Real-Space Construction Of Topological Insulators And Topological Superconductors

Topological quantum matter has been a hot research direction in the field of condensed matter physics for the past ten years.According to the symmetry,dimension,and other conditions of the system,a variety of rich topological states have been proposed one after another,which greatly expands people’s understanding of quantum states.With the deepening of research,some researchers began to systematically classify,search and diagnose topological states,and strive to describe the properties of a certain type of topological states under a unified theoretical framework.For this research purpose,different methods and means have been proposed.This paper focuses on and discusses the method of real-space construction,and shows its important role and characteristics in the study of topological states protected by crystalline symmetry.Firstly,we briefly review the development of topological states in Chapter 1,including basic concepts,important models,and classification.Then,in Chapter 2,we introduce the complete procedure of the real-space construction method,using the previous classification work of non-magnetic topological crystalline insulators as an example.In Chapter 3,we present the complete classification and diagnosis of magnetic topological states.Based on the previously developed real-space construction method,we have made a complete classification of magnetic topological crystalline insulators.For each magnetic space group,we give the real-space construction and classification of the topological insulators,and on this basis,find out all the topological invariants protected by magnetic symmetries and the corresponding non-trivial surface states.At the same time,we combine the symmetry-based indicator theory to establish a quantitative mapping between the symmetry indicators and the topological invariants,which greatly simplifies the process of diagnosing topological states.It is worth mentioning that as a byproduct of the quantitative mapping,we found all the Weyl semimetals in magnetic space groups that can be diagnosed by symmetry indicators.In Chapter 4,we use real-space construction method to treat the classification of two-dimensional DIII class topological crystalline superconductors.We consider both the topological superconductors protected by wallpaper groups and those protected by layer groups,give a relatively complete classification,and discuss the corresponding boundary states.Among the topological superconducting states by real-space construction,we found several of them have never been discussed in previous works,and their boundary states are not easy to see directly.Our method provides a real-space image for understanding topological crystalline superconductors and provides some guidance for searching topological superconductor materials.