Due to the lack of the translational symmetry, calculating the energy spectrum of an incommensurate system has always been a theoretical challenge. Here, we propose a natural approach to generalize the energy band theory to the incommensurate systems without reliance on the commensurate approximation, thus providing a comprehensive energy spectrum theory of the incommensurate systems. Except for a truncation dependent weighting factor, the formulae of this theory are formally almost identical to that of the Bloch electrons, making it particularly suitable for complex incommensurate structures. To illustrate the application of this theory, we give three typical examples: one-dimensional bichromatic and trichromatic incommensurate potential model, as well as a moiré quasicrystal. Our theory establishes a fundamental framework for understanding the incommensurate systems.

中文翻译：

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不通约系统的能谱理论

由于缺乏平移对称性，计算非通约系统的能谱一直是一个理论挑战。在这里，我们提出了一种自然的方法，将能带理论推广到不可通约系统，而不依赖于可通约近似，从而提供了不可通约系统的综合能谱理论。除了截断相关的权重因子外，该理论的公式在形式上几乎与布洛赫电子的公式相同，这使得它特别适合复杂的不可通约结构。为了说明该理论的应用，我们给出了三个典型的例子：一维双色和三色不可通约势模型以及莫尔准晶体。我们的理论为理解不通约系统建立了一个基本框架。