{"id":137730,"date":"2025-08-11T19:37:15","date_gmt":"2025-08-11T19:37:15","guid":{"rendered":"https:\/\/www.europesays.com\/us\/137730\/"},"modified":"2025-08-11T19:37:15","modified_gmt":"2025-08-11T19:37:15","slug":"strongly-interacting-hofstadter-states-in-magic-angle-twisted-bilayer-graphene","status":"publish","type":"post","link":"https:\/\/www.europesays.com\/us\/137730\/","title":{"rendered":"Strongly interacting Hofstadter states in magic-angle twisted bilayer graphene"},"content":{"rendered":"<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"1.\">\n<p class=\"c-article-references__text\" id=\"ref-CR1\">Hofstadter, D. R. Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Phys. Rev. B <b>14<\/b>, 2239\u20132249 (1976).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 1\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Energy%20levels%20and%20wave%20functions%20of%20Bloch%20electrons%20in%20rational%20and%20irrational%20magnetic%20fields&amp;journal=Phys.%20Rev.%20B&amp;volume=14&amp;pages=2239-2249&amp;publication_year=1976&amp;author=Hofstadter%2CDR\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"2.\">\n<p class=\"c-article-references__text\" id=\"ref-CR2\">Wannier, G. H. A result not dependent on rationality for Bloch electrons in a magnetic field. Phys. Status Solidi B <b>88<\/b>, 757\u2013765 (1978).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 2\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=A%20result%20not%20dependent%20on%20rationality%20for%20Bloch%20electrons%20in%20a%20magnetic%20field&amp;journal=Phys.%20Status%20Solidi%20B&amp;volume=88&amp;pages=757-765&amp;publication_year=1978&amp;author=Wannier%2CGH\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"3.\">\n<p class=\"c-article-references__text\" id=\"ref-CR3\">Wang, L. et al. Evidence for a fractional fractal quantum Hall effect in graphene superlattices. Science <b>350<\/b>, 1231\u20131234 (2015).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=3445901\" aria-label=\"MathSciNet reference 3\" target=\"_blank\">MathSciNet<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 3\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Evidence%20for%20a%20fractional%20fractal%20quantum%20Hall%20effect%20in%20graphene%20superlattices&amp;journal=Science&amp;volume=350&amp;pages=1231-1234&amp;publication_year=2015&amp;author=Wang%2CL\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"4.\">\n<p class=\"c-article-references__text\" id=\"ref-CR4\">Spanton, E. M. et al. Observation of fractional Chern insulators in a van der Waals heterostructure. Science <b>360<\/b>, 62\u201366 (2018).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 4\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Observation%20of%20fractional%20Chern%20insulators%20in%20a%20van%20der%20Waals%20heterostructure&amp;journal=Science&amp;volume=360&amp;pages=62-66&amp;publication_year=2018&amp;author=Spanton%2CEM\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"5.\">\n<p class=\"c-article-references__text\" id=\"ref-CR5\">Bistritzer, R. &amp; MacDonald, A. H. Moir\u00e9 bands in twisted double-layer graphene. Proc. Natl Acad. Sci. USA <b>108<\/b>, 12233\u201312237 (2011).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 5\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Moir%C3%A9%20bands%20in%20twisted%20double-layer%20graphene&amp;journal=Proc.%20Natl%20Acad.%20Sci.%20USA&amp;volume=108&amp;pages=12233-12237&amp;publication_year=2011&amp;author=Bistritzer%2CR&amp;author=MacDonald%2CAH\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"6.\">\n<p class=\"c-article-references__text\" id=\"ref-CR6\">Cao, Y. et al. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices. Nature <b>556<\/b>, 80\u201384 (2018).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 6\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Correlated%20insulator%20behaviour%20at%20half-filling%20in%20magic-angle%20graphene%20superlattices&amp;journal=Nature&amp;volume=556&amp;pages=80-84&amp;publication_year=2018&amp;author=Cao%2CY\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"7.\">\n<p class=\"c-article-references__text\" id=\"ref-CR7\">Cao, Y. et al. Unconventional superconductivity in magic-angle graphene superlattices. Nature <b>556<\/b>, 43\u201350 (2018).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 7\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Unconventional%20superconductivity%20in%20magic-angle%20graphene%20superlattices&amp;journal=Nature&amp;volume=556&amp;pages=43-50&amp;publication_year=2018&amp;author=Cao%2CY\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"8.\">\n<p class=\"c-article-references__text\" id=\"ref-CR8\">Yankowitz, M. et al. Tuning superconductivity in twisted bilayer graphene. Science <b>363<\/b>, eaav1910 (2019).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 8\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Tuning%20superconductivity%20in%20twisted%20bilayer%20graphene&amp;journal=Science&amp;volume=363&amp;publication_year=2019&amp;author=Yankowitz%2CM\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"9.\">\n<p class=\"c-article-references__text\" id=\"ref-CR9\">Lu, X. et al. Superconductors, orbital magnets and correlated states in magic-angle bilayer graphene. Nature <b>574<\/b>, 653\u2013657 (2019).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 9\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Superconductors%2C%20orbital%20magnets%20and%20correlated%20states%20in%20magic-angle%20bilayer%20graphene&amp;journal=Nature&amp;volume=574&amp;pages=653-657&amp;publication_year=2019&amp;author=Lu%2CX\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"10.\">\n<p class=\"c-article-references__text\" id=\"ref-CR10\">Sharpe, A. L. et al. Emergent ferromagnetism near three-quarters filling in twisted bilayer graphene. Science <b>365<\/b>, 605\u2013608 (2019).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 10\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Emergent%20ferromagnetism%20near%20three-quarters%20filling%20in%20twisted%20bilayer%20graphene&amp;journal=Science&amp;volume=365&amp;pages=605-608&amp;publication_year=2019&amp;author=Sharpe%2CAL\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"11.\">\n<p class=\"c-article-references__text\" id=\"ref-CR11\">Serlin, M. et al. Intrinsic quantized anomalous Hall effect in a moir\u00e9 heterostructure. Science <b>367<\/b>, 900\u2013903 (2020).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 11\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Intrinsic%20quantized%20anomalous%20Hall%20effect%20in%20a%20moir%C3%A9%20heterostructure&amp;journal=Science&amp;volume=367&amp;pages=900-903&amp;publication_year=2020&amp;author=Serlin%2CM\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"12.\">\n<p class=\"c-article-references__text\" id=\"ref-CR12\">Xie, Y. et al. Fractional Chern insulators in magic-angle twisted bilayer graphene. Nature <b>600<\/b>, 439\u2013443 (2021).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 12\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Fractional%20Chern%20insulators%20in%20magic-angle%20twisted%20bilayer%20graphene&amp;journal=Nature&amp;volume=600&amp;pages=439-443&amp;publication_year=2021&amp;author=Xie%2CY\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"13.\">\n<p class=\"c-article-references__text\" id=\"ref-CR13\">Saito, Y. et al. Hofstadter subband ferromagnetism and symmetry-broken Chern insulators in twisted bilayer graphene. Nat. Phys. <b>17<\/b>, 478\u2013481 (2021).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 13\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Hofstadter%20subband%20ferromagnetism%20and%20symmetry-broken%20Chern%20insulators%20in%20twisted%20bilayer%20graphene&amp;journal=Nat.%20Phys.&amp;volume=17&amp;pages=478-481&amp;publication_year=2021&amp;author=Saito%2CY\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"14.\">\n<p class=\"c-article-references__text\" id=\"ref-CR14\">Yu, J. et al. Correlated Hofstadter spectrum and flavour phase diagram in magic-angle twisted bilayer graphene. Nat. Phys. <b>18<\/b>, 825\u2013831 (2022).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 14\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Correlated%20Hofstadter%20spectrum%20and%20flavour%20phase%20diagram%20in%20magic-angle%20twisted%20bilayer%20graphene&amp;journal=Nat.%20Phys.&amp;volume=18&amp;pages=825-831&amp;publication_year=2022&amp;author=Yu%2CJ\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"15.\">\n<p class=\"c-article-references__text\" id=\"ref-CR15\">Polshyn, H. et al. Topological charge density waves at half-integer filling of a moir\u00e9 superlattice. Nat. Phys. <b>18<\/b>, 42\u201347 (2021).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 15\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Topological%20charge%20density%20waves%20at%20half-integer%20filling%20of%20a%20moir%C3%A9%20superlattice&amp;journal=Nat.%20Phys.&amp;volume=18&amp;pages=42-47&amp;publication_year=2021&amp;author=Polshyn%2CH\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"16.\">\n<p class=\"c-article-references__text\" id=\"ref-CR16\">He, M. et al. Symmetry-broken Chern insulators in twisted double bilayer graphene. Nano Lett. <b>23<\/b>, 11066\u201311072 (2023).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 16\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Symmetry-broken%20Chern%20insulators%20in%20twisted%20double%20bilayer%20graphene&amp;journal=Nano%20Lett.&amp;volume=23&amp;pages=11066-11072&amp;publication_year=2023&amp;author=He%2CM\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"17.\">\n<p class=\"c-article-references__text\" id=\"ref-CR17\">Park, J. M., Cao, Y., Watanabe, K., Taniguchi, T. &amp; Jarillo-Herrero, P. Flavour Hund\u2019s coupling, Chern gaps and charge diffusivity in moir\u00e9 graphene. Nature <b>592<\/b>, 43\u201348 (2021).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 17\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Flavour%20Hund%E2%80%99s%20coupling%2C%20Chern%20gaps%20and%20charge%20diffusivity%20in%20moir%C3%A9%20graphene&amp;journal=Nature&amp;volume=592&amp;pages=43-48&amp;publication_year=2021&amp;author=Park%2CJM&amp;author=Cao%2CY&amp;author=Watanabe%2CK&amp;author=Taniguchi%2CT&amp;author=Jarillo-Herrero%2CP\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"18.\">\n<p class=\"c-article-references__text\" id=\"ref-CR18\">Nuckolls, K. P. et al. Strongly correlated Chern insulators in magic-angle twisted bilayer graphene. Nature <b>588<\/b>, 610\u2013615 (2020).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 18\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Strongly%20correlated%20Chern%20insulators%20in%20magic-angle%20twisted%20bilayer%20graphene&amp;journal=Nature&amp;volume=588&amp;pages=610-615&amp;publication_year=2020&amp;author=Nuckolls%2CKP\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"19.\">\n<p class=\"c-article-references__text\" id=\"ref-CR19\">Choi, Y. et al. Correlation-driven topological phases in magic-angle twisted bilayer graphene. Nature <b>589<\/b>, 536\u2013541 (2021).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 19\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Correlation-driven%20topological%20phases%20in%20magic-angle%20twisted%20bilayer%20graphene&amp;journal=Nature&amp;volume=589&amp;pages=536-541&amp;publication_year=2021&amp;author=Choi%2CY\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"20.\">\n<p class=\"c-article-references__text\" id=\"ref-CR20\">Das, I. et al. Symmetry-broken Chern insulators and Rashba-like Landau-level crossings in magic-angle bilayer graphene. Nat. Phys. <b>17<\/b>, 710\u2013714 (2021).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"21.\">\n<p class=\"c-article-references__text\" id=\"ref-CR21\">Wu, S., Zhang, Z., Watanabe, K., Taniguchi, T. &amp; Andrei, E. Y. Chern insulators, van Hove singularities and topological flat bands in magic-angle twisted bilayer graphene. Nat. Mater. <b>20<\/b>, 488\u2013494 (2021).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 21\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Chern%20insulators%2C%20van%20Hove%20singularities%20and%20topological%20flat%20bands%20in%20magic-angle%20twisted%20bilayer%20graphene&amp;journal=Nat.%20Mater.&amp;volume=20&amp;pages=488-494&amp;publication_year=2021&amp;author=Wu%2CS&amp;author=Zhang%2CZ&amp;author=Watanabe%2CK&amp;author=Taniguchi%2CT&amp;author=Andrei%2CEY\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"22.\">\n<p class=\"c-article-references__text\" id=\"ref-CR22\">Wang, X. &amp; Vafek, O. Theory of correlated Chern insulators in twisted bilayer graphene. Phys. Rev. X <b>14<\/b>, 021042 (2023).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"23.\">\n<p class=\"c-article-references__text\" id=\"ref-CR23\">Jain, J. K. Composite-fermion approach for the fractional quantum Hall effect. Phys. Rev. Lett. <b>63<\/b>, 199\u2013202 (1989).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 23\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Composite-fermion%20approach%20for%20the%20fractional%20quantum%20Hall%20effect&amp;journal=Phys.%20Rev.%20Lett.&amp;volume=63&amp;pages=199-202&amp;publication_year=1989&amp;author=Jain%2CJK\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"24.\">\n<p class=\"c-article-references__text\" id=\"ref-CR24\">Polski, R. et al. Hierarchy of symmetry breaking correlated phases in twisted bilayer graphene. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/2205.05225\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/2205.05225\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/2205.05225<\/a> (2022).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"25.\">\n<p class=\"c-article-references__text\" id=\"ref-CR25\">He, M. et al. Dynamically tunable moir\u00e9 exciton Rydberg states in a monolayer semiconductor on twisted bilayer graphene. Nat. Mater. <b>23<\/b>, 224\u2013229 (2024).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"26.\">\n<p class=\"c-article-references__text\" id=\"ref-CR26\">Tschirhart, C. L. et al. Imaging orbital ferromagnetism in a moir\u00e9 Chern insulator. Science <b>372<\/b>, 1323\u20131327 (2021).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 26\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Imaging%20orbital%20ferromagnetism%20in%20a%20moir%C3%A9%20Chern%20insulator&amp;journal=Science&amp;volume=372&amp;pages=1323-1327&amp;publication_year=2021&amp;author=Tschirhart%2CCL\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"27.\">\n<p class=\"c-article-references__text\" id=\"ref-CR27\">Arora, H. S. et al. Superconductivity in metallic twisted bilayer graphene stabilized by WSe2. Nature <b>583<\/b>, 379\u2013384 (2020).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 27\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Superconductivity%20in%20metallic%20twisted%20bilayer%20graphene%20stabilized%20by%20WSe2&amp;journal=Nature&amp;volume=583&amp;pages=379-384&amp;publication_year=2020&amp;author=Arora%2CHS\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"28.\">\n<p class=\"c-article-references__text\" id=\"ref-CR28\">Lin, J.-X. et al. Spin-orbit\u2013driven ferromagnetism at half moir\u00e9 filling in magic-angle twisted bilayer graphene. Science <b>375<\/b>, 437\u2013441 (2022).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 28\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Spin-orbit%E2%80%93driven%20ferromagnetism%20at%20half%20moir%C3%A9%20filling%20in%20magic-angle%20twisted%20bilayer%20graphene&amp;journal=Science&amp;volume=375&amp;pages=437-441&amp;publication_year=2022&amp;author=Lin%2CJ-X\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"29.\">\n<p class=\"c-article-references__text\" id=\"ref-CR29\">Stepanov, P. et al. Competing zero-field Chern insulators in superconducting twisted bilayer graphene. Phys. Rev. Lett. <b>127<\/b>, 197701 (2021).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 29\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Competing%20zero-field%20Chern%20insulators%20in%20superconducting%20twisted%20bilayer%20graphene&amp;journal=Phys.%20Rev.%20Lett.&amp;volume=127&amp;publication_year=2021&amp;author=Stepanov%2CP\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"30.\">\n<p class=\"c-article-references__text\" id=\"ref-CR30\">Grover, S. et al. Chern mosaic and Berry-curvature magnetism in magic-angle graphene. Nat. Phys. <b>18<\/b>, 885\u2013892 (2022).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 30\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Chern%20mosaic%20and%20Berry-curvature%20magnetism%20in%20magic-angle%20graphene&amp;journal=Nat.%20Phys.&amp;volume=18&amp;pages=885-892&amp;publication_year=2022&amp;author=Grover%2CS\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"31.\">\n<p class=\"c-article-references__text\" id=\"ref-CR31\">Tseng, C.-C. et al. Anomalous Hall effect at half filling in twisted bilayer graphene. Nat. Phys. <b>18<\/b>, 1038\u20131042 (2022).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 31\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Anomalous%20Hall%20effect%20at%20half%20filling%20in%20twisted%20bilayer%20graphene&amp;journal=Nat.%20Phys.&amp;volume=18&amp;pages=1038-1042&amp;publication_year=2022&amp;author=Tseng%2CC-C\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"32.\">\n<p class=\"c-article-references__text\" id=\"ref-CR32\">Streda, P. Quantised Hall effect in a two-dimensional periodic potential. J. Phys. C <b>15<\/b>, L1299 (1982).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 32\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Quantised%20Hall%20effect%20in%20a%20two-dimensional%20periodic%20potential&amp;journal=J.%20Phys.%20C&amp;volume=15&amp;publication_year=1982&amp;author=Streda%2CP\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"33.\">\n<p class=\"c-article-references__text\" id=\"ref-CR33\">Kwan, Y. H. et al. Kekul\u00e9 spiral order at all nonzero integer fillings in twisted bilayer graphene. Phys. Rev. X <b>11<\/b>, 041063 (2021).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 33\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Kekul%C3%A9%20spiral%20order%20at%20all%20nonzero%20integer%20fillings%20in%20twisted%20bilayer%20graphene&amp;journal=Phys.%20Rev.%20X&amp;volume=11&amp;publication_year=2021&amp;author=Kwan%2CYH\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"34.\">\n<p class=\"c-article-references__text\" id=\"ref-CR34\">Nuckolls, K. P. et al. Quantum textures of the many-body wavefunctions in magic-angle graphene. Nature <b>620<\/b>, 525\u2013532 (2023).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 34\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Quantum%20textures%20of%20the%20many-body%20wavefunctions%20in%20magic-angle%20graphene&amp;journal=Nature&amp;volume=620&amp;pages=525-532&amp;publication_year=2023&amp;author=Nuckolls%2CKP\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"35.\">\n<p class=\"c-article-references__text\" id=\"ref-CR35\">Uri, A. et al. Mapping the twist-angle disorder and Landau levels in magic-angle graphene. Nature <b>581<\/b>, 47\u201352 (2020).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 35\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Mapping%20the%20twist-angle%20disorder%20and%20Landau%20levels%20in%20magic-angle%20graphene&amp;journal=Nature&amp;volume=581&amp;pages=47-52&amp;publication_year=2020&amp;author=Uri%2CA\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"36.\">\n<p class=\"c-article-references__text\" id=\"ref-CR36\">Bolotin, K. I., Ghahari, F., Shulman, M. D., Stormer, H. L. &amp; Kim, P. Observation of the fractional quantum Hall effect in graphene. Nature <b>462<\/b>, 196\u2013199 (2009).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 36\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Observation%20of%20the%20fractional%20quantum%20Hall%20effect%20in%20graphene&amp;journal=Nature&amp;volume=462&amp;pages=196-199&amp;publication_year=2009&amp;author=Bolotin%2CKI&amp;author=Ghahari%2CF&amp;author=Shulman%2CMD&amp;author=Stormer%2CHL&amp;author=Kim%2CP\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"37.\">\n<p class=\"c-article-references__text\" id=\"ref-CR37\">Du, X., Skachko, I., Duerr, F., Luican, A. &amp; Andrei, E. Y. Fractional quantum Hall effect and insulating phase of Dirac electrons in graphene. Nature <b>462<\/b>, 192\u2013195 (2009).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 37\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Fractional%20quantum%20Hall%20effect%20and%20insulating%20phase%20of%20Dirac%20electrons%20in%20graphene&amp;journal=Nature&amp;volume=462&amp;pages=192-195&amp;publication_year=2009&amp;author=Du%2CX&amp;author=Skachko%2CI&amp;author=Duerr%2CF&amp;author=Luican%2CA&amp;author=Andrei%2CEY\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"38.\">\n<p class=\"c-article-references__text\" id=\"ref-CR38\">Ghahari, F., Zhao, Y., Cadden-Zimansky, P., Bolotin, K. &amp; Kim, P. Measurement of the \u03bd\u2009=\u20091\/3 fractional quantum Hall energy gap in suspended graphene. Phys. Rev. Lett. <b>106<\/b>, 046801 (2011).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 38\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Measurement%20of%20the%20%CE%BD%E2%80%89%3D%E2%80%891%2F3%20fractional%20quantum%20Hall%20energy%20gap%20in%20suspended%20graphene&amp;journal=Phys.%20Rev.%20Lett.&amp;volume=106&amp;publication_year=2011&amp;author=Ghahari%2CF&amp;author=Zhao%2CY&amp;author=Cadden-Zimansky%2CP&amp;author=Bolotin%2CK&amp;author=Kim%2CP\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"39.\">\n<p class=\"c-article-references__text\" id=\"ref-CR39\">Dean, C. R. et al. Multicomponent fractional quantum Hall effect in graphene. Nat. Phys. <b>7<\/b>, 693\u2013696 (2011).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 39\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Multicomponent%20fractional%20quantum%20Hall%20effect%20in%20graphene&amp;journal=Nat.%20Phys.&amp;volume=7&amp;pages=693-696&amp;publication_year=2011&amp;author=Dean%2CCR\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"40.\">\n<p class=\"c-article-references__text\" id=\"ref-CR40\">Feldman, B. E., Krauss, B., Smet, J. H. &amp; Yacoby, A. Unconventional sequence of fractional quantum Hall states in suspended graphene. Science <b>337<\/b>, 1196\u20131199 (2012).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 40\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Unconventional%20sequence%20of%20fractional%20quantum%20Hall%20states%20in%20suspended%20graphene&amp;journal=Science&amp;volume=337&amp;pages=1196-1199&amp;publication_year=2012&amp;author=Feldman%2CBE&amp;author=Krauss%2CB&amp;author=Smet%2CJH&amp;author=Yacoby%2CA\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"41.\">\n<p class=\"c-article-references__text\" id=\"ref-CR41\">Kou, A. et al. Electron-hole asymmetric integer and fractional quantum Hall effect in bilayer graphene. Science <b>345<\/b>, 55\u201357 (2014).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 41\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Electron-hole%20asymmetric%20integer%20and%20fractional%20quantum%20Hall%20effect%20in%20bilayer%20graphene&amp;journal=Science&amp;volume=345&amp;pages=55-57&amp;publication_year=2014&amp;author=Kou%2CA\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"42.\">\n<p class=\"c-article-references__text\" id=\"ref-CR42\">Maher, P. et al. Tunable fractional quantum Hall phases in bilayer graphene. Science <b>345<\/b>, 61\u201364 (2014).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 42\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Tunable%20fractional%20quantum%20Hall%20phases%20in%20bilayer%20graphene&amp;journal=Science&amp;volume=345&amp;pages=61-64&amp;publication_year=2014&amp;author=Maher%2CP\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"43.\">\n<p class=\"c-article-references__text\" id=\"ref-CR43\">Li, J. I. A. et al. Even-denominator fractional quantum Hall states in bilayer graphene. Science <b>358<\/b>, 648\u2013652 (2017).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 43\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Even-denominator%20fractional%20quantum%20Hall%20states%20in%20bilayer%20graphene&amp;journal=Science&amp;volume=358&amp;pages=648-652&amp;publication_year=2017&amp;author=Li%2CJIA\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"44.\">\n<p class=\"c-article-references__text\" id=\"ref-CR44\">Zibrov, A. A. et al. Tunable interacting composite fermion phases in a half-filled bilayer-graphene Landau level. Nature <b>549<\/b>, 360\u2013364 (2017).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 44\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Tunable%20interacting%20composite%20fermion%20phases%20in%20a%20half-filled%20bilayer-graphene%20Landau%20level&amp;journal=Nature&amp;volume=549&amp;pages=360-364&amp;publication_year=2017&amp;author=Zibrov%2CAA\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"45.\">\n<p class=\"c-article-references__text\" id=\"ref-CR45\">Huang, K. et al. Valley isospin controlled fractional quantum Hall states in bilayer graphene. Phys. Rev. X <b>12<\/b>, 031019 (2022).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 45\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Valley%20isospin%20controlled%20fractional%20quantum%20Hall%20states%20in%20bilayer%20graphene&amp;journal=Phys.%20Rev.%20X&amp;volume=12&amp;publication_year=2022&amp;author=Huang%2CK\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"46.\">\n<p class=\"c-article-references__text\" id=\"ref-CR46\">Haldane, F. D. M. Fractional quantization of the Hall effect: a hierarchy of incompressible quantum fluid states. Phys. Rev. Lett. <b>51<\/b>, 605\u2013608 (1983).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=710280\" aria-label=\"MathSciNet reference 46\" target=\"_blank\">MathSciNet<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 46\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Fractional%20quantization%20of%20the%20Hall%20effect%3A%20a%20hierarchy%20of%20incompressible%20quantum%20fluid%20states&amp;journal=Phys.%20Rev.%20Lett.&amp;volume=51&amp;pages=605-608&amp;publication_year=1983&amp;author=Haldane%2CFDM\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"47.\">\n<p class=\"c-article-references__text\" id=\"ref-CR47\">Roy, R. Band geometry of fractional topological insulators. Phys. Rev. B <b>90<\/b>, 165139 (2014).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 47\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Band%20geometry%20of%20fractional%20topological%20insulators&amp;journal=Phys.%20Rev.%20B&amp;volume=90&amp;publication_year=2014&amp;author=Roy%2CR\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"48.\">\n<p class=\"c-article-references__text\" id=\"ref-CR48\">Parker, D. et al. Field-tuned and zero-field fractional Chern insulators in magic angle graphene. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/2112.13837\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/2112.13837\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/2112.13837<\/a> (2021).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"49.\">\n<p class=\"c-article-references__text\" id=\"ref-CR49\">Wang, N. et al. Quantum-metric-induced nonlinear transport in a topological antiferromagnet. Nature <b>621<\/b>, 487\u2013492 (2023).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 49\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Quantum-metric-induced%20nonlinear%20transport%20in%20a%20topological%20antiferromagnet&amp;journal=Nature&amp;volume=621&amp;pages=487-492&amp;publication_year=2023&amp;author=Wang%2CN\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"50.\">\n<p class=\"c-article-references__text\" id=\"ref-CR50\">Gao, A. et al. Quantum metric nonlinear Hall effect in a topological antiferromagnetic heterostructure. Science <b>381<\/b>, 181\u2013186 (2023).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 50\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Quantum%20metric%20nonlinear%20Hall%20effect%20in%20a%20topological%20antiferromagnetic%20heterostructure&amp;journal=Science&amp;volume=381&amp;pages=181-186&amp;publication_year=2023&amp;author=Gao%2CA\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"51.\">\n<p class=\"c-article-references__text\" id=\"ref-CR51\">Luu, T. T. &amp; W\u00f6rner, H. J. Measurement of the Berry curvature of solids using high-harmonic spectroscopy. Nat. Commun. <b>9<\/b>, 916 (2018).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 51\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Measurement%20of%20the%20Berry%20curvature%20of%20solids%20using%20high-harmonic%20spectroscopy&amp;journal=Nat.%20Commun.&amp;volume=9&amp;publication_year=2018&amp;author=Luu%2CTT&amp;author=W%C3%B6rner%2CHJ\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"52.\">\n<p class=\"c-article-references__text\" id=\"ref-CR52\">Sch\u00fcler, M. et al. Local Berry curvature signatures in dichroic angle-resolved photoelectron spectroscopy from two-dimensional materials. Sci. Adv. <b>6<\/b>, eaay2730 (2020).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 52\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Local%20Berry%20curvature%20signatures%20in%20dichroic%20angle-resolved%20photoelectron%20spectroscopy%20from%20two-dimensional%20materials&amp;journal=Sci.%20Adv.&amp;volume=6&amp;publication_year=2020&amp;author=Sch%C3%BCler%2CM\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"53.\">\n<p class=\"c-article-references__text\" id=\"ref-CR53\">de Juan, F., Grushin, A. G., Morimoto, T. &amp; Moore, J. E. Quantized circular photogalvanic effect in Weyl semimetals. Nat. Commun. <b>8<\/b>, 15995 (2017).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 53\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Quantized%20circular%20photogalvanic%20effect%20in%20Weyl%20semimetals&amp;journal=Nat.%20Commun.&amp;volume=8&amp;publication_year=2017&amp;author=Juan%2CF&amp;author=Grushin%2CAG&amp;author=Morimoto%2CT&amp;author=Moore%2CJE\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n","protected":false},"excerpt":{"rendered":"Hofstadter, D. R. Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Phys.&hellip;\n","protected":false},"author":3,"featured_media":137731,"comment_status":"","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[25],"tags":[2267,2266,2271,2270,16443,834,2265,2268,2269,492,11839,159,2264,67,132,68],"class_list":{"0":"post-137730","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-physics","8":"tag-atomic","9":"tag-classical-and-continuum-physics","10":"tag-complex-systems","11":"tag-condensed-matter-physics","12":"tag-electronic-properties-and-materials","13":"tag-general","14":"tag-mathematical-and-computational-physics","15":"tag-molecular","16":"tag-optical-and-plasma-physics","17":"tag-physics","18":"tag-quantum-hall","19":"tag-science","20":"tag-theoretical","21":"tag-united-states","22":"tag-unitedstates","23":"tag-us"},"share_on_mastodon":{"url":"https:\/\/pubeurope.com\/@us\/115011823196700605","error":""},"_links":{"self":[{"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/posts\/137730","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/comments?post=137730"}],"version-history":[{"count":0,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/posts\/137730\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/media\/137731"}],"wp:attachment":[{"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/media?parent=137730"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/categories?post=137730"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/tags?post=137730"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}