**Topological Insulators**

**Chapter 1**

Provisional chapter

**Analysis of Topological Material Surfaces**

We provide a systematic analysis of the boundary condition for the edge state, which is a ubiquitous feature in topological phases of matter. We show how to characterize the boundary condition, and how the edge state spectrum depends on it, with several examples, including 2d topological insulator and 3d Weyl semimetal. We also demonstrate the

DOI: 10.5772/intechopen.74934

Keywords: topological insulator, Weyl semimetal, boundary condition, lattice fermion

Study of topological phases of matter has been a hot topic in condensed-matter physics for recent years [1]. An importance of topological aspects of materials themselves was already noticed around the discovery of quantum Hall effect (QHE) in early 1980s. QHE is universally observed in a two-dimensional system, but it requires a strong magnetic field, which breaks time-reversal symmetry. A breakthrough after 20 years was the discovery of quantum spin Hall effect (QSHE), which actually demonstrates that a topological phase is possible even without breaking time-reversal symmetry. This opens a new window of the research on

A universal feature of topological phases is the bulk/edge correspondence [2]: once the bulk wave function has a topologically nontrivial configuration; there exists a gapless edge state

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

Analysis of Topological Material Surfaces

edge-of-edge state localized at the intersection of boundaries.

topological insulators (TIs) and topological superconductors (TSCs).

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.74934

Taro Kimura

Taro Kimura

Abstract

1. Introduction

#### **Analysis of Topological Material Surfaces** Analysis of Topological Material Surfaces

#### Taro Kimura Taro Kimura

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.74934

### Abstract

We provide a systematic analysis of the boundary condition for the edge state, which is a ubiquitous feature in topological phases of matter. We show how to characterize the boundary condition, and how the edge state spectrum depends on it, with several examples, including 2d topological insulator and 3d Weyl semimetal. We also demonstrate the edge-of-edge state localized at the intersection of boundaries.

DOI: 10.5772/intechopen.74934

Keywords: topological insulator, Weyl semimetal, boundary condition, lattice fermion
