無義拓樸中之穩定保護無隙邊緣態

國立臺灣大學理學院物理學研究所 碩士論文

Department of Physics College of Science

National Taiwan University Master Thesis

無義拓樸中之穩定保護無隙邊緣態

Stably Protected Gapless Edge States in Trivial Topology

陳允中

Yun­Chung Chen

指導教授: 高英哲 博士 Advisor: Ying­Jer Kao, Ph.D.

中華民國 112 年 1 月 January, 2023

國立臺灣大學碩士學位論文

口試委員會審定書

無義拓樸中之穩定保護無隙邊緣態

Stably Protected Gapless Edge States in Trivial Topology

本論文係陳允中君（R10222002）在國立臺灣大學物理學

研究所完成之碩士學位論文，於民國 112 年 1 月 4 日承下列

考試委員審查通過及口試及格，特此證明

致謝

這份論文能夠完成，最感謝的就是我的指導教授高英哲老師。因 為在大學時做的是宇宙學領域，跟凝態理論完全沒有交集，所以一開 始有點徬徨。但高英哲老師從我比較熟悉的軸子出發，帶我進入拓樸 絕緣體的世界裡，並讓我認識了謝長澤、黃一平老師和張博堯老師。

高英哲老師及這三位老師給我非常多的建議，告訴我可以去尋找那

些可行的方向，並在研究有初步的結果時給了我很多寶貴的意見，使

得我更加理解研究內容。另外，林育平學長也在這個研究中幫助我很

多，除了不斷地指引我方向，也告訴我哪些是比較有趣的議題，讓我

比較有個目標。學長也很辛苦地幫我潤飾了英文的語句，這方面真的

很感謝。我也想謝謝陳昱學學長，總是很有熱情的陪我討論物理問

題，並以學長的身分告訴我很多做研究該注意的事情。最後，我想要

謝謝一直支持我的女朋友廖品瑜，以及我的家人，還有在實驗室陪我

聊天打球的同學及學弟們，讓我的研究生生涯充實了不少。

摘要

拓樸絕緣體是一種具有整體性拓樸或邊界性異常的一系列材料。

其中，整體­邊界對應關係是拓樸絕緣體中很重要的特徵，包括陳絕 緣體和 Z

拓樸絕緣體。這些拓樸絕緣體具有威爾森迴圈光譜環繞的 特性，並且無法被構造出對稱局域萬尼爾函數。然而，最近的研究指 出這些整體性拓樸特徵並不一定能反映出邊界的保護無隙態。本論文 第一個提出穩定保護邊界態也可以在沒有前述整體性拓樸的情況下出 現。我們發現這些「無義拓樸絕緣體」具有穩定的多胞性拓樸。特別 地是，本論文模型中的穩定保護無隙態並不是被晶體對稱性保護而是 被鏡像反對稱給保護。另外，我們也給出了拓樸不變量以及確認了糾 纏光譜中的環繞。本論文因此得到了即使原子絕緣體也可以有穩定無 隙邊界態的結論。

關鍵字：拓樸絕緣體、保護無隙邊緣態、鏡像反對稱、萬尼爾函數、

纏結光譜、威爾森迴圈、多胞性

Abstract

Topological insulator is a class of materials that exhibits nontrivial bulk topology or boundary anomalies. The associated bulk­boundary correspon­

dence serves as an important feature of topological insulators, including Chern insulators and Z

topological insulators. These topological insulators have spectral flow in the Wilson­loop spectrum and encounter Wannier obstruc­

tion. However, recent studies show that the bulk topological features may not imply the existence of protected gapless boundary states. In this thesis, we first provide an example where the stably protected gapless edge states arise without the aforementioned bulk topological features. We show that this trivialized topological insulator belongs to the first example of systems with stable multicellularity and non­delta­like Wannier functions. The gap­

less edge states in the model are not protected by the crystalline symmetry but the mirror antisymmetry. In addition, we identify the associated topolog­

ical invariant and the spectral flow in the entanglement spectrum. This thesis thus clarifies that even the atomic insulators can host stably protected gapless edge states.

Keywords: topological insulator, protected gapless edge states, mirror anti­

symmetry, Wannier function, entanglement spectrum, Wilson loop, multicel­

lularity

Contents

Page

口試委員審定書 i

致謝 iii

摘要 v

Abstract vii

Contents ix

Chapter 1 Introduction 1

1.1 Background and Motivation . . . 1

1.2 Outline of the Thesis . . . 2

Chapter 2 Characterization of Topological Insulators 5 2.1 K theory . . . 6

2.2 Entanglement Spectrum . . . 10

2.3 Wilson­loop Spectrum . . . 12

2.3.1 Theory . . . 12

2.3.2 Interpolation Theorem . . . 15

2.4 Wannier Function . . . 16

2.4.1 Basic Properties . . . 17

2.4.2 Wannier Obstruction . . . 19

Chapter 3 Trivialized Topological Insulator 25 3.1 Model . . . 25

3.2 Mirror Antisymmetry . . . 27

3.3 Wannier Function . . . 29

3.3.2 Exponentially Localized Wannier Function . . . 31 3.3.3 Multicellularity . . . 34 3.4 Entanglement Spectrum . . . 35

Chapter 4 Summary and Outlook 37

References 39

List of Figures

1.1 Different types of topological insulators under the bulk­boundary corre­

spondence. The arrow indicates the direction of the correspondence. . . . 3 2.1 (a) Entanglement spectrum and (b) entanglement energy of the Chern in­

sulator. . . 11 2.2 Illustration of obtaining the Wilson­loop spectrum. . . 13 2.3 (a) Wilson­loop spectrum of a Chern insulator with Chern number C = 2.

(b) Wilson­loop spectrum of aZ2topological insulator. . . 14 2.4 Illustration of the interpolation theorem. . . 15 3.1 Protected gapless edge states in the TTI. Here we consider the model (3.2)

with m = 1.0 and B_z = −λSO = 0.7. (a) and (c) show the Wilson­

loop spectra as a function of k_xand k_y, respectively. (b) and (d) show the protected gapless edge states (red) from nanoribbon energy spectra in the x and y­direction, respectively. . . . 29 3.2 (a) det[s_nm(k)] of the TTI. Here we set the parameters m = 1.0 and

B_z = −λSO = 0.7. Since det[s_nm(k)] > 0 for all k in the Brillouin zone, there is no obstruction to constructing the exponentially localized Wannier functions. (b) The exponential localization of the fist valence band Wannier function W₁^v(r). The probability of the Wannier wavefunc­

tion exponentially approaches to zero when r is large. The other Wannier functions also show the same decaying behavior. . . 32 3.3 Wannier functions of the TTI around the Wanner centers. Each plot cor­

responds to a single Wannier function. Here we plot only the nine neigh­

boring atoms, although the results are more extended. The size of the red spot is proportional to the probability of the Wannier function. The blue arrows represent the magnitude and direction of the spins in the x­y plane. (a) and (b) illustrate one of the valence band Wannier functions

|W0,1^v ⟩ projected to orbital A and B, respectively. (c) and (d) are the cor­

responding conduction band Wannier function|W0,1^c ⟩ projected to orbital B and A, respectively. The Wannier functions in (a) and (c), (b) and (d)

3.4 Entanglement spectrum of the TTI. Here m = 1.0 and B_z =−λSO = 0.7.

(a) shows the entanglement spectrum. The entanglement cut is parallel to the x­axis. (b) shows the entanglement energies. . . . 35

List of Tables

2.1 Periodic table of topological insulators classified by internal symmetries. 8 2.2 Classifying spaces for the ten symmetry classes. . . 9

Chapter 1 Introduction

1.1 Background and Motivation

Topological insulator (TI) is a class of materials that exhibits nontrivial bulk topology and boundary phenomena [1–7]. The prototypical examples are the strong TIs protected by internal symmetries. They host the protected gapless boundary states which are robust under disorders. Strong TIs are well studied and can be completely classified by K theory [8–11]. These include Chern insulators and Z₂topological insulators [1,12]. In addition to internal symmetries, the topology can also be protected by crystalline symmetries, and one can define the topological crystalline insulators (TCIs) [13–18]. An example in this class is the three­dimensional mirror Chern insulator, which hosts gapless surface states under the protection of mirror symmetry [13,15,16]. Contrary to the bulk­boundary correspondence (BBC) of strong TIs, the gapless states can arise only at mirror­invariant surfaces, while all other surfaces are gapped.

Recently, a new class of TCIs with quite different BBC, the higher­order topological insulators (HOTIs), has been identified [18–23]. HOTIs host protected boundary states that live on lower­dimensional surfaces. By definition, a n­th order HOTI has (d− n)­

dimensional protected surface states. HOTIs have been observed in many models and realized in solid­state, photonic, and phononic systems [19, 20, 22–36]. Another exam­

ple is the multipole­moment insulators with quantized fractional corner charges (FCCs) [20,21,37–43]. They do not host midgap corner states in general, although the total charge accumulations at the corners are quantized. Note that HOTIs and multipole­moment insu­

lators are TCIs in general, where the crystalline symmetry (or some combination of crys­

talline symmetry and internal symmetry) is necessary to protect the boundary features.

It is natural to ask whether there is also a general BBC between the bulk topology and surface responses in different kinds of TCI. Here, the concept of topology must first be defined. A recent definition utilizes the idea of Wannierizability of the occupied bands, including topological quantum chemistry (TQC) [44–46] and symmetry indicators (SIs) [17, 47, 48]. In this type of definition, the occupied space of a topological insulator cannot be represented by exponentially localized symmetric Wannier functions [44, 45]. One can understand it physically as a topological insulator cannot be deformed adiabatically into an atomic insulator with exponentially localized symmetric Wannier functions. We adopt this strict definition throughout this thesis.

In contrast, the classification by K theory can only offer the relative topology. An example of this difference is the Su­Schrieffer­Hegger (SSH) model. Although the trivial and the nontrivial phase of the model are Wannierizable [49], a mismatch of the atomic and Wannier centers turns the nontrivial phase into an obstructed atomic insulator (OAI) [20, 37, 44]. Even though the nontrivial phase is considered trivial in the theory of topological quantum chemistry, OAIs can show nontrivial physics such as the quantized FCCs [20,37].

Surprisingly, the converse is also true as recent studies have identified a class of non­

Wannierizable systems that are considered relatively trivial in K theory, known as fragile topological systems [50–63]. Opposed to the stable topological systems with robust bulk­

boundary correspondence, fragile topological systems can be trivialized by adding trivial atomic bands into the occupied space [50]. Although protected gapless boundary states may still appear in fragile topological systems [14, 58, 59, 62, 64, 65], they already offer examples where the bulk topology does not necessarily imply the protected gapless surface states.

1.2 Outline of the Thesis

In this thesis, we consider the converse question: is it also possible to find systems hosting stably protected gapless surface states, yet without Wannier obstruction? We show that the answer is positive and construct an example of such systems, termed the trivialized topological insulator (TTI). The TTI, although considered trivial in the theory of topologi­

Gapless Wilson-loop spectrum

Gapless edge spectrum

Strong TIs

Strong TIs, Stable TCIs, Fragile TIs

Stable TCIs, Fragile TIs

Trivialized TIs

Figure 1.1: Different types of topological insulators under the bulk­boundary correspon­

dence. The arrow indicates the direction of the correspondence.

cal quantum chemistry, host stably protected gapless edge states. The absence of Wannier obstruction further implies that the Wilson­loop spectrum does not exhibit nontrivial wind­

ing. Interestingly, the TTI in this thesis is not protected by crystalline symmetries but a combination of mirror (M) and chiral (S) symmetries, or mirror antisymmetry ( ˜M). Here we note that the TTI in this thesis serves as a counterexample to the interpolation theorem between the Wilson­loop and edge energy spectra [18, 20, 66–72], see Fig. 1.1.

Two­dimensional strong TIs, such as Chern insulators and Z2topological insulators, must host the robust one­to­one bulk­boundary correspondence. However, as we have dis­

cussed, TCIs or fragile TIs can exhibit protected winding in Wilson­loop spectra yet still with corresponding gapped surface states. TTIs, contrary to the above cases, host stably protected gapless edge states yet without Wannier obstruction and Wilson­loop winding.

It is worth emphasizing that the Wannier functions of TTIs still cannot be deformed adi­

abatically into delta functions localized on the atoms. In this perspective, TTIs belong to a new class of non­delta­function insulators with multicellularity [73–75]. For the non­

delta­function insulators in these recent studies, either there is no protected edge state or the conditionally robust surface states exist under the delicate topology [74]. The topology can be trivialized by adding trivial atomic bands into either the occupied or unoccupied space. In contrast, TTIs host stably protected gapless edge states and the topology cannot be trivialized by adding trivial atomic bands. TTIs thus present more promising applica­

tions in real materials or photonic systems.

This thesis is organized as follows. We begin with an introduction of some basic topological theories in Chap. 2. In Sec. 2.1, we review the basics of K theory and the ten­

fold way classification. In Sec. 2.2, we introduce the concept of entanglement spectrum and show its closed relation with the edge spectrum. In Sec. 2.3, we discuss the Wilson­

loop spectrum and its useful applications, in addition with the interpolation theorem which relates the Wilson­loop spectrum to the edge spectrum. In Sec. 2.4, we show that how the Wannier function can successfully characterize topological insulators with crystalline symmetries. The key concepts are the Wyckoff positions and their corresponding band representations.

After the introduction in Chap. 2, we turn to the main theme in this thesis: the triv­

ialized topological insulator, which is our focus in Chap. 3. In Sec. 3.1, we construct the model of the TTI from the Z₂ TI by adding the perturbed terms that trivialize the Z₂ index but preserve mirror antisymmetry. In Sec. 3.2, we show that mirror antisymme­

try can protect the gapless edge states which serve as the domain walls between the bulk and the vacuum. Besides, we construct the topological invariant associated with mirror antisymmetry and show that it is nontrivial for the TTI. We verify our theory by numeri­

cally calculating the edge spectra. It is found that the Wilson­loop spectra are gapped and therefore the TTI serves as a counterexample of the interpolation theorem.

In Sec. 3.3, we explicitly calculate the Wannier functions and show that they are mir­

ror antisymmetric and exponentially localized. Going into the flat­band limit, we prove that the Wanner function cannot be deformed adiabatically into the delta function. There­

fore, the TTI is the first example of the stable multicellularity in two dimensions. In Sec. 3.4, we further calculate the entanglement spectrum to confirm the existence of the nontrivial entanglement in the system which leads to the gapless edge states. In Chap. 4, we conclude and discuss the results in this thesis. Several materials in this thesis have been adopted from materials published in arXiv:2206.04858. All the numerical results in this thesis are performed using the P_YTHTB package [76].

Chapter 2 Characterization of Topological Insulators

How to characterize topological insulators has been a long­standing problem. Stud­

ies had focused from specific topological invariants to K theory which can completely classify the internal­symmetry protected TIs or even TCIs with simple crystalline sym­

metries [9–11, 77]. For example, Chern insulators which live in two dimensions can be identified by the Chern number which gives a Z classification in Altland­Zirnbauer class A. Another example is the Z₂ topological insulator which is protected by time­reversal symmetry and has a Z₂ classification in Altland­Zirnbauer class AII.

After the appearance of the ten­fold way which classifies the strong TIs, studies had turned the focus to the TCIs. Here, K theory is not as successful as usual since it is trou­

blesome to generalize the mathematical theory to 230 space groups. In this period, there are two power tools appeared which can indicate the underlying topology: entanglement spectrum [78–82] and Wilson­loop spectrum [20, 23, 51, 54, 55, 62, 65–71, 83–85]. The topology is often exhibited as the winding patterns in these spectra, and both theories are successful for the inversion­symmetry (I) protected topological insulators. However, people started to realize that they are not really powerful as there are many systems that cannot be fully captured by both spectra. One of the main shortcoming is that the winding patterns cannot clearly distinguish topological phases protected by different symmetries.

Therefore, both tools serve as the auxiliary tools to characterize the topology nowadays . The modern theory of TCIs lies at the heart of Wannier functions, including the theory of topological quantum chemistry and symmetry indicators [17, 44]. By their definition, the occupied space of a topological insulator cannot be spanned by symmetric and ex­

ponentially localized Wannier functions. The physical picture is clear as a topological insulator cannot be deformed adiabatically to an atomic insulator which has symmetric and exponentially localized Wannier functions. By showing that the representations of the occupied bands cannot be spanned by any of the atomic bands with different Wyckoff positions, one can identify the topology in the systems. In this way, all the TCIs protected by 230 space groups and even the 1421 magnetic space groups are classified [86, 87].

There are even new classes of topological systems identified along with the theory.

The first one is the OAI which has obstructed Wannier centers that do not live on the atoms [21, 44]. Although they are considered trivial in the theory of Wannier obstruction, one should note that it is still impossible to deform adiabatically to the unobstructed trivial atomic insulators without closing the band gap. The SSH model and multipole­moment insulators all fall into this class. Another example is the fragile TIs [50]. Although they are considered topological because of the Wannier obstruction, the topology can be trivialized by adding specific trivial atomic bands. Therefore, they are considered trivial in K theory and often without protected gapless edge states, contrary to the stable TCIs. The complete understanding of the fragile TIs is still an ongoing research topic.

2.1 K theory

Consider the non­interacting fermionic systems. Here, we also assume the trans­

lational symmetry to simplify the discussions although it is not necessary. The second quantized tight­binding Hamiltonian can be written into the Bloch form:

H =ˆ X

k

c^†_i,k H(k)

ijc_j,k, (2.1)

where

c_j,k = 1

√N_t X

R

e^ik·Rc_j,R (2.2)

is the electron annihilation operator. Here, i, j = 1, ..., 2N labels the basis orbitals and spins, R labels the unit cell position, and Nt is the total number of the unit cells. We use the convention that the Bloch Hamiltonian H(k) is periodic under a translation of a

reciprocal lattice vector G:

H(k) = H(k + G). (2.3)

The intra­cell eigenstates are defined by:

H(k)|u^l(k)⟩ = E^l(k)|u^l(k)⟩, (2.4)

where l = 1, ..., 2N labels the band indices. E^l(k) is the eigenenergy. Note that one can generically choose a smooth and periodic gauge for the eigenstates if the total Chern number is zero. Considering the half­filling band insulators, the intra­cell states can be decomposed into the valence states|u^lv⟩ and the conduction states |u^lc⟩, where l = 1, ..., N.

The Bloch state is given by|ψ^l(k)⟩ = e^ik·r|u^l(k)⟩.

K theory classifies the different topological phases with internal symmetry, where the symmetry operator only acts on the internal basis index j but not the lattice index R.

The simplest and most common internal symmetry is the unitary symmetry, and the Hamil­

tonian has an associated conserved current. Most importantly, the Hamiltonian commutes with the unitary symmetry operator and we can bring it into the block­diagonal form. It follows that the Hilbert space is a direct sum of these eigensubspaces, and there is nothing interesting about it.

However, something interesting happens when we consider the anti­unitary symme­

try or antisymmetry, where the Hamiltonian anti­commutes with the symmetry operator.

Actually, there are only ten possible different symmetry combinations which are com­

posed of three elementary symmetries: time­reversal symmetry (T )

T H(k)T⁻¹ = H(−k), (2.5)

particle­hole symmetry (P)

PH(k)P⁻¹ =−H(−k), (2.6)

and chiral symmetry (C)

CH(k)C⁻¹ =−H(k). (2.7)

Note thatT and P are anti­unitary and C is unitary. Finally, from Eq. (2.7), it is clear that

Table 2.1: Periodic table of topological insulators classified by internal symmetries.

class T² P² C² d = 0 1 2 3 4 5 6 7

A Z 0 Z 0 Z 0 Z 0

AIII 1 0 Z 0 Z 0 Z 0 Z

AI 1 Z 0 0 0 2Z 0 Z2 Z2

BDI 1 1 1 Z2 Z 0 0 0 2Z 0 Z2

D 1 Z2 Z2 Z 0 0 0 2Z 0

DIII −1 1 1 0 Z2 Z2 Z 0 0 0 2Z

AII −1 2Z 0 Z2 Z2 Z 0 0 0

CII −1 −1 1 0 2Z 0 Z2 Z2 Z 0 0

C −1 0 0 2Z 0 Z2 Z2 Z 0

CI 1 −1 1 0 0 0 2Z 0 Z2 Z2 Z

C = PT .

In Tab. 2.1, we list all the possible classifications of topological phases with ten Altland­Zirnbauer (AZ) symmetry classes in all dimensions [10, 11]. Notice that the clas­

sification is periodic under d→ d + 8, and this is why it is called the periodic table. The ten symmetry classes are differentiated by the existence of three basic symmetries. 1 and

−1 denote the square of the symmetry operators, which can be minus for T and P. Blank space represents that there is no such symmetry in this class. One can easily notice the pe­

riodic behavior in this table, and this is associated with the Bott periodicity in the Clifford algebra. For each topological insulator, we can compute the quantized topological invari­

ant. For example, theZ classifications with (without) chiral symmetry are given by the winding (Chern) number, and theZ2 classifications are given by the Chern­Simons and Fu­Kane invariants. For the detailed mathematical formulas, see for example Ref. [11].

One approach to obtain this periodic table is to construct the homotopy group of the classifying space. The classifying space can be constructed using the Q matrix:

Q(k) = 1− 2P (k), (2.8)

where P (k) =P

i∈o.c.c|uⁱ(k)⟩⟨uⁱ(k)| is the projector to the occupied space. The Q ma­

trix has the elegant property that it is like the original Hamiltonian but squares to 1 with spectrum E(k) =±1. The eigenstates are exactly the same as the original Hamiltonian.

With the imposed symmetries, we can obtain the classifying spaces from Eq. 2.8, using the Clifford algebra in the low­energy Dirac theory. The result for different symmetry

Table 2.2: Classifying spaces for the ten symmetry classes.

class classifying space

A S

nU (N )/(U (N− n) × U(n))

AIII U (N )

AI S

nO(N )/(O(N − n) × O(n))

BDI O(N )

D O(2N )/U (N )

DIII U (N )/Sp(N )

AII S

nSp(N )/(Sp(N − n) × Sp(n))

CII Sp(N )

C Sp(2N )/U (N )

CI U (N )/O(N )

classes are listed in Tab. 2.2 [10, 11]. The capital N represents the total number of bands.

The classification is then given by the d­th homotopty group of the classifying space as N → ∞.

Here, we emphasize again that this classification is still valid under disorders in which the translational symmetry is broken. The nontrivial phases in this periodic table are thus called the strong TIs as the phases are robust under disorders. In contrast, the weak TIs are the phases that are vulnerable under disorders that break the translationail symmetry in cer­

tain directions. These weak TIs are in fact equivalent to the stacking of lower­dimensional strong TIs into higher dimensions. As a result, the classification of weak TIs follows di­

rectly from the lower­dimensional strong TIs.

Finally, for the strong TIs, there exists a relation called the bulk­boundary correspon­

dence which connects the topological invariant to the protected gapless boundary states.

The robust boundary states can be explained through the domain walls that exist at the boundary between the trivial and topological phases [2, 11]. In the low­energy Dirac the­

ory, the mass term changes sign at the boundary such that the solution of the corresponding wave equation describes a domain­wall configuration with linear dispersion. Mathemat­

ically, the topological defect protected by internal symmetry has the same classification as the strong TI, given by K theory. Furthermore, the index theorem exactly relates the number of zero­mode boundary states to the topological invariant of the low­energy Dirac Hamiltonian, which can be expressed as a differential operator [11]. This robust corre­

spondence leads to the most important feature of the topological insulator as the surfaces

can be conducting although the bulk is an insulator. This triggers a huge interest in topo­

logical electronics which may lead to useful applications in our real world.

2.2 Entanglement Spectrum

Entanglement spectrum (ES) is first proposed to study fractional quantum Hall sys­

tems where the ES shows richer phenomena than the entanglement entropy which is merely a number [78]. The ES is defined using the idea of bipartition of the system. Given the ground­state wave function, one can perform the Schmidt decomposition:

|ψ0⟩ =X

i

e^−εi

√Z|iL⟩ ⊗ |iR⟩. (2.9)

It is clear that if one tries to measure left half of the system, the probability is e^−2εi/Z for the state|iL⟩. The quantity εiis called the entanglement energy (or entanglement spectrum in the field of strongly interacting systems). Another equivalent definition is using the reduced density matrix, where the entanglement energy corresponds to the eigenvalues of Tr_Lρ. Then the ES is defined by:

ξ_i = 1

1 + e^2εi. (2.10)

Notice that the ES can only take values between 0 and 1.

For the non­interacting systems, it is shown that the ES can be directly related to the eigenvalues of the two­particle reduced correlation function (C_L)^αβ_ij =⟨c^†i,αc_j,β⟩ [88].

Here, the indices i, j∈ {L} denote the lattice sites in the left half of the system, and α, β represent the orbital degrees of freedom. We can further do the Fourier transform of the reduced correlation function:

(C_L)^αβ_ij (k_y, k_z, ...) = 1 L_x

X

kx

e^ikx(i−j)P_αβ^∗ (k) (2.11)

such that the ES, which is the eigenspectrum of the (C_L)^αβ_ij (k_y, k_z, ...), is also a function of (ky, kz, ...). Now it is clear that the ES is just the flat­band limit (with a 1/2 factor and a shift of center) of the original system with open boundary condition in x­direction, see

(a) (b)

Figure 2.1: (a) Entanglement spectrum and (b) entanglement energy of the Chern insulator.

Eq. 2.8. Especially, the existence of the protected boundary states can often be detected with the gapless spectrum or midgap states in ES [79–81, 89]. For example, for the Chern insulator, the spectral flow can be clearly seen in the ES, see Fig. 2.1. This correspondence works even when the gapless edge states originate from the weak TIs, or midgap corner states in the multipole­moment insulators [82, 90]. This comes from the fact that they are OAIs in which the entanglement cut must cut through the dominant hoppings between the atoms.

Notice that the gapless states and midgap states contribute most of the entanglement entropy since they are near the center 1/2. If there exist certain symmetries which protect the midgap states in the ES, the system cannot be adiabatically transformed to a trivial one with zero entanglement entropy. However, the midgap states in the ES can appear even when there is no gapless edge state in the energy spectrum [51,81,91], such as in the fragile TIs. It happens if the protecting crystalline symmetry (say, inversion symmetry) cannot protect the gapless edge states. The spectral flow of the ES, on the other hand, gives an indication to the protected gapless edge states [81]. It is represented by the middle bands which continuously flow from the valence bands to the conduction bands in the entanglement energy spectrum. In the case of strong TIs, the spectral flow necessarily appears due to the presence of protected gapless edge states [80, 81].

2.3 Wilson­loop Spectrum

Wilson loop, although originated from the lattice gauge theory in quantum chromo­

dynamics (QCD) [92], has become a powerful tool in various types of gauge theories nowadays. Since the Bloch states describe a fiber bundle in the k space, the Wilson loop also successfully characterize the topological properties in TIs [20,23,51,54,55,62,65,66, 70, 71, 83, 84]. The hybrid Wannier center [67–69, 85], although has a different physical meaning, is actually equivalent to the Wilson­loop spectrum [20, 23].

2.3.1 Theory

The definition of the Wilson loop is

W(l) = Pe^−iHlA·dk (2.12)

where l is the loop we integrate along, andP represents the path ordering of the integral.

Anm ≡ ⟨un(k)|∂kum(k)⟩ is the non­Abelian Berry connection, where n, m are the band indices, given a group of bands N . SinceW(l) is a N ×N matrix, one usually diagonalizes this matrix to obtain the useful information from it. The Wilson­loop integral can also be written into the product form:

W(l)n0,nR−1 = X

n1,n2,...nR−1

⟨un0(k₀)|un1(k₁)⟩⟨un1(k₁)|un2(k₂)⟩...⟨unR−1(k_R−1)|unR(k_R)⟩, (2.13) where k_n are the points that discretize the loop l, and the number R → ∞. From this equation, it is easy to see that the Wilson­loop calculates the phases accumulated in a loop of parallel transport. For the simplest case where one only considers a single band with N = 1, the Wilson loop just gives the Berry phase around this loop l:

W(l) = e^−iHlA·dk = e^iθ. (2.14)

Although in general the loop can be chosen arbitrarily, there is a conventional and convenient way to define this loop in the Brillouin zone (BZ). In 1D, the loop is defined

Figure 2.2: Illustration of obtaining the Wilson­loop spectrum.

as the entire BZ because its topology is S¹. When the system has dimension d > 1, the loop is integrated along one of the reciprocal lattice vectors G₁ in the BZ. Then the Wilson loopWG1 can be regarded as a periodic function of k_i with period G_i , i ̸= 1.

After diagonalizing it, we obtain the eigenvalues of the Wilson loop, usually labeled as e^{iθG1,n({ki})}, where n labels the different eigenvalues. θ_G1,n({ki}) is known as the Wilson­

loop spectrum since it is a function of ki and exhibits like the band structures in d− 1 dimensions with total N numbers of bands. Note that it also corresponds to the hybrid Wannier centers [67–69, 85], where the Wannier functions are maximally­localized in one direction but periodic in all the other directions. One can also regard θ_G1,n({ki}) as the

“Wannier bands” but have periodicity θG1,n({ki}) = θG1,n({ki}) + 2π. The algorithm is summarized in Fig. 2.2.

The topological properties of the Wilson loop are manifested through the winding patterns in the Wilson­loop spectrum. The prototypical example is the Chern insulator which is known to have an obstruction of defining a smooth and periodic gauge on the BZ torus due to its non­zero Chern number [93, 94]. This obstruction can be exactly related to the chiral winding number in the Wilson­loop spectrum [23, 67, 83], see Fig. 2.3 (a). It is easy to see that with an adiabatic deformation, it is impossible to trivialize the winding pattern since the periodicity in θ and k should be preserved.

0 2𝜋

𝑘

2𝜋 4𝜋

𝜃(𝑘

)

0 2𝜋

𝑘

2𝜋 4𝜋

𝜃(𝑘

)

(a) (b)

Figure 2.3: (a) Wilson­loop spectrum of a Chern insulator with Chern number C = 2. (b) Wilson­loop spectrum of aZ2 topological insulator.

Similarly, for the case of Z₂ TI in 2D, the Wilson­loop spectrum exhibits relative winding although the chiral winding number is zero totally [67, 84], see Fig. 2.3 (b). The relative winding is protected by the time­reversal symmetry as long as the system does not go into a phase transition. This can be shown by considering the constraints due toT :

{θG1,n(k₂)} = {θG1,n(−k2)}, (2.15)

where the whole Wilson­loop spectrum is symmetric with respect to k₂. There is another constraint: the Wannier bands are degenerate at the high­symmetry points k₂ = 0, π. This Kramers degeneracy enforces the relative winding pattern as long as the relative winding is an odd number such that there exists an odd number of pairs at the high­symmetry points. By breaking the time­reversal symmetry, there is no constraint on the Wilson­

loop spectrum and the relative winding can be adiabatically deformed into the trivial one.

Therefore, we can directly conclude theT ­protected Z2 classification by examining the Wilson­loop spectrum only.

Finally, for the topological crystalline insulators, the Wilson­loop spectra show much more complicated behaviors in general. However, for most of the TCIs, the Wilson­loop spectra still host the protected winding patterns which directly indicate the underlying topology, includingI­protected fragile TCIs [55,83], mirror Chern insulators [23], axion

𝒲_𝒏(𝑘_∥)

0 2𝜋

2𝜋 𝜃_𝒏

𝑘_∥

Edge

𝒏ෝ

0

2𝜋 𝐸

𝑘_∥

Edge spectrum Wilson-loop spectrum

Figure 2.4: Illustration of the interpolation theorem.

TIs [69], nonsymmorphic TCIs [70, 71].

2.3.2 Interpolation Theorem

Interestingly, there is also a “bulk­boundary interpolation” between the Wilson­loop spectrum and the edge spectrum. To be more specific, the Wilson­loop spectrum can be continuously deformed into the edge energy spectrum through an interpolation [18, 66], see Fig. 2.4. If the Wilson­loop spectrum shows the gapless or winding behavior, the energy spectrum is often gapless because of this interpolation. There are many topological systems which explicitly realize this interpolation [18, 20, 67–70, 72].

This interpolation can be shown by first noticing that the Wilson­loop spectrum is exactly the same as the hybrid Wannier centers, which are the eigenvalues of the operator P₁(k_i)x₁P₁(k_i), where P₁(k_i) = R2π

0 P (k)dk₁/2π. Notice that x₁ acts like a potential V (x₁) = x₁ in the x₁ direction. We can thus continuously deform this potential into the step function V (x₁) = Θ(x₁) which has a sharp change at x₁ = 0. This potential configuration can be viewed as the boundary between the bulk (which has a flat energy band in this construction) and the vacuum. If the boundary shows nontrivial behavior

in the gapless spectrum due to the domain walls, it is expected that the hybrid Wannier centers which describe a linear potential also show the same gapless spectra, although they must be periodic contrary to the energy spectra.

However, there are loopholes in this interpolation. Although the interpolation is a continuously deformation, it does not guarantee that the gapless feature is preserved un­

der this interpolation. Specifically, there have been examples with gapless Wilson­loop winding but without the gapless surface states, such as theI­protected fragile TCIs and axion TIs [55, 69, 83]. The key point is that the inversion symmetry itself cannot protect the gapless edge states although it can protect the winding in the Wilson­loop spectrum.

Nevertheless, it is claimed that the gapless edge spectra can still correspond to the gapless Wilson­loop spectra [69]. In this thesis, we given an explicit example showing that this is also not true in general, see Sec. 3.2.

A stronger version of the interpolation theorem which preserves the gapless features is proposed with the requirement that the interpolation must be preserved under the protect­

ing symmetries [71]. Under this constraint, the interpolation Hamiltonian (the potential) is also preserved under the protecting symmetries. If the gapless surface states are protected under the symmetries, the hybrid Wannier bands must also exhibit the gapless spectra.

However, this requirement makes the interpolation theorem somewhat useless: first, for most of the symmetries (including chiral, particle­hole and inversion symmetries), this requirement is not satisfied. Second, given a topological system, we usually don’t know the protecting symmetry at the first place, so one cannot be sure whether the interpolation preserves the gapless spectra. If we can figure out the underlying protecting symmetries, the existence of the corresponding protected gapless edge states can usually be known, without referring to the Wilson­loop spectrum.

2.4 Wannier Function

If all the electrons are strictly localized on the atoms without any hoppings or interac­

tions between, the ground state is composed of the product of all these electrons’ orbitals.

This is called the trivial atomic insulator [50], where the systems cannot conduct since the

single­particle wavefunctions are all localized.

When we turn on the hoppings between the orbitals of different atoms, the many­body ground state wavefunction can still be described by the product of the so­called Wannier functions, which are linear combinations of the orbitals of different atoms. Therefore, it serves as an effective orbital of which the product is the ground state wavefunction. Con­

trary to the Bloch state which is in the momentum space, the Wannier function lives in the position base. The construction of the Wannier function can thus give us an effective picture of the hybridized orbitals. Moreover, it is realized recently that the Wannier func­

tion can be a powerful tool to characterize the topological insulators. The idea relies on the Wannier obstruction which states that the Wannier functions cannot be exponentially localized due to the underlying topology.

2.4.1 Basic Properties

Given the Bloch state|ψn(k)⟩ = e^ik·r|un(k)⟩, the Wannier function is defined by:

|Wn,R⟩ = V (2π)^d

Z

BZ

d^dke^−ik·R|ψn(k)⟩, (2.16)

where d is the dimension of the system, and V is the volume of the unit cell. The index R labels the position of the Wannier function. It is clear from the definition that the Wannier function is simply the inverse Fourier transform of the Bloch state. Oppositely, one can obtain the Bloch state from the Wannier function:

|ψn(k)⟩ =X

R

e^ik·R|Wn,R⟩. (2.17)

From the definition, we can simply prove several important properties of the Wannier function. First, they are translational images of one another,

W_n,R(r) = W_n,0(r− R). (2.18)

Therefore, the Wannier functions indeed constitute the original lattice and serve as the

effective orbitals at each lattice site. Second, they form an orthonormal set,

⟨Wn,R|Wm,R^′⟩ = δnmδ_RR′. (2.19)

Finally, the Wannier functions form a complete set, P =˜ V

(2π)^d X

n

Z

BZ

d^dk|ψn(k)⟩⟨ψn(k)| =X

n

X

R

|Wn,R⟩⟨Wn,R| (2.20)

where ˜P is the many­body ground state projector.

However, although the Wannier function serves as an effective picture of the atomic orbitals, it is not well­defined. One can in general perform a k­dependent U (N ) gauge transformation on the Bloch states:

|ψ^′n(k)⟩ =X

m

U_nm(k)|ψm(k)⟩, (2.21)

and the resultant Wannier functions

|W_n,R^′ ⟩ = V (2π)^d

Z

BZ

d^dke^−ik·R|ψ^′_n(k)⟩ (2.22)

in general have different shapes with the original ones. This is the gauge degree of free­

dom of the Bloch states. Since the Wannier functions are not gauge­invariant, they don’t have any physical meaning at all if we arbitrarily choose a random gauge. The most common gauge is obtained by maximally localizing the Wannier functions with minimal spread [95, 96]. In this way, the Wannier functions mostly resemble the atomic orbitals as they are well­localized. In general, the maximally­localized Wannier functions can only be found by numerically minimizing the spread iteratively. Interestingly, in 1D, the task be­

comes quite easy as the maximally­localized Wannier functions are eigenfunctions of the operator ˜P x ˜P [95,96]. Notice that if the projector equals to the identity, the Wannier func­

tions become the eigenfunction of the position operator, which is the delta function. By projecting position operator into the occupied space, one can thus obtain the maximally­

localized Wannier functions. This procedure cannot be used in higher dimension since in general ˜P x ˜P , ˜P y ˜P , ... don’t commute with each other. In addition, in 1D, this simply implies that all the systems are Wannier representable: it is always possible to find the

Wannier functions that are exponentially localized in 1D. Therefore, there is no Wannier obstruction in 1D.

Although the Wannier functions are not gauge­invariant, one can simply prove that the total Wannier center at R = 0

X

n

¯

rn=X

n

⟨Wn,0|r|Wn,0⟩ (2.23)

is a gauge­invariant quantity. Therefore, even though the shape of the Wannier functions can change arbitrarily, the center of them is always fixed. This is an important property and we are going to use it to construct the topological theory of TCIs in the next section.

2.4.2 Wannier Obstruction

To understand the reason that leads to the Wannier obstruction, we first study the Chern insulator which is a paradigmatic example. It is well known that the Bloch bands with nontrivial Chern number cannot have smooth and periodic gauge in the Brillouin zone (BZ) [93, 94]. It can be understood by examining the Stokes theorem:

1 2π

I

Tr A· dk = 1 2π

Z

BZ

d²k Tr F_xy =C, (2.24)

where the loop integral integrates around the BZ torus, and C is the Chern number. If the Berry connection is periodic and smooth in the BZ, the integrals along k_x = 0, 2π and ky = 0, 2π should be cancelled out, respectively. The nonvanishing values of the integrals thus imply that there exist singularities of the Berry connection inside the BZ. If we do the inverse Fourier transform to obtain the Wannier functions (Eq. 2.16), the singularities will lead to the power­law decay of the Wannier functions. The reason is exactly the same as the correlation function: the gapless ground state has the power­law decay behavior rather than the exponential decay due to the singularities (gapless points) [97].

It is proved that all the systems with trivial Chern class can be constructed with ex­

ponentially localized Wannier functions [96, 98]. This reflects the fact that the Chern in­

sulator can exist without any symmetry protection. By going into the symmetry­protected

topological insulators, we need to require that the Wannier functions are also symmetric respect to the original Hamiltonian. In other words, the gauge we choose to construct the Wannier functions must also respect the symmetry of the Hamiltonian. The first identified example with such kind of Wannier obstruction is theZ2 TI protected byT symmetry. If one requires that the Wannier functions must form Kramers’ pairs, then there is no way to construct the exponentially localized Wannier function due to the singularities [99].

The Wannier obstruction is then found to be extremely powerful with the considera­

tion of crystalline symmetries. The topological insulators, in the strict definition, can be defined as the systems with Wannier obstruction. However, one still needs a theory to quickly and systematically identify whether the occupied space encounters Wannier ob­

struction. There are two main theories which develop approximately in the same time:

topological quantum chemistry (TQC) [44–46] and symmetry indicators (SIs) [17, 47].

Although they use different approaches to identify the topological insulators, the central idea behind them is still the Wannier obstruction.

In terms of TQC, it considers the so­called elementary band representations (EBRs).

The band representations (BRs) are basically the Bloch states that are induced by the basis orbitals. The EBRs are defined as the BRs then cannot be written as a direct sum of other BRs. It is found that the EBRs are induced from the orbitals with irreducible representa­

tions at all of the maximal Wyckoff positions for most of the space groups [44, 45]. By using the graph­theoretical method which helps to construct all the compatibility relations in the band structures, one can identify the topological insulators which are not BRs. An important theorem of TQC is that if an EBR is disconnected in the band structures, one of the isolated group of bands must be topological since an EBR cannot be written as a sum of two BRs. For example, the topology of the Kane­Mele model can be easily obtained without any knowledge of the details of the Bloch states [44, 45]. Since the complete the­

ory contains a lot of mathematical language, we will not go into the full details of TQC.

Rather, we will focus on the theory of SIs which are more elegant and have a more phys­

ical picture. Nevertheless, compared to TQC, it has some limits and deficiencies as we will discuss in the final remarks.

The idea first comes from the fact that the Z2 index in 3D withT and I symmetry

can be simplified into a product formula [7]:

(−1)^ν =Y

i N /2Y

m=1

ξ_2m(Γ_i), (2.25)

where ν is the Z2 index, ξ_2m(Γ_i) are the parity eigenvalues of the occupied bands 2m (do not count for the time­reversal pairs) at time­reversal invariant momentum Γ_i. The index i counts for all the time­reversal invariant momenta. A similar formula also applies for theZ2 index in 2D withT and I symmetry. More interestingly, there also exists a similar formula for the Chern insulator with Cnsymmetry [100], although in this case the formula can only detect the Chern numberC modulo n. The motivation is to generalize this formula to all 230 space groups with or without spin­orbit coupling and time­reversal symmetry.

The theory of SIs uses the information of the little group irreducible representations (irreps) at all the high­symmetry momenta to detect the topology. The little groups are the nontrivial subgroups of the space group that leave the high­symmetry momenta invariant, up to a translation. The SI is defined by the quotient group,

X_BS≡ {BS}

{AI}. (2.26)

Here,{BS} stands for all the possible band structures that satisfy the compatibility con­

ditions at all the high­symmetry momenta. One can write the sets of{BS} as an Abelian groupZ^dBS. In contrast,{AI} represents all the possible sets of atomic insulators induced from different maximal Wyckoff positions, which is also an Abelian group Z^dAI. The maximal Wyckoff positions are the points in real space that are left invariant (up to a translation) under the nontrivial subgroups (which is called the site­symmetry groups) of the space group. Surprisingly, it is found that d_BS = d_AIand X_BSis always a finite group for 230 space groups [17]. From the definition, it is clear that the nontrivial indices of SIs indicate the Wannier obstruction of which the occupied bands cannot be written as a sum of atomic insulators.

Let’s illustrate the theory of SIs in a simple example in 2D with I symmetry. First, the high­symmetry momenta are Γ = (0, 0), X = (π, 0), Y = (0, π) and M = (π, π).

These high­symmetry momenta are invariant under the the C₂ rotation. In this case, the little group is the same as the point subgroup C₂of the space group P 2. There are only two irreps: plus (+) or minus (−). Now we can construct the set {BS}. Since at each high­

symmetry momentum, the irreps can be (+) or (−), there are total 8 degrees of freedom, which are non­negative integers. But remember that the total band number should be the same for all these four high­symmetry momenta, therefore we have finally 8 + 1− 4 = 5 independent degrees of freedom. Here, following the same procedure in K theory, we generalize the non­negative integers toZ with total band number N → ∞. It follows that {BS} = Z⁵ and d_BS= 5.

Now, we construct the set {AI}. Similar to the high­symmetry momenta, the four maximal Wyckoff positions are a = (0, 0), b = (1/2, 0), c = (0, 1/2) and d = (1/2, 1/2) in the unit of a lattice constant. These are the points that are left invariant under the site­

symmetry group C₂, up to a translation. For each Wyckoff position, we can induce the band representations from two irreps of C2: plus (+) or minus (−), which correspond to s or p orbitals, respectively. The irreps at the four high­symmetry momenta induced from the four Wyckoff positions can be obtained by doing the Fourier transform:

as: (+, +, +, +), b_s: (+,−, +, −), c_s: (+, +,−, −), d_s: (+,−, −, +),

(2.27)

where the order is (Γ, X, Y, M ). These correspond to the BRs induced from the s or­

bitals, and the irreps of the associated p orbitals are the opposite signs of them. Although there seems to be 8 degrees of freedom totally, one should note that there are not linearly­

independent by observing the sum:

w_s⊕ wp = (+−, +−, +−, +−) (2.28)

are all the same for w = a, b, c, d. It is evident that there are only 8 + 1− 4 = 5 degrees of freedom left over. Therefore,{AI} = Z⁵ and d_AI = 5. Note that this result confirms our previous discussions that dBS= dAIin general.

Now we are going to compute the SI for the space group P 2. We can simply observe that the band structures with irreps:

t₁ = (+, +, +,−), t2 = (+, +,−, +), t₃ = (+,−, +, +), t₄ = (−, +, +, +),

(2.29)

and all the opposite signs of them cannot be induced from the atomic orbitals. These band structures cannot be obtained by summing over the atomic insulators (2.27) with integer coefficients. It must involve coefficients of±1/2. Therefore, these band structures cor­

respond to the systems with Wannier obstruction. In this case, the topology is manifested through the band­inversion which exchanges the irreps at one of the high­symmetry mo­

mentum. Therefore, they correspond to the Chern insulators with Chern numberC = 1.

What happens if we arbitrarily choose two of the band structures t_i, t_j and add them together? It is clear that if i ̸= j, the sum ti ⊕ tj can be written as a sum of two atomic insulators. If i = j, for example, i = 1, the sum t₁ ⊕ t1 = b_s ⊕ cs ⊕ ds⊖ as. Notice that although the coefficients are all integers, there exists a coefficient with negative sign.

Therefore, the band structure t₁⊕ t1 is also topological and encounters Wannier obstruc­

tion. However, contrary to the previous case, by adding the atomic band a_s, the band structure t1⊕ t1⊕ as = b_s⊕ cs⊕ dsbecomes an atomic insulator. One can conclude that the system is actually fragile topological rather than stably topological.

From these discussions, it is clear that only when the coefficients involve ±1/2 do the system has a nontrivial symmetry indicator. Since adding two topological bands leads to the integer coefficients, we conclude that XBS ≡ {BS}/{AI} = Z2 for the P 2 space group. The symmetry indicator can be further written into the simple formula in Eq. (2.25), where ξ_2mshould be replaced with ξ_msince there is no time­reversal symmetry in general and we should multiply it over all the occupied bands. The index ν =C modulo 2 is called the parity of the Chern number. In addition, there might exist band structures that can be written as a sum that includes negative integers. These cases correspond to the fragile topological bands. By generalizing this discussion into all the 230 space groups, one can

construct the full symmetry indicators with or without significant spin­orbit coupling or time­reversal symmetry, see Tab. III, IV, X, XI in Ref. [17].

There are some remarks and caveats regarding the theory of SIs. First, the SIs only provide the sufficient condition for the Wannier obstruction, but not necessary. Therefore, not all the topological insulators with Wannier obstruction can be detected using SIs. For example, for systems without any crystalline symmetries, the SIs become useless. For the Chern insulators with C_n symmetry, the SIs can only detect the Chern number modulo n. The topology or the Wannier obstruction in these cases is then hidden in the Wilson­

loop spectrum which cannot be reflected by the irreps at the high­symmetry momenta.

Nevertheless, the theory of TQC provides some additional information on the topology as an EBR cannot be written as a sum of two BRs. Therefore, even if the irreps are identical to the trivial ones, one can still detect the topology if the disconnected group of bands together form an EBR [44, 45].

Second, the theory doesn’t assume that the systems are insulators. It is possible that the bands are connected at some points outside the high­symmetry momenta and the systems become semimetals. For example, in symmetry class AI, all the nontrivial SIs are incompatible with insulators but rather the symmetry­protected topological semimet­

als [101]. The TCIs in this class that used to be considered topological have been proved to be only fragile topological [14, 58, 62, 64, 65].

Third, the theory of SIs does not provide any information on the bulk­boundary cor­

respondence. Nevertheless, for systems in symmetry class AII, the relation between the anomalous gapless surface states (and also gapless hinge states) and SIs has been well studied [48, 102]. Contrary to the symmetry class AI, all the nontrivial SIs in class AII are compatible with the topological insulators.

Chapter 3 Trivialized Topological Insulator

The theory of Wannier obstruction is not the end of the story. It is found that the del­

icate topology can exist without Wannier obstruction [73, 74]. Here, the delicate topology can be trivialized by adding trivial atomic bands either to the occupied space or unoccu­

pied space. Although the Wannier functions are exponentially localized, they still cannot be continuously deformed into the delta functions which describe the trivial atomic insu­

lators. It is termed the multicellularity of the Wannier functions [73]. In this chapter, we show that there exist systems without Wannier obstruction, but with stable topology con­

trary to delicate topology. In addition, the gapless edge states in these systems are stably protected and robust, in contrast to the conditionally­robust gapless edge states in delicate TIs which require the sharp boundary condition. The “trivialized topological insulators”

host no Wilson­loop winding and Wannier obstruction. However, they can still be charac­

terized by the entanglement spectrum and the crystalline K theory. The TTI thus serves as an interesting example in which the stably protected gapless edge states exist without Wannier obstruction and Wilson­loop winding.

3.1 Model

Our point of departure is the 2D time­reversal (T )­invariant Z2topological insulators [1, 103]. While this model is chosen specifically to exemplify our analysis, it should be noted that similar results can also be produced in other models. Consider a four­band