论文标题
磁场中的Bloch电子与Apollonian圆圈包装有什么关系？
What do Bloch Electrons in a Magnetic Field have to do with Apollonian packing of Circles ?
论文作者
论文摘要
积分的Apollonian包装，具有整数曲率的圆圈的包装，每个圆圈都与其他三个相互切线圆相切，可在磁场中编码二维Bloch电子的能量谱的分形结构，称为“ Hofstadter Butterfly”。在这个apollonian-butterfly-connection中，被称为$ \ Mathcal {abc} $，圆的整数曲率包含曲折形式，蝴蝶图的拓扑量子数 - 霍尔电导率的量子。这两个分形的嵌套特性是根据阿波罗尼亚组和共形变换来描述的。 $ \ Mathcal {ABC} $展开，因为描述蝴蝶递归的保形图与描述Apollonian包装中圆圈嵌套的保形图有关。在Farey Tree层次结构中扮演核心角色的所有尺度上，将蝴蝶映射到Apollonian，这揭示了如何在磁场Bloch电子的量子力学中交织在一起的几何和数字理论。
Integral Apollonian packing, the packing of circles with integer curvatures, where every circle is tangent to three other mutually tangent circles, is shown to encode the fractal structure of the energy spectrum of two-dimensional Bloch electrons in a magnetic field, known as the "Hofstadter butterfly". In this Apollonian-Butterfly-Connection, dubbed as $\mathcal{ABC}$, the integer curvatures of the circles contain in a convoluted form, the topological quantum numbers of the butterfly graph -- the quanta of the Hall conductivity. Nesting properties of these two fractals are described in terms of the Apollonian group and the conformal transformations. The $\mathcal{ABC}$ unfolds as the conformal maps describing butterfly recursions are related to the conformal maps describing nesting of circles in the Apollonian packing. Mapping of butterflies to Apollonian at all scales where Farey tree hierarchy plays the central role, reveals how geometry and number theory are intertwined in the quantum mechanics of Bloch electrons in a magnetic field.
