标题： 拓扑能带系统和有限尺寸效应 显示英文标题

标题： Topological Band Systems and Finite Size Effects

摘要： 最近关于拓扑绝缘体的发现正在促进理论和实验研究。 在本论文中，回顾了拓扑绝缘体和量子霍尔效应的基本概念，重点讨论了边缘态及其能带结构。 描述了具有磁通穿过的晶格模型，并介绍了霍夫斯塔特模型，适用于有无局域无序的有限系统。 给出了基于冷原子在合成维度的光学晶格中的实验程序概述。 为了了解这些冷原子的小系统在多大程度上模拟拓扑绝缘体的行为，提供了某些有限尺寸效应的分析，并使用微扰理论推导了能带结构中的能隙开启。 显示英文摘要

摘要： The recent discoveries about topological insulators have been promoting theoretical and experimental research. In this dissertation, the basic concepts of topological insulators and the Quantum Hall Effect are reviewed focusing the discussion on edge states and their band structure. Lattice models with pierced magnetism are described and the Hofstadter model is presented for bounded systems with and without an in-site disorder. An overview of the experimental procedure based on cold atoms in optical lattices with synthetic dimensions is given. In order to understand to what extent these small systems of cold atoms mimic the behaviour of a topological insulator, an analysis of some finite size effects is provided and a deduction of the gap opening in the band structure is presented using perturbation theory.