关于Hofstadter蝶形谱：Chern-Simons理论，子带间隙映射，IQHE和FQHE标记
On Hofstadter butterfly spectrum: Chern-Simons theory, subband gap mapping, IQHE and FQHE labelling
摘要：
磁场以几种方式影响布洛赫能带结构。首先，它打破布洛赫带磁子带或朗道能级加宽到磁布洛赫带。在能量标度的中心部分中的子带的所得组与整数量子霍尔效应（IQHE）相关联。然后在高场下，它分别改变剩余的最低和最高子带的积分态密度，这可以与分数量子霍尔效应（Fractional Quantum Hall Effect，简称FWHE）相关联。在这里，我们采用麦克斯韦陈-西蒙斯规范理论制定的子带隙映射算法，并构建蝴蝶轮廓的霍夫施塔特频谱。确定了频谱中负责IQHE的两个区域。在非常高的磁场的最高和最低的子带的影响，分别由磁场引起的重组的集成状态密度在每个子带。两个子带中的每一个的所得变换负责所述CQHE。因此，在能量尺度的中心区域，由间隙映射定义的子带的主要组负责IQHE。传达磁谱的迭代性质的最高和最低子带的精细结构是由磁场对每个相应子带中的积分态密度进行分级缩放和重构的结果，并且负责磁谱的精细结构。
英文摘要：
The magnetic field affects the Bloch band structure in a couple of ways. First it breaks the Bloch band into magnetic subbands or the Landau levels are broadened into magnetic Bloch bands. The resulting group of subbands in the central portion of the energy scale is associated with the integer quantum Hall effect (IQHE). Then at high fields it changes the integrated density of states of the remaining lowest and topmost subband, respectively, which can be associated with fractional quantum Hall effect (FQHE). Here, we employ the Maxwell Chern-Simons gauge theory to formulate the subband-gap mapping algorithm and to construct the butterfly profile of the Hofstadter spectrum. The two regions in the spectrum responsible for the IQHE are identified. At very high magnetic fields the highest and lowest subband are affected by magnetic-field induced restructuring of the integrated density of states in each subband, respectively. The resulting transformation of each of the two subband is responsible for the FQHE. Thus, in the central regions of the energy scale, the principal group of subbands defined by the gap mapping is responsible for the IQHE. The fine structure of the topmost and lowest subband, which convey an iterative nature of the magnetic spectrum is a result of a hierarchical scaling and restructuring by the magnetic fields on the integrated density of states in each respective subband and is responsible for the FQHE.
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