Hofstadter和Fibonacci蝴蝶拓扑phase diagram和带电导的高维准晶方法：符号序列、Sturmian编码和所有磁通量的自相似规则
A higher-dimensional quasicrystalline approach to the Hofstadter and Fibonacci butterflies topological phase diagram and band conductance: symbolic sequences, Sturmian coding and self-similar rules at all magnetic fluxes
摘要：
本文利用一种生成准晶结构的几何方法--切割投影法，推导了晶格中量子霍尔效应的拓扑性质，这些性质用Hofstadter蝶形量子相图中的陈数来描述。在此基础上，我们提供了一个几何统一的方法来计算所有通量下的Hofstadter拓扑相图。然后，我们表明，对于任何通量，带电导遵循两个字母的符号序列。其结果是，带电导在不同的流量服从膨胀/紧缩规则的观察到建立准晶体。带电导符号序列由通量的Sturmian编码给出，并且可以通过考虑圆映射、台球或环面上的轨迹来找到。从而提供了一种简单、快速的方法来获得任意磁通量下的陈数。这种方法使之前观察到的斐波那契和哈珀势（也称为几乎马蒂厄算子问题）或其他三角势之间的拓扑等价性，以及与Farey序列和树的关系合理化。
英文摘要：
The topological properties of the quantum Hall effect in a crystalline lattice, described by Chern numbers of the Hofstadter butterfly quantum phase diagram, are deduced by using a geometrical method to generate the structure of quasicrystals: the cut and projection method. Based on this, we provide a geometric unified approach to the Hofstadter topological phase diagram at all fluxes. Then we show that for any flux, the bands conductance follow a two letter symbolic sequence . As a result, bands conductance at different fluxes obey inflation/deflation rules as the ones observed to build quasicrystals. The bands conductance symbolic sequences are given by the Sturmian coding of the flux and can be found by considering a circle map, a billiard or trajectories on a torus. Simple and fast techniques are thus provided to obtain Chern numbers at any magnetic flux. This approach rationalize the previously observed topological equivalences between the Fibonacci and Harper potentials (also known as the almost Mathieu operator problem) or with other trigonometric potential, as well as the relationship with Farey sequences and trees.
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