2. Preliminaries: bulk, edge, and topology

In this section, we provide several preliminary aspects of topological materials. In particular, we show simple models, effectively describing the bulk of topological system, and discuss the role of topology thereof.

#### 2.1. Bulk system

We start with a simple two-band Hamiltonian in two dimensions,

$$H\_{\rm 2d} = -i\sigma\_1 \frac{\partial}{\partial \mathbf{x}\_1} - i\sigma\_2 \frac{\partial}{\partial \mathbf{x}\_2} + m\sigma\_3 \tag{1}$$

where Pauli matrices are defined <sup>σ</sup><sup>1</sup> <sup>¼</sup> 0 1 1 0 , <sup>σ</sup><sup>2</sup> <sup>¼</sup> <sup>0</sup> �<sup>i</sup> i 0 , <sup>σ</sup><sup>3</sup> <sup>¼</sup> 1 0 0 �1 . This is a simple effective model for 2d Integer QHE, classified into the 2d class A system according to the 10-fold way classification of TIs and TSCs [8, 9]. In order to investigate the band structure of this system, we consider the Bloch wave function Ψ p ! x ! <sup>¼</sup> <sup>e</sup>i p!�<sup>x</sup> ! ψp ! x ! , and the corresponding Hamiltonian acting on ψ<sup>p</sup> ! x ! , simply denoted by <sup>ψ</sup> below, is given by

<sup>1</sup> See also related works [5–7] for the boundary condition analysis of topological materials.

Analysis of Topological Material Surfaces http://dx.doi.org/10.5772/intechopen.74934 5

$$\mathcal{H}\_{2d}\left(\overrightarrow{p}\right) = p\_1\sigma\_1 + p\_2\sigma\_2 + m\sigma\_3 = \begin{pmatrix} m & \Delta^\*\left(\overrightarrow{p}\right) \\\\ \Delta\left(\overrightarrow{p}\right) & -m \end{pmatrix}.\tag{2}$$

We obtain two eigenvalues e� p !� � ¼ � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ! � � � � � � 2 þ m<sup>2</sup> r . The eigenstate, parametrized by a complex number ξ∈ C, is accordingly obtained as

$$\psi = \frac{1}{\sqrt{1 + |\xi|^2}} \begin{pmatrix} 1 \\ \xi \end{pmatrix} \quad \text{with} \quad \xi = \frac{\Delta \begin{pmatrix} \vec{p} \\ \vec{p} \end{pmatrix}}{\epsilon + m} = \frac{\epsilon - m}{\Delta \begin{pmatrix} \vec{p} \end{pmatrix}^\*}. \tag{3}$$

We remark that the parameter ξ becomes singular ξ ! ∞ at p !¼ 0. At this point, we have to reparametrize the eigenstate with ξ�<sup>1</sup> instead of ξ. This means that ξ is not a global, but just a local coordinate, and the eigenstate is given by an element of CP<sup>1</sup> in this model.

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

$$\mathcal{A} = \psi^\dagger(\mathrm{id})\psi = -\mathrm{Im}\frac{\xi^\* d\xi}{1 + |\xi|^2}, \qquad \mathcal{F} = d\mathcal{A} = i \frac{d\xi^\* d\xi}{\left(1 + |\xi|^2\right)^2} \tag{4}$$

where we use the differential form notation in the momentum space, d ¼ ∂=∂pi � �dpi , namely 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> , and the curvature is two-form <sup>F</sup> <sup>¼</sup> <sup>F</sup>12dp<sup>1</sup> dp<sup>2</sup> . Under the momentum-dependent transformation, ξ ! e iϕ p ! ð Þξ (not an overall 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 <sup>C</sup>P<sup>1</sup> <sup>¼</sup> <sup>S</sup><sup>3</sup> =S<sup>1</sup> , and interpreted as a consequence of the particle number conservation of each eigenstate which holds under the adiabatic process.

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 momentum space,

$$\nu\_{\rm 2d} = \frac{1}{2\pi} \int dp\_1 dp\_2 F\_{12} = \frac{1}{2} \operatorname{sgn}\left(m\right) \tag{5}$$

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

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

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

In this section, we provide several preliminary aspects of topological materials. In particular, we show simple models, effectively describing the bulk of topological system, and discuss the

> ∂ ∂x<sup>1</sup> � iσ<sup>2</sup>

1 0 

! x !

simple effective model for 2d Integer QHE, classified into the 2d class A system according to the 10-fold way classification of TIs and TSCs [8, 9]. In order to investigate the band structure

∂ ∂x<sup>2</sup>

, <sup>σ</sup><sup>2</sup> <sup>¼</sup> <sup>0</sup> �<sup>i</sup> i 0 

> p ! x !

, simply denoted by ψ below, is given by

þ mσ<sup>3</sup> (1)

0 �1 

. This is a

, and the

, <sup>σ</sup><sup>3</sup> <sup>¼</sup> 1 0

<sup>¼</sup> <sup>e</sup>i p !�<sup>x</sup> ! ψp ! x !

predictions.

4 Heterojunctions and Nanostructures

call the edge-of-edge state.

role of topology thereof.

2.1. Bulk system

1

2. Preliminaries: bulk, edge, and topology

where Pauli matrices are defined <sup>σ</sup><sup>1</sup> <sup>¼</sup> 0 1

corresponding Hamiltonian acting on ψ<sup>p</sup>

We start with a simple two-band Hamiltonian in two dimensions,

of this system, we consider the Bloch wave function Ψ

See also related works [5–7] for the boundary condition analysis of topological materials.

H2d ¼ �iσ<sup>1</sup>

<sup>2</sup> See a textbook on this topic, e.g., [10] for more details.

it would be a topological invariant, but with a discontinuous point at m ¼ 0, which is the gapless (sign changing) point m ¼ 0. Typically the topological number takes an integer value, but ν2d does not. The reason why we obtain a half integer value is that we take a specific slice of the mass parameter in the total parameter space of the three-parameter Hamiltonian (2).

To explain this let us consider the 3d system as follows,

$$\mathcal{H}\_{\text{3d}}\left(\stackrel{\rightarrow}{p}\right) = p\_1\sigma\_1 + p\_2\sigma\_2 + p\_3\sigma\_3 = \begin{pmatrix} p\_3 & \Lambda^\*(p) \\ \Lambda(p) & -p\_3 \end{pmatrix} \tag{6}$$

continuously deformed to a trivial system ν3d ¼ 0 by definition. This explains the topological stability of the WSM. If we want to obtain a topologically trivial situation, we need pairannihilation of the Weyl points having opposite topological invariants: ν3d ¼ þð Þþ � 1 ð Þ¼ 1 0.

So far, we have discussed the bulk system, and the material boundary is not yet considered. Let us show a simple argument to incorporate the boundary of the system. If we have a material which has nontrivial topology, the vacuum, outside of the material, should be topologically trivial. Otherwise they cannot be topologically distinguished. As explained above, in order to obtain the topology change in the 2d system, we need the mass parameter whose sign is flipped at the boundary. For this purpose we impose a simple spatial dependence on the mass parameter as m xð Þ¼ <sup>1</sup> ϑx<sup>1</sup> with a positive slope ϑ > 0, giving rise to the sign flip at x<sup>1</sup> ¼ 0,

> ∂ ∂x<sup>1</sup>

where we exchange Pauli matrices compared with the previous one to simplify the expression. Since x1-dependence remains in this system, we do not consider the momentum basis in

state of the 2d class A system. See Figure 3 for numerical plot of the spectrum. In general, we obtain the zero mode localized on the topological material boundary from the mass term

Figure 2. The topological invariant distinguishes topologically different situations. The green and red spheres show the monopole with topological charge ν3d ¼ þ1 and ν3d ¼ �1, respectively. We need pair annihilation to eliminate the mono-

<sup>þ</sup> <sup>p</sup>2σ<sup>3</sup> <sup>¼</sup> <sup>p</sup><sup>2</sup>

<sup>2</sup><sup>ϑ</sup> <sup>p</sup> , <sup>b</sup><sup>a</sup>

† h i <sup>¼</sup> 1, so that it is interpreted as a creation/annihilation

� � ¼ � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2

ffiffiffiffiffiffi <sup>2</sup><sup>ϑ</sup> <sup>p</sup> <sup>b</sup><sup>a</sup> �p<sup>2</sup>

ffiffiffiffiffiffi <sup>2</sup><sup>ϑ</sup> <sup>p</sup> <sup>b</sup><sup>a</sup> †

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

† <sup>¼</sup> ð Þ <sup>ϑ</sup>x<sup>1</sup> � ð Þ <sup>∂</sup>=∂x<sup>1</sup> <sup>=</sup> ffiffiffiffiffiffi

� � <sup>¼</sup> <sup>p</sup><sup>2</sup> (gapless), which is the chiral edge

<sup>2</sup> <sup>þ</sup> <sup>2</sup>ϑ<sup>n</sup> <sup>p</sup> for <sup>n</sup> <sup>≥</sup> 1 (gapped),

(8)

7

. The off-diagonal

<sup>2</sup><sup>ϑ</sup> <sup>p</sup> , obeying

!

so that the boundary is the plane x<sup>1</sup> ¼ 0 [12]. Then the Hamiltonian takes a form of

� � <sup>¼</sup> m xð Þ<sup>1</sup> <sup>σ</sup><sup>1</sup> � <sup>i</sup>σ<sup>2</sup>

element is given by an operator <sup>b</sup><sup>a</sup> <sup>¼</sup> ð Þ <sup>ϑ</sup>x<sup>1</sup> <sup>þ</sup> ð Þ <sup>∂</sup>=∂x<sup>1</sup> <sup>=</sup> ffiffiffiffiffiffi

operator. Then the energy spectrum is given by e<sup>n</sup> p<sup>2</sup>

while the zero mode dispersion is given by e<sup>0</sup> p<sup>2</sup>

x1-direction, while the momentum in x2-direction is now denoted by p<sup>2</sup>

ℋ2d x1; p<sup>2</sup>

the commutation relation <sup>b</sup>a;b<sup>a</sup>

poles.

See Figure 2.

2.2. Edge state

which is known as an effective Hamiltonian of the WSM. This Hamiltonian is simply obtained from the 2d system (2) by replacing the mass parameter with another momentum p3. We apply essentially the same analysis to this 3d system as 2d, and we obtain the genuine topological invariant:

$$\nu\_{3d} = \frac{1}{4\pi} \int\_{S^2} d\vec{S} \cdot \vec{B} = \frac{1}{2\pi} \int dp\_1 dp\_2 F\_{12} \Big|\_{p\_3 > 0} - \frac{1}{2\pi} \int dp\_1 dp\_2 F\_{12} \Big|\_{p\_3 < 0} = 1 \tag{7}$$

where the "magnetic field" is defined as Bi <sup>¼</sup> <sup>1</sup> <sup>2</sup> eijkFjk, namely B ! ¼∇ ! � A ! . This means that the gapless point (also called the Weyl point) plays a role as the magnetic monopole in the momentum space. As shown in Figure 1, the 2d invariant ν2d is related to the 3d invariant through taking a constant p3, identified with the mass m, which covers either upper or lower half of the monopole fluxes. This explains why the 2d invariant can be a half-integer, although the 3d invariant takes an integer value. We remark that, in this case, one cannot consider welldefined Berry phase, since the current 3d system is gapless in which the adiabatic process does not make sense. However, the topological invariant still plays a role to discuss stability of the Weyl point: Since a system having a nontrivial topological number, say ν3d 6¼ 0, cannot be

Figure 1. Monopole at Weyl point in the momentum space. The monopole charge is an integer-valued topological invariant ν3d. The 2d invariant ν2d is obtained at a constant p3ð Þ ! m plane, which covers either upper or lower half of the fluxes, so that ν2d is given by a half of ν3d.

continuously deformed to a trivial system ν3d ¼ 0 by definition. This explains the topological stability of the WSM. If we want to obtain a topologically trivial situation, we need pairannihilation of the Weyl points having opposite topological invariants: ν3d ¼ þð Þþ � 1 ð Þ¼ 1 0. See Figure 2.

#### 2.2. Edge state

it would be a topological invariant, but with a discontinuous point at m ¼ 0, which is the gapless (sign changing) point m ¼ 0. Typically the topological number takes an integer value, but ν2d does not. The reason why we obtain a half integer value is that we take a specific slice of the mass parameter in the total parameter space of the three-parameter Hamiltonian (2).

<sup>¼</sup> <sup>p</sup>1σ<sup>1</sup> <sup>þ</sup> <sup>p</sup>2σ<sup>2</sup> <sup>þ</sup> <sup>p</sup>3σ<sup>3</sup> <sup>¼</sup> <sup>p</sup><sup>3</sup> <sup>Δ</sup><sup>∗</sup>ð Þ<sup>p</sup>

which is known as an effective Hamiltonian of the WSM. This Hamiltonian is simply obtained from the 2d system (2) by replacing the mass parameter with another momentum p3. We apply essentially the same analysis to this 3d system as 2d, and we obtain the genuine topological

dp1dp<sup>2</sup> <sup>F</sup><sup>12</sup> <sup>p</sup>3><sup>0</sup> � <sup>1</sup>

gapless point (also called the Weyl point) plays a role as the magnetic monopole in the momentum space. As shown in Figure 1, the 2d invariant ν2d is related to the 3d invariant through taking a constant p3, identified with the mass m, which covers either upper or lower half of the monopole fluxes. This explains why the 2d invariant can be a half-integer, although the 3d invariant takes an integer value. We remark that, in this case, one cannot consider welldefined Berry phase, since the current 3d system is gapless in which the adiabatic process does not make sense. However, the topological invariant still plays a role to discuss stability of the Weyl point: Since a system having a nontrivial topological number, say ν3d 6¼ 0, cannot be

Figure 1. Monopole at Weyl point in the momentum space. The monopole charge is an integer-valued topological invariant ν3d. The 2d invariant ν2d is obtained at a constant p3ð Þ ! m plane, which covers either upper or lower half of

� � � � 2π ð

<sup>2</sup> eijkFjk, namely B

ΔðÞ � p p<sup>3</sup> � �

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

! ¼∇ ! � A !

� � � � p3<0 (6)

¼ 1 (7)

. This means that the

To explain this let us consider the 3d system as follows,

H3d p !� �

<sup>ν</sup>3d <sup>¼</sup> <sup>1</sup> 4π ð S2 d S ! � B ! ¼ 1 2π ð

6 Heterojunctions and Nanostructures

the fluxes, so that ν2d is given by a half of ν3d.

where the "magnetic field" is defined as Bi <sup>¼</sup> <sup>1</sup>

invariant:

So far, we have discussed the bulk system, and the material boundary is not yet considered. Let us show a simple argument to incorporate the boundary of the system. If we have a material which has nontrivial topology, the vacuum, outside of the material, should be topologically trivial. Otherwise they cannot be topologically distinguished. As explained above, in order to obtain the topology change in the 2d system, we need the mass parameter whose sign is flipped at the boundary. For this purpose we impose a simple spatial dependence on the mass parameter as m xð Þ¼ <sup>1</sup> ϑx<sup>1</sup> with a positive slope ϑ > 0, giving rise to the sign flip at x<sup>1</sup> ¼ 0, so that the boundary is the plane x<sup>1</sup> ¼ 0 [12]. Then the Hamiltonian takes a form of

$$\mathcal{R}\_{2d}(\mathbf{x}\_1, p\_2) = m(\mathbf{x}\_1)\sigma\_1 - \mathrm{i}\sigma\_2\\\frac{\partial}{\partial \mathbf{x}\_1} + p\_2\sigma\_3 = \begin{pmatrix} p\_2 & \sqrt{2}\boldsymbol{\hat{s}}\boldsymbol{\hat{a}}^\dagger\\\sqrt{2}\boldsymbol{\hat{s}}\boldsymbol{\hat{a}} & -p\_2 \end{pmatrix} \tag{8}$$

where we exchange Pauli matrices compared with the previous one to simplify the expression. Since x1-dependence remains in this system, we do not consider the momentum basis in x1-direction, while the momentum in x2-direction is now denoted by p<sup>2</sup> . The off-diagonal element is given by an operator <sup>b</sup><sup>a</sup> <sup>¼</sup> ð Þ <sup>ϑ</sup>x<sup>1</sup> <sup>þ</sup> ð Þ <sup>∂</sup>=∂x<sup>1</sup> <sup>=</sup> ffiffiffiffiffiffi <sup>2</sup><sup>ϑ</sup> <sup>p</sup> , <sup>b</sup><sup>a</sup> † <sup>¼</sup> ð Þ <sup>ϑ</sup>x<sup>1</sup> � ð Þ <sup>∂</sup>=∂x<sup>1</sup> <sup>=</sup> ffiffiffiffiffiffi <sup>2</sup><sup>ϑ</sup> <sup>p</sup> , obeying the commutation relation <sup>b</sup>a;b<sup>a</sup> † h i <sup>¼</sup> 1, so that it is interpreted as a creation/annihilation operator. Then the energy spectrum is given by e<sup>n</sup> p<sup>2</sup> � � ¼ � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2 <sup>2</sup> <sup>þ</sup> <sup>2</sup>ϑ<sup>n</sup> <sup>p</sup> for <sup>n</sup> <sup>≥</sup> 1 (gapped), while the zero mode dispersion is given by e<sup>0</sup> p<sup>2</sup> � � <sup>¼</sup> <sup>p</sup><sup>2</sup> (gapless), which is the chiral edge state of the 2d class A system. See Figure 3 for numerical plot of the spectrum. In general, we obtain the zero mode localized on the topological material boundary from the mass term

Figure 2. The topological invariant distinguishes topologically different situations. The green and red spheres show the monopole with topological charge ν3d ¼ þ1 and ν3d ¼ �1, respectively. We need pair annihilation to eliminate the monopoles.

Figure 3. The dispersion relation of the edge state with ϑ ¼ 1. We find a gapless chiral mode specific to the 2d class A TI. The gapped spectra are interpreted as bulk contributions.

with a spatial profile, which is known as the domain-wall fermion. See, for example, [13] for more details.

Let us see the topological structure of the lattice model. Applying the same procedure to the

Figure 4. The mass dependence of the 2d topological invariant ν2d for the lattice model (10). Topology change occurs at the gapless points m ¼ �4, � 2, 0. Change of the invariant corresponds to the monopole charge þ1, � 2, þ 1 associated

> 8 ><

> >:

where the momentum integral is taken over the Brillouin zone. In contrast to the continuum effective model, we have integer valued topological invariants in this case. This is essentially related to the anomaly of (2 + 1)-dimensional Dirac system, known as the parity anomaly. However, it is also known that the lattice regularization naively gives rise to an anomaly-free system: The gapless points have to appear as a pair, so that each anomalous contribution is canceled with each other [15, 16]. Actually the present model (10) has four gapless points in the parameter space: <sup>p</sup>1; <sup>p</sup>2; <sup>m</sup> � � <sup>¼</sup> ð Þ <sup>0</sup>; <sup>0</sup>; <sup>0</sup> , ð Þ <sup>π</sup>; <sup>0</sup>; �<sup>2</sup> , 0ð Þ ; <sup>π</sup>; �<sup>2</sup> , ð Þ <sup>π</sup>; <sup>π</sup>; �<sup>4</sup> . Each gapless point plays basically the same role as that discussed in the continuum model with the monopole

We can similarly consider a lattice model for 3d WSM system. We consider the Hamiltonian

2

0 ð Þ m > 0 or m < �4

(11)

9

<sup>2</sup> ðþ1 � 2þ 1Þ ¼ 0 for m > 0,

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

<sup>2</sup> ð Þ¼ �1 þ 2 � 1 0

�1 ð Þ �2 < m < 0 þ1 ð Þ �4 < m < �2

<sup>2</sup> ð Þ¼ �<sup>1</sup> <sup>þ</sup> <sup>2</sup> <sup>þ</sup> <sup>1</sup> 1 for �<sup>4</sup> <sup>&</sup>lt; <sup>m</sup> <sup>&</sup>lt; �2, and <sup>1</sup>

<sup>σ</sup><sup>2</sup> <sup>∇</sup><sup>2</sup> � <sup>∇</sup>†

!� � <sup>¼</sup> <sup>σ</sup><sup>1</sup> cos <sup>p</sup><sup>1</sup> � cos <sup>p</sup><sup>2</sup> <sup>þ</sup> <sup>c</sup> � � <sup>þ</sup> <sup>σ</sup><sup>2</sup> sin <sup>p</sup><sup>2</sup> <sup>þ</sup> <sup>σ</sup><sup>3</sup> sin <sup>p</sup>3, (13)

2 � � � <sup>i</sup>

2

<sup>σ</sup><sup>3</sup> <sup>∇</sup><sup>3</sup> � <sup>∇</sup>†

3 � �: (12)

Hamiltonian (10), we obtain the topological invariant as follows [14]:

charge <sup>þ</sup>1 or �1. Thus we immediately obtain <sup>ν</sup>2d <sup>¼</sup> <sup>1</sup>

<sup>1</sup> � <sup>∇</sup><sup>2</sup> � <sup>∇</sup>† <sup>2</sup> <sup>þ</sup> <sup>2</sup><sup>c</sup> � � � <sup>i</sup>

<sup>σ</sup><sup>1</sup> <sup>∇</sup><sup>1</sup> <sup>þ</sup> <sup>∇</sup>†

The corresponding Bloch Hamiltonian is given by

Hlat 3d p

<sup>2</sup> ð Þ¼ �<sup>1</sup> � <sup>2</sup> <sup>þ</sup> <sup>1</sup> 1 for �<sup>2</sup> <sup>&</sup>lt; <sup>m</sup> <sup>&</sup>lt; 0, <sup>1</sup>

for m < �4. See Figure 4.

with each gapless point.

defined on a 3d lattice

Hlat 3d <sup>¼</sup> <sup>1</sup> 2

1

dp1dp<sup>2</sup> F<sup>12</sup> ¼

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

#### 2.3. Lattice system

Since the electron lives on a lattice in the material, studying the lattice model is important to understand the actual behavior of the electron. Let us introduce the Hamiltonian describing the electron on a lattice

$$H\_{2\text{d}}^{\text{lat}} = -\frac{\dot{\mathbf{i}}}{2}\sigma\_1 \left(\nabla\_1 - \nabla\_1^\dagger\right) - \frac{\dot{\mathbf{i}}}{2}\sigma\_2 \left(\nabla\_2 - \nabla\_2^\dagger\right) + \sigma\_3 \left(m + 2 - \frac{1}{2}\left(\nabla\_1 + \nabla\_1^\dagger\right) - \frac{1}{2}\left(\nabla\_2 + \nabla\_2^\dagger\right)\right) \tag{9}$$

where we define the difference operator ∇1,2ψ<sup>n</sup> ! ¼ ψ<sup>n</sup> !þ <sup>e</sup> ! <sup>1</sup>, <sup>2</sup> � ψ<sup>n</sup> ! with the unit vector e ! <sup>1</sup>,<sup>2</sup> in n<sup>1</sup> and n2-direction. Then the corresponding Bloch Hamiltonian is given by

$$\mathcal{H}\_{\text{2d}}^{\text{lat}}\left(\overrightarrow{p}\right) = \sigma\_1 \sin p\_1 + \sigma\_2 \sin p\_2 + \sigma\_3 \left(m + 2 - \cos p\_1 - \cos p\_2\right). \tag{10}$$

Periodicity p1,<sup>2</sup> � p1,<sup>2</sup> þ 2π reflects the lattice structure: The momentum is restricted to the Brillouin zone p1,<sup>2</sup> ∈½ � 0; 2π . The spectrum is given by e p !� � <sup>¼</sup> � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin p<sup>1</sup> � �<sup>2</sup> <sup>þ</sup> sin <sup>p</sup><sup>2</sup> � �<sup>2</sup> <sup>þ</sup> <sup>m</sup> <sup>þ</sup> <sup>2</sup> � cos <sup>p</sup><sup>1</sup> � cos <sup>p</sup><sup>2</sup> � �<sup>2</sup> <sup>q</sup> , which has four gapless points p !¼ ð Þ <sup>0</sup>; <sup>0</sup> at m ¼ 0, p !¼ ð Þ <sup>π</sup>; <sup>0</sup> and 0ð Þ ; <sup>π</sup> at <sup>m</sup> ¼ �2, <sup>p</sup> !¼ ð Þ <sup>π</sup>; <sup>π</sup> at <sup>m</sup> ¼ �4. Expanding the momentum around p !¼ ð Þ <sup>0</sup>; <sup>0</sup> , one can see the effective Hamiltonian (2) is obtained. If expanding the momentum around p !¼ ð Þ <sup>π</sup>; <sup>0</sup> instead, we similarly obtain the Hamiltonian (2), but we have to replace p<sup>1</sup> ! �p1.

Analysis of Topological Material Surfaces http://dx.doi.org/10.5772/intechopen.74934 9

Figure 4. The mass dependence of the 2d topological invariant ν2d for the lattice model (10). Topology change occurs at the gapless points m ¼ �4, � 2, 0. Change of the invariant corresponds to the monopole charge þ1, � 2, þ 1 associated with each gapless point.

Let us see the topological structure of the lattice model. Applying the same procedure to the Hamiltonian (10), we obtain the topological invariant as follows [14]:

$$\nu\_{\rm 2d} = \frac{1}{2\pi} \int\_{\rm BZ} dp\_1 dp\_2 F\_{12} = \begin{cases} 0 & (m > 0 \text{ or } m < -4) \\ -1 & (-2 < m < 0) \\ +1 & (-4 < m < -2) \end{cases} \tag{11}$$

where the momentum integral is taken over the Brillouin zone. In contrast to the continuum effective model, we have integer valued topological invariants in this case. This is essentially related to the anomaly of (2 + 1)-dimensional Dirac system, known as the parity anomaly. However, it is also known that the lattice regularization naively gives rise to an anomaly-free system: The gapless points have to appear as a pair, so that each anomalous contribution is canceled with each other [15, 16]. Actually the present model (10) has four gapless points in the parameter space: <sup>p</sup>1; <sup>p</sup>2; <sup>m</sup> � � <sup>¼</sup> ð Þ <sup>0</sup>; <sup>0</sup>; <sup>0</sup> , ð Þ <sup>π</sup>; <sup>0</sup>; �<sup>2</sup> , 0ð Þ ; <sup>π</sup>; �<sup>2</sup> , ð Þ <sup>π</sup>; <sup>π</sup>; �<sup>4</sup> . Each gapless point plays basically the same role as that discussed in the continuum model with the monopole charge <sup>þ</sup>1 or �1. Thus we immediately obtain <sup>ν</sup>2d <sup>¼</sup> <sup>1</sup> <sup>2</sup> ðþ1 � 2þ 1Þ ¼ 0 for m > 0, 1 <sup>2</sup> ð Þ¼ �<sup>1</sup> � <sup>2</sup> <sup>þ</sup> <sup>1</sup> 1 for �<sup>2</sup> <sup>&</sup>lt; <sup>m</sup> <sup>&</sup>lt; 0, <sup>1</sup> <sup>2</sup> ð Þ¼ �<sup>1</sup> <sup>þ</sup> <sup>2</sup> <sup>þ</sup> <sup>1</sup> 1 for �<sup>4</sup> <sup>&</sup>lt; <sup>m</sup> <sup>&</sup>lt; �2, and <sup>1</sup> <sup>2</sup> ð Þ¼ �1 þ 2 � 1 0 for m < �4. See Figure 4.

We can similarly consider a lattice model for 3d WSM system. We consider the Hamiltonian defined on a 3d lattice

$$H\_{\rm 3d}^{\rm lat} = \frac{1}{2}\sigma\_1 \left(\nabla\_1 + \nabla\_1^\dagger - \nabla\_2 - \nabla\_2^\dagger + 2c\right) - \frac{i}{2}\sigma\_2 \left(\nabla\_2 - \nabla\_2^\dagger\right) - \frac{i}{2}\sigma\_3 \left(\nabla\_3 - \nabla\_3^\dagger\right). \tag{12}$$

The corresponding Bloch Hamiltonian is given by

with a spatial profile, which is known as the domain-wall fermion. See, for example, [13] for

Figure 3. The dispersion relation of the edge state with ϑ ¼ 1. We find a gapless chiral mode specific to the 2d class A TI.

Since the electron lives on a lattice in the material, studying the lattice model is important to understand the actual behavior of the electron. Let us introduce the Hamiltonian describing

� � <sup>þ</sup> <sup>σ</sup><sup>3</sup> <sup>m</sup> <sup>þ</sup> <sup>2</sup> � <sup>1</sup>

! ¼ ψ<sup>n</sup> !þ <sup>e</sup> ! <sup>1</sup>, <sup>2</sup> � ψ<sup>n</sup>

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

!¼ ð Þ <sup>0</sup>; <sup>0</sup> , one can see the effective Hamiltonian (2) is obtained. If expanding the

!¼ ð Þ <sup>π</sup>; <sup>0</sup> instead, we similarly obtain the Hamiltonian (2), but we have

Periodicity p1,<sup>2</sup> � p1,<sup>2</sup> þ 2π reflects the lattice structure: The momentum is restricted

to the Brillouin zone p1,<sup>2</sup> ∈½ � 0; 2π . The spectrum is given by e p

<sup>2</sup> <sup>∇</sup><sup>1</sup> <sup>þ</sup> <sup>∇</sup>† 1 � � � <sup>1</sup>

� � � �

! with the unit vector e

� �: (10)

, which has four gapless points p

!¼ ð Þ <sup>π</sup>; <sup>π</sup> at <sup>m</sup> ¼ �4. Expanding the momentum

<sup>2</sup> <sup>∇</sup><sup>2</sup> <sup>þ</sup> <sup>∇</sup>† 2

> ! <sup>1</sup>,<sup>2</sup> in n<sup>1</sup>

> > !� � ¼

!¼ ð Þ <sup>0</sup>; <sup>0</sup>

(9)

more details.

2.3. Lattice system

8 Heterojunctions and Nanostructures

the electron on a lattice

<sup>σ</sup><sup>1</sup> <sup>∇</sup><sup>1</sup> � <sup>∇</sup>†

The gapped spectra are interpreted as bulk contributions.

where we define the difference operator ∇1,2ψ<sup>n</sup>

Hlat 2d p !� �

1 � � � <sup>i</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

� �<sup>2</sup> <sup>q</sup>

!¼ ð Þ <sup>π</sup>; <sup>0</sup> and 0ð Þ ; <sup>π</sup> at <sup>m</sup> ¼ �2, <sup>p</sup>

� �<sup>2</sup> <sup>þ</sup> <sup>m</sup> <sup>þ</sup> <sup>2</sup> � cos <sup>p</sup><sup>1</sup> � cos <sup>p</sup><sup>2</sup>

2

<sup>σ</sup><sup>2</sup> <sup>∇</sup><sup>2</sup> � <sup>∇</sup>†

and n2-direction. Then the corresponding Bloch Hamiltonian is given by

2

Hlat 2d ¼ � <sup>i</sup> 2

�

sin p<sup>1</sup>

at m ¼ 0, p

around p

� �<sup>2</sup> <sup>þ</sup> sin <sup>p</sup><sup>2</sup>

momentum around p

to replace p<sup>1</sup> ! �p1.

$$\mathcal{H}\_{\rm 3d}^{\rm dat} \left( \overrightarrow{p} \right) = \sigma\_1 \left( \cos p\_1 - \cos p\_2 + c \right) + \sigma\_2 \sin p\_2 + \sigma\_3 \sin p\_3 \tag{13}$$

and the spectrum yields e p !� � ¼ � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos <sup>p</sup><sup>1</sup> � cos <sup>p</sup><sup>2</sup> <sup>þ</sup> <sup>c</sup> � �<sup>2</sup> <sup>þ</sup> sin <sup>p</sup><sup>2</sup> � �<sup>2</sup> <sup>þ</sup> sin <sup>p</sup><sup>3</sup> � �<sup>2</sup> <sup>q</sup> . The parameter c tunes the gapless Weyl points as follows

$$\mathbf{x}\_{1}(p\_{1}, p\_{2}, p\_{3}) = \begin{cases} \left(\cos^{-1}(1-c), 0, 0\right) & \& \left(\cos^{-1}(1+c), 0, \pi\right) & (0 < c < 2) \\ \left(\cos^{-1}(-1-c), \pi, 0\right) & \& \left(\cos^{-1}(1+c), \pi, \pi\right) & (-2 < c < 0) \\ \mathbf{n}/\mathbf{a} & & (|c| > 2) \end{cases} \tag{14}$$

<sup>γ</sup><sup>5</sup> <sup>¼</sup> <sup>1</sup> <sup>0</sup>

0 �1

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

matrix. The spectrum is simply obtained as eð Þ¼� p

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

tion, which is a consequence of <sup>S</sup><sup>3</sup> fibration of <sup>H</sup>P<sup>1</sup> <sup>¼</sup> <sup>S</sup><sup>7</sup>

which is known as the instanton number,

mass with another momentum m ! p5,

equivalent to the hemisphere integral of S<sup>4</sup>

3. Boundary condition analysis

3.1. Operator formalism

operator [3, 4, 17, 18]

q

degenerated. We have a similar eigenvector to (3) as follows,

1 ξ

the eigenvector is a 2 � 4 matrix due to the degeneracy, namely ψ ¼ ψ<sup>1</sup> ψ<sup>2</sup>

<sup>ν</sup>4d ¼ � <sup>1</sup>

8π<sup>2</sup> ð

and thus the integral over the 5d momentum space, instead of 4d, gives rise to

<sup>ν</sup>5d ¼ � <sup>1</sup>

8π<sup>2</sup> ð S4

which implies the SU(2) monopole, called the Wu-Yang monopole, at the origin in the momentum space. Then the 4d momentum integral performed to obtain the 4d invariant ν4d is

In order to discuss the boundary condition, we start with a first order Hermitian differential

� �, and the off-diagonal element is given by <sup>Δ</sup>ð Þ¼ <sup>p</sup> <sup>p</sup> � <sup>σ</sup><sup>∈</sup> <sup>H</sup> with <sup>σ</sup> <sup>¼</sup> <sup>i</sup> <sup>σ</sup>

We remark that this Hamiltonian is a 4 � 4 matrix, such that each element shows a 2 � 2

� � with <sup>ξ</sup> <sup>¼</sup> <sup>Δ</sup>

Currently each component shows a 2 � 2 matrix, which takes a value in quaternion H, so that

is a four vector. For a degenerated system, we can define non-Abelian analog of the Berry

4d system is given by the four-dimensional momentum integral of the second Chern class,

Actually the instanton configuration obtained here, by solving a matrix equation, is closely related to the ADHM construction. See [12] for more details. We again obtain a half-integer topological invariant. The reason is totally parallel with the previous case. To obtain an integer valued topological invariant, we consider the 5d uplift, the 5d WSM, obtained by replacing the

Tr<sup>F</sup> <sup>∧</sup> <sup>F</sup> <sup>¼</sup> <sup>1</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j j p <sup>2</sup> <sup>þ</sup> <sup>m</sup><sup>2</sup>

<sup>ε</sup> <sup>þ</sup> <sup>m</sup> <sup>¼</sup> <sup>ε</sup> � <sup>m</sup>

q

<sup>a</sup> ð Þ id ψ<sup>b</sup> for a, b ¼ 1, 2. In this case, we obtain an SU(2) valued Berry connec-

2

=S<sup>3</sup>

H5dð Þ¼ p p � γ þ p5γ5, (18)

, which provides a half of the 5d invariant.

TrF ∧ F ¼ 1, (19)

!; 1 � �. 11

, and each state is doubly

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

<sup>Δ</sup>† <sup>∈</sup> <sup>H</sup>: (16)

� �, where each <sup>ψ</sup>1, <sup>2</sup>

. The topological invariant for the

signð Þ m : (17)

The band spectrum is shown in Figure 5 at p<sup>3</sup> ¼ 0 and c ¼ 1. We can see two Weyl points at p1; p<sup>2</sup> � � ¼ �ð Þ <sup>π</sup>=2; <sup>0</sup> . We will study the boundary condition of this model in Section 3.3.

#### 2.4. Higher-dimensional system

So far we have considered a simple system in two and three dimensions. We can even discuss such a topological structure in the momentum space of more involved systems. In this section we discuss a higher-dimensional generalization of the system discussed above. Dimensional reduction of this system gives rise to several interesting situations in 2d and 3d.

We consider a four-band model defined in four spatial dimensions, which is a natural higherdimensional generalization of (2),

$$\mathcal{H}\_{\mathsf{4d}}(p) = p \cdot \boldsymbol{\gamma} + m \boldsymbol{\gamma}\_{5} = \begin{pmatrix} m & \boldsymbol{\Delta}(p)^{\dagger} \\ \boldsymbol{\Delta}(p) & -m \end{pmatrix} \quad \text{with} \quad \boldsymbol{\Delta}(p) = p \cdot \boldsymbol{\sigma} = \begin{pmatrix} p\_{4} + ip\_{3} & p\_{2} + ip\_{1} \\ -p\_{2} + ip\_{1} & p\_{4} - ip\_{3} \end{pmatrix} \tag{15}$$

where we use the gamma matrices defined as <sup>γ</sup><sup>k</sup> <sup>¼</sup> <sup>0</sup> �iσ<sup>k</sup> σ<sup>k</sup> 0 � � for <sup>k</sup> <sup>¼</sup> <sup>1</sup>; <sup>2</sup>; 3, <sup>γ</sup><sup>4</sup> <sup>¼</sup> <sup>0</sup> <sup>1</sup> 1 0 � �,

Figure 5. The energy spectrum of the lattice WSM model (13) with p<sup>3</sup> ¼ 0 and c ¼ 1. There exist two gapless Weyl points at p1; p<sup>2</sup> � � ¼ �ð Þ <sup>π</sup>=2; <sup>0</sup> . The parameter <sup>c</sup> characterizes the distance between the Weyl points.

#### Analysis of Topological Material Surfaces http://dx.doi.org/10.5772/intechopen.74934 11

<sup>γ</sup><sup>5</sup> <sup>¼</sup> <sup>1</sup> <sup>0</sup> 0 �1 � �, and the off-diagonal element is given by <sup>Δ</sup>ð Þ¼ <sup>p</sup> <sup>p</sup> � <sup>σ</sup><sup>∈</sup> <sup>H</sup> with <sup>σ</sup> <sup>¼</sup> <sup>i</sup> <sup>σ</sup> !; 1 � �. We remark that this Hamiltonian is a 4 � 4 matrix, such that each element shows a 2 � 2 matrix. The spectrum is simply obtained as eð Þ¼� p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j j p <sup>2</sup> <sup>þ</sup> <sup>m</sup><sup>2</sup> q , and each state is doubly degenerated. We have a similar eigenvector to (3) as follows,

$$\psi = \frac{1}{\sqrt{1 + |\xi|^2}} \begin{pmatrix} 1 \\ \xi \end{pmatrix} \quad \text{with} \quad \xi = \frac{\Delta}{\varepsilon + m} = \frac{\varepsilon - m}{\Delta^\dagger} \in \mathbb{H}. \tag{16}$$

Currently each component shows a 2 � 2 matrix, which takes a value in quaternion H, so that the eigenvector is a 2 � 4 matrix due to the degeneracy, namely ψ ¼ ψ<sup>1</sup> ψ<sup>2</sup> � �, where each <sup>ψ</sup>1, <sup>2</sup> is a four vector. For a degenerated system, we can define non-Abelian analog of the Berry connection <sup>A</sup>ab <sup>¼</sup> <sup>ψ</sup>† <sup>a</sup> ð Þ id ψ<sup>b</sup> for a, b ¼ 1, 2. In this case, we obtain an SU(2) valued Berry connection, which is a consequence of <sup>S</sup><sup>3</sup> fibration of <sup>H</sup>P<sup>1</sup> <sup>¼</sup> <sup>S</sup><sup>7</sup> =S<sup>3</sup> . The topological invariant for the 4d system is given by the four-dimensional momentum integral of the second Chern class, which is known as the instanton number,

$$\nu\_{\rm 4d} = -\frac{1}{8\pi^2} \left[ \mathrm{Tr} \mathcal{F} \wedge \mathcal{F} = \frac{1}{2} \mathrm{sign}(m). \tag{17}$$

Actually the instanton configuration obtained here, by solving a matrix equation, is closely related to the ADHM construction. See [12] for more details. We again obtain a half-integer topological invariant. The reason is totally parallel with the previous case. To obtain an integer valued topological invariant, we consider the 5d uplift, the 5d WSM, obtained by replacing the mass with another momentum m ! p5,

$$\mathcal{H}\_{\rm 5d}(p) = p \cdot \gamma + p\_{\rm 5} \gamma\_{\rm 5'} \tag{18}$$

and thus the integral over the 5d momentum space, instead of 4d, gives rise to

$$\nu\_{\rm 5d} = -\frac{1}{8\pi^2} \int\_{S^4} \text{Tr}\mathcal{F} \wedge \mathcal{F} = 1,\tag{19}$$

which implies the SU(2) monopole, called the Wu-Yang monopole, at the origin in the momentum space. Then the 4d momentum integral performed to obtain the 4d invariant ν4d is equivalent to the hemisphere integral of S<sup>4</sup> , which provides a half of the 5d invariant.

#### 3. Boundary condition analysis

#### 3.1. Operator formalism

and the spectrum yields e p

10 Heterojunctions and Nanostructures

p1; p2; p<sup>3</sup> � � <sup>¼</sup>

2.4. Higher-dimensional system

dimensional generalization of (2),

<sup>H</sup>4dð Þ¼ <sup>p</sup> <sup>p</sup> � <sup>γ</sup> <sup>þ</sup> <sup>m</sup>γ<sup>5</sup> <sup>¼</sup> <sup>m</sup> <sup>Δ</sup>ð Þ<sup>p</sup>

p1; p<sup>2</sup>

at p1; p<sup>2</sup>

!� �

eter c tunes the gapless Weyl points as follows

8 ><

>:

¼ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

� �<sup>2</sup> <sup>q</sup>

� �<sup>2</sup> <sup>þ</sup> sin <sup>p</sup><sup>3</sup>

with <sup>Δ</sup>ð Þ¼ <sup>p</sup> <sup>p</sup> � <sup>σ</sup> <sup>¼</sup> <sup>p</sup><sup>4</sup> <sup>þ</sup> ip<sup>3</sup> <sup>p</sup><sup>2</sup> <sup>þ</sup> ip<sup>1</sup>

σ<sup>k</sup> 0 � �

�p<sup>2</sup> þ ip<sup>1</sup> p<sup>4</sup> � ip<sup>3</sup> � �

for <sup>k</sup> <sup>¼</sup> <sup>1</sup>; <sup>2</sup>; 3, <sup>γ</sup><sup>4</sup> <sup>¼</sup> <sup>0</sup> <sup>1</sup>

. The param-

(14)

(15)

,

1 0 � �

cos <sup>p</sup><sup>1</sup> � cos <sup>p</sup><sup>2</sup> <sup>þ</sup> <sup>c</sup> � �<sup>2</sup> <sup>þ</sup> sin <sup>p</sup><sup>2</sup>

cos �<sup>1</sup>ð Þ <sup>1</sup> � <sup>c</sup> ; <sup>0</sup>; <sup>0</sup> � � & cos �<sup>1</sup>ð Þ <sup>1</sup> <sup>þ</sup> <sup>c</sup> ; <sup>0</sup>; <sup>π</sup> � � ð Þ <sup>0</sup> <sup>&</sup>lt; <sup>c</sup> <sup>&</sup>lt; <sup>2</sup> cos �<sup>1</sup>ð Þ �<sup>1</sup> � <sup>c</sup> ; <sup>π</sup>; <sup>0</sup> � � & cos �<sup>1</sup>ð Þ <sup>1</sup> <sup>þ</sup> <sup>c</sup> ; <sup>π</sup>; <sup>π</sup> � � ð Þ �<sup>2</sup> <sup>&</sup>lt; <sup>c</sup> <sup>&</sup>lt; <sup>0</sup> n=a ð Þ jcj > 2

The band spectrum is shown in Figure 5 at p<sup>3</sup> ¼ 0 and c ¼ 1. We can see two Weyl points at

So far we have considered a simple system in two and three dimensions. We can even discuss such a topological structure in the momentum space of more involved systems. In this section we discuss a higher-dimensional generalization of the system discussed above. Dimensional

We consider a four-band model defined in four spatial dimensions, which is a natural higher-

Figure 5. The energy spectrum of the lattice WSM model (13) with p<sup>3</sup> ¼ 0 and c ¼ 1. There exist two gapless Weyl points

� � ¼ �ð Þ <sup>π</sup>=2; <sup>0</sup> . The parameter <sup>c</sup> characterizes the distance between the Weyl points.

� � ¼ �ð Þ <sup>π</sup>=2; <sup>0</sup> . We will study the boundary condition of this model in Section 3.3.

reduction of this system gives rise to several interesting situations in 2d and 3d.

†

ΔðÞ � p m

where we use the gamma matrices defined as <sup>γ</sup><sup>k</sup> <sup>¼</sup> <sup>0</sup> �iσ<sup>k</sup>

!

In order to discuss the boundary condition, we start with a first order Hermitian differential operator [3, 4, 17, 18]

$$
\hat{D} = -i\sigma \frac{\partial}{\partial \mathbf{x}}.\tag{20}
$$

X N

<sup>n</sup> �iσ∇ψ<sup>n</sup> � � <sup>¼</sup> <sup>X</sup>

D E requires that the surface term should vanish:

eigenvalues 1, 0 with the corresponding eigenvectors

1 �e<sup>i</sup><sup>θ</sup> � � <sup>¼</sup> <sup>1</sup> �e<sup>i</sup><sup>θ</sup>

> ψ � � � xL,R

Since <sup>P</sup><sup>ψ</sup> <sup>¼</sup> <sup>ψ</sup>, the current in <sup>x</sup>3-direction vanishes at the boundary, <sup>J</sup><sup>3</sup> <sup>¼</sup> <sup>ψ</sup>†

formalism. The integral over the spacetime M of the Lagrangian defines the action

one-parameter family of the solution to the boundary condition (23),

condition for the normal component of the current as expected.

P

3.2. Lagrangian formalism

N

iσ∇† ϕn � �†

where <sup>ϕ</sup><sup>0</sup> and <sup>ψ</sup><sup>N</sup>þ<sup>1</sup> are considered as auxiliary fields. The self-conjugacy condition <sup>ϕ</sup>

<sup>0</sup>σψ<sup>1</sup> � <sup>ϕ</sup>†

suggests that we have to take care of the locality and continuum limit of the system.

The periodic boundary condition <sup>ϕ</sup><sup>n</sup>þ<sup>N</sup> <sup>¼</sup> <sup>ϕ</sup>n, <sup>ψ</sup><sup>n</sup>þ<sup>N</sup> <sup>¼</sup> <sup>ψ</sup><sup>n</sup> is a simple solution to this. The other possibility is that each term independently vanishes, corresponding to the open boundary condition. This means that the lattice system is similarly considered as the continuum system, and the open boundary condition is imposed by (23). We remark that for the lattice system the surface term (27) is not given by the on-site term, but involving a hopping to the next site. This

Let us consider an example with σ ¼ σ3. Then the matrix M should be a linear combination of σ1,2. Since the operator P has a zero eigenvalue, the determinant should vanish detP ¼ 0, which leads to

<sup>M</sup> <sup>¼</sup> <sup>σ</sup><sup>1</sup> cos <sup>θ</sup> <sup>þ</sup> <sup>σ</sup><sup>2</sup> sin <sup>θ</sup> <sup>¼</sup> <sup>0</sup> <sup>e</sup>�i<sup>θ</sup>

obeying <sup>M</sup>† <sup>¼</sup> <sup>M</sup> and <sup>M</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup> with two eigenvalues �1. It is also expressed as <sup>M</sup> <sup>¼</sup> <sup>σ</sup>1e<sup>i</sup>θσ<sup>3</sup>

� � and <sup>P</sup>

We remark <sup>σ</sup>3<sup>P</sup> <sup>¼</sup> <sup>P</sup><sup>σ</sup>3, <sup>σ</sup>3<sup>P</sup> <sup>¼</sup> <sup>P</sup>σ<sup>3</sup> where <sup>P</sup> <sup>¼</sup> <sup>1</sup> � <sup>P</sup> obeying PP <sup>¼</sup> PP <sup>¼</sup> 0. Thus we obtain a

<sup>∝</sup> <sup>1</sup> eiθ

<sup>P</sup><sup>ψ</sup> <sup>¼</sup> <sup>ψ</sup>†σ3PP<sup>ψ</sup> <sup>¼</sup> 0. In other words, the open boundary condition is interpreted as a vanishing

We explain how to derive a proper boundary condition for a given system with the Lagrangian

<sup>i</sup> <sup>θ</sup>�<sup>π</sup> ð Þ<sup>2</sup> <sup>σ</sup><sup>3</sup> . Thus the operator <sup>P</sup> <sup>¼</sup> <sup>P</sup>† � � turns out to be a projection operator <sup>P</sup><sup>2</sup> <sup>¼</sup> <sup>P</sup> having

e<sup>i</sup><sup>θ</sup> 0 !

> 1 eiθ � �

� �: (30)

<sup>ψ</sup><sup>n</sup> <sup>þ</sup> <sup>ϕ</sup>†

<sup>0</sup>ð Þ <sup>i</sup><sup>σ</sup> <sup>ψ</sup><sup>1</sup> � <sup>ϕ</sup>†

<sup>N</sup>σψ<sup>N</sup>þ<sup>1</sup> <sup>¼</sup> <sup>0</sup>: (27)

<sup>N</sup>ð Þ <sup>i</sup><sup>σ</sup> <sup>ψ</sup><sup>N</sup>þ<sup>1</sup> (26)

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

, (28)

¼ 0: (29)

<sup>σ</sup>3<sup>ψ</sup> <sup>¼</sup> <sup>ψ</sup>†

Pσ<sup>3</sup>

� � � Db latψ D E <sup>¼</sup> 13

n¼1

ϕ†

n¼1 ϕ†

Db latϕ � � �ψ

3.1.2. Example

¼ σ2e

Now we put a Pauli matrix σ, but we can consider a generic Hermitian matrix. Considering the inner product in a finite size system defined on the interval x∈ ½ � xL; xR , we obtain

$$
\begin{split}
\left\langle\phi\middle|\hat{D}\psi\right\rangle &= \int\_{\mathbf{x}\_{\perp}}^{\mathbf{x}\_{\mathbb{R}}} d\mathbf{x} \phi(\mathbf{x})^{\dagger} \left(-i\sigma \frac{\partial}{\partial \mathbf{x}} \psi(\mathbf{x})\right) \\&= \phi(\mathbf{x})^{\dagger}(-i\sigma)\psi(\mathbf{x})\Big|\_{\mathbf{x}\_{\mathbb{L}}}^{\mathbf{x}\_{\mathbb{R}}} + \int\_{\mathbf{x}\_{\mathbb{L}}}^{\mathbf{x}\_{\mathbb{R}}} d\mathbf{x} \left(-i\sigma \frac{\partial}{\partial \mathbf{x}} \phi(\mathbf{x})\right)^{\dagger}\psi(\mathbf{x}) \\&= \phi(\mathbf{x})^{\dagger}(-i\sigma)\psi(\mathbf{x})\Big|\_{\mathbf{x}\_{\mathbb{L}}}^{\mathbf{x}\_{\mathbb{R}}} + \left\langle\widehat{D}\phi\middle|\psi\right\rangle.
\end{split}
\tag{21}
$$

The Hermitian condition ϕ � � � Dbψ D E <sup>¼</sup> <sup>D</sup><sup>b</sup> � � �ϕψ D E implies that the surface term should vanish

$$\left.\phi(\mathbf{x})^{\dagger}\sigma\psi(\mathbf{x})\right|\_{\mathbf{x}\_{L}}^{\times\mathbb{R}} = \phi(\mathbf{x}\_{R})^{\dagger}\sigma\psi(\mathbf{x}\_{R}) - \phi(\mathbf{x}\_{L})^{\dagger}\sigma\psi(\mathbf{x}\_{L}) = \mathbf{0},\tag{22}$$

which gives rise to two possibilities:


In particular, the open boundary condition 2 has the following solution

$$\left.P\psi\right|\_{\mathbf{x}\_{l,R}} = \left.P\phi\right|\_{\mathbf{x}\_{l,R}} = \mathbf{0} \quad \text{where} \quad P = \frac{\mathbf{1} - M}{2} \tag{23}$$

with the matrix M satisfying M† σ þ σM ¼ 0, since ψ ¼ Mψ, ϕ ¼ Mϕ at the boundary, then

$$
\phi^\dagger \sigma \psi = \phi^\dagger \sigma M \psi = -\phi^\dagger M^\dagger \sigma \psi = -\phi^\dagger \sigma \psi \quad \Rightarrow \quad \phi^\dagger \sigma \psi = 0. \tag{24}
$$

In general we can apply different matrices ML,R for xL and xR, but here we assume ML,R ¼ M for simplicity, namely the same boundary condition for xL,R. We remark that the condition (23) is specific to the operator choice (20). We have to derive the corresponding boundary condition case by case. We will show a generic formulation of the boundary condition using the Lagrangian formalism in Section 3.2.

#### 3.1.1. Lattice system

Let us apply the similar argument to the lattice system defined on a one-dimensional interval n∈f g 1;…; N . We introduce an analogous difference operator to (20) as

$$
\dot{D}\_{\text{lat}} = -i\sigma \nabla \tag{25}
$$

where <sup>∇</sup>ψ<sup>n</sup> <sup>¼</sup> <sup>ψ</sup><sup>n</sup>þ<sup>1</sup> � <sup>ψ</sup><sup>n</sup> and <sup>∇</sup>†ψ<sup>n</sup> <sup>¼</sup> <sup>ψ</sup><sup>n</sup>�<sup>1</sup> � <sup>ψ</sup>n. In this case, the inner product <sup>ϕ</sup> � � � Db latψ D E is given by

#### Analysis of Topological Material Surfaces http://dx.doi.org/10.5772/intechopen.74934 13

$$\sum\_{n=1}^{N} \phi\_n^\dagger \left(-i\sigma \nabla \psi\_n\right) = \sum\_{n=1}^{N} \left(i\sigma \nabla^\dagger \phi\_n\right)^\dagger \psi\_n + \phi\_0^\dagger (i\sigma) \psi\_1 - \phi\_N^\dagger (i\sigma) \psi\_{N+1} \tag{26}$$

where <sup>ϕ</sup><sup>0</sup> and <sup>ψ</sup><sup>N</sup>þ<sup>1</sup> are considered as auxiliary fields. The self-conjugacy condition <sup>ϕ</sup> � � � Db latψ D E <sup>¼</sup> Db latϕ � � �ψ D E requires that the surface term should vanish:

$$
\phi\_0^\dagger \sigma \psi\_1 - \phi\_N^\dagger \sigma \psi\_{N+1} = 0.\tag{27}
$$

The periodic boundary condition <sup>ϕ</sup><sup>n</sup>þ<sup>N</sup> <sup>¼</sup> <sup>ϕ</sup>n, <sup>ψ</sup><sup>n</sup>þ<sup>N</sup> <sup>¼</sup> <sup>ψ</sup><sup>n</sup> is a simple solution to this. The other possibility is that each term independently vanishes, corresponding to the open boundary condition. This means that the lattice system is similarly considered as the continuum system, and the open boundary condition is imposed by (23). We remark that for the lattice system the surface term (27) is not given by the on-site term, but involving a hopping to the next site. This suggests that we have to take care of the locality and continuum limit of the system.

#### 3.1.2. Example

Db ¼ �iσ

inner product in a finite size system defined on the interval x∈ ½ � xL; xR , we obtain

ð Þ �iσ ψð Þx

ð Þ �iσ ψð Þx

¼ ϕð Þ xR †

¼ Db � � �ϕψ D E

1. Periodic boundary condition: ϕð Þ¼ xR ϕð Þ xL and ψð Þ¼ xR ψð Þ xL

<sup>P</sup><sup>ψ</sup> xL,R <sup>¼</sup> <sup>P</sup><sup>ϕ</sup> � � �

†

� xL,R

M†

In particular, the open boundary condition 2 has the following solution

<sup>σ</sup>M<sup>ψ</sup> ¼ �ϕ†

n∈f g 1;…; N . We introduce an analogous difference operator to (20) as

† �i<sup>σ</sup>

� � � xR xL þ ðxR xL

� � � xR xL þ Dbϕ � � �ψ D E :

σψð Þ� xR ϕð Þ xL

σψð Þ¼ xR 0 and ϕð Þ xL

σψ ¼ �ϕ†

In general we can apply different matrices ML,R for xL and xR, but here we assume ML,R ¼ M for simplicity, namely the same boundary condition for xL,R. We remark that the condition (23) is specific to the operator choice (20). We have to derive the corresponding boundary condition case by case. We will show a generic formulation of the boundary condition using the Lagrang-

Let us apply the similar argument to the lattice system defined on a one-dimensional interval

where <sup>∇</sup>ψ<sup>n</sup> <sup>¼</sup> <sup>ψ</sup><sup>n</sup>þ<sup>1</sup> � <sup>ψ</sup><sup>n</sup> and <sup>∇</sup>†ψ<sup>n</sup> <sup>¼</sup> <sup>ψ</sup><sup>n</sup>�<sup>1</sup> � <sup>ψ</sup>n. In this case, the inner product <sup>ϕ</sup>

dxϕð Þx

ϕ � � � Dbψ D E

The Hermitian condition ϕ

12 Heterojunctions and Nanostructures

which gives rise to two possibilities:

2. Open boundary condition: ϕð Þ xR

with the matrix M satisfying M†

ian formalism in Section 3.2.

3.1.1. Lattice system

given by

ϕ†

σψ <sup>¼</sup> <sup>ϕ</sup>†

¼ ðxR xL

¼ ϕð Þx †

¼ ϕð Þx †

� � � Dbψ D E

ϕð Þx † σψð Þx � � � xR xL

Now we put a Pauli matrix σ, but we can consider a generic Hermitian matrix. Considering the

∂ ∂x ψð Þx � �

∂ ∂x

dx �iσ

†

†

σ þ σM ¼ 0, since ψ ¼ Mψ, ϕ ¼ Mϕ at the boundary, then

σψ ) <sup>ϕ</sup>†

Db lat ¼ �iσ∇ (25)

<sup>¼</sup> 0 where <sup>P</sup> <sup>¼</sup> <sup>1</sup> � <sup>M</sup>

σψð Þ¼ xL 0

∂ ∂x ϕð Þx � �†

implies that the surface term should vanish

: (20)

ψð Þx

σψð Þ¼ xL 0, (22)

<sup>2</sup> (23)

σψ ¼ 0: (24)

� � � Db latψ D E

is

(21)

Let us consider an example with σ ¼ σ3. Then the matrix M should be a linear combination of σ1,2. Since the operator P has a zero eigenvalue, the determinant should vanish detP ¼ 0, which leads to

$$M = \sigma\_1 \cos \theta + \sigma\_2 \sin \theta = \begin{pmatrix} 0 & e^{-i\theta} \\ e^{i\theta} & 0 \end{pmatrix} \tag{28}$$

obeying <sup>M</sup>† <sup>¼</sup> <sup>M</sup> and <sup>M</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup> with two eigenvalues �1. It is also expressed as <sup>M</sup> <sup>¼</sup> <sup>σ</sup>1e<sup>i</sup>θσ<sup>3</sup> ¼ σ2e <sup>i</sup> <sup>θ</sup>�<sup>π</sup> ð Þ<sup>2</sup> <sup>σ</sup><sup>3</sup> . Thus the operator <sup>P</sup> <sup>¼</sup> <sup>P</sup>† � � turns out to be a projection operator <sup>P</sup><sup>2</sup> <sup>¼</sup> <sup>P</sup> having eigenvalues 1, 0 with the corresponding eigenvectors

$$P\begin{pmatrix} 1 \\ -\varepsilon^{i\theta} \end{pmatrix} = \begin{pmatrix} 1 \\ -\varepsilon^{i\theta} \end{pmatrix} \qquad \text{and} \qquad P\begin{pmatrix} 1 \\ \varepsilon^{i\theta} \end{pmatrix} = 0. \tag{29}$$

We remark <sup>σ</sup>3<sup>P</sup> <sup>¼</sup> <sup>P</sup><sup>σ</sup>3, <sup>σ</sup>3<sup>P</sup> <sup>¼</sup> <sup>P</sup>σ<sup>3</sup> where <sup>P</sup> <sup>¼</sup> <sup>1</sup> � <sup>P</sup> obeying PP <sup>¼</sup> PP <sup>¼</sup> 0. Thus we obtain a one-parameter family of the solution to the boundary condition (23),

$$\left.\psi\right|\_{x\_{l,R}} \propto \begin{pmatrix} 1\\ e^{i\theta} \end{pmatrix}. \tag{30}$$

Since <sup>P</sup><sup>ψ</sup> <sup>¼</sup> <sup>ψ</sup>, the current in <sup>x</sup>3-direction vanishes at the boundary, <sup>J</sup><sup>3</sup> <sup>¼</sup> <sup>ψ</sup>† <sup>σ</sup>3<sup>ψ</sup> <sup>¼</sup> <sup>ψ</sup>† Pσ<sup>3</sup> <sup>P</sup><sup>ψ</sup> <sup>¼</sup> <sup>ψ</sup>†σ3PP<sup>ψ</sup> <sup>¼</sup> 0. In other words, the open boundary condition is interpreted as a vanishing condition for the normal component of the current as expected.

#### 3.2. Lagrangian formalism

We explain how to derive a proper boundary condition for a given system with the Lagrangian formalism. The integral over the spacetime M of the Lagrangian defines the action

$$S = \int\_{\mathcal{M}} \mathcal{L}(\phi, \partial \phi) \,. \tag{31}$$

3.3. 3d Weyl semimetal

3.3.1. Continuum system

which is normalized as Ð <sup>∞</sup>

�α p !

eigenvalue equation, given as follows

e p

length. In this case, the exponential factor e

due to the factor e

4

Hamiltonian (6) with a slight modification

ℋ3d p !; x<sup>3</sup>

domain x<sup>3</sup> > 0. In this case, since the current operator is defined as J

ψ p !; x<sup>3</sup> � � <sup>¼</sup>

<sup>0</sup> dx3ψ†

the replacement p<sup>3</sup> ! iα, we can apply almost the same analysis.

!� � <sup>¼</sup> <sup>p</sup><sup>1</sup> cos <sup>θ</sup> <sup>þ</sup> <sup>p</sup><sup>2</sup> sin <sup>θ</sup>, <sup>α</sup> <sup>p</sup>

e α � �

This is of course equivalent to the boundary condition discussed in Section 3.1.2.

Actually it is written using an SO(2) transformation with the relation <sup>e</sup><sup>2</sup> <sup>þ</sup> <sup>α</sup><sup>2</sup> <sup>¼</sup> <sup>p</sup>

<sup>¼</sup> cos <sup>θ</sup> sin <sup>θ</sup> � sin θ cos θ � � p<sup>1</sup>

We show the spectrum of the edge state depending on the boundary condition with the bulk spectrum in Figure 6. We remark that the edge state cannot be defined in the whole

ð Þ<sup>x</sup><sup>3</sup> , where the parameter α p

condition, corresponding to the zero current condition (36), that we impose is4

ψ† σ3ψ � � � x3¼0

The eigenstate satisfying the condition (38) is parameterized by a single phase factor

Let us apply the argument discussed above to the WSM system. We consider the effective

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

We put a boundary only at x<sup>3</sup> ¼ 0 for simplicity, so that the system is defined on a positive

ffiffiffiffiffiffiffiffiffiffiffiffi α p ! <sup>r</sup> � �<sup>e</sup>

eigenstate is localized on the boundary x<sup>3</sup> ¼ 0 and exponentially decay into the bulk x<sup>3</sup> > 0

�α p !

dence, instead of the plane wave factor eip3x<sup>3</sup> used for the bulk analysis. In other words, the current analysis of the edge state uses Laplace basis instead of Fourier basis. Therefore, under

Then the spectrum and the inverse penetration length of the edge state are obtained from the

�α p ! ð Þ<sup>x</sup><sup>3</sup> <sup>1</sup> eiθ

ψ ¼ 1, and the normalizability requires α p

p2

∂ ∂x<sup>3</sup>

> ! <sup>¼</sup> <sup>ψ</sup>† <sup>σ</sup>

¼ 0: (38)

!� � plays a role as the inverse penetration

!� � ¼ �p<sup>1</sup> sin <sup>θ</sup> <sup>þ</sup> <sup>p</sup><sup>2</sup> cos <sup>θ</sup>: (40)

! � � � � � � 2 ,

� �: (41)

ð Þ<sup>x</sup><sup>3</sup> is responsible for the x3-direction depen-

� � (39)

: (37)

Analysis of Topological Material Surfaces http://dx.doi.org/10.5772/intechopen.74934 15

! ψ, the boundary

!� � <sup>&</sup>gt; 0. This

If the system has a continuous local symmetry, the action may be invariant under the infinitesimal deviation of the field <sup>ϕ</sup> ! <sup>ϕ</sup> <sup>þ</sup> <sup>e</sup>ð Þ<sup>x</sup> <sup>φ</sup><sup>3</sup> :

$$\begin{split} 0 &= \delta \mathcal{S} \\ &= \int\_{\mathcal{M}} \left( \frac{\partial \mathcal{E}}{\partial \phi} - \partial\_{\mu} \left( \frac{\partial \mathcal{E}}{\partial (\partial\_{\mu} \phi)} \right) \right) \epsilon(\mathbf{x}) \phi - \int\_{\mathcal{M}} \epsilon(\mathbf{x}) \partial\_{\mu} \left( \frac{\partial \mathcal{E}}{\partial (\partial\_{\mu} \phi)} \phi \right) + \int\_{\mathcal{M}} \partial\_{\mu} \left( \frac{\partial \mathcal{E}}{\partial (\partial\_{\mu} \phi)} \epsilon(\mathbf{x}) \phi \right) . \end{split} \tag{32}$$

The first term vanishes due to the Euler-Lagrange equation of motion for the bulk, <sup>∂</sup><sup>ℒ</sup> <sup>∂</sup><sup>ϕ</sup> � ∂<sup>μ</sup> ∂ℒ ∂ ∂ð Þ μϕ � � ¼ 0. The vanishing condition for the second term implies the current J <sup>μ</sup> <sup>¼</sup> <sup>∂</sup><sup>ℒ</sup> ∂ ∂ð Þ μϕ <sup>φ</sup>, satisfying the conservation law ∂μJ <sup>μ</sup> <sup>¼</sup> 0, a.k.a. the Nöther current. The third term is a surface contribution which plays a role in the system with the boundary. The invariance of the action is thus rephrased as

$$0 = \int\_{\partial \mathcal{M}} \epsilon(\mathbf{x}) n \cdot \mathbf{J} \tag{33}$$

where n is the normal vector defined as Ð <sup>M</sup>∂μV<sup>μ</sup> <sup>¼</sup> <sup>Ð</sup> <sup>∂</sup>M<sup>n</sup> � <sup>V</sup> with an arbitrary vector field <sup>V</sup><sup>μ</sup> and the boundary of the manifold denoted by ∂M. This ends up with the condition such that the normal component of the current should vanish at the boundary

$$n \cdot J \Big|\_{\partial \mathcal{M}} = 0. \tag{34}$$

This seems physically reasonable and consistent with the previous argument in Section 3.1.2 because at the boundary there is no ingoing and outgoing current.

Furthermore, this zero current condition can be modified by taking into account the additional surface contribution to the action

$$\mathcal{S}\_{\mathsf{B}} = \int\_{\partial \mathcal{M}} \mathcal{L}\_{\mathsf{B}}(\phi) \tag{35}$$

where we assume the boundary d.o.f. is not dynamical (not including the derivative ∂ϕ). Then the condition (34) becomes

$$
\left[n \cdot J + \frac{\partial \mathcal{L}\_{\mathsf{B}}}{\partial \phi} \varphi\right]\_{\partial \mathcal{M}} = 0. \tag{36}
$$

This characterizes the boundary condition. In the following, we consider several examples to see how the boundary condition plays a role in the topological materials.

<sup>3</sup> This is an assumption. In general, the action itself is not invariant under the deviation.

#### 3.3. 3d Weyl semimetal

S ¼ ð M

:

ð M eð Þx ∂<sup>μ</sup>

0 ¼ ð ∂M

> n � J � � �

S<sup>B</sup> ¼ ð ∂M L<sup>B</sup> ϕ

n � J þ

see how the boundary condition plays a role in the topological materials.

This is an assumption. In general, the action itself is not invariant under the deviation.

the normal component of the current should vanish at the boundary

because at the boundary there is no ingoing and outgoing current.

The first term vanishes due to the Euler-Lagrange equation of motion for the bulk, <sup>∂</sup><sup>ℒ</sup>

¼ 0. The vanishing condition for the second term implies the current J

contribution which plays a role in the system with the boundary. The invariance of the action is

<sup>M</sup>∂μV<sup>μ</sup> <sup>¼</sup> <sup>Ð</sup>

and the boundary of the manifold denoted by ∂M. This ends up with the condition such that

This seems physically reasonable and consistent with the previous argument in Section 3.1.2

Furthermore, this zero current condition can be modified by taking into account the additional

where we assume the boundary d.o.f. is not dynamical (not including the derivative ∂ϕ). Then

∂L<sup>B</sup> <sup>∂</sup><sup>ϕ</sup> <sup>φ</sup> � �

This characterizes the boundary condition. In the following, we consider several examples to

∂M

eð Þx φ �

imal deviation of the field <sup>ϕ</sup> ! <sup>ϕ</sup> <sup>þ</sup> <sup>e</sup>ð Þ<sup>x</sup> <sup>φ</sup><sup>3</sup>

! !

∂ℒ ∂ ∂μϕ � �

0 ¼ δS

∂ℒ <sup>∂</sup><sup>ϕ</sup> � <sup>∂</sup><sup>μ</sup>

14 Heterojunctions and Nanostructures

satisfying the conservation law ∂μJ

where n is the normal vector defined as Ð

surface contribution to the action

the condition (34) becomes

3

¼ ð M

∂ℒ ∂ ∂ð Þ μϕ � �

thus rephrased as

If the system has a continuous local symmetry, the action may be invariant under the infinites-

∂ℒ ∂ ∂μϕ � �<sup>φ</sup> !

þ ð M ∂μ

<sup>μ</sup> <sup>¼</sup> 0, a.k.a. the Nöther current. The third term is a surface

eð Þx n � J (33)

<sup>∂</sup><sup>M</sup> <sup>¼</sup> <sup>0</sup>: (34)

� � (35)

¼ 0: (36)

<sup>∂</sup>M<sup>n</sup> � <sup>V</sup> with an arbitrary vector field <sup>V</sup><sup>μ</sup>

L ϕ; ∂ϕ � �: (31)

∂ℒ ∂ ∂μϕ � � <sup>e</sup>ð Þ<sup>x</sup> <sup>φ</sup> !

: (32)

<sup>∂</sup><sup>ϕ</sup> � ∂<sup>μ</sup>

<sup>μ</sup> <sup>¼</sup> <sup>∂</sup><sup>ℒ</sup> ∂ ∂ð Þ μϕ <sup>φ</sup>,

#### 3.3.1. Continuum system

Let us apply the argument discussed above to the WSM system. We consider the effective Hamiltonian (6) with a slight modification

$$\mathcal{H}\_{3d}\left(\overrightarrow{p}, \mathbf{x}\_3\right) = p\_1\sigma\_1 + p\_2\sigma\_2 - i\sigma\_3 \frac{\partial}{\partial \mathbf{x}\_3}.\tag{37}$$

We put a boundary only at x<sup>3</sup> ¼ 0 for simplicity, so that the system is defined on a positive domain x<sup>3</sup> > 0. In this case, since the current operator is defined as J ! <sup>¼</sup> <sup>ψ</sup>† <sup>σ</sup> ! ψ, the boundary condition, corresponding to the zero current condition (36), that we impose is4

$$
\left.\psi^\dagger \sigma\_3 \psi\right|\_{x\circ =0} = 0.\tag{38}
$$

The eigenstate satisfying the condition (38) is parameterized by a single phase factor

$$
\psi\left(\overrightarrow{p}, \mathbf{x}\_3\right) = \sqrt{\alpha\left(\overrightarrow{p}\right)} e^{-\alpha\left(\overrightarrow{p}\right)\mathbf{x}\_3} \begin{pmatrix} 1 \\ e^{i\theta} \end{pmatrix} \tag{39}
$$

which is normalized as Ð <sup>∞</sup> <sup>0</sup> dx3ψ†<sup>ψ</sup> <sup>¼</sup> 1, and the normalizability requires <sup>α</sup> <sup>p</sup> !� � <sup>&</sup>gt; 0. This eigenstate is localized on the boundary x<sup>3</sup> ¼ 0 and exponentially decay into the bulk x<sup>3</sup> > 0 due to the factor e �α p ! ð Þ<sup>x</sup><sup>3</sup> , where the parameter α p !� � plays a role as the inverse penetration length. In this case, the exponential factor e �α p ! ð Þ<sup>x</sup><sup>3</sup> is responsible for the x3-direction dependence, instead of the plane wave factor eip3x<sup>3</sup> used for the bulk analysis. In other words, the current analysis of the edge state uses Laplace basis instead of Fourier basis. Therefore, under the replacement p<sup>3</sup> ! iα, we can apply almost the same analysis.

Then the spectrum and the inverse penetration length of the edge state are obtained from the eigenvalue equation, given as follows

$$\mathbf{e}\left(\stackrel{\rightarrow}{p}\right) = p\_1 \cos \theta + p\_2 \sin \theta, \qquad a\left(\stackrel{\rightarrow}{p}\right) = -p\_1 \sin \theta + p\_2 \cos \theta. \tag{40}$$

Actually it is written using an SO(2) transformation with the relation <sup>e</sup><sup>2</sup> <sup>þ</sup> <sup>α</sup><sup>2</sup> <sup>¼</sup> <sup>p</sup> ! � � � � � � 2 ,

$$
\begin{pmatrix} \epsilon \\ \alpha \end{pmatrix} = \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix} \begin{pmatrix} p\_1 \\ p\_2 \end{pmatrix}. \tag{41}
$$

We show the spectrum of the edge state depending on the boundary condition with the bulk spectrum in Figure 6. We remark that the edge state cannot be defined in the whole

<sup>4</sup> This is of course equivalent to the boundary condition discussed in Section 3.1.2.

momentum space due to the normalizability condition α p !� � <sup>&</sup>gt; 0. Such a bounded spectrum associated with the WSM edge state is called the Fermi arc, and the boundary condition parameter, a relative phase factor, parameterizes the direction of the arc. Accordingly the current similarly behaves as J1; J2; J<sup>3</sup> ð Þ∝ ð Þ cos θ; sin θ; 0 .

#### 3.3.2. Lattice system

Let us apply a similar analysis to the lattice model for 3d WSM. We consider the lattice model (13) with a boundary at n<sup>3</sup> ¼ 1, defined on the positive n<sup>3</sup> region, n<sup>3</sup> ≥ 1,

$$\mathcal{H}\_{\text{3d}}^{\text{lat}}\left(\overrightarrow{p},n\_{3}\right) = \begin{pmatrix} 0 & \Delta\left(\overrightarrow{p}\right)^{\*}\\ \Delta\left(\overrightarrow{p}\right) & 0 \end{pmatrix} - \frac{i}{2}\sigma\_{3}\left(\nabla\_{3} - \nabla\_{3}^{\dagger}\right) \tag{42}$$

with a complex parameter

$$
\Delta \left( \stackrel{\rightarrow}{p} \right) = \cos p\_1 - \cos p\_2 + c + i \sin p\_{2'} \tag{43}
$$

Dψ p !; n<sup>3</sup>

where we define α~ p

3.4. 2d topological insulator

the Hamiltonian (37), we obtain

3.4.1. Continuum system

out α~ p

� � <sup>¼</sup> 0 with <sup>D</sup> <sup>¼</sup>

! ð Þ�<sup>1</sup> �β p ! ð Þ

e p

α~ p

which has an analogous expression as (41) using SO(2) rotation

e α~ � �

Figure 7. The boundary condition dependence of the edge state spectrum e p

addition to the bulk spectrum (orange and blue), the parameter c is taken to be c ¼ 1.

!� � <sup>¼</sup> cos <sup>θ</sup>Re<sup>Δ</sup> <sup>p</sup>

!� � ¼ � sin <sup>θ</sup>Re<sup>Δ</sup> <sup>p</sup>

continuum model under the correspondence <sup>p</sup>1; <sup>p</sup>2; <sup>α</sup> � � \$ ð Þ ReΔ;ImΔ; <sup>α</sup><sup>~</sup> .

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

At this moment, it is obvious that the spectrum of the current lattice model is parallel with the

We show the spectrum of the edge state in Figure 7, in particular, its boundary condition dependence. Figure 8 shows constant energy slices of the spectrum. We can see the so-called Fermi arc, which connects two bulk Weyl points. As discussed for the continuum model, the boundary condition parameter plays a role as a rotation angle in the momentum space.

The 2d class A TI is given by the dimensional reduction of the 3d WSM. Replacing p<sup>2</sup> ! m in

� � ReΔ

!� � <sup>¼</sup> <sup>β</sup> <sup>p</sup>

!� � <sup>&</sup>gt; 0. The solution is then obtained as

iα~ p

0 B@ !� � � <sup>e</sup> <sup>p</sup>

Δ p

<sup>2</sup> . Since we consider the situation 0 < β p

!� � <sup>þ</sup> sin <sup>θ</sup>Im<sup>Δ</sup> <sup>p</sup>

!� � <sup>þ</sup> cos <sup>θ</sup>Im<sup>Δ</sup> <sup>p</sup>

ImΔ

!� � <sup>Δ</sup> <sup>p</sup>

!� � �iα<sup>~</sup> <sup>p</sup>

!� �<sup>∗</sup>

!� � � <sup>e</sup> <sup>p</sup>

!� �

!� � (46)

!� � (47)

� �: (48)

!� � for <sup>θ</sup> <sup>¼</sup> <sup>π</sup>4, <sup>π</sup>=3, <sup>5</sup>π=3, <sup>2</sup><sup>π</sup> (green), in

1

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

CA (45)

17

!� � <sup>&</sup>lt; 1, it turns

which is analogous to the model in the continuum Δ p !� � � <sup>p</sup><sup>1</sup> � ip2. We keep an explicit <sup>n</sup>3 dependence of the system to deal with the boundary condition. According to the discussion in Sections 3.1.1 and 3.1.2, we consider the edge state consistent with the boundary condition as

$$
\psi\left(\overrightarrow{p}, n\_3\right) = \beta\left(\overrightarrow{p}\right)^{n\_3 - 1} \begin{pmatrix} 1 \\ e^{i\theta} \end{pmatrix} \tag{44}
$$

where β p !� � is a real parameter, corresponding to the penetration depth, and the normalizability requires ∣β p !� �<sup>∣</sup> <sup>&</sup>lt; 1. In particular, we consider the situation 0 <sup>&</sup>lt; <sup>β</sup> <sup>p</sup> !� � <sup>&</sup>lt; 1 for the moment: The negative β p !� � solution is interpreted as a doubling counterpart of the positive one. The eigenvalue equation ℋlat 3d p !; n<sup>3</sup> � �<sup>ψ</sup> <sup>p</sup> !; n<sup>3</sup> � � <sup>¼</sup> <sup>e</sup> <sup>p</sup> !� �<sup>ψ</sup> <sup>p</sup> !; n<sup>3</sup> � � leads to

Figure 6. The boundary condition dependence of the edge state spectrum for θ ¼ 0, π=4, π=2, 3π=4, π with the bulk spectrum. The parameter θ plays a role as a rotation angle in the momentum space.

Analysis of Topological Material Surfaces http://dx.doi.org/10.5772/intechopen.74934 17

$$\mathcal{D}\psi\left(\overrightarrow{p},\eta\_{3}\right) = 0 \quad \text{with} \quad \mathcal{D} = \begin{pmatrix} i\ddot{\alpha}\left(\overrightarrow{p}\right) - \epsilon\left(\overrightarrow{p}\right) & \Delta\left(\overrightarrow{p}\right)^{\*} \\ \Delta\left(\overrightarrow{p}\right) & -i\ddot{\alpha}\left(\overrightarrow{p}\right) - \epsilon\left(\overrightarrow{p}\right) \end{pmatrix} \tag{45}$$

where we define α~ p !� � <sup>¼</sup> <sup>β</sup> <sup>p</sup> ! ð Þ�<sup>1</sup> �β p ! ð Þ <sup>2</sup> . Since we consider the situation 0 < β p !� � <sup>&</sup>lt; 1, it turns out α~ p !� � <sup>&</sup>gt; 0. The solution is then obtained as

$$\hat{\mathbf{e}}\left(\vec{p}\right) = \cos\theta \text{Re}\Delta\left(\vec{p}\right) + \sin\theta \text{Im}\Delta\left(\vec{p}\right) \tag{46}$$

$$\tilde{\alpha}\left(\vec{p}\right) = -\sin\theta \text{Re}\Delta\left(\vec{p}\right) + \cos\theta \text{Im}\Delta\left(\vec{p}\right) \tag{47}$$

which has an analogous expression as (41) using SO(2) rotation

$$
\begin{pmatrix} \epsilon \\ \check{\alpha} \end{pmatrix} = \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix} \begin{pmatrix} \text{Re}\Delta \\ \text{Im}\Delta \end{pmatrix}. \tag{48}
$$

At this moment, it is obvious that the spectrum of the current lattice model is parallel with the continuum model under the correspondence <sup>p</sup>1; <sup>p</sup>2; <sup>α</sup> � � \$ ð Þ ReΔ;ImΔ; <sup>α</sup><sup>~</sup> .

We show the spectrum of the edge state in Figure 7, in particular, its boundary condition dependence. Figure 8 shows constant energy slices of the spectrum. We can see the so-called Fermi arc, which connects two bulk Weyl points. As discussed for the continuum model, the boundary condition parameter plays a role as a rotation angle in the momentum space.

#### 3.4. 2d topological insulator

#### 3.4.1. Continuum system

momentum space due to the normalizability condition α p

(13) with a boundary at n<sup>3</sup> ¼ 1, defined on the positive n<sup>3</sup> region, n<sup>3</sup> ≥ 1,

0 B@

¼

ψ p !; n<sup>3</sup> � �

Δ p !� �

which is analogous to the model in the continuum Δ p

ψ p !; n<sup>3</sup> � �

spectrum. The parameter θ plays a role as a rotation angle in the momentum space.

current similarly behaves as J1; J2; J<sup>3</sup> ð Þ∝ ð Þ cos θ; sin θ; 0 .

ℋlat 3d p !; n<sup>3</sup> � �

3.3.2. Lattice system

16 Heterojunctions and Nanostructures

with a complex parameter

where β p

requires ∣β p

negative β p

!� �

value equation ℋlat

!� �

!� �

3d p !; n<sup>3</sup> � � !� �

associated with the WSM edge state is called the Fermi arc, and the boundary condition parameter, a relative phase factor, parameterizes the direction of the arc. Accordingly the

Let us apply a similar analysis to the lattice model for 3d WSM. We consider the lattice model

0 Δ p

dependence of the system to deal with the boundary condition. According to the discussion in Sections 3.1.1 and 3.1.2, we consider the edge state consistent with the boundary condition as

¼ β p

Figure 6. The boundary condition dependence of the edge state spectrum for θ ¼ 0, π=4, π=2, 3π=4, π with the bulk

∣ < 1. In particular, we consider the situation 0 < β p

¼ e p !� � ψ p !; n<sup>3</sup> � �

Δ p !� � !� �<sup>∗</sup>

1 CA � <sup>i</sup> 2

!� �

eiθ � �

!� �<sup>n</sup>3�<sup>1</sup> 1

is a real parameter, corresponding to the penetration depth, and the normalizability

solution is interpreted as a doubling counterpart of the positive one. The eigen-

leads to

<sup>σ</sup><sup>3</sup> <sup>∇</sup><sup>3</sup> � <sup>∇</sup>†

¼ cos p<sup>1</sup> � cos p<sup>2</sup> þ c þ i sin p2, (43)

!� �

3

0

> 0. Such a bounded spectrum

� � (42)

� p<sup>1</sup> � ip2. We keep an explicit n3-

< 1 for the moment: The

(44)

The 2d class A TI is given by the dimensional reduction of the 3d WSM. Replacing p<sup>2</sup> ! m in the Hamiltonian (37), we obtain

Figure 7. The boundary condition dependence of the edge state spectrum e p !� � for <sup>θ</sup> <sup>¼</sup> <sup>π</sup>4, <sup>π</sup>=3, <sup>5</sup>π=3, <sup>2</sup><sup>π</sup> (green), in addition to the bulk spectrum (orange and blue), the parameter c is taken to be c ¼ 1.

ψ p1; x<sup>3</sup> � � <sup>¼</sup>

� � <sup>¼</sup> <sup>p</sup><sup>1</sup> cos <sup>θ</sup> <sup>þ</sup> <sup>m</sup> sin <sup>θ</sup>, <sup>α</sup> <sup>p</sup><sup>1</sup>

where the inverse penetration depth α p<sup>1</sup>

e p<sup>1</sup>

we obtain the solution

edge mode.

3.4.2. Lattice system

this case, we have two options,

<sup>p</sup><sup>1</sup> ! <sup>m</sup><sup>1</sup> : <sup>H</sup>ð Þ<sup>1</sup>

<sup>p</sup><sup>2</sup> ! <sup>m</sup><sup>2</sup> : <sup>H</sup>ð Þ<sup>2</sup>

e<sup>1</sup> p2; p<sup>3</sup> � � ¼ �

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

and the corresponding spectra are given as follows:

q

q

ffiffiffiffiffiffiffiffiffiffiffiffi α p<sup>1</sup> <sup>q</sup> � �<sup>e</sup>

Figure 9 shows the boundary condition dependence of the edge state spectrum. Replacement p<sup>2</sup> ! m corresponds to take a section at p<sup>2</sup> ¼ m, and the 3d Fermi arc is reduced to the 2d chiral

Similarly, we consider the dimensional reduction of the lattice Hamiltonian of 3d WSM (13). In

cos <sup>m</sup><sup>1</sup> � cos <sup>p</sup><sup>2</sup> <sup>þ</sup> <sup>c</sup> � �<sup>2</sup> <sup>þ</sup> sin <sup>p</sup><sup>2</sup>

cos <sup>p</sup><sup>1</sup> � cos <sup>m</sup><sup>2</sup> <sup>þ</sup> <sup>c</sup> � �<sup>2</sup> <sup>þ</sup> ð Þ sin <sup>m</sup><sup>2</sup>

Figure 9. The boundary condition dependence of the edge state for the 2d system. The dimensional reduction corre-

sponds to taking a section at p<sup>2</sup> ¼ m. The horizontal axis in the bottom panel shows the momentum p1.

�<sup>α</sup> <sup>p</sup>ð Þ<sup>1</sup> <sup>x</sup><sup>3</sup> <sup>1</sup>

eiθ

2d <sup>¼</sup> <sup>σ</sup><sup>1</sup> cos <sup>m</sup><sup>1</sup> � cos <sup>p</sup><sup>2</sup> <sup>þ</sup> <sup>c</sup> � � <sup>þ</sup> <sup>σ</sup><sup>2</sup> sin <sup>p</sup><sup>2</sup> <sup>þ</sup> <sup>σ</sup><sup>3</sup> sin <sup>p</sup><sup>3</sup> (52)

2d <sup>¼</sup> <sup>σ</sup><sup>1</sup> cos <sup>p</sup><sup>1</sup> � cos <sup>m</sup><sup>2</sup> <sup>þ</sup> <sup>c</sup> � � <sup>þ</sup> <sup>σ</sup><sup>2</sup> sin <sup>m</sup><sup>2</sup> <sup>þ</sup> <sup>σ</sup><sup>3</sup> sin <sup>p</sup><sup>3</sup> (53)

� �<sup>2</sup> <sup>þ</sup> sin <sup>p</sup><sup>3</sup>

� �<sup>2</sup>

<sup>2</sup> <sup>þ</sup> sin <sup>p</sup><sup>3</sup> � �<sup>2</sup> (54)

(55)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

� � has to be positive due to the normalizability. Then

� � ¼ �p<sup>1</sup> sin <sup>θ</sup> <sup>þ</sup> <sup>m</sup> cos <sup>θ</sup>: (51)

� � (50)

Analysis of Topological Material Surfaces http://dx.doi.org/10.5772/intechopen.74934 19

Figure 8. The Fermi arc at (a) zero energy ε p ! <sup>¼</sup> 0 and (b) finite energy <sup>E</sup> <sup>p</sup> ! = 0.3 with <sup>θ</sup> <sup>¼</sup> <sup>π</sup>=5, <sup>3</sup>π=5, <sup>π</sup>, <sup>7</sup>π=5, <sup>9</sup>π=5. The red dot and shaded region show the bulk contribution. The last panels show Fermi arcs with various values of the parameter θ.

$$\mathcal{H}\_{2d}(p\_1, \mathbf{x}\_3) = p\_1 \sigma\_1 + m \sigma\_2 - i \sigma\_3 \frac{\partial}{\partial \mathbf{x}\_3}. \tag{49}$$

After this dimensional reduction, we can apply totally the same analysis to this model discussed in Section 3.3: we consider the localized edge state satisfying the boundary condition

Analysis of Topological Material Surfaces http://dx.doi.org/10.5772/intechopen.74934 19

$$
\psi(p\_1, \mathbf{x}\_3) = \sqrt{\alpha(p\_1)} e^{-\alpha(p\_1)\mathbf{x}\_3} \begin{pmatrix} 1 \\ e^{i\theta} \end{pmatrix} \tag{50}
$$

where the inverse penetration depth α p<sup>1</sup> � � has to be positive due to the normalizability. Then we obtain the solution

$$\epsilon(p\_1) = p\_1 \cos \theta + m \sin \theta, \qquad a(p\_1) = -p\_1 \sin \theta + m \cos \theta. \tag{51}$$

Figure 9 shows the boundary condition dependence of the edge state spectrum. Replacement p<sup>2</sup> ! m corresponds to take a section at p<sup>2</sup> ¼ m, and the 3d Fermi arc is reduced to the 2d chiral edge mode.

#### 3.4.2. Lattice system

ℋ2d p1; x<sup>3</sup>

!

Figure 8. The Fermi arc at (a) zero energy ε p

18 Heterojunctions and Nanostructures

parameter θ.

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

The red dot and shaded region show the bulk contribution. The last panels show Fermi arcs with various values of the

¼ 0 and (b) finite energy E p

After this dimensional reduction, we can apply totally the same analysis to this model discussed in Section 3.3: we consider the localized edge state satisfying the boundary condition

∂ ∂x<sup>3</sup>

!

: (49)

= 0.3 with θ ¼ π=5, 3π=5, π, 7π=5, 9π=5.

Similarly, we consider the dimensional reduction of the lattice Hamiltonian of 3d WSM (13). In this case, we have two options,

$$p\_1 \to m\_1: \quad \mathcal{H}\_{2d}^{(1)} = \sigma\_1 \left(\cos m\_1 - \cos p\_2 + c\right) + \sigma\_2 \sin p\_2 + \sigma\_3 \sin p\_3 \tag{52}$$

$$p\_2 \to \mathfrak{m}\_2 : \quad \mathcal{H}\_{2d}^{(2)} = \sigma\_1 \left(\cos p\_1 - \cos m\_2 + c\right) + \sigma\_2 \sin m\_2 + \sigma\_3 \sin p\_3 \tag{53}$$

and the corresponding spectra are given as follows:

$$\epsilon\_1(p\_2, p\_3) = \pm \sqrt{\left(\cos m\_1 - \cos p\_2 + c\right)^2 + \left(\sin p\_2\right)^2 + \left(\sin p\_3\right)^2} \tag{54}$$

$$\epsilon\_2(p\_1, p\_3) = \pm \sqrt{\left(\cos p\_1 - \cos m\_2 + c\right)^2 + \left(\sin m\_2\right)^2 + \left(\sin p\_3\right)^2} \tag{55}$$

Figure 9. The boundary condition dependence of the edge state for the 2d system. The dimensional reduction corresponds to taking a section at p<sup>2</sup> ¼ m. The horizontal axis in the bottom panel shows the momentum p1.

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 spectrum survives for the whole region of the parameter θ.

4. Edge-of-edge state

[4]. See also related works [20–24].

the zero current condition (34), is now given by

since the current operator is given by <sup>J</sup><sup>μ</sup> <sup>¼</sup> <sup>ψ</sup>†

P4ψ � � � x4¼0

<sup>M</sup><sup>5</sup> <sup>¼</sup> <sup>0</sup> <sup>U</sup>†

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

<sup>4</sup> and also

where the matrix M4, <sup>5</sup> obeys M†

given by

to e<sup>2</sup> <sup>5</sup> <sup>þ</sup> <sup>α</sup><sup>2</sup>

ψ p<sup>1</sup>;2;3;<sup>5</sup>; x<sup>4</sup> � � <sup>¼</sup> <sup>e</sup>

> <sup>5</sup> ¼ p ! � � � � � � 2 <sup>þ</sup> <sup>p</sup><sup>2</sup>

ψ† γ4ψ � � � x4¼0

¼ 0 P5ψ

5 U<sup>5</sup> 0 !

1 þ U<sup>4</sup>

� �<sup>χ</sup> <sup>p</sup><sup>1</sup>;2;3;<sup>4</sup>

� � � x5¼0

, M<sup>4</sup> ¼ � <sup>1</sup>

4.1. 5d Weyl semimetal

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

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

<sup>¼</sup> <sup>0</sup> <sup>ψ</sup>†

γ5ψ � � � x5¼0

<sup>a</sup>γ<sup>a</sup> þ γaMa ¼ 0 for a ¼ 4, 5. Explicitly we have

<sup>U</sup><sup>4</sup> <sup>þ</sup> <sup>U</sup>†

�U<sup>4</sup> <sup>þ</sup> <sup>U</sup>†

� � <sup>¼</sup> <sup>e</sup>

2

� �, <sup>ψ</sup> <sup>p</sup><sup>1</sup>;2;3;<sup>4</sup>; <sup>x</sup><sup>5</sup>

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

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

γμψ. These conditions are rephrased as

<sup>4</sup> <sup>U</sup><sup>4</sup> � <sup>U</sup>†

!

<sup>4</sup> �U<sup>4</sup> � <sup>U</sup>†

4

�α5ð Þ<sup>p</sup> <sup>x</sup><sup>5</sup> <sup>1</sup>

U<sup>5</sup>

� �<sup>ξ</sup> <sup>p</sup><sup>1</sup>;2;3;<sup>4</sup>

� �:

(59)

4

<sup>¼</sup> 0 with <sup>P</sup>4,<sup>5</sup> <sup>¼</sup> <sup>1</sup> � <sup>M</sup>4,<sup>5</sup>

¼ 0, (56)

Analysis of Topological Material Surfaces http://dx.doi.org/10.5772/intechopen.74934 21

<sup>2</sup> (57)

, (58)

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> )–(f<sup>0</sup> ) show 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> and p1, respectively. The blue region is the bulk, and the orange line is the edge state spectrum.
