Open access peer-reviewed chapter

Analysis of Topological Material Surfaces

By Taro Kimura

Submitted: October 17th 2017Reviewed: February 5th 2018Published: March 23rd 2018

DOI: 10.5772/intechopen.74934

Downloaded: 1207

Abstract

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

Keywords

  • topological insulator
  • Weyl semimetal
  • boundary condition
  • lattice fermion

1. Introduction

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

A universal feature of topological phases is the bulk/edge correspondence [2]: once the bulk wave function has a topologically nontrivial configuration; there exists a gapless edge state localized at the boundary. Such an edge state is topologically protected, and thus is robust against any perturbations as long as respecting symmetry of the system. In practice, the edge state plays a significant role in detection of topological phases since it can be directly observed in experiments using angle-resolved photo-emission spectroscopy (ARPES). Therefore the boundary condition dependence of the edge state is expected to provide experimentally useful predictions.

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

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,

H2d=iσ1x1iσ2x2+mσ3E1

where Pauli matrices are defined σ1=0110, σ2=0ii0, σ3=1001. 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 Ψpx=eipxψpx, and the corresponding Hamiltonian acting on ψpx, simply denoted by ψbelow, is given by

H2dp=p1σ1+p2σ2+mσ3=mΔpΔpm.E2

We obtain two eigenvalues ϵ±p=±p2+m2. The eigenstate, parametrized by a complex number ξC, is accordingly obtained as

ψ=11+ξ21ξwithξ=Δpϵ+m=ϵmΔp.E3

We remark that the parameter ξbecomes singular ξat p=0. At this point, we have to reparametrize the eigenstate with ξ1instead of ξ. This means that ξis not a global, but just a local coordinate, and the eigenstate is given by an element of CP1in 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 eigenstate2

A=ψidψ=Imξ1+ξ2,F=dA=idξ1+ξ22E4

where we use the differential form notation in the momentum space, d=/pidpi, namely the Berry connection is one-form A=A1dp1+A2dp2, and the curvature is two-form F=F12dp1dp2. Under the momentum-dependent transformation, ξepξ(not an overall phase rotation of the eigenstate ψ), the connection behaves as AA/1+ξ2. 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 S1fibration of CP1=S3/S1, 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,

ν2d=12πdp1dp2F12=12sgnmE5

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 it would be a topological invariant, but with a discontinuous point at m=0, which is the gapless (sign changing) point m=0. Typically the topological number takes an integer value, but ν2ddoes 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,

H3dp=p1σ1+p2σ2+p3σ3=p3ΔpΔpp3E6

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:

ν3d=14πS2dSB=12πdp1dp2F12p3>012πdp1dp2F12p3<0=1E7

where the “magnetic field” is defined as Bi=12ϵijkFjk, namely B=×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 ν2dis 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 well-defined 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 ν3d0, cannot be continuously deformed to a trivial system ν3d=0by definition. This explains the topological stability of the WSM. If we want to obtain a topologically trivial situation, we need pair-annihilation of the Weyl points having opposite topological invariants: ν3d=+1+1=0. See Figure 2.

Figure 1.

Monopole at Weyl point in the momentum space. The monopole charge is an integer-valued topological invariant ν 3 d . The 2d invariant ν 2 d is obtained at a constant p 3 → m plane, which covers either upper or lower half of the fluxes, so that ν 2 d is given by a half of ν 3 d .

Figure 2.

The topological invariant distinguishes topologically different situations. The green and red spheres show the monopole with topological charge ν 3 d = + 1 and ν 3 d = − 1 , respectively. We need pair annihilation to eliminate the monopoles.

2.2. Edge state

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 mx1=ϑx1with a positive slope ϑ>0, giving rise to the sign flip at x1=0, so that the boundary is the plane x1=0[12]. Then the Hamiltonian takes a form of

2dx1p2=mx1σ1iσ2x1+p2σ3=p22ϑâ2ϑâp2E8

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 p2. The off-diagonal element is given by an operator â=ϑx1+/x1/2ϑ, â=ϑx1/x1/2ϑ, obeying the commutation relation ââ=1, so that it is interpreted as a creation/annihilation operator. Then the energy spectrum is given by ϵnp2=±p22+2ϑnfor n1(gapped), while the zero mode dispersion is given by ϵ0p2=p2(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 with a spatial profile, which is known as the domain-wall fermion. See, for example, [13] for more details.

Figure 3.

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

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

H2dlat=i2σ111i2σ222+σ3m+2121+1122+2E9

where we define the difference operator 1,2ψn=ψn+e1,2ψnwith the unit vector e1,2in n1and n2-direction. Then the corresponding Bloch Hamiltonian is given by

H2dlatp=σ1sinp1+σ2sinp2+σ3m+2cosp1cosp2.E10

Periodicity p1,2p1,2+2πreflects the lattice structure: The momentum is restricted to the Brillouin zone p1,202π. The spectrum is given by ϵp=±sinp12+sinp22+m+2cosp1cosp22, which has four gapless points p=00at m=0, p=π0and 0πat m=2, p=ππat m=4. Expanding the momentum around p=00, one can see the effective Hamiltonian (2) is obtained. If expanding the momentum around p=π0instead, we similarly obtain the Hamiltonian (2), but we have to replace p1p1.

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]:

ν2d=12πBZdp1dp2F12=0m>0orm<412<m<0+14<m<2E11

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: p1p2m=0,0,0, π02, 0π2, ππ4. Each gapless point plays basically the same role as that discussed in the continuum model with the monopole charge +1or 1. Thus we immediately obtain ν2d=12+12+1=0for m>0, 1212+1=1for 2<m<0, 121+2+1=1for 4<m<2, and 121+21=0for m<4. See Figure 4.

Figure 4.

The mass dependence of the 2d topological invariant ν 2 d for the lattice model (10). Topology change occurs at the gapless points m = − 4 , − 2 , 0 . Change of the invariant corresponds to the monopole charge + 1 , − 2 , + 1 associated with each gapless point.

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

H3dlat=12σ11+122+2ci2σ222i2σ333.E12

The corresponding Bloch Hamiltonian is given by

H3dlatp=σ1cosp1cosp2+c+σ2sinp2+σ3sinp3,E13

and the spectrum yields ϵp=±cosp1cosp2+c2+sinp22+sinp32. The parameter ctunes the gapless Weyl points as follows

p1p2p3=cos11c,0,0&cos11+c0π0<c<2cos11cπ0&cos11+cππ2<c<0n/ac>2E14

The band spectrum is shown in Figure 5 at p3=0and c=1. We can see two Weyl points at p1p2=±π/20. We will study the boundary condition of this model in Section 3.3.

Figure 5.

The energy spectrum of the lattice WSM model (13) with p 3 = 0 and c = 1 . There exist two gapless Weyl points at p 1 p 2 = ± π / 2 0 . The parameter c characterizes the distance between the Weyl points.

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 higher-dimensional generalization of (2),

H4dp=pγ+mγ5=mΔpΔpmwithΔp=pσ=p4+ip3p2+ip1p2+ip1p4ip3E15

where we use the gamma matrices defined as γk=0iσkσk0for k=1,2,3, γ4=0110, γ5=1001, and the off-diagonal element is given by Δp=pσHwith σ=iσ1. We remark that this Hamiltonian is a 4×4matrix, such that each element shows a 2×2matrix. The spectrum is simply obtained as ϵp=±p2+m2, and each state is doubly degenerated. We have a similar eigenvector to (3) as follows,

ψ=11+ξ21ξwithξ=Δε+m=εmΔH.E16

Currently each component shows a 2×2matrix, which takes a value in quaternion H, so that the eigenvector is a 2×4matrix due to the degeneracy, namely ψ=ψ1ψ2, where each ψ1,2is a four vector. For a degenerated system, we can define non-Abelian analog of the Berry connection Aab=ψaidψbfor a,b=1,2. In this case, we obtain an SU(2) valued Berry connection, which is a consequence of S3fibration of HP1=S7/S3. 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,

ν4d=18π2TrFF=12signm.E17

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 mp5,

H5dp=pγ+p5γ5,E18

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

ν5d=18π2S4TrFF=1,E19

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 ν4dis equivalent to the hemisphere integral of S4, which provides a half of the 5d invariant.

3. Boundary condition analysis

3.1. Operator formalism

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

D̂=x.E20

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 xxLxR, we obtain

ϕ|D̂ψ=xLxRdxϕxxψx=ϕxψxxLxR+xLxRdxxϕxψx=ϕxψxxLxR+D̂ϕ|ψ.E21

The Hermitian condition ϕ|D̂ψ=D̂|ϕψimplies that the surface term should vanish

ϕxσψxxLxR=ϕxRσψxRϕxLσψxL=0,E22

which gives rise to two possibilities:

  1. Periodic boundary condition: ϕxR=ϕxLand ψxR=ψxL

  2. Open boundary condition: ϕxRσψxR=0and ϕxLσψxL=0

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

xL,R=xL,R=0whereP=1M2E23

with the matrix Msatisfying Mσ+σM=0, since ψ=, ϕ=at the boundary, then

ϕσψ=ϕσMψ=ϕMσψ=ϕσψϕσψ=0.E24

In general we can apply different matrices ML,Rfor xLand xR, but here we assume ML,R=Mfor 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 n1N. We introduce an analogous difference operator to (20) as

D̂lat=E25

where ψn=ψn+1ψnand ψn=ψn1ψn. In this case, the inner product ϕ|D̂latψis given by

n=1Nϕnψn=n=1Nϕnψn+ϕ0ψ1ϕNψN+1E26

where ϕ0and ψN+1are considered as auxiliary fields. The self-conjugacy condition ϕ|D̂latψ=D̂latϕ|ψrequires that the surface term should vanish:

ϕ0σψ1ϕNσψN+1=0.E27

The periodic boundary condition ϕn+N=ϕn, ψn+N=ψnis 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

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

M=σ1cosθ+σ2sinθ=0ee0,E28

obeying M=Mand M2=1with two eigenvalues ±1. It is also expressed as M=σ1eiθσ3=σ2eiθπ2σ3. Thus the operator P=Pturns out to be a projection operator P2=Phaving eigenvalues 1,0with the corresponding eigenvectors

P1e=1eandP1e=0.E29

We remark σ3P=Pσ3, σ3P=Pσ3where P=1Pobeying PP=PP=0. Thus we obtain a one-parameter family of the solution to the boundary condition (23),

ψ|xL,R1e.E30

Since Pψ=ψ, the current in x3-direction vanishes at the boundary, J3=ψσ3ψ=ψPσ3Pψ=ψσ3PPψ=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 Mof the Lagrangian defines the action

S=MLϕϕ.E31

If the system has a continuous local symmetry, the action may be invariant under the infinitesimal deviation of the field ϕϕ+ϵxφ3:

0=δS=M∂ℒϕμ∂ℒμϕϵxφMϵxμ∂ℒμϕφ+Mμ∂ℒμϕϵxφ.E32

The first term vanishes due to the Euler-Lagrange equation of motion for the bulk, ∂ℒϕμ∂ℒμϕ=0. The vanishing condition for the second term implies the current Jμ=∂ℒμϕφ, satisfying the conservation law μJμ=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=MϵxnJE33

where nis the normal vector defined as MμVμ=MnVwith an arbitrary vector field Vμand 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

nJ|M=0.E34

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

SB=MLBϕE35

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

nJ+LBϕφM=0.E36

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.

3.3. 3d Weyl semimetal

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

3dpx3=p1σ1+p2σ2iσ3x3.E37

We put a boundary only at x3=0for simplicity, so that the system is defined on a positive domain x3>0. In this case, since the current operator is defined as J=ψσψ, the boundary condition, corresponding to the zero current condition (36), that we impose is4

ψσ3ψ|x3=0=0.E38

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

ψpx3=αpeαpx31eE39

which is normalized as 0dx3ψψ=1, and the normalizability requires αp>0. This eigenstate is localized on the boundary x3=0and exponentially decay into the bulk x3>0due to the factor eαpx3, where the parameter αpplays a role as the inverse penetration length. In this case, the exponential factor eαpx3is responsible for the x3-direction dependence, instead of the plane wave factor eip3x3used 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 p3, 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

ϵp=p1cosθ+p2sinθ,αp=p1sinθ+p2cosθ.E40

Actually it is written using an SO(2) transformation with the relation ϵ2+α2=p2,

ϵα=cosθsinθsinθcosθp1p2.E41

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 momentum space due to the normalizability condition αp>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 J1J2J3cosθsinθ0.

Figure 6.

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

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 n3=1, defined on the positive n3region, n31,

3dlatpn3=0ΔpΔp0i2σ333E42

with a complex parameter

Δp=cosp1cosp2+c+isinp2,E43

which is analogous to the model in the continuum Δpp1±ip2. We keep an explicit n3-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

ψpn3=βpn311eE44

where βpis a real parameter, corresponding to the penetration depth, and the normalizability requires βp<1. In particular, we consider the situation 0<βp<1for the moment: The negative βpsolution is interpreted as a doubling counterpart of the positive one. The eigenvalue equation 3dlatpn3ψpn3=ϵpψpn3leads to

Dψpn3=0withD=iα˜pϵpΔpΔpiα˜pϵpE45

where we define α˜p=βp1βp2. Since we consider the situation 0<βp<1, it turns out α˜p>0. The solution is then obtained as

ϵp=cosθReΔp+sinθImΔpE46
α˜p=sinθReΔp+cosθImΔpE47

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

ϵα˜=cosθsinθsinθcosθReΔImΔ.E48

At this moment, it is obvious that the spectrum of the current lattice model is parallel with the continuum model under the correspondence p1p2αReΔImΔα˜.

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.

Figure 7.

The boundary condition dependence of the edge state spectrum ϵ p → for θ = π 4 , π / 3 , 5 π / 3 , 2 π (green), in addition to the bulk spectrum (orange and blue), the parameter c is taken to be c = 1 .

Figure 8.

The Fermi arc at (a) zero energy ε p → = 0 and (b) finite energy ɛ p → = 0.3 with θ = π / 5 , 3 π / 5 , π , 7 π / 5 , 9 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 .

3.4. 2d topological insulator

3.4.1. Continuum system

The 2d class A TI is given by the dimensional reduction of the 3d WSM. Replacing p2min the Hamiltonian (37), we obtain

2dp1x3=p1σ1+mσ2iσ3x3.E49

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

ψp1x3=αp1eαp1x31eE50

where the inverse penetration depth αp1has to be positive due to the normalizability. Then we obtain the solution

ϵp1=p1cosθ+msinθ,αp1=p1sinθ+mcosθ.E51

Figure 9 shows the boundary condition dependence of the edge state spectrum. Replacement p2mcorresponds to take a section at p2=m, and the 3d Fermi arc is reduced to the 2d chiral edge mode.

Figure 9.

The boundary condition dependence of the edge state for the 2d system. The dimensional reduction corresponds to taking a section at p 2 = m . The horizontal axis in the bottom panel shows the momentum p 1 .

3.4.2. Lattice system

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

p1m1:H2d1=σ1cosm1cosp2+c+σ2sinp2+σ3sinp3E52
p2m2:H2d2=σ1cosp1cosm2+c+σ2sinm2+σ3sinp3E53

and the corresponding spectra are given as follows:

ϵ1p2p3=±cosm1cosp2+c2+sinp22+sinp32E54
ϵ2p1p3=±cosp1cosm2+c2+sinm22+sinp32E55

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 p1m1, which is equivalent to topologically trivial state. Actually the edge state is almost embedded, and indistinguishable with the bulk spectrum, in particular, for θ=5π/7, 9π/7. On the other hand, we obtain a single chiral edge state from the reduction p2m2, indicating topologically nontrivial state. We can see an edge state spectrum survives for the whole region of the parameter θ.

Figure 10.

The boundary condition dependence of the 2d lattice system (52) and (53) with c = 1 . (a)–(f) and (a′)–(f′) show the spectra obtained from the reduction p 1 → m 1 = π / 2 + 0.5 and p 2 → m 2 = 0.2 . The horizontal axes are the momenta p 2 and p 1 , respectively. The blue region is the bulk, and the orange line is the edge state spectrum.

4. Edge-of-edge state

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, 21, 22, 23, 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 x4=0and x5=0. The boundary condition, namely the zero current condition (34), is now given by

ψγ4ψ|x4=0=0ψγ5ψ|x5=0=0,E56

since the current operator is given by Jμ=ψγμψ. These conditions are rephrased as

P4ψ|x4=0=0P5ψ|x5=0=0withP4,5=1M4,52E57

where the matrix M4,5obeys Maγa+γaMa=0for a=4,5. Explicitly we have

M5=0U5U50,M4=12U4+U4U4U4U4+U4U4U4,E58

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

ψp1,2,3,5x4=eα4px41U41+U4χp1,2,3,4,ψp1,2,3,4x5=eα5px51U5ξp1,2,3,4.E59

In particular, the edge state localized at x5=0is apparently similar to the 3d case (39), just replacing the phase factor eU(1) with U5U(2). The eigenvalue equation H5dψ=ɛψleads to ϵ52+α52=p2+p42and also

iα5ϵ5+p4iσpU5ξ=0.E60

Decomposing U5=eiθ5V5with eiθ5U(1) and V5SU2, we consider the SU(2) transformation p4iσpV5=p4iσp. Then we have

α5sinθ5ϵ5cosθ5+p4ξ=0,E61
α5cosθ5+ϵ5sinθ5σpξ=0.E62

Diagonalizing σp, which is equivalent to the 3d Hamiltonian (6), as σpξ±=±p2ξ±, we obtain the spectrum and the inverse penetration depth as follows:

ϵ5p=p4cosθ5±p2sinθ5,α5p=p4sinθ5±p2cosθ5,E63

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

ϵ5α5=cosθ5sinθ5sinθ5cosθ5p4±p2.E64

We can solve the boundary condition and obtain the spectrum for the boundary at x4=0in a similar way.

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

P4ψ|x4,5=0=P5ψ|x4,5=0=0.E65

A solution to this condition is given by

ψp1,2,3x4x5=eα4px4α5px51U41+U4χpE66

with U51U41+U4χp=0, which is covariant under U(2) transformation U4U5χWU4WWU5Wwith WU2. To have a nontrivial solution, they should obey U51U41+U4=0. For example, a simple choice is U4U5=σ3σ2, and the corresponding solution is χT=1i. Then we obtain the spectrum of the edge-of-edge state ϵp=p1, α4p=p3, α5p=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 p4p5m0as shown in Figure 11, then we obtain the 3d chiral (class AIII) TI

H3dAIIIp=pγ+mγ4,E67

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 x 1 -direction.

where the gamma matrices are chosen as γ=τ2σ, γ4=τ11, γ5=τ31, and Pauli matrices σsand τsact on the spin space and the sublattice space AB, respectively. This Hamiltonian has a chiral symmetry with respect to the sublattice structure H3dAIIIγ5=0. We can apply a similar analysis as before. The edge-of-edge state is in this case given by

ψp1x2x3=eα2p1x2α3p1x31+iσ3U3iσ31iσ3U3ξp1E68

with the compatibility condition

det1+U2iσ2+iσ3U3Uiσ1U3+iσ1iσ2U3U2iσ3U2U3=0E69

where U2,3U(2) parameterize the boundary condition. We consider the following choice satisfying the compatibility condition U2=σ2cosϕ+isinϕ, U3=icosϕσ3sinϕ. Then we obtain the spectra of the edge state localized at x2=0and x3, and the edge-of-edge state localized at their intersection

ϵ2p1p3=±p12+p32+mcosϕ2,E70
ϵ3p1p2=±p12+p22+msinϕ2,E71
ϵeoep1=p1.E72

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 x2=0and x3=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.

Acknowledgments

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).

Notes

  • See also related works [5, 6, 7] for the boundary condition analysis of topological materials.
  • See a textbook on this topic, e.g., [10] for more details.
  • This is an assumption. In general, the action itself is not invariant under the deviation.
  • This is of course equivalent to the boundary condition discussed in Section 3.1.2.

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Taro Kimura (March 23rd 2018). Analysis of Topological Material Surfaces, Heterojunctions and Nanostructures, Vasilios N. Stavrou, IntechOpen, DOI: 10.5772/intechopen.74934. Available from:

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