拓扑新材料研究前沿的理论基础导引Ⅱ:多种拓扑材料

INTRODUCTION TO BASICS OF NEW TOPOLOGICAL MATERIAL RESEARCH Ⅱ:VARIOUS TOPOLOGICAL MATERIALS

本专题旨在为感兴趣且学有余力的本科生和低年级研究生提供一条从大学物理到当前二维、三维拓扑新材料研究前沿的学习路径。专题包含两篇,本文是第二篇。基于第一篇回顾过的各种量子霍尔效应及其中的拓扑不变量,本篇介绍多种新材料,包括石墨烯、拓扑绝缘体和拓扑半金属。重点将放在晶格结构的紧束缚模型处理,以蜂窝六角晶格和方晶格为例推导出描述量子反常霍尔效应的两带模型,介绍能带结构与拓扑不变量(陈数、拓扑映射度等)的联系及其所导致的体-边对应关系(bulk-boundary correspondence)。其中自旋轨道耦合(spin-orbitcoupling)的写法、哈密顿量的傅里叶(Fourier)变换、能带结构的求解、拓扑不变量及边缘态的计算都有助于夯实初学者的理论基础。另外本篇还将简述不同种类的拓扑半金属的能带结构及其表面态。

We aim to provide a way for interested undergraduate and postgraduate students to launch from college physics to access the cutting edge of the current international research on two-dimensional and three-dimensional new topological materials.This topic contains two parts,this paper is the second part.Based on the quantum Hall effects(integer,fractional,anomalous,spin)reviewed in the first paper of this topic,we give a brief introduction in this paper to several new topological materials:graphene,topological insulators and topological semimetals.As the examples of the two-band model describing the quantum anomalous Hall effects,we discuss the tight-binding model in graphene and square lattices,with emphasis placed on the relationship between energy bands and topological invariants,as well as the resulting bulk-boundary correspondence.The reader is suggested to strengthen the capability of theoretical calculation by paying attention to spin-orbit coupling,Fourier transform of Hamiltonian,solving energy bands,and the calculation of topological invariants and edge states.Finally a brief introduction to the energy bands and the corresponding surface states are also presented for different types of semimetals.