无面积律纠缠的分形格上的Laughlin拓扑
Laughlin topology on fractal lattices without area law entanglement
摘要：
最近在分形格点上构造了Laughlin态，并利用准空穴的电荷统计和编织统计证实了这些态具有Laughlin型拓扑。在这里，我们调查的密度，相关性和纠缠性质的分形晶格来自Sierpinski三角形的目的，以确定相似性和差异相比，二维系统，并调查是否有各种探头的拓扑工作的分形晶格。类似于二维系统，我们发现，连接粒子-粒子关联函数大致指数衰减与在二维平面上测量的晶格位置之间的距离，但值也取决于当地的环境。相反的二维系统，我们发现，纠缠熵不遵循面积定律，如果一个定义的面积是最近邻键的数量，通过边缘的选定的子系统。考虑两个键交叉的边缘的双分区，我们发现一个接近对数标度的纠缠熵的网站在子系统中的数量。这也意味着拓扑纠缠熵不能使用Kitaev-Preskill或Levin-Wen方法提取。研究了不同双分块的纠缠谱，发现纠缠能隙以下的态数与二维晶格上的Laughlin态数相同，具有鲁棒性。
英文摘要：
Laughlin states have recently been constructed on fractal lattices, and the charge and braiding statistics of the quasiholes were used to confirm that these states have Laughlin type topology. Here, we investigate density, correlation, and entanglement properties of the states on a fractal lattice derived from a Sierpinski triangle with the purpose of identifying similarities and differences compared to two-dimensional systems and with the purpose of investigating whether various probes of topology work for fractal lattices. Similarly to two-dimensional systems, we find that the connected particle-particle correlation function decays roughly exponentially with the distance between the lattice sites measured in the two-dimensional plane, but the values also depend on the local environment. Contrary to two-dimensional systems, we find that the entanglement entropy does not follow the area law if one defines the area to be the number of nearest neighbor bonds that cross the edge of the selected subsystem. Considering bipartitions with two bonds crossing the edge, we find a close to logarithmic scaling of the entanglement entropy with the number of sites in the subsystem. This also means that the topological entanglement entropy cannot be extracted using the Kitaev-Preskill or the Levin-Wen methods. Studying the entanglement spectrum for different bipartitions, we find that the number of states below the entanglement gap is robust and the same as for Laughlin states on two-dimensional lattices.
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