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首页> 外文期刊>Physical review >Self-similar occurrence of massless Dirac particles in graphene under a magnetic field
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Self-similar occurrence of massless Dirac particles in graphene under a magnetic field

机译：磁场下石墨烯中无质量Dirac粒子的自相似出现

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Intricate interplay between the periodicity of the lattice structure and that of the cyclotron motion gives rise to a well-known self-similar fractal structure of the energy eigenvalue, known as the Hofstadter butterfly, for an electron moving in lattice under magnetic field. Connected with the n = 0 Landau level, the central band of the Hofstadter butterfly is especially interesting in the honeycomb lattice. While the entire Hofstadter butterfly can be in principle obtained by solving Harper's equations numerically, the weak-field limit, most relevant for experiment, is intractable owing to the fact that the size of the Hamiltonian matrix, which needs to be diagonalized, diverges. In this paper, we develop an effective Hamiltonian method that can be used to provide an accurate analytic description of the central Hofstadter band in the weak-field regime. One of the most important discoveries obtained in this work is that massless Dirac particles always exist inside the central Hofstadter band no matter how small the magnetic flux may become. In other words, with its bandwidth broadened by the lattice effect, the n = 0 Landau level contains massless Dirac particles within itself. In fact, by carefully analyzing the self-similar recursive pattern of the central Hofstadter band, we conclude that massless Dirac particles should occur under arbitrary magnetic field. As a corollary, the central Hofstadter band also contains a self-similar structure of recursive Landau levels associated with such massless Dirac particles. To assess the experimental feasibility of observing massless Dirac particles inside the central Hofstadter band, we compute the width of the central Hofstadter band as a function of magnetic field in the weak-field regime.
机译：晶格结构的周期性与回旋加速器运动的周期性之间的复杂相互作用产生了众所周知的能量本征值的自相似分形结构，即电子在磁场下在晶格中移动的霍夫施塔特蝴蝶。 Hofstadter蝴蝶的中央带与n = 0的Landau平面相连，在蜂窝格中特别有趣。虽然原则上可以通过数值求解Harper方程来获得整个Hofstadter蝴蝶，但由于需要对角线化的汉密尔顿矩阵的大小不同，所以与实验最相关的弱场限制是棘手的。在本文中，我们开发了一种有效的哈密顿方法，可用于提供弱场状态下中央霍夫施塔特谱带的准确分析描述。这项工作获得的最重要发现之一是，无论磁通量有多小，无质量的狄拉克粒子始终存在于霍夫施塔特中心带内。换句话说，随着其带宽因晶格效应而变宽，n = 0的朗道能级在其内部包含无质量的狄拉克粒子。实际上，通过仔细分析中心霍夫斯塔特带的自相似递归模式，我们得出结论，无质量的狄拉克粒子应在任意磁场下发生。作为推论，霍夫斯塔特中心带也包含与这种无质量狄拉克粒子相关的递归朗道能级的自相似结构。为了评估观察中心霍夫施塔特带内无质量狄拉克粒子的实验可行性，我们计算了中心霍夫施塔特带的宽度与弱场状态下磁场的函数关系。

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