1. Introduction

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

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.

localized at the boundary. Such an edge state is topologically protected, and thus is robust against any perturbations as long as respecting symmetry of the system. In practice, the edge state plays a significant role in detection of topological phases since it can be directly observed in experiments using angle-resolved photo-emission spectroscopy (ARPES). Therefore the boundary condition dependence of the edge state is expected to provide experimentally useful predictions.

H2d p

We obtain two eigenvalues e� p

number ξ∈ C, is accordingly obtained as

!� � <sup>¼</sup> <sup>p</sup>1σ<sup>1</sup> <sup>þ</sup> <sup>p</sup>2σ<sup>2</sup> <sup>þ</sup> <sup>m</sup>σ<sup>3</sup> <sup>¼</sup>

r

1 ξ

local coordinate, and the eigenstate is given by an element of CP<sup>1</sup> in this model.

where we use the differential form notation in the momentum space, d ¼ ∂=∂pi

. Under the momentum-dependent transformation, ξ ! e

phase rotation of the eigenstate <sup>ψ</sup>), the connection behaves as <sup>A</sup> ! <sup>A</sup> � <sup>d</sup>ϕ<sup>=</sup> <sup>1</sup> <sup>þ</sup> j j <sup>ξ</sup> <sup>2</sup> � �. This is a U(1) gauge transformation, which is local in momentum space, and the curvature is invariant under this transformation by itself. This U(1) structure is directly related to the S<sup>1</sup> fibration of

An important point is that we can construct the topological invariant from the Berry connection and curvature (4). For the 2d system, it is given as an integral of the curvature over the

dp1dp<sup>2</sup> <sup>F</sup><sup>12</sup> <sup>¼</sup> <sup>1</sup>

which is called the TKNN number, which computes the Hall conductivity of the system [11]. We remark that it is invariant under the continuous deformation of the mass parameter, so that

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ! � � � � � � 2 þ m<sup>2</sup>

� � with <sup>ξ</sup> <sup>¼</sup>

reparametrize the eigenstate with ξ�<sup>1</sup> instead of ξ. This means that ξ is not a global, but just a

Since this system is gapped, we can neglect the transition between lower and upper bands as long as we consider the adiabatic process. Under such a process, we can consider the Berry

<sup>1</sup> <sup>þ</sup> j j <sup>ξ</sup> <sup>2</sup> , <sup>F</sup> <sup>¼</sup> <sup>d</sup><sup>A</sup> <sup>¼</sup> <sup>i</sup> <sup>d</sup>ξ<sup>∗</sup>

, and interpreted as a consequence of the particle number conservation of each

!� � ¼ �

<sup>ψ</sup> <sup>¼</sup> <sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> <sup>þ</sup> j j <sup>ξ</sup> <sup>2</sup>

We remark that the parameter ξ becomes singular ξ ! ∞ at p

connection and curvature defined from the gapped eigenstate<sup>2</sup>

the Berry connection is one-form <sup>A</sup> <sup>¼</sup> <sup>A</sup>1dp<sup>1</sup> <sup>þ</sup> <sup>A</sup>2dp<sup>2</sup>

eigenstate which holds under the adiabatic process.

See a textbook on this topic, e.g., [10] for more details.

ð Þ id <sup>ψ</sup> ¼ �Im <sup>ξ</sup><sup>∗</sup>d<sup>ξ</sup>

<sup>ν</sup>2d <sup>¼</sup> <sup>1</sup> 2π ð

<sup>A</sup> <sup>¼</sup> <sup>ψ</sup>†

<sup>F</sup> <sup>¼</sup> <sup>F</sup>12dp<sup>1</sup>

<sup>C</sup>P<sup>1</sup> <sup>¼</sup> <sup>S</sup><sup>3</sup>

2

dp<sup>2</sup>

=S<sup>1</sup>

momentum space,

q

m Δ<sup>∗</sup> p

!� � �<sup>m</sup>

<sup>e</sup> <sup>þ</sup> <sup>m</sup> <sup>¼</sup> <sup>e</sup> � <sup>m</sup> Δ p

Δ p

Δ p !� �

0 B@ !� �

1

. The eigenstate, parametrized by a complex

dξ

CA: (2)

5

Analysis of Topological Material Surfaces http://dx.doi.org/10.5772/intechopen.74934

!� �<sup>∗</sup> : (3)

!¼ 0. At this point, we have to

<sup>1</sup> <sup>þ</sup> j j <sup>ξ</sup> <sup>2</sup> � �<sup>2</sup> (4)

� �dpi

ð Þξ (not an overall

, and the curvature is two-form

iϕ p !

<sup>2</sup> sgn ð Þ <sup>m</sup> (5)

, namely

In this article, we provide a systematic analysis of the boundary condition of topological material surfaces, including TIs and also Weyl semimetals (WSMs) [3, 4].<sup>1</sup> In Section 2, we discuss some preliminaries on the band topology of TI and WSM. We explain how one can obtain topological invariants from the band spectrum. In Section 3, we provide a systematic study of the boundary condition. We show how to obtain and characterize the boundary condition for a given Lagrangian or Hamiltonian. Then we apply this analysis to the edge state of 2d TI and 3d WSM both in the continuum effective model and the discretized lattice model. In Section 4, we extend the analysis to the situation with two boundaries in different directions. We demonstrate the existence of the edge state localized at the intersection of surfaces, that we call the edge-of-edge state.
