论文标题
准晶体中局部状态的垂直空间会计
Perpendicular space accounting of localized states in a quasicrystal
论文作者
论文摘要
准晶体可以描述为较高维度周期性晶格的截面的投影。晶格的图像指向被投影的尺寸（称为垂直空间），带有有关真实空间晶格的局部结构的宝贵信息。在本文中，我们表明垂直空间投影可用于分析准晶体的基本激发。特别是，我们考虑在二维Penrose晶格上的顶点紧密结合模型，并使用垂直空间图像研究了严格局部状态的特性。我们的方法在此模型中重现了先前报道的六种局部状态的频率。我们还计算了不同局部状态之间的重叠，并表明独立于其他四种类型的类型-5和6型局部状态的数量是黄金比$τ=（1+ \ sqrt {5}}）/2 $的因子。在同一站点周围支持的同一类型-5或类型6的两个方向被证明与其他类型的添加相关。我们还通过对垂直空间中所有晶格的耗尽表明，Penrose晶格中的任何点要么在至少一个局部状态下支持，要么被当地几何形状禁止托管严格的局部局部状态。
Quasicrystals can be described as projections of sections of higher dimensional periodic lattices into real space. The image of the lattice points in the projected out dimensions, called the perpendicular space, carries valuable information about the local structure of the real space lattice. In this paper, we show that perpendicular space projection can be used to analyze the elementary excitations of a quasicrystal. In particular, we consider the vertex tight binding model on the two dimensional Penrose lattice and investigate the properties of strictly localized states using perpendicular space images. Our method reproduces the previously reported frequencies for the six types of localized states in this model. We also calculate the overlaps between different localized states and show that the number of type-5 and type-6 localized states which are independent from the four other types is a factor of golden ratio $τ=(1+\sqrt{5})/2$ higher than previously reported values. Two orientations of the same type-5 or type-6 which are supported around the same site are shown to be linearly dependent with the addition of other types. We also show through exhaustion of all lattice sites in perpendicular space that any point in the Penrose lattice is either in the support of at least one localized state or is forbidden by local geometry to host a strictly localized state.
