{"title":"Critical magnetic flux for Weyl points in the three-dimensional Hofstadter model","authors":"Pierpaolo Fontana, Andrea Trombettoni","doi":"10.1103/physrevb.110.045121","DOIUrl":null,"url":null,"abstract":"We investigate the band structure of the three-dimensional Hofstadter model on cubic lattices, with an isotropic magnetic field oriented along the diagonal of the cube with flux <math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi mathvariant=\"normal\">Φ</mi><mo>=</mo><mn>2</mn><mi>π</mi><mspace width=\"0.28em\"></mspace><mi>m</mi><mo>/</mo><mi>n</mi></mrow></math>, where <math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow></math> are coprime integers. Using reduced exact diagonalization in momentum space, we show that, at fixed <math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>m</mi></math>, there exists an integer <math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>n</mi><mo>(</mo><mi>m</mi><mo>)</mo></mrow></math> associated with a specific value of the magnetic flux, that we denote by <math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><msub><mi mathvariant=\"normal\">Φ</mi><mi>c</mi></msub><mrow><mo>(</mo><mi>m</mi><mo>)</mo></mrow><mo>≡</mo><mn>2</mn><mi>π</mi><mspace width=\"0.28em\"></mspace><mi>m</mi><mo>/</mo><mi>n</mi><mrow><mo>(</mo><mi>m</mi><mo>)</mo></mrow></mrow></math>, separating two different regimes. The first one, for fluxes <math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi mathvariant=\"normal\">Φ</mi><mo><</mo><msub><mi mathvariant=\"normal\">Φ</mi><mi>c</mi></msub><mrow><mo>(</mo><mi>m</mi><mo>)</mo></mrow></mrow></math>, is characterized by complete band overlaps, while the second one, for <math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi mathvariant=\"normal\">Φ</mi><mo>></mo><msub><mi mathvariant=\"normal\">Φ</mi><mi>c</mi></msub><mrow><mo>(</mo><mi>m</mi><mo>)</mo></mrow></mrow></math>, features isolated band-touching points in the density of states and Weyl points between the <math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>m</mi><mi>th</mi></mrow></math> and the <math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mo>(</mo><mi>m</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></math>-th bands. In the Hasegawa gauge, the minimum of the <math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mo>(</mo><mi>m</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></math>-th band abruptly moves at the critical flux <math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><msub><mi mathvariant=\"normal\">Φ</mi><mi>c</mi></msub><mrow><mo>(</mo><mi>m</mi><mo>)</mo></mrow></mrow></math> from <math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><msub><mi>k</mi><mi>z</mi></msub><mo>=</mo><mn>0</mn></mrow></math> to <math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><msub><mi>k</mi><mi>z</mi></msub><mo>=</mo><mi>π</mi></mrow></math>. We then argue that the limit for large <math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>m</mi></math> of <math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><msub><mi mathvariant=\"normal\">Φ</mi><mi>c</mi></msub><mrow><mo>(</mo><mi>m</mi><mo>)</mo></mrow></mrow></math> exists and it is finite: <math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><msub><mo form=\"prefix\" movablelimits=\"true\">lim</mo><mrow><mi>m</mi><mo>→</mo><mi>∞</mi></mrow></msub><msub><mi mathvariant=\"normal\">Φ</mi><mi>c</mi></msub><mrow><mo>(</mo><mi>m</mi><mo>)</mo></mrow><mo>≡</mo><msub><mi mathvariant=\"normal\">Φ</mi><mi>c</mi></msub></mrow></math>. Our estimate is <math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><msub><mi mathvariant=\"normal\">Φ</mi><mi>c</mi></msub><mo>/</mo><mn>2</mn><mi>π</mi><mo>=</mo><mn>0.1296</mn><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></math>. Based on the values of <math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>n</mi><mo>(</mo><mi>m</mi><mo>)</mo></mrow></math> determined for integers <math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>m</mi><mo>≤</mo><mn>60</mn></mrow></math>, we propose a mathematical conjecture for the form of <math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><msub><mi mathvariant=\"normal\">Φ</mi><mi>c</mi></msub><mrow><mo>(</mo><mi>m</mi><mo>)</mo></mrow></mrow></math> to be used in the large-<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>m</mi></math> limit. The asymptotic critical flux obtained using this conjecture is <math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><msubsup><mi mathvariant=\"normal\">Φ</mi><mi>c</mi><mrow><mo>(</mo><mi>conj</mi><mo>)</mo></mrow></msubsup><mo>/</mo><mn>2</mn><mi>π</mi><mo>=</mo><mn>7</mn><mo>/</mo><mn>54</mn></mrow></math>.","PeriodicalId":20082,"journal":{"name":"Physical Review B","volume":null,"pages":null},"PeriodicalIF":3.7000,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Physical Review B","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1103/physrevb.110.045121","RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"Physics and Astronomy","Score":null,"Total":0}
引用次数: 0
Abstract
We investigate the band structure of the three-dimensional Hofstadter model on cubic lattices, with an isotropic magnetic field oriented along the diagonal of the cube with flux , where are coprime integers. Using reduced exact diagonalization in momentum space, we show that, at fixed , there exists an integer associated with a specific value of the magnetic flux, that we denote by , separating two different regimes. The first one, for fluxes , is characterized by complete band overlaps, while the second one, for , features isolated band-touching points in the density of states and Weyl points between the and the -th bands. In the Hasegawa gauge, the minimum of the -th band abruptly moves at the critical flux from to . We then argue that the limit for large of exists and it is finite: . Our estimate is . Based on the values of determined for integers , we propose a mathematical conjecture for the form of to be used in the large- limit. The asymptotic critical flux obtained using this conjecture is .
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