Consider the tight binding model of graphene, sharply terminated along an edge l parallel to a direction of translational symmetry of the underlying period lattice. We classify such edges l into those of “zigzag type” and those of “armchair type”, generalizing the classical zigzag and armchair edges. We prove that zero energy / flat band edge states arise for edges of zigzag type, but never for those of armchair type. We exhibit explicit formulas for flat band edge states when they exist. We produce strong evidence for the existence of dispersive (non flat) edge state curves of nonzero energy for most l.

中文翻译：

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离散蜂窝、有理边和边缘状态

考虑石墨烯的紧密结合模型，沿着平行于底层周期晶格平移对称方向的边缘l急​​剧终止。我们将此类边l分为“锯齿型”边和“扶手椅型”边，概括了经典的锯齿型边和扶手椅型边。我们证明，锯齿形边缘会出现零能量/平带边缘态，但扶手椅类型的边缘不会出现零能量/平带边缘态。当平带边缘态存在时，我们展示了它们的明确公式。我们提供了强有力的证据，证明大多数l存在非零能量的色散（非平坦）边缘状态曲线。