标题： 从准周期性到科莫托蝴蝶的完全着色 显示英文标题

标题： From Quasiperiodicity to a Complete Coloring of the Kohmoto Butterfly

摘要： Kohmoto模型的谱产生了一个分形相图，称为Kohmoto蝴蝶。 蝴蝶包含了所有周期性Kohmoto哈密顿量的谱，其指标不变量备受关注。 由于势的不连续性，拓扑方法——如陈数——定义不清，因此无法提供指标不变量。 这封信克服了这一障碍，并提供了Kohmoto模型指标的完整分类。 我们的方法将Kohmoto蝴蝶编码为一个谱树图，通过周期谱反映准周期性质。 这产生了相图的完整着色和对其他谱蝴蝶的新视角。 显示英文摘要

摘要： The spectra of the Kohmoto model give rise to a fractal phase diagram, known as the Kohmoto butterfly. The butterfly encapsulates the spectra of all periodic Kohmoto Hamiltonians, whose index invariants are sought after. Topological methods - such as Chern numbers - are ill defined due to the discontinuous potential, and hence fail to provide index invariants. This Letter overcomes that obstacle and provides a complete classification of the Kohmoto model indices. Our approach encodes the Kohmoto butterfly as a spectral tree graph, reflecting the quasiperiodic nature via the periodic spectra. This yields a complete coloring of the phase diagram and a new perspective on other spectral butterflies.