{"title":"三维霍夫斯塔德模型中韦尔点的临界磁通量","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":"{\"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}","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
摘要
我们研究了立方晶格上三维霍夫斯塔德模型的带状结构,该模型的各向同性磁场沿立方体对角线方向,磁通量为Φ=2πm/n,其中 m、n 为共整数。利用动量空间中的简化精确对角法,我们证明了在固定的 m 处,存在一个与特定磁通量值相关的整数 n(m),我们用 Φc(m)≡2πm/n(m) 表示,它将两种不同的状态区分开来。第一种是通量 Φ<Φc(m),其特征是完全的带重叠;第二种是通量 Φ>Φc(m),其特征是状态密度中孤立的带接触点以及第 m 和 (m+1)-th 带之间的韦尔点。在长谷川规中,第(m+1)-带的最小值在临界通量Φc(m)处从kz=0突然移动到kz=π。然后我们论证了 Φc(m)在大 m 时的极限是存在的,而且是有限的:limm→∞Φc(m)≡Φc。我们的估计值为 Φc/2π=0.1296(1)。根据对整数 m≤60 所确定的 n(m)值,我们对大 m 极限中使用的 Φc(m) 形式提出了一个数学猜想。利用这一猜想得到的渐近临界通量为 Φc(conj)/2π=7/54。
Critical magnetic flux for Weyl points in the three-dimensional Hofstadter model
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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