4. Edge-of-edge state

We can follow the analysis discussed in Section 3.3.2 for the current system. Figure 10 shows the boundary condition dependence of the edge state spectrum. These behaviors are consistent with the continuum model in the vicinity of the would-be gapless points. Such a dependence of the boundary condition has been recently predicted to be observed in monolayer silicene/ germanene/stanene nanoribbons [19]. We remark that we obtain the edge state with positive and negative chiralities from the reduction p<sup>1</sup> ! m1, which is equivalent to topologically trivial state. Actually the edge state is almost embedded, and indistinguishable with the bulk spectrum, in particular, for θ ¼ 5π=7, 9π=7. On the other hand, we obtain a single chiral edge state from the reduction p<sup>2</sup> ! m2, indicating topologically nontrivial state. We can see an edge state

Figure 10. The boundary condition dependence of the 2d lattice system (52) and (53) with c ¼ 1. (a)–(f) and (a<sup>0</sup>

and p1, respectively. The blue region is the bulk, and the orange line is the edge state spectrum.

the spectra obtained from the reduction p<sup>1</sup> ! m<sup>1</sup> ¼ π=2 þ 0:5 and p<sup>2</sup> ! m<sup>2</sup> ¼ 0:2. The horizontal axes are the momenta p<sup>2</sup>

)–(f<sup>0</sup> ) show

spectrum survives for the whole region of the parameter θ.

20 Heterojunctions and Nanostructures

So far we have examined situations with a single boundary with the boundary condition. In general we can impose another boundary in the different direction, and a different boundary condition. In this section we consider a generic situation involving two boundaries with two different conditions. Then an intersection of two boundary plays a role of "edge-of-edge" and we study the corresponding edge-of-edge state localized on such an intersecting boundaries [4]. See also related works [20–24].

#### 4.1. 5d Weyl semimetal

As discussed in Section 3.1, the boundary condition is characterized by the projection (23), so that the degrees of freedom of the boundary state should be a half of the original one. This implies that, if we impose two boundary conditions, we will have a quarter of the original d.o.f. Therefore, to obtain physical degrees of freedom at the edge-of-edge, we have to start with a four-component system or more. For this purpose, we start with the 5d WSM system discussed in Section 2.4 by introducing boundaries at x<sup>4</sup> ¼ 0 and x<sup>5</sup> ¼ 0. The boundary condition, namely the zero current condition (34), is now given by

$$
\left.\psi^\dagger \gamma\_4 \psi\right|\_{\mathbf{x}\_4=0} = \mathbf{0} \qquad \left.\psi^\dagger \gamma\_5 \psi\right|\_{\mathbf{x}\_5=0} = \mathbf{0},\tag{56}
$$

since the current operator is given by <sup>J</sup><sup>μ</sup> <sup>¼</sup> <sup>ψ</sup>† γμψ. These conditions are rephrased as

$$P\_4\psi\Big|\_{\mathfrak{x}=0} = 0 \quad P\_5\psi\Big|\_{\mathfrak{x}=0} = 0 \quad \text{with} \quad P\_{4,5} = \frac{1 - M\_{4,5}}{2} \tag{57}$$

where the matrix M4, <sup>5</sup> obeys M† <sup>a</sup>γ<sup>a</sup> þ γaMa ¼ 0 for a ¼ 4, 5. Explicitly we have

$$M\_5 = \begin{pmatrix} 0 & U\_5^\dagger \\ U\_5 & 0 \end{pmatrix}' \qquad M\_4 = -\frac{1}{2} \begin{pmatrix} U\_4 + U\_4^\dagger & U\_4 - U\_4^\dagger \\ -U\_4 + U\_4^\dagger & -U\_4 - U\_4^\dagger \end{pmatrix}' \tag{58}$$

where U4, <sup>5</sup> are elements of U(2). A solution to these conditions localized at the boundary is given by

$$
\psi\left(p\_{1,2,3,5}, \mathbf{x}\_4\right) = e^{-\alpha \mathbf{z}(p)\mathbf{x}\_4} \begin{pmatrix} 1 - \mathcal{U}\_4 \\ \mathbb{1} + \mathcal{U}\_4 \end{pmatrix} \chi\left(p\_{1,2,3,4}\right), \quad \psi\left(p\_{1,2,3,4}, \mathbf{x}\_5\right) = e^{-\alpha \mathbf{z}(p)\mathbf{x}\_5} \begin{pmatrix} 1 \\ \mathcal{U}\_5 \end{pmatrix} \xi\left(p\_{1,2,3,4}\right). \tag{59}
$$

In particular, the edge state localized at x<sup>5</sup> ¼ 0 is apparently similar to the 3d case (39), just replacing the phase factor ei<sup>θ</sup> <sup>∈</sup> U(1) with <sup>U</sup><sup>5</sup> <sup>∈</sup> U(2). The eigenvalue equation <sup>H</sup>5d<sup>ψ</sup> <sup>¼</sup> <sup>E</sup><sup>ψ</sup> leads to e<sup>2</sup> <sup>5</sup> <sup>þ</sup> <sup>α</sup><sup>2</sup> <sup>5</sup> ¼ p ! � � � � � � 2 <sup>þ</sup> <sup>p</sup><sup>2</sup> <sup>4</sup> and also

$$\left[\left(i\boldsymbol{\alpha}\_{5}-\boldsymbol{\epsilon}\_{5}\right)+\left(\boldsymbol{p}\_{4}-\boldsymbol{i}\ \overrightarrow{\boldsymbol{\sigma}}\ \overrightarrow{\boldsymbol{p}}\right)\mathcal{U}\_{5}\right]\boldsymbol{\xi}=\boldsymbol{0}.\tag{60}$$

HAIII 3d p !� � <sup>¼</sup><sup>p</sup>

Hamiltonian has a chiral symmetry with respect to the sublattice structure HAIII

can apply a similar analysis as before. The edge-of-edge state is in this case given by

where the gamma matrices are chosen as γ

ψ p1; x2; x<sup>3</sup> � � <sup>¼</sup> <sup>e</sup>

<sup>2</sup>ð Þþ <sup>i</sup>σ<sup>2</sup> <sup>i</sup>σ3U<sup>3</sup> � <sup>U</sup>†

e<sup>2</sup> p1; p<sup>3</sup> � � ¼ �

e<sup>3</sup> p1; p<sup>2</sup> � � ¼ �

matrices σ<sup>0</sup>

s and τ<sup>0</sup>

with the compatibility condition

det <sup>1</sup> <sup>þ</sup> <sup>U</sup>†

localized at their intersection

Acknowledgements

boundary intersection, propagating in x1-direction.

! � <sup>γ</sup>

!¼ <sup>τ</sup><sup>2</sup> <sup>⊗</sup> <sup>σ</sup>

�α<sup>2</sup> <sup>p</sup>ð Þ<sup>1</sup> <sup>x</sup>2�α<sup>3</sup> <sup>p</sup>ð Þ<sup>1</sup> <sup>x</sup><sup>3</sup> <sup>1</sup> <sup>þ</sup> <sup>i</sup>σ3U<sup>3</sup>

where U2, <sup>3</sup> ∈ U(2) parameterize the boundary condition. We consider the following choice satisfying the compatibility condition U<sup>2</sup> ¼ σ<sup>2</sup> cos ϕ þ i sin ϕ, U<sup>3</sup> ¼ i cos ϕ � σ<sup>3</sup> sin ϕ. Then we obtain the spectra of the edge state localized at x<sup>2</sup> ¼ 0 and x3, and the edge-of-edge state

> p2 <sup>1</sup> <sup>þ</sup> <sup>p</sup><sup>2</sup>

p2 <sup>1</sup> <sup>þ</sup> <sup>p</sup><sup>2</sup>

Here both edge states are gapped, while only the edge-of-edge state is gapless. This is a suitable situation for experimental detection of the edge-of-edge state because we have to distinguish it from the spectra of the edge states at x<sup>2</sup> ¼ 0 and x<sup>3</sup> ¼ 0. The reason why we obtain the gapped edge states seems that the symmetry protecting the edge state is weakly broken due to the boundary condition, which is analogous to the TI/ferromagnet junction, etc.

The author would like to thank Koji Hashimoto and Xi Wu for an enlightening collaboration on the boundary condition analysis of topological materials, which materializes this article.

Figure 11. Dimensional reduction from 5d WSM to 3d chiral TI. There exists the edge-of-edge state localized at the

q

q

eeoe p<sup>1</sup>

ð Þ <sup>i</sup>σ<sup>1</sup> <sup>U</sup><sup>3</sup> <sup>þ</sup> <sup>i</sup>σ<sup>1</sup> � <sup>i</sup>σ2U<sup>3</sup> � <sup>U</sup>†

� � <sup>¼</sup> 0 (69)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>3</sup> <sup>þ</sup> <sup>m</sup> cos <sup>ϕ</sup> � �<sup>2</sup>

<sup>2</sup> <sup>þ</sup> <sup>m</sup> sin <sup>ϕ</sup> � �<sup>2</sup>

s act on the spin space ð Þ ↑; ↓ and the sublattice space ð Þ A; B , respectively. This

iσ3ð Þ 1 � iσ3U<sup>3</sup> � �<sup>ξ</sup> <sup>p</sup><sup>1</sup>

! <sup>þ</sup>mγ4, (67)

<sup>2</sup>ð Þ� <sup>i</sup>σ<sup>3</sup> <sup>U</sup>†

� � ¼ �p1: (72)

<sup>2</sup>U<sup>3</sup>

, (70)

, (71)

!, <sup>γ</sup><sup>4</sup> <sup>¼</sup> <sup>τ</sup><sup>1</sup> <sup>⊗</sup> <sup>1</sup>, <sup>γ</sup><sup>5</sup> <sup>¼</sup> <sup>τ</sup><sup>3</sup> <sup>⊗</sup> <sup>1</sup>, and Pauli

Analysis of Topological Material Surfaces http://dx.doi.org/10.5772/intechopen.74934 23

3d ; γ<sup>5</sup> � � <sup>¼</sup> 0. We

� � (68)

Decomposing <sup>U</sup><sup>5</sup> <sup>¼</sup> <sup>e</sup><sup>i</sup>θ5V<sup>5</sup> with ei<sup>θ</sup><sup>5</sup> <sup>∈</sup> U(1) and <sup>V</sup><sup>5</sup> <sup>∈</sup>SU 2ð Þ, we consider the SU(2) transformation p<sup>4</sup> � i σ ! � <sup>p</sup> ! � �V<sup>5</sup> <sup>¼</sup> <sup>p</sup><sup>0</sup> <sup>4</sup> � i σ ! �<sup>p</sup> !0 . Then we have

$$\left(a\_5 \sin \theta\_5 - \epsilon\_5 \cos \theta\_5 + p\_4'\right)\xi = 0,\tag{61}$$

$$\left(\alpha\mathfrak{s}\cos\theta\mathfrak{s} + \mathfrak{e}\mathfrak{s}\sin\theta\mathfrak{s} - \overrightarrow{\sigma}\cdot\overrightarrow{p'}\right)\xi = 0.\tag{62}$$

Diagonalizing σ ! �<sup>p</sup> !0 , which is equivalent to the 3d Hamiltonian (6), as σ ! �<sup>p</sup> !0 � �ξ� <sup>¼</sup> � ffiffiffiffiffiffiffiffiffiffi p !0 � � � � � � <sup>2</sup> r ξ�, we obtain the spectrum and the inverse penetration depth as follows:

$$\mathbf{e}\_5(p) = p\_4' \cos \theta\_5 \pm \sqrt{\left|\overline{\vec{p}}'\right|^2} \sin \theta\_5, \qquad a\_5(p) = -p\_4' \sin \theta\_5 \pm \sqrt{\left|\overline{\vec{p}}'\right|^2} \cos \theta\_5. \tag{63}$$

which is written using an SO(2) transformation as before,

$$
\begin{pmatrix} \epsilon\_5\\ \alpha\_5 \end{pmatrix} = \begin{pmatrix} \cos\theta\_5 & \sin\theta\_5\\ -\sin\theta\_5 & \cos\theta\_5 \end{pmatrix} \begin{pmatrix} p'\_4\\ \pm\sqrt{\left|\vec{p'}\right|^2} \end{pmatrix}.\tag{64}
$$

We can solve the boundary condition and obtain the spectrum for the boundary at x<sup>4</sup> ¼ 0 in a similar way.

Let us then consider a compatible boundary condition for the localized edge-of-edge state

$$\left.P\_{4}\psi\right|\_{x\_{4,\frac{\nu}{8}}=0} = P\_{5}\psi\Big|\_{x\_{4,\frac{\nu}{8}}=0} = 0.\tag{65}$$

A solution to this condition is given by

$$
\psi\left(p\_{1,2,3}, \mathbf{x}\_4, \mathbf{x}\_5\right) = e^{-a\_4(p)\mathbf{x}\_4 - a\_5(p)\mathbf{x}\_5} \begin{pmatrix} 1 - \mathcal{U}\_4 \\ 1 + \mathcal{U}\_4 \end{pmatrix} \chi(p) \tag{66}
$$

with ½ � U5ð Þ� 1 � U<sup>4</sup> ð Þ 1 þ U<sup>4</sup> χð Þ¼ p 0, which is covariant under U(2) transformation ð Þ! <sup>U</sup>4; <sup>U</sup>5; <sup>χ</sup> WU4W† ; WU5W† ; <sup>W</sup><sup>χ</sup> � � with <sup>W</sup> <sup>∈</sup> U 2ð Þ. To have a nontrivial solution, they should obey ½U5ð Þ� 1 � U<sup>4</sup> ð Þ 1 þ U<sup>4</sup> � ¼ 0. For example, a simple choice is ð Þ¼ U4; U<sup>5</sup> ð Þ σ3; σ<sup>2</sup> , and the corresponding solution is <sup>χ</sup><sup>T</sup> <sup>¼</sup> ð Þ <sup>1</sup> <sup>i</sup> . Then we obtain the spectrum of the edge-of-edge state eð Þ¼� p p1, α4ð Þ¼ p p3, α5ð Þ¼ p p2.

#### 4.2. 3d chiral topological insulator

We discuss dimensional reduction of the edge-of-edge state in the 5d WSM to a more realistic 3d system. Replacing p4; p<sup>5</sup> � � ! ð Þ <sup>m</sup>; <sup>0</sup> as shown in Figure 11, then we obtain the 3d chiral (class AIII) TI

$$\mathcal{H}\_{\text{3d}}^{\text{AIII}} \left( \overrightarrow{p} \right) = \overrightarrow{p} \cdot \overrightarrow{\gamma} + m \gamma\_{\text{4}} \tag{67}$$

where the gamma matrices are chosen as γ !¼ <sup>τ</sup><sup>2</sup> <sup>⊗</sup> <sup>σ</sup> !, <sup>γ</sup><sup>4</sup> <sup>¼</sup> <sup>τ</sup><sup>1</sup> <sup>⊗</sup> <sup>1</sup>, <sup>γ</sup><sup>5</sup> <sup>¼</sup> <sup>τ</sup><sup>3</sup> <sup>⊗</sup> <sup>1</sup>, and Pauli matrices σ<sup>0</sup> s and τ<sup>0</sup> s act on the spin space ð Þ ↑; ↓ and the sublattice space ð Þ A; B , respectively. This Hamiltonian has a chiral symmetry with respect to the sublattice structure HAIII 3d ; γ<sup>5</sup> � � <sup>¼</sup> 0. We can apply a similar analysis as before. The edge-of-edge state is in this case given by

$$\psi(p\_1, \mathbf{x}\_2, \mathbf{x}\_3) = e^{-a\_2(p\_1)\mathbf{x}\_2 - a\_3(p\_1)\mathbf{x}\_3} \begin{pmatrix} 1 + i\sigma\_3 \mathbf{U}\_3 \\ i\sigma\_3 (\mathbf{1} - i\sigma\_3 \mathbf{U}\_3) \end{pmatrix} \xi(p\_1) \tag{68}$$

with the compatibility condition

ð Þþ iα<sup>5</sup> � e<sup>5</sup> p<sup>4</sup> � i σ

tion p<sup>4</sup> � i σ

Diagonalizing σ

ffiffiffiffiffiffiffiffiffiffi p !0 � � � � � � <sup>2</sup> r

similar way.

�

! � <sup>p</sup> ! � �

22 Heterojunctions and Nanostructures

! �<sup>p</sup> !0

e5ð Þ¼ p p<sup>0</sup>

A solution to this condition is given by

state eð Þ¼� p p1, α4ð Þ¼ p p3, α5ð Þ¼ p p2.

4.2. 3d chiral topological insulator

3d system. Replacing p4; p<sup>5</sup>

(class AIII) TI

ð Þ! <sup>U</sup>4; <sup>U</sup>5; <sup>χ</sup> WU4W†

<sup>4</sup> cos θ<sup>5</sup> �

ffiffiffiffiffiffiffiffiffiffi p !0 � � � � � � 2

> P4ψ � � � x4,5¼0

> > ¼ e

ψ p<sup>1</sup>;2;<sup>3</sup>; x4; x<sup>5</sup> � �

; WU5W†

r

which is written using an SO(2) transformation as before,

e5 α5 � �

V<sup>5</sup> ¼ p<sup>0</sup>

<sup>4</sup> � i σ ! �<sup>p</sup> !0

h i

. Then we have

α<sup>5</sup> sin θ<sup>5</sup> � e<sup>5</sup> cos θ<sup>5</sup> þ p<sup>0</sup>

!0 � �

ξ�, we obtain the spectrum and the inverse penetration depth as follows:

<sup>¼</sup> cos <sup>θ</sup><sup>5</sup> sin <sup>θ</sup><sup>5</sup> � sin θ<sup>5</sup> cos θ<sup>5</sup>

sin θ5, α5ð Þ¼� p p<sup>0</sup>

� � <sup>p</sup><sup>0</sup>

We can solve the boundary condition and obtain the spectrum for the boundary at x<sup>4</sup> ¼ 0 in a

¼ P5ψ � � � x4, <sup>5</sup>¼0

with ½ � U5ð Þ� 1 � U<sup>4</sup> ð Þ 1 þ U<sup>4</sup> χð Þ¼ p 0, which is covariant under U(2) transformation

should obey ½U5ð Þ� 1 � U<sup>4</sup> ð Þ 1 þ U<sup>4</sup> � ¼ 0. For example, a simple choice is ð Þ¼ U4; U<sup>5</sup> ð Þ σ3; σ<sup>2</sup> , and the corresponding solution is <sup>χ</sup><sup>T</sup> <sup>¼</sup> ð Þ <sup>1</sup> <sup>i</sup> . Then we obtain the spectrum of the edge-of-edge

We discuss dimensional reduction of the edge-of-edge state in the 5d WSM to a more realistic

�α4ð Þ<sup>p</sup> <sup>x</sup>4�α5ð Þ<sup>p</sup> <sup>x</sup><sup>5</sup> <sup>1</sup> � <sup>U</sup><sup>4</sup>

; <sup>W</sup><sup>χ</sup> � � with <sup>W</sup> <sup>∈</sup> U 2ð Þ. To have a nontrivial solution, they

� � ! ð Þ <sup>m</sup>; <sup>0</sup> as shown in Figure 11, then we obtain the 3d chiral

1 þ U<sup>4</sup> � �

Let us then consider a compatible boundary condition for the localized edge-of-edge state

α<sup>5</sup> cos θ<sup>5</sup> þ e<sup>5</sup> sin θ5� σ

Decomposing <sup>U</sup><sup>5</sup> <sup>¼</sup> <sup>e</sup><sup>i</sup>θ5V<sup>5</sup> with ei<sup>θ</sup><sup>5</sup> <sup>∈</sup> U(1) and <sup>V</sup><sup>5</sup> <sup>∈</sup>SU 2ð Þ, we consider the SU(2) transforma-

! � <sup>p</sup> ! � �

U<sup>5</sup>

4

! �<sup>p</sup>

, which is equivalent to the 3d Hamiltonian (6), as σ

� �<sup>ξ</sup> <sup>¼</sup> <sup>0</sup>, (61)

<sup>4</sup> sin θ<sup>5</sup> �

4

ffiffiffiffiffiffiffiffiffiffi p !0 � � � � � � <sup>2</sup> r

1

�

0 B@ ξ ¼ 0: (60)

ξ ¼ 0: (62)

ffiffiffiffiffiffiffiffiffiffi p !0 � � � � � � 2

r

! �<sup>p</sup> !0 � �

cos θ5, (63)

CA: (64)

χð Þp (66)

¼ 0: (65)

ξ� ¼

$$\det\left[\mathbb{1} + \mathcal{U}\_2^\dagger(\dot{\boldsymbol{\sigma}}\_2) + \dot{\boldsymbol{\sigma}}\_3 \mathcal{U}\_3 - \mathcal{U}^\dagger(\dot{\boldsymbol{\sigma}}\_1)\mathcal{U}\_3 + \dot{\boldsymbol{\sigma}}\_1 - \dot{\boldsymbol{\sigma}}\_2 \mathcal{U}\_3 - \mathcal{U}\_2^\dagger(\dot{\boldsymbol{\sigma}}\_3) - \mathcal{U}\_2^\dagger \mathcal{U}\_3\right] = 0\tag{69}$$

where U2, <sup>3</sup> ∈ U(2) parameterize the boundary condition. We consider the following choice satisfying the compatibility condition U<sup>2</sup> ¼ σ<sup>2</sup> cos ϕ þ i sin ϕ, U<sup>3</sup> ¼ i cos ϕ � σ<sup>3</sup> sin ϕ. Then we obtain the spectra of the edge state localized at x<sup>2</sup> ¼ 0 and x3, and the edge-of-edge state localized at their intersection

$$\epsilon\_2(p\_1, p\_3) = \pm \sqrt{p\_1^2 + p\_3^2 + \left(m \cos \phi\right)^2},\tag{70}$$

$$
\epsilon\_3(p\_1, p\_2) = \pm \sqrt{p\_1^2 + p\_2^2 + \left(m \sin \phi\right)^2},
\tag{71}
$$

$$
\epsilon\_{\rm eoe}(p\_1) = -p\_1. \tag{72}
$$

Here both edge states are gapped, while only the edge-of-edge state is gapless. This is a suitable situation for experimental detection of the edge-of-edge state because we have to distinguish it from the spectra of the edge states at x<sup>2</sup> ¼ 0 and x<sup>3</sup> ¼ 0. The reason why we obtain the gapped edge states seems that the symmetry protecting the edge state is weakly broken due to the boundary condition, which is analogous to the TI/ferromagnet junction, etc.

Figure 11. Dimensional reduction from 5d WSM to 3d chiral TI. There exists the edge-of-edge state localized at the boundary intersection, propagating in x1-direction.

#### Acknowledgements

The author would like to thank Koji Hashimoto and Xi Wu for an enlightening collaboration on the boundary condition analysis of topological materials, which materializes this article. The work of the author was supported in part by Keio Gijuku Academic Development Funds, JSPS Grant-in-Aid for Scientific Research (no. JP17K18090), MEXT-Supported Program for the Strategic Research Foundation at Private Universities "Topological Science" (no. S1511006), JSPS Grant-in-Aid for Scientific Research on Innovative Areas "Topological Materials Science" (no. JP15H05855), and "Discrete Geometric Analysis for Materials Design" (no. JP17H06462).

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