstructured":"Sinelnik, A. D. et al. Experimental observation of intrinsic light localization in photonic icosahedral quasicrystals. Adv. Opt. Mater. 8, 2001170 (2020).","journal-title":"Adv. Opt. Mater."},{"key":"2662_CR29","doi-asserted-by":"crossref","first-page":"725","DOI":"10.1038\/s41567-020-0906-9","volume":"16","author":"L Balents","year":"2020","unstructured":"Balents, L., Dean, C. R., Efetov, D. K. & Young, A. F. Superconductivity and strong correlations in moir\u00e9 flat bands. Nat. Phys. 16, 725\u2013733 (2020).","journal-title":"Nat. Phys."},{"key":"2662_CR30","doi-asserted-by":"crossref","first-page":"1265","DOI":"10.1038\/s41563-020-00840-0","volume":"19","author":"EY Andrei","year":"2020","unstructured":"Andrei, E. Y. & MacDonald, A. H. Graphene bilayers with a twist. Nat. Mater. 19, 1265\u20131275 (2020).","journal-title":"Nat. Mater."},{"key":"2662_CR31","doi-asserted-by":"crossref","first-page":"2863","DOI":"10.1103\/PhysRevLett.69.2863","volume":"69","author":"SR White","year":"1992","unstructured":"White, S. R. Density matrix formulation for quantum renormalization groups. Phys. Rev. Lett. 69, 2863\u20132866 (1992).","journal-title":"Phys. Rev. Lett."},{"key":"2662_CR32","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1103\/RevModPhys.77.259","volume":"77","author":"U Schollw\u00f6ck","year":"2005","unstructured":"Schollw\u00f6ck, U. The density-matrix renormalization group. Rev. Mod. Phys. 77, 259\u2013315 (2005).","journal-title":"Rev. Mod. Phys."},{"key":"2662_CR33","doi-asserted-by":"crossref","unstructured":"Szab\u00f3, A. & Schneider, U. Mixed spectra and partially extended states in a two-dimensional quasiperiodic model. Phys. Rev. B 101, 014205 (2020).","DOI":"10.1103\/PhysRevB.101.014205"},{"key":"2662_CR34","unstructured":"Gon\u00e7alves, T. S., Gon\u00e7alves, M., Ribeiro, P. & Amorim, B. Topological phase transitions for any taste in 2D quasiperiodic systems. Preprint at https:\/\/arxiv.org\/abs\/2212.08024 (2022)."},{"key":"2662_CR35","doi-asserted-by":"crossref","first-page":"176401","DOI":"10.1103\/PhysRevLett.131.176401","volume":"131","author":"X-C Zhou","year":"2023","unstructured":"Zhou, X.-C., Wang, Y., Poon, T.-F. J., Zhou, Q. & Liu, X.-J. Exact new mobility edges between critical and localized states. Phys. Rev. Lett. 131, 176401 (2023).","journal-title":"Phys. Rev. Lett."},{"key":"2662_CR36","doi-asserted-by":"crossref","first-page":"017102","DOI":"10.1088\/0256-307X\/41\/1\/017102","volume":"41","author":"T Liu","year":"2024","unstructured":"Liu, T. & Xia, X. Predicted critical state based on invariance of the Lyapunov exponent in dual spaces. Chin. Phys. Lett. 41, 017102 (2024).","journal-title":"Chin. Phys. Lett."},{"key":"2662_CR37","doi-asserted-by":"crossref","first-page":"134202","DOI":"10.1103\/PhysRevB.87.134202","volume":"87","author":"S Iyer","year":"2013","unstructured":"Iyer, S., Oganesyan, V., Refael, G. & Huse, D. A. Many-body localization in a quasiperiodic system. Phys. Rev. B 87, 134202 (2013).","journal-title":"Phys. Rev. B"},{"key":"2662_CR38","doi-asserted-by":"crossref","first-page":"041601","DOI":"10.1103\/PhysRevA.92.041601","volume":"92","author":"R Mondaini","year":"2015","unstructured":"Mondaini, R. & Rigol, M. Many-body localization and thermalization in disordered Hubbard chains. Phys. Rev. A 92, 041601 (2015).","journal-title":"Phys. Rev. A"},{"key":"2662_CR39","doi-asserted-by":"crossref","first-page":"4595","DOI":"10.1073\/pnas.1800589115","volume":"115","author":"M \u017dnidari\u010d","year":"2018","unstructured":"\u017dnidari\u010d, M. & Ljubotina, M. Interaction instability of localization in quasiperiodic systems. Proc. Natl Acad. Sci. USA 115, 4595\u20134600 (2018).","journal-title":"Proc. Natl Acad. Sci. USA"},{"key":"2662_CR40","doi-asserted-by":"crossref","first-page":"032039","DOI":"10.1103\/PhysRevResearch.1.032039","volume":"1","author":"S Xu","year":"2019","unstructured":"Xu, S., Li, X., Hsu, Y.-T., Swingle, B. & Das Sarma, S. Butterfly effect in interacting Aubry-Andre model: thermalization, slow scrambling, and many-body localization. Phys. Rev. Res. 1, 032039 (2019).","journal-title":"Phys. Rev. Res."},{"key":"2662_CR41","doi-asserted-by":"crossref","first-page":"104203","DOI":"10.1103\/PhysRevB.100.104203","volume":"100","author":"E V. H. Doggen","year":"2019","unstructured":"V. H. Doggen, E. & Mirlin, A. D. Many-body delocalization dynamics in long Aubry-Andr\u00e9 quasiperiodic chains. Phys. Rev. B 100, 104203 (2019).","journal-title":"Phys. Rev. B"},{"key":"2662_CR42","doi-asserted-by":"crossref","first-page":"146601","DOI":"10.1103\/PhysRevLett.128.146601","volume":"128","author":"DD Vu","year":"2022","unstructured":"Vu, D. D., Huang, K., Li, X. & Das Sarma, S. Fermionic many-body localization for random and quasiperiodic systems in the presence of short- and long-range interactions. Phys. Rev. Lett. 128, 146601 (2022).","journal-title":"Phys. Rev. Lett."},{"key":"2662_CR43","doi-asserted-by":"crossref","first-page":"214201","DOI":"10.1103\/PhysRevB.104.214201","volume":"104","author":"AS Aramthottil","year":"2021","unstructured":"Aramthottil, A. S., Chanda, T., Sierant, P. & Zakrzewski, J. Finite-size scaling analysis of the many-body localization transition in quasiperiodic spin chains. Phys. Rev. B 104, 214201 (2021).","journal-title":"Phys. Rev. B"},{"key":"2662_CR44","doi-asserted-by":"crossref","first-page":"080602","DOI":"10.1103\/PhysRevLett.126.080602","volume":"126","author":"Y Wang","year":"2021","unstructured":"Wang, Y., Cheng, C., Liu, X.-J. & Yu, D. Many-body critical phase: extended and nonthermal. Phys. Rev. Lett. 126, 080602 (2021).","journal-title":"Phys. Rev. Lett."},{"key":"2662_CR45","doi-asserted-by":"crossref","first-page":"161106","DOI":"10.1103\/PhysRevB.89.161106","volume":"89","author":"YE Kraus","year":"2014","unstructured":"Kraus, Y. E., Zilberberg, O. & Berkovits, R. Enhanced compressibility due to repulsive interaction in the Harper model. Phys. Rev. B 89, 161106 (2014).","journal-title":"Phys. Rev. B"},{"key":"2662_CR46","doi-asserted-by":"crossref","first-page":"010","DOI":"10.21468\/SciPostPhys.1.1.010","volume":"1","author":"P Naldesi","year":"2016","unstructured":"Naldesi, P., Ercolessi, E. & Roscilde, T. Detecting a many-body mobility edge with quantum quenches. SciPost Phys. 1, 010 (2016).","journal-title":"SciPost Phys."},{"key":"2662_CR47","doi-asserted-by":"crossref","first-page":"174203","DOI":"10.1103\/PhysRevB.101.174203","volume":"101","author":"T Cookmeyer","year":"2020","unstructured":"Cookmeyer, T., Motruk, J. & Moore, J. E. Critical properties of the ground-state localization-delocalization transition in the many-particle Aubry-Andr\u00e9 model. Phys. Rev. B 101, 174203 (2020).","journal-title":"Phys. Rev. B"},{"key":"2662_CR48","doi-asserted-by":"crossref","first-page":"036803","DOI":"10.1103\/PhysRevLett.126.036803","volume":"126","author":"DD Vu","year":"2021","unstructured":"Vu, D. D. & Das Sarma, S. Moir\u00e9 versus Mott: incommensuration and interaction in one-dimensional bichromatic lattices. Phys. Rev. Lett. 126, 036803 (2021).","journal-title":"Phys. Rev. Lett."},{"key":"2662_CR49","unstructured":"Oliveira, R., Gon\u00e7alves, M., Ribeiro, P., Castro, E. V. & Amorim, B. Incommensurability-induced enhancement of superconductivity in one dimensional critical systems. Preprint at https:\/\/arxiv.org\/abs\/2303.17656 (2023)."},{"key":"2662_CR50","doi-asserted-by":"crossref","first-page":"014211","DOI":"10.1103\/PhysRevB.109.014211","volume":"109","author":"M Gon\u00e7alves","year":"2024","unstructured":"Gon\u00e7alves, M., Pixley, J. H., Amorim, B., Castro, E. V. & Ribeiro, P. Short-range interactions are irrelevant at the quasiperiodicity-driven Luttinger liquid to Anderson glass transition. Phys. Rev. B 109, 014211 (2024).","journal-title":"Phys. Rev. B"},{"key":"2662_CR51","doi-asserted-by":"crossref","first-page":"762","DOI":"10.1038\/s41586-023-06294-z","volume":"620","author":"A Uri","year":"2023","unstructured":"Uri, A. et al. Superconductivity and strong interactions in a tunable moir\u00e9 quasicrystal. Nature 620, 762\u2013767 (2023).","journal-title":"Nature"},{"key":"2662_CR52","doi-asserted-by":"crossref","first-page":"1405","DOI":"10.1103\/RevModPhys.83.1405","volume":"83","author":"MA Cazalilla","year":"2011","unstructured":"Cazalilla, M. A., Citro, R., Giamarchi, T., Orignac, E. & Rigol, M. One dimensional bosons: from condensed matter systems to ultracold gases. Rev. Mod. Phys. 83, 1405\u20131466 (2011).","journal-title":"Rev. Mod. Phys."},{"key":"2662_CR53","doi-asserted-by":"crossref","first-page":"115135","DOI":"10.1103\/PhysRevB.84.115135","volume":"84","author":"T Mishra","year":"2011","unstructured":"Mishra, T., Carrasquilla, J. & Rigol, M. Phase diagram of the half-filled one-dimensional t-v-$${V}^{{\\prime} }$$ model. Phys. Rev. B 84, 115135 (2011).","journal-title":"Phys. Rev. B"},{"key":"2662_CR54","doi-asserted-by":"crossref","first-page":"4371","DOI":"10.1142\/S0217979210056335","volume":"24","author":"S-J Gu","year":"2010","unstructured":"Gu, S.-J. Fidelity approach to quantum phase transitions. Int. J. Mod. Phys. B 24, 4371\u20134458 (2010).","journal-title":"Int. J. Mod. Phys. B"},{"key":"2662_CR55","doi-asserted-by":"crossref","first-page":"014418","DOI":"10.1103\/PhysRevB.91.014418","volume":"91","author":"G Sun","year":"2015","unstructured":"Sun, G., Kolezhuk, A. K. & Vekua, T. Fidelity at Berezinskii-Kosterlitz-Thouless quantum phase transitions. Phys. Rev. B 91, 014418 (2015).","journal-title":"Phys. Rev. B"},{"key":"2662_CR56","doi-asserted-by":"crossref","first-page":"146404","DOI":"10.1103\/PhysRevLett.110.146404","volume":"110","author":"W DeGottardi","year":"2013","unstructured":"DeGottardi, W., Sen, D. & Vishveshwara, S. Majorana fermions in superconducting 1D systems having periodic, quasiperiodic, and disordered potentials. Phys. Rev. Lett. 110, 146404 (2013).","journal-title":"Phys. Rev. Lett."},{"key":"2662_CR57","doi-asserted-by":"crossref","first-page":"235121","DOI":"10.1103\/PhysRevB.101.235121","volume":"101(23)","author":"Y-Z Chou","year":"2020","unstructured":"Chou, Y.-Z., Fu, Y., Wilson, J. H., K\u00f6nig, E. J. & Pixley, J. H. Magic-angle semimetals with chiral symmetry. Phys. Rev. B 101(23), 235121 (2020).","journal-title":"Phys. Rev. B"},{"key":"2662_CR58","doi-asserted-by":"publisher","unstructured":"Fishman, M., White, S. R. & Stoudenmire, E. M. Codebase release 0.3 for ITensor. SciPost Phys. Codebases https:\/\/doi.org\/10.21468\/SciPostPhysCodeb.4-r0.3 (2022).","DOI":"10.21468\/SciPostPhysCodeb.4-r0.3"},{"key":"2662_CR59","doi-asserted-by":"publisher","unstructured":"Fishman, M., White, S. R. & Stoudenmire, E. M. The ITensor software library for tensor network calculations. SciPost Phys. Codebases https:\/\/doi.org\/10.21468\/SciPostPhysCodeb.4 (2022).","DOI":"10.21468\/SciPostPhysCodeb.4"},{"key":"2662_CR60","doi-asserted-by":"publisher","unstructured":"Gon\u00e7alves, M. et al. Data of figures in arXiv:2305.03800. Zenodo https:\/\/doi.org\/10.5281\/zenodo.8082294 (2023).","DOI":"10.5281\/zenodo.8082294"}],"updated-by":[{"DOI":"10.1038\/s41567-025-02824-w","type":"correction","label":"Correction","source":"publisher","updated":{"date-parts":[[2025,2,17]],"date-time":"2025-02-17T00:00:00Z","timestamp":1739750400000}}],"container-title":["Nature Physics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.nature.com\/articles\/s41567-024-02662-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.nature.com\/articles\/s41567-024-02662-2","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.nature.com\/articles\/s41567-024-02662-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,2,18]],"date-time":"2025-02-18T03:50:34Z","timestamp":1739850634000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.nature.com\/articles\/s41567-024-02662-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,10,11]]},"references-count":60,"journal-issue":{"issue":"12","published-print":{"date-parts":[[2024,12]]}},"alternative-id":["2662"],"URL":"https:\/\/doi.org\/10.1038\/s41567-024-02662-2","relation":{"has-preprint":[{"id-type":"doi","id":"10.21203\/rs.3.rs-3059689\/v1","asserted-by":"object"}]},"ISSN":["1745-2473","1745-2481"],"issn-type":[{"value":"1745-2473","type":"print"},{"value":"1745-2481","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,10,11]]},"assertion":[{"value":"13 June 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 September 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 October 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 February 2025","order":4,"name":"change_date","label":"Change Date","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"Correction","order":5,"name":"change_type","label":"Change Type","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"A Correction to this paper has been published:","order":6,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"https:\/\/doi.org\/10.1038\/s41567-025-02824-w","URL":"https:\/\/doi.org\/10.1038\/s41567-025-02824-w","order":7,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare no competing interests.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}]}}