Tianhong Lu, Luiz H. Santos
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{"title":"Fractional Chern Insulators in Twisted BilayerMoTe2: A Composite Fermion Perspective","authors":"Tianhong Lu, Luiz H. Santos","doi":"10.1103/physrevlett.133.186602","DOIUrl":null,"url":null,"abstract":"The discovery of fractional Chern insulators (FCIs) in twisted bilayer <mjx-container ctxtmenu_counter=\"59\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"(2 0 1)\"><mjx-mrow><mjx-msub data-semantic-children=\"0,1\" data-semantic- data-semantic-owns=\"0 1\" data-semantic-role=\"unknown\" data-semantic-speech=\"upper M o upper T e 2\" data-semantic-type=\"subscript\"><mjx-mrow><mjx-mi data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"unknown\" data-semantic-type=\"identifier\"><mjx-c noic=\"true\" style=\"padding-top: 0.657em;\">M</mjx-c><mjx-c noic=\"true\" style=\"padding-top: 0.657em;\">o</mjx-c><mjx-c noic=\"true\" style=\"padding-top: 0.657em;\">T</mjx-c><mjx-c style=\"padding-top: 0.657em;\">e</mjx-c></mjx-mi></mjx-mrow><mjx-script style=\"vertical-align: -0.15em;\"><mjx-mrow size=\"s\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c>2</mjx-c></mjx-mn></mjx-mrow></mjx-script></mjx-msub></mjx-mrow></mjx-math></mjx-container> has sparked significant interest in fractional topological matter without external magnetic fields. Unlike the flat dispersion of Landau levels, moiré electronic states are influenced by lattice effects within a nanometer-scale superlattice. This Letter examines the impact of these lattice effects on the topological phases in twisted bilayer <mjx-container ctxtmenu_counter=\"60\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"(2 0 1)\"><mjx-mrow><mjx-msub data-semantic-children=\"0,1\" data-semantic- data-semantic-owns=\"0 1\" data-semantic-role=\"unknown\" data-semantic-speech=\"upper M o upper T e 2\" data-semantic-type=\"subscript\"><mjx-mrow><mjx-mi data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"unknown\" data-semantic-type=\"identifier\"><mjx-c noic=\"true\" style=\"padding-top: 0.657em;\">M</mjx-c><mjx-c noic=\"true\" style=\"padding-top: 0.657em;\">o</mjx-c><mjx-c noic=\"true\" style=\"padding-top: 0.657em;\">T</mjx-c><mjx-c style=\"padding-top: 0.657em;\">e</mjx-c></mjx-mi></mjx-mrow><mjx-script style=\"vertical-align: -0.15em;\"><mjx-mrow size=\"s\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c>2</mjx-c></mjx-mn></mjx-mrow></mjx-script></mjx-msub></mjx-mrow></mjx-math></mjx-container>, uncovering a family of FCIs with Abelian anyonic quasiparticles. Using a composite fermion approach, we identify a sequence of FCIs with fractional Hall conductivities <mjx-container ctxtmenu_counter=\"61\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"(35 (5 0 (4 1 3 2)) 6 (34 (30 8 (29 9 10 (28 11 (27 (26 12 25 13) 14 15) 16)) 17) 33 (32 18 (31 (21 19 20) 22 23) 24)))\"><mjx-mrow data-semantic-children=\"5,34\" data-semantic-content=\"6\" data-semantic- data-semantic-owns=\"5 6 34\" data-semantic-role=\"equality\" data-semantic-speech=\"sigma Subscript x y Baseline equals left bracket upper C divided by left parenthesis 2 upper C plus 1 right parenthesis right bracket left parenthesis e squared divided by h right parenthesis\" data-semantic-type=\"relseq\"><mjx-msub data-semantic-children=\"0,4\" data-semantic- data-semantic-owns=\"0 4\" data-semantic-parent=\"35\" data-semantic-role=\"greekletter\" data-semantic-type=\"subscript\"><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"5\" data-semantic-role=\"greekletter\" data-semantic-type=\"identifier\"><mjx-c>𝜎</mjx-c></mjx-mi></mjx-mrow><mjx-script style=\"vertical-align: -0.15em;\"><mjx-mrow data-semantic-annotation=\"clearspeak:simple;clearspeak:unit\" data-semantic-children=\"1,2\" data-semantic-content=\"3\" data-semantic- data-semantic-owns=\"1 3 2\" data-semantic-parent=\"5\" data-semantic-role=\"implicit\" data-semantic-type=\"infixop\" size=\"s\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"4\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>𝑥</mjx-c></mjx-mi><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"infixop,\" data-semantic-parent=\"4\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"4\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>𝑦</mjx-c></mjx-mi></mjx-mrow></mjx-script></mjx-msub><mjx-mo data-semantic- data-semantic-operator=\"relseq,=\" data-semantic-parent=\"35\" data-semantic-role=\"equality\" data-semantic-type=\"relation\" space=\"4\"><mjx-c>=</mjx-c></mjx-mo><mjx-mspace></mjx-mspace><mjx-mrow data-semantic-added=\"true\" data-semantic-annotation=\"clearspeak:unit\" data-semantic-children=\"30,32\" data-semantic-content=\"33\" data-semantic- data-semantic-owns=\"30 33 32\" data-semantic-parent=\"35\" data-semantic-role=\"implicit\" data-semantic-type=\"infixop\" space=\"4\"><mjx-mrow data-semantic-added=\"true\" data-semantic-children=\"29\" data-semantic-content=\"8,17\" data-semantic- data-semantic-owns=\"8 29 17\" data-semantic-parent=\"34\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"30\" data-semantic-role=\"open\" data-semantic-type=\"fence\" style=\"vertical-align: -0.02em;\"><mjx-c>[</mjx-c></mjx-mo><mjx-mrow data-semantic-added=\"true\" data-semantic-children=\"9,28\" data-semantic-content=\"10\" data-semantic- data-semantic-owns=\"9 10 28\" data-semantic-parent=\"30\" data-semantic-role=\"division\" data-semantic-type=\"infixop\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"29\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>𝐶</mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\"infixop,/\" data-semantic-parent=\"29\" data-semantic-role=\"division\" data-semantic-type=\"operator\"><mjx-c>/</mjx-c></mjx-mo><mjx-mrow data-semantic-added=\"true\" data-semantic-children=\"27\" data-semantic-content=\"11,16\" data-semantic- data-semantic-owns=\"11 27 16\" data-semantic-parent=\"29\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"28\" data-semantic-role=\"open\" data-semantic-type=\"fence\" style=\"vertical-align: -0.02em;\"><mjx-c>(</mjx-c></mjx-mo><mjx-mrow data-semantic-added=\"true\" data-semantic-children=\"26,15\" data-semantic-content=\"14\" data-semantic- data-semantic-owns=\"26 14 15\" data-semantic-parent=\"28\" data-semantic-role=\"addition\" data-semantic-type=\"infixop\"><mjx-mrow data-semantic-added=\"true\" data-semantic-annotation=\"clearspeak:simple;clearspeak:unit\" data-semantic-children=\"12,13\" data-semantic-content=\"25\" data-semantic- data-semantic-owns=\"12 25 13\" data-semantic-parent=\"27\" data-semantic-role=\"implicit\" data-semantic-type=\"infixop\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"26\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c>2</mjx-c></mjx-mn><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"infixop,\" data-semantic-parent=\"26\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"26\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>𝐶</mjx-c></mjx-mi></mjx-mrow><mjx-mo data-semantic- data-semantic-operator=\"infixop,+\" data-semantic-parent=\"27\" data-semantic-role=\"addition\" data-semantic-type=\"operator\" space=\"3\"><mjx-c>+</mjx-c></mjx-mo><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"27\" data-semantic-role=\"integer\" data-semantic-type=\"number\" space=\"3\"><mjx-c>1</mjx-c></mjx-mn></mjx-mrow><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"28\" data-semantic-role=\"close\" data-semantic-type=\"fence\" style=\"vertical-align: -0.02em;\"><mjx-c>)</mjx-c></mjx-mo></mjx-mrow></mjx-mrow><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"30\" data-semantic-role=\"close\" data-semantic-type=\"fence\" style=\"vertical-align: -0.02em;\"><mjx-c>]</mjx-c></mjx-mo></mjx-mrow><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"infixop,\" data-semantic-parent=\"34\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-added=\"true\" data-semantic-children=\"31\" data-semantic-content=\"18,24\" data-semantic- data-semantic-owns=\"18 31 24\" data-semantic-parent=\"34\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"32\" data-semantic-role=\"open\" data-semantic-type=\"fence\" style=\"vertical-align: -0.02em;\"><mjx-c>(</mjx-c></mjx-mo><mjx-mrow data-semantic-added=\"true\" data-semantic-children=\"21,23\" data-semantic-content=\"22\" data-semantic- data-semantic-owns=\"21 22 23\" data-semantic-parent=\"32\" data-semantic-role=\"division\" data-semantic-type=\"infixop\"><mjx-msup data-semantic-children=\"19,20\" data-semantic- data-semantic-owns=\"19 20\" data-semantic-parent=\"31\" data-semantic-role=\"latinletter\" data-semantic-type=\"superscript\"><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"21\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>𝑒</mjx-c></mjx-mi></mjx-mrow><mjx-script style=\"vertical-align: 0.363em;\"><mjx-mrow size=\"s\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"21\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c>2</mjx-c></mjx-mn></mjx-mrow></mjx-script></mjx-msup><mjx-mo data-semantic- data-semantic-operator=\"infixop,/\" data-semantic-parent=\"31\" data-semantic-role=\"division\" data-semantic-type=\"operator\"><mjx-c>/</mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"31\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>ℎ</mjx-c></mjx-mi></mjx-mrow><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"32\" data-semantic-role=\"close\" data-semantic-type=\"fence\" style=\"vertical-align: -0.02em;\"><mjx-c>)</mjx-c></mjx-mo></mjx-mrow></mjx-mrow></mjx-mrow></mjx-math></mjx-container> linked to partial filling <mjx-container ctxtmenu_counter=\"62\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"(2 0 1)\"><mjx-msub data-semantic-children=\"0,1\" data-semantic- data-semantic-owns=\"0 1\" data-semantic-role=\"greekletter\" data-semantic-speech=\"nu Subscript normal h\" data-semantic-type=\"subscript\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"greekletter\" data-semantic-type=\"identifier\"><mjx-c>𝜈</mjx-c></mjx-mi><mjx-script style=\"vertical-align: -0.15em;\"><mjx-mrow size=\"s\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>h</mjx-c></mjx-mi></mjx-mrow></mjx-script></mjx-msub></mjx-math></mjx-container> of holes of the topmost moiré valence band. These states emerge from incompressible composite fermion bands of Chern number <mjx-container ctxtmenu_counter=\"63\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"0\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper C\" data-semantic-type=\"identifier\"><mjx-c>𝐶</mjx-c></mjx-mi></mjx-math></mjx-container> within a complex Hofstadter spectrum. This approach explains FCIs with Hall conductivities <mjx-container ctxtmenu_counter=\"64\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"(22 (5 0 (4 1 3 2)) 6 (21 (20 (18 7 (17 8 9 10) 11) 19 (14 12 13)) 15 16))\"><mjx-mrow data-semantic-children=\"5,21\" data-semantic-content=\"6\" data-semantic- data-semantic-owns=\"5 6 21\" data-semantic-role=\"equality\" data-semantic-speech=\"sigma Subscript x y Baseline equals left parenthesis 2 divided by 3 right parenthesis e squared divided by h\" data-semantic-type=\"relseq\"><mjx-msub data-semantic-children=\"0,4\" data-semantic- data-semantic-owns=\"0 4\" data-semantic-parent=\"22\" data-semantic-role=\"greekletter\" data-semantic-type=\"subscript\"><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"5\" data-semantic-role=\"greekletter\" data-semantic-type=\"identifier\"><mjx-c>𝜎</mjx-c></mjx-mi></mjx-mrow><mjx-script style=\"vertical-align: -0.15em;\"><mjx-mrow data-semantic-annotation=\"clearspeak:simple;clearspeak:unit\" data-semantic-children=\"1,2\" data-semantic-content=\"3\" data-semantic- data-semantic-owns=\"1 3 2\" data-semantic-parent=\"5\" data-semantic-role=\"implicit\" data-semantic-type=\"infixop\" size=\"s\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"4\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>𝑥</mjx-c></mjx-mi><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"infixop,\" data-semantic-parent=\"4\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"4\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>𝑦</mjx-c></mjx-mi></mjx-mrow></mjx-script></mjx-msub><mjx-mo data-semantic- data-semantic-operator=\"relseq,=\" data-semantic-parent=\"22\" data-semantic-role=\"equality\" data-semantic-type=\"relation\" space=\"4\"><mjx-c>=</mjx-c></mjx-mo><mjx-mrow data-semantic-added=\"true\" data-semantic-children=\"20,16\" data-semantic-content=\"15\" data-semantic- data-semantic-owns=\"20 15 16\" data-semantic-parent=\"22\" data-semantic-role=\"division\" data-semantic-type=\"infixop\" space=\"4\"><mjx-mrow data-semantic-added=\"true\" data-semantic-annotation=\"clearspeak:unit\" data-semantic-children=\"18,14\" data-semantic-content=\"19\" data-semantic- data-semantic-owns=\"18 19 14\" data-semantic-parent=\"21\" data-semantic-role=\"implicit\" data-semantic-type=\"infixop\"><mjx-mrow data-semantic-added=\"true\" data-semantic-children=\"17\" data-semantic-content=\"7,11\" data-semantic- data-semantic-owns=\"7 17 11\" data-semantic-parent=\"20\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"18\" data-semantic-role=\"open\" data-semantic-type=\"fence\" style=\"vertical-align: -0.02em;\"><mjx-c>(</mjx-c></mjx-mo><mjx-mrow data-semantic-added=\"true\" data-semantic-children=\"8,10\" data-semantic-content=\"9\" data-semantic- data-semantic-owns=\"8 9 10\" data-semantic-parent=\"18\" data-semantic-role=\"division\" data-semantic-type=\"infixop\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"17\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c>2</mjx-c></mjx-mn><mjx-mo data-semantic- data-semantic-operator=\"infixop,/\" data-semantic-parent=\"17\" data-semantic-role=\"division\" data-semantic-type=\"operator\"><mjx-c>/</mjx-c></mjx-mo><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"17\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c>3</mjx-c></mjx-mn></mjx-mrow><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"18\" data-semantic-role=\"close\" data-semantic-type=\"fence\" style=\"vertical-align: -0.02em;\"><mjx-c>)</mjx-c></mjx-mo></mjx-mrow><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"infixop,\" data-semantic-parent=\"20\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\"><mjx-c></mjx-c></mjx-mo><mjx-msup data-semantic-children=\"12,13\" data-semantic- data-semantic-owns=\"12 13\" data-semantic-parent=\"20\" data-semantic-role=\"latinletter\" data-semantic-type=\"superscript\"><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"14\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>𝑒</mjx-c></mjx-mi></mjx-mrow><mjx-script style=\"vertical-align: 0.363em;\"><mjx-mrow size=\"s\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"14\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c>2</mjx-c></mjx-mn></mjx-mrow></mjx-script></mjx-msup></mjx-mrow><mjx-mo data-semantic- data-semantic-operator=\"infixop,/\" data-semantic-parent=\"21\" data-semantic-role=\"division\" data-semantic-type=\"operator\"><mjx-c>/</mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"21\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>ℎ</mjx-c></mjx-mi></mjx-mrow></mjx-mrow></mjx-math></mjx-container> and <mjx-container ctxtmenu_counter=\"65\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math breakable=\"true\" data-semantic-children=\"5,21\" data-semantic-content=\"6\" data-semantic- data-semantic-owns=\"5 6 21\" data-semantic-role=\"equality\" data-semantic-speech=\"sigma Subscript x y Baseline equals left parenthesis 3 divided by 5 right parenthesis e squared divided by h\" data-semantic-structure=\"(22 (5 0 (4 1 3 2)) 6 (21 (20 (18 7 (17 8 9 10) 11) 19 (14 12 13)) 15 16))\" data-semantic-type=\"relseq\"><mjx-msub data-semantic-children=\"0,4\" data-semantic- data-semantic-owns=\"0 4\" data-semantic-parent=\"22\" data-semantic-role=\"greekletter\" data-semantic-type=\"subscript\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"5\" data-semantic-role=\"greekletter\" data-semantic-type=\"identifier\"><mjx-c>𝜎</mjx-c></mjx-mi><mjx-script style=\"vertical-align: -0.15em;\"><mjx-mrow data-semantic-annotation=\"clearspeak:simple;clearspeak:unit\" data-semantic-children=\"1,2\" data-semantic-content=\"3\" data-semantic- data-semantic-owns=\"1 3 2\" data-semantic-parent=\"5\" data-semantic-role=\"implicit\" data-semantic-type=\"infixop\" size=\"s\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"4\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>𝑥</mjx-c></mjx-mi><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"infixop,\" data-semantic-parent=\"4\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"4\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>𝑦</mjx-c></mjx-mi></mjx-mrow></mjx-script></mjx-msub><mjx-break size=\"4\"></mjx-break><mjx-mo data-semantic- data-semantic-operator=\"relseq,=\" data-semantic-parent=\"22\" data-semantic-role=\"equality\" data-semantic-type=\"relation\"><mjx-c>=</mjx-c></mjx-mo><mjx-mrow data-semantic-added=\"true\" data-semantic-children=\"20,16\" data-semantic-content=\"15\" data-semantic- data-semantic-owns=\"20 15 16\" data-semantic-parent=\"22\" data-semantic-role=\"division\" data-semantic-type=\"infixop\" space=\"4\"><mjx-mrow data-semantic-added=\"true\" data-semantic-annotation=\"clearspeak:unit\" data-semantic-children=\"18,14\" data-semantic-content=\"19\" data-semantic- data-semantic-owns=\"18 19 14\" data-semantic-parent=\"21\" data-semantic-role=\"implicit\" data-semantic-type=\"infixop\"><mjx-mrow data-semantic-added=\"true\" data-semantic-children=\"17\" data-semantic-content=\"7,11\" data-semantic- data-semantic-owns=\"7 17 11\" data-semantic-parent=\"20\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"18\" data-semantic-role=\"open\" data-semantic-type=\"fence\" style=\"vertical-align: -0.02em;\"><mjx-c>(</mjx-c></mjx-mo><mjx-mrow data-semantic-added=\"true\" data-semantic-children=\"8,10\" data-semantic-content=\"9\" data-semantic- data-semantic-owns=\"8 9 10\" data-semantic-parent=\"18\" data-semantic-role=\"division\" data-semantic-type=\"infixop\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"17\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c>3</mjx-c></mjx-mn><mjx-mo data-semantic- data-semantic-operator=\"infixop,/\" data-semantic-parent=\"17\" data-semantic-role=\"division\" data-semantic-type=\"operator\"><mjx-c>/</mjx-c></mjx-mo><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"17\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c>5</mjx-c></mjx-mn></mjx-mrow><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"18\" data-semantic-role=\"close\" data-semantic-type=\"fence\" style=\"vertical-align: -0.02em;\"><mjx-c>)</mjx-c></mjx-mo></mjx-mrow><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"infixop,\" data-semantic-parent=\"20\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\"><mjx-c></mjx-c></mjx-mo><mjx-msup data-semantic-children=\"12,13\" data-semantic- data-semantic-owns=\"12 13\" data-semantic-parent=\"20\" data-semantic-role=\"latinletter\" data-semantic-type=\"superscript\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"14\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>𝑒</mjx-c></mjx-mi><mjx-script style=\"vertical-align: 0.363em;\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"14\" data-semantic-role=\"integer\" data-semantic-type=\"number\" size=\"s\"><mjx-c>2</mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-mrow><mjx-mo data-semantic- data-semantic-operator=\"infixop,/\" data-semantic-parent=\"21\" data-semantic-role=\"division\" data-semantic-type=\"operator\"><mjx-c>/</mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"21\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>ℎ</mjx-c></mjx-mi></mjx-mrow></mjx-math></mjx-container> at fractional fillings <mjx-container ctxtmenu_counter=\"66\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math breakable=\"true\" data-semantic-children=\"2,7\" data-semantic-content=\"3\" data-semantic- data-semantic-owns=\"2 3 7\" data-semantic-role=\"equality\" data-semantic-speech=\"nu Subscript normal h Baseline equals 2 divided by 3\" data-semantic-structure=\"(8 (2 0 1) 3 (7 4 5 6))\" data-semantic-type=\"relseq\"><mjx-msub data-semantic-children=\"0,1\" data-semantic- data-semantic-owns=\"0 1\" data-semantic-parent=\"8\" data-semantic-role=\"greekletter\" data-semantic-type=\"subscript\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"greekletter\" data-semantic-type=\"identifier\"><mjx-c>𝜈</mjx-c></mjx-mi><mjx-script style=\"vertical-align: -0.15em;\"><mjx-mrow size=\"s\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>h</mjx-c></mjx-mi></mjx-mrow></mjx-script></mjx-msub><mjx-break size=\"4\"></mjx-break><mjx-mo data-semantic- data-semantic-operator=\"relseq,=\" data-semantic-parent=\"8\" data-semantic-role=\"equality\" data-semantic-type=\"relation\"><mjx-c>=</mjx-c></mjx-mo><mjx-mrow data-semantic-added=\"true\" data-semantic-children=\"4,6\" data-semantic-content=\"5\" data-semantic- data-semantic-owns=\"4 5 6\" data-semantic-parent=\"8\" data-semantic-role=\"division\" data-semantic-type=\"infixop\" space=\"4\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c>2</mjx-c></mjx-mn><mjx-mo data-semantic- data-semantic-operator=\"infixop,/\" data-semantic-parent=\"7\" data-semantic-role=\"division\" data-semantic-type=\"operator\"><mjx-c>/</mjx-c></mjx-mo><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c>3</mjx-c></mjx-mn></mjx-mrow></mjx-math></mjx-container> and <mjx-container ctxtmenu_counter=\"67\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math breakable=\"true\" data-semantic-children=\"2,7\" data-semantic-content=\"3\" data-semantic- data-semantic-owns=\"2 3 7\" data-semantic-role=\"equality\" data-semantic-speech=\"nu Subscript normal h Baseline equals 3 divided by 5\" data-semantic-structure=\"(8 (2 0 1) 3 (7 4 5 6))\" data-semantic-type=\"relseq\"><mjx-msub data-semantic-children=\"0,1\" data-semantic- data-semantic-owns=\"0 1\" data-semantic-parent=\"8\" data-semantic-role=\"greekletter\" data-semantic-type=\"subscript\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"greekletter\" data-semantic-type=\"identifier\"><mjx-c>𝜈</mjx-c></mjx-mi><mjx-script style=\"vertical-align: -0.15em;\"><mjx-mrow size=\"s\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>h</mjx-c></mjx-mi></mjx-mrow></mjx-script></mjx-msub><mjx-break size=\"4\"></mjx-break><mjx-mo data-semantic- data-semantic-operator=\"relseq,=\" data-semantic-parent=\"8\" data-semantic-role=\"equality\" data-semantic-type=\"relation\"><mjx-c>=</mjx-c></mjx-mo><mjx-mrow data-semantic-added=\"true\" data-semantic-children=\"4,6\" data-semantic-content=\"5\" data-semantic- data-semantic-owns=\"4 5 6\" data-semantic-parent=\"8\" data-semantic-role=\"division\" data-semantic-type=\"infixop\" space=\"4\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c>3</mjx-c></mjx-mn><mjx-mo data-semantic- data-semantic-operator=\"infixop,/\" data-semantic-parent=\"7\" data-semantic-role=\"division\" data-semantic-type=\"operator\"><mjx-c>/</mjx-c></mjx-mo><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c>5</mjx-c></mjx-mn></mjx-mrow></mjx-math></mjx-container> observed in experiments, and uncovers other fractal FCI states. The Hofstadter spectrum reveals new phenomena, distinct from Landau levels, including a higher-order Van Hove singularity (HOVHS) at half-filling, leading to novel quantum phase transitions. This Letter offers a comprehensive framework for understanding FCIs in transition metal dichalcogenide moiré systems and highlights mechanisms for topological quantum criticality.","PeriodicalId":20069,"journal":{"name":"Physical review letters","volume":"33 1","pages":""},"PeriodicalIF":8.1000,"publicationDate":"2024-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Physical review letters","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1103/physrevlett.133.186602","RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"PHYSICS, MULTIDISCIPLINARY","Score":null,"Total":0}
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