Topological charges of three-dimensional Dirac semimetals with rotation symmetry

Topological charges of three-dimensional Dirac semimetals with rotation symmetry

Bohm-Jung Yang RIKEN, Center for Emergent Matter Science, Wako, Saitama, 351-0198, Japan Department of Physics and Astronomy, Seoul National University, Seoul 151-747, Korea Center for Correlated Electron Systems, Institute for Basic Science (IBS), Seoul 151-747, Korea    Takahiro Morimoto RIKEN, Center for Emergent Matter Science, Wako, Saitama, 351-0198, Japan Condensed Matter Theory Laboratory, RIKEN, Wako, Saitama, 351-0198, Japan    Akira Furusaki RIKEN, Center for Emergent Matter Science, Wako, Saitama, 351-0198, Japan Condensed Matter Theory Laboratory, RIKEN, Wako, Saitama, 351-0198, Japan
July 7, 2019
Abstract

In general, the stability of a band crossing point indicates the presence of a quantized topological number associated with it. In particular, the recent discovery of three-dimensional Dirac semimetals in NaBi and CdAs demonstrates that a Dirac point with four-fold degeneracy can be stable as long as certain crystalline symmetries are supplemented in addition to the time-reversal and inversion symmetries. However, the topological charges associated with NaBi and CdAs are not clarified yet. In this work, we identify the topological charge of three-dimensional Dirac points. It is found that although the simultaneous presence of the time-reversal and inversion symmetries forces the net chiral charge to vanish, a Dirac point can carry another quantized topological charge when an additional rotation symmetry is considered. Two different classes of Dirac semimetals are identified depending on the nature of the rotation symmetries. First, the conventional symmorphic rotational symmetry which commutes with the inversion gives rise to the class I Dirac semimetals having a pair of Dirac points on the rotation axes. Since the topological charges of each pair of Dirac points have the opposite sign, a pair-creation or a pair-annihilation is required to change the number of Dirac points in the momentum space. On the other hand, the class II Dirac semimetals possess a single isolated Dirac point at a time-reversal invariant momentum, which is protected by a screw rotation. The non-symmorphic nature of screw rotations allows the anti-commutation relation between the rotation and inversion symmetries, which enables to circumvent the doubling of the number of Dirac points and create a single Dirac point at the Brillouin zone boundary.

pacs:
75.10.Pq,75.10.Jm,64.70.Tg

I Introduction

After the discovery of graphene, a class of materials, dubbed Dirac semimetals, have come to the fore of condensed matter research. In general, a Dirac semimetal has several Fermi points around which pseudo-relativistic linear dispersion relation is realized. This pseudo-relativistic energy dispersion forces the density of states on the Fermi level to vanish without opening of an energy gap, which is the unique property of Dirac semimetals distinct from ordinary metals or insulators Graphene (). In particular, the recent theoretical prediction Na3Bi_DFT (); Cd3As2_DFT () and the experimental confirmation Na3Bi_Exp1 (); Na3Bi_Exp2 (); Na3Bi_Exp3 (); Cd3As2_Exp1 (); Cd3As2_Exp2 (); Cd3As2_Exp3 (); Cd3As2_Exp4 () of three-dimensional (3D) Dirac semimetals in NaBi and CdAs demonstrate that there are a variety of materials realizing Dirac semimetals in both two dimensions and three dimensions. Such a diversity of Dirac materials requires us to find a systematic way to characterize and classify them.

In general, the stability of nodal points in a Dirac semimetal has topological origin. This is because there is no characteristic energy scale, such as the Fermi energy or the energy gap, characterizing the perturbative stability of the system. For instance, a nodal point in graphene carries a quantized pseudo-spin winding number that is defined on a loop encircling the Dirac point Graphene (). On the other hand, a nodal point in 3D Weyl semimetals is endowed with a Chern number defined on a two-dimensional (2D) closed surface surrounding the Weyl point Wan (). The presence of such a quantized topological charge carried by a nodal point guarantees its stability, hence a nodal point can be annihilated only by colliding with another nodal point with the opposite topological charge as long as the symmetry of the system is preserved. Recently, there have been several theoretical studies which attempt to classify the topological invariants of nodal points, and to extend the concept of topological band theory to gapless systems such as semimetals and nodal superconductors Horava (); Beri (); nodalSC1 (); nodalSC2 (); nodalSC3 (); Manes (); Matsuura (); Zhao1 (); Zhao2 (); Chiu (); BJYang (); Morimoto_Z2 (); FanZhang (); Young1 (); Young2 (); Sigrist (); nonsymmorphicTI (). However, in our opinion, a proper definition of the topological charge of Dirac points in NaBi or CdAs has not been given so far.

Dirac points in NaBi or CdAs are protected by the time-reversal (), the inversion (), and the rotation symmetries Na3Bi_DFT (); Cd3As2_DFT (); BJYang (). The experimental observation of Dirac points in these systems demonstrates their stability, hence the presence of topological invariants associated with them. Moreover, the theoretical observation of pair-annihilation and pair-creation of Dirac points BJYang (); TPT () indicates that the topological charges of the two Dirac points should have the opposite sign. Then the question is what the nature of the topological charge is associated with the Dirac points. Considering the two-dimensionality of the sphere surrounding a Dirac point, the natural candidate is either a Chern number similar to the case of Weyl semimetals, or a invariant associated with symmetry satisfying . However, the simultaneous presence of the time-reversal and the inversion symmetries forces the Berry curvature to be zero at each momentum, hence the Chern number of the Dirac point, which is basically the integral of the Berry curvature, also vanishes. Moreover, a Dirac point can carry a topological charge only in the presence of SU(2) spin rotation symmetry together with time-reversal and inversion symmetries satisfying as shown in Ref. Morimoto_Z2, . These indicate that a special care is required to find the topological charge of a Dirac point in NaBi or CdAs, which should obviously be distinct from the monopole charge of Weyl semimetals.

In this work, we will show that the Dirac points in NaBi or CdAs are characterized by topological invariants of zero-dimensional subsystems defined on the rotation axis. Since the rotation eigenvalue is a good quantum number on the rotation axis, a zero-dimensional topological invariant can be defined by comparing the rotation eigenvalues of the valence and conduction bands at two points enclosing a Dirac point. We find that the nature of 3D Dirac semimetals strongly depends on the nature of the rotation symmetry. Namely, the ordinary symmorphic rotation symmetry commuting with the inversion symmetry always creates a pair of Dirac points having the opposite topological charges, and generates class I Dirac semimetals. Both NaBi and CdAs belong to this class.

On the other hand, we find a different class of Dirac semimetals when the system has a screw rotation symmetry. In general, the presence of non-symmorphic symmetries, such as screw rotations and glide mirror symmetries, guarantees a nontrivial band connection at the Brillouin zone boundary Zak1 (); Zak2 (); Zak3 (); Parameswaran (). Also, it is proposed that when the double space group of non-symmorphic crystals satisfies certain conditions, a Dirac point can be realized at the Brillouin zone boundary Young1 (). Consistent with these results, our theoretical study shows that when the band degeneracy at the zone boundary is compatible with the time-reversal and inversion symmetries, a single isolated Dirac point can be created on the rotation axis. Based on this observation, we define a class II Dirac semimetal which is protected by a screw rotation symmetry and the inversion, which are mutually anti-commuting, in addition to the time-reversal symmetry. The partial translation associated with a screw rotation adds a U(1) phase to the rotation eigenvalue, which varies on the rotation axis. This projective nature of a screw rotation enables to circumvent the doubling of Dirac points, and create a single isolated Dirac point at a time-reversal invariant momentum at the Brillouin zone boundary.

The rest of the paper is organized in the following way. We describe the general idea to define a topological charge in systems with rotation symmetry in Sec. II. Based on this general idea, class I Dirac semimetals are defined and systematically classified in Sec. III. In particular, we show that the doubling of Dirac points is unavoidable in class I Dirac semimetals. Sec. IV is about the nontrivial band connection generated by non-symmorphic screw rotation symmetries. Here we show that the partial translation associated with a screw rotation induces a momentum dependent U(1) phase factor to the rotation eigenvalue, which enables to circumvent the fermion number doubling and protects a single Dirac point at the Brillouin zone boundary. Based on the discussion in Sec. IV, class II Dirac semimetals are defined and systematically classified in Sec. V. We present the conclusion and discussion in Sec. VI. In the Appendix, we prove that there is no stable Dirac semimetal in systems only with the time-reversal and inversion symmetries based on theory approach. The classification of Dirac semimetals in invariant systems shown in the main text is also confirmed by using the theory approach. Finally, we present a short discussion about the stability of 2D Dirac semimetals protected by two-fold screw rotations.

Ii General idea: role of rotational symmetry in symmorphic crystals

In general, electronic systems having only the time-reversal () and inversion () symmetries cannot support a stable Dirac point with a quantized topological charge Murakami (); Wan (); Burkov (). In Appendix A, we have revisited this known fact in a different perspective and proved it by using theory approach. Thus additional crystalline symmetries play a crucial role to stabilize Dirac semimetals realized in NaBi and CdAs. Here we consider the role of the additional rotation symmetry () in addition to and . For convenience, we first focus on 3D crystals with a symmorphic space group symmetry in which the point group can be completely separable from pure translation operations. Also we choose the axis as the axis for rotation with indicating the discrete rotation angle of (). Under the operation of the symmetry, the Hamiltonian satisfies

(1)

where is obtained from rotation of , i.e., . The symmetry operators satisfy

(2)

and

(3)

where we have considered the fact that an electron is a spin-1/2 particle.

Figure 1: Local geometry around a Dirac point at sitting on the rotation axis. The sphere surrounds a Dirac point at the center. and mark the points on the sphere crossing the rotation axis.

Now let us explain the general idea of how to determine the topological charge of a Dirac point locating at a generic point on the rotation axis. To determine the topological charge, we first consider a sphere in the momentum space surrounding the Dirac point at in a symmetric way as shown in Fig. 1. Namely, the center of the sphere sits on the rotation axis. At every point on the sphere, the Hamiltonian is invariant under the compound antiunitary symmetry satisfying . Moreover, the intersection of the axis and the sphere consists of a north pole at and a south pole at , which are invariant under the rotation. We define the topological charge of the Dirac point from the topological numbers associated with these two points. Since the Hamiltonian commutes with the rotation operator at these points,

(4)

hence can be block-diagonalized in the eigenspace of with the eigenvalues given by

(5)

Since the symmetry is not satisfied in each eigenspace with a given in general, (one exceptional case is shown in Sec. III.2), each diagonal block of belongs to the symmetry class A in terms of the Altland-Zirnbauer classification scheme AZ (), hence carries an integer topological number . Here or indicates the topological invariant of a zero-dimensional system belonging to the symmetry class A, which is defined as

(6a)
(6b)

where denotes the number of conduction bands (c) or valence bands (v) with the eigenvalue at the momentum . It is worth to note that a trivial conduction or valence band with a constant energy can always be added to each sector, so that the sum can be changed freely. Therefore the nontrivial topological number in the sector is determined by the difference

(7)

Next let us consider possible constraints to the allowed values. For a gapped system defined on the sphere, the number of conduction bands and that of valence bands are constants independent of the momentum on the sphere, which leads to the following constraint

(8)

Moreover, since symmetry imposes additional constraints between different values, the number of independent topological invariants depends on the details of the symmetry as shown in Sec. III and V. However, as long as a Dirac point possesses a nonzero value, it guarantees the stability of the relevant Dirac point. Hence the set of nonzero can be considered as a topological invariant characterizing a stable Dirac point.

Iii Class I Dirac semimetals

Class I Dirac semimetals are protected by the ordinary rotation symmetry commuting with an inversion symmetry, i.e.,

(9)

Let us consider the eigenstate of the operator with the eigenvalue . The symmetry requires

(10)

from which we find is an eigenstate of with the eigenvalue . Therefore when a state with the eigenvalue is occupied (or unoccupied), there should be another occupied (or unoccupied) state with the eigenvalue , which leads to the constraint

(11)

For further analysis, we distinguish two cases based on the parity of as shown below.

Figure 2: Constraints on the rotation eigenvalues (a) for even , (b) for odd . The (black) solid circle indicates a unit circle in the complex plane, and each (red) dot on the circle denotes . Two dots connected by a dotted line are related by the symmetry, hence only one of them is independent.

iii.1 symmetric systems with even

Due to the symmetry, the eigenspaces with the eigenvalues and can be paired as with . [See Fig. 2 (a).] Since interchanges eigenspaces within each pair, is not a symmetry in each eigenspace separately, and each eigenspace belongs to the symmetry class A. Therefore an integer topological invariant defined in Eq. (6a) can be computed in each eigenspace with . Considering the constraints shown in Eq. (II) and Eq. (11), we conclude that the topological charge of the Dirac point is given by

(12)

Therefore, the topological charge of a Dirac point with or symmetry is an element of or , respectively. At the same time, it implies that a invariant system cannot support a stable Dirac point.

iii.2 symmetric systems with odd

When is odd, the symmetry pairs the eigenspaces of in a slightly different way as compared to even cases as shown in Fig. 2 (b). At first, we find pairs of eigenspaces

(13)

On the other hand, the remaining eigenspace with the eigenvalue is invariant under the symmetry, hence belongs to class AII. Thus, in a block-diagonalized Hamiltonian , there are blocks belonging to class A and an extra block with the eigenvalue belonging to class AII. In both symmetry classes, zero dimensional systems have an integer topological invariant, which is defined as the difference in the number of conduction bands and that of valence bands. Hence as in the even case, the topological charge can be defined as

(14)

in each eigenspace with the eigenvalue . The constraints to the topological numbers are

(15)
(16)

Thus the independent topological charge for a Dirac point is given by

(17)

Therefore a Dirac point with symmetry has an integer () topological charge.

Topological charge
Not allowed
Table 1: Summary of topological charges of class I Dirac semimetals.

iii.3 Applications: classification of stable Dirac points in 4-band systems

Let us apply the general theory developed above to minimal 4-band models, and classify stable Dirac points. In a 4-band model, a pair of doubly degenerate bands cross at a Dirac point which we assume to sit on the Fermi level. On the rotation axis , each band is assigned with a quantum number . Since the pair of degenerate bands should have different rotation eigenvalues to generate a stable Dirac point, each band with the rotation eigenvalue satisfies . Namely, a band which is below (above) the Fermi level at the momentum should be above (below) the Fermi level at the momentum to have a Dirac point in between.

iii.3.1 symmetric systems

and are the only allowed eigenvalues. Due to the symmetry, a pair of eigenstates are always degenerate locally at each momentum , hence and . Then a 4-band model can be constructed by introducing two pairs of eigenstates and , where indicates the valence band (v) or the conduction band (c), respectively. It is straightforward to show that ( 0, 1), because if one state is occupied, among , , the other state is unoccupied. Therefore and there is no stable Dirac point with a nontrivial topological invariant in systems with symmetry.

iii.3.2 symmetric systems

Possible eigenvalues are , , . Due to the symmetry, and form degenerate pairs, where . Thus . Since , we have

(18)

Hence there is only one independent topological number, . Since the topological charge of a Dirac point can be nonzero only when the valence and conduction bands have different rotation eigenvalues, a 4-band model can be constructed by using a basis where and . Since and for 4-band models, the Dirac point has a nonzero topological invariant

(19)

iii.3.3 symmetric systems

Possible eigenvalues are , , , . Due to the symmetry, and form degenerate pairs at each momentum, thus and . Since ,

(20)

hence there is only one independent topological number, . Since the topological charge of a Dirac point can be nonzero only when the valence and conduction bands have different rotation eigenvalues, a 4-band model with Dirac points can be constructed by using a basis where and . Since and for 4-band models, the Dirac point has a nonzero topological invariant

(21)

iii.3.4 symmetric systems

In the presence of a rotation symmetry, , , and form degenerate pairs. Thus , , . Considering , we can find two independent topological numbers , which, for instance, can be defined as,

(22)

However, for convenience, we can also use to indicate the topological charge in which . A 4-band model can be constructed by choosing two different pairs of eigenstates such as

(23)

For a given 4-band model, a nonzero topological number can be assigned if is the eigenvalue of one of the four bands. Whereas if is the eigenvalue of the other two states which are not included in the 4-band model. Therefore the topological charges of the system are in the form of

(24)

for each case shown in Eq. (III.3.4), respectively. Then the corresponding are

(25)

respectively.

4-band model Materials
Not allowed
NaBi [Na3Bi_DFT, ], strained TlN [DFT_TlN, ]
CdAs [Cd3As2_DFT, ]
Table 2: Topological charges of class I Dirac semimetals for 4-band models and relevant materials.

iii.4 Fermion number doubling in class I Dirac semimetals

Up to now, we have described how to determine the topological charge of a single Dirac point. Now let us compare the topological charges of two Dirac points at the momenta and . Due to the inversion symmetry satisfying , the eigenstates at and satisfy the following relationship,

(26)

which means that if there is an eigenstate with the eigenvalue at , there should be a degenerate eigenstate with the same eigenvalue at . This imposes the following condition of

(27)

It is to be noted that the north (south) pole at and the south (north) pole at are interchanged under the inversion symmetry. Thus we obtain

(28)

Since the net topological charge of the two Dirac points related by the inversion symmetry is zero, we obtain the following conclusions.

  • The total topological charge of two Dirac points within the first Brillouin zone should be zero in each angular momentum channel (), i.e.,

    (29)

    where labels Dirac points. It is worth to note that this is nothing but the Nielsen-Ninomiya theorem NM1 (); NM2 () for three-dimensional Dirac semimetals.

  • A stable Dirac point with a nontrivial topological charge cannot exist at a time-reversal invariant momentum (TRIM) where modulo a reciprocal lattice vector due to the relationship

    (30)

Iv Screw rotations and a single Dirac point

iv.1 Projective symmetry and circumventing fermion number doubling

It is worth to note that the doubling of the number of Dirac points in class I Dirac semimetals results from the commutation relation as discussed in Sec. III.4. This means that it may be possible to avoid the doubling of the Dirac points, once the commutation relation is violated, i.e., . However, the presence of a single Dirac point on the rotation axis brings about a more fundamental problem, when the periodic structure of the system is considered. This is because the band crossing (or a nonzero topological charge of a Dirac point) requires that the valence band and the conduction band should have distinct eigenvalues, whereas the lattice periodicity requires the continuity of the eigenstate and its relevant eigenvalues as described in Fig. 3. Therefore the presence of a single Dirac point or an odd number of Dirac points on the rotation axis sounds unphysical, when the rotation symmetry exists along a line satisfying the periodic boundary condition.

One possible way to circumvent the contradiction is when the rotation symmetry is realized projectively. Namely, if the rotation eigenvalue is well-defined only up to an additional phase factor, it is possible to create a single Dirac point compatible with the lattice periodicity by adjusting the phase degrees of freedom on the rotation axis. In fact, a screw rotation is such an example of projective symmetry, which can support a single isolated Dirac point as discussed in detail below.

Figure 3: Band structure along the rotation axis ( axis) of (a) a class I Dirac semimetal and (b) a class II Dirac semimetal. and are the rotation eigenvalues of each doubly degenerate band. A band crossing requires . In class I (II) Dirac semimetals, the band crossing condition and the periodicity of the eigenstates are compatible (incompatible) when are constant on the rotation axis.

A screw rotation () is a non-symmorphic symmetry operation composed of an ordinary rotation () followed by a partial lattice translation () parallel to the rotation axis. Here is the unit lattice translation along the axis assuming that the screw axis is parallel to it. Schematic figures describing all possible screw rotations in 3D crystals are shown in Fig. 4. Let us note that, in many crystals, the screw rotation axis does not pass the reference point of the point group symmetry, which is invariant under point group operations of the lattice. In this case, the partial translation associated with the screw rotation also includes in-plane translation components perpendicular to the screw axis direction. Generally, can be compactly represented as

(31)

where

(32)

In the real space, transforms the spatial coordinates in the following way,

(33)

where

(34)

Now we consider the combination of a screw rotation and the inversion symmetry. At first, we see

thus

(35)

Similarly,

thus

(36)

Let us note that in general. Equations (35) and (36) clearly show that generally due to the partial translation , thus there is a chance to avoid the doubling of the Dirac points.

In the presence of the screw rotation symmetry , the Bloch Hamiltonian on the rotation axis () satisfies

(37)

where

(38)

Here the minus sign stems from the spin 1/2 nature of electrons. Therefore all bands on the axis can be labeled by the eigenvalues of given by

(39)

where is an eigenvalue of defined in Eq. (5).

Figure 4: Schematic figure describing all possible screw rotations in 3D crystals. In each figure, the system is periodic under the lattice translation along the vertical direction. indicates a counterclockwise rotation combined with a partial translation . (a) symmetry. (b) symmetry (). (c) symmetry (). (d) symmetry ().

It is worth to note that the eigenvalue of the screw rotation is not but which varies along the rotation axis. Therefore through the variation of this additional phase factor, it may be possible to satisfy the condition for the band crossing to create a Dirac point and the periodicity (or the continuity) of the eigenvalues, simultaneously, even in the presence of a single Dirac point. In fact, the assignment of non-quantized quantum numbers to fermions, such as varying in the momentum space, is one way to get around the fermion doubling problem, as pointed out by Nielsen and Ninomiya in their seminal work NM1 (); NM2 ().

iv.2 Screw rotations and band connections at the zone boundary

The momentum dependence of screw rotation eigenvalues shown in Eq. (IV.1) induces nontrivial band connections between different eigenstates at the Brillouin zone boundary. For instance, if the system has periodicity along the axis, we find that

(40)

thus the eigenstate with the eigenvalue should be smoothly connected to the other eigenstate with the eigenvalue () at the Brillouin zone boundary () as shown in Fig. 5. This naturally gives rise to a band crossing point at the Brillouin zone boundary. If this band connection is compatible with the and symmetries, a single Dirac point can be realized at the Brillouin zone boundary.

Figure 5: The band connection required by a screw rotation when the system is periodic along the direction.

The symmetry requires that the state with the eigenvalue should be locally degenerate with the other state with the eigenvalue at each . Similarly, we can expect the degeneracy between two states with the eigenvalues and , respectively. Here the important point is that the screw rotation requires a nontrivial band connection between and , similar to the relation shown in Eq. (IV.2). Namely,

(41)

which gives

(42)

thus

(43)

Since is an integer, this condition can be satisfied only in systems with , , symmetries.

Let us note that, in 3D crystals, the periodicity along the direction can be longer than although the system is periodic under the translation by along the direction, unless is a primitive lattice vector. (For instance, it happens in the face centered cubic lattice.) Generally, when the system is periodic along the axis with an integer ,

(44)

thus the eigenstate with the eigenvalue should be smoothly connected to the other eigenstate with the eigenvalue () at the Brillouin zone boundary (). Considering the symmetry and following the same procedure that we have used to derive Eq. (43), we obtain

(45)

For example, when the system is periodic (), Eq. (45) can also be satisfied in systems with , , , symmetries. However, in systems with and symmetries, a shift merely maps an eigenstate into itself, hence nontrivial band connection at the Brillouin zone boundary is not expected. On the other hand, when the system is periodic (), Eq. (45) can be satisfied in systems with symmetry where . However, in the case of and symmetries, a shift connects an eigenstate with itself. Also in the symmetric case, a shift is simply equivalent to a shift, which is already considered before. Hence only the systems with and can support a nontrivial band connection at the Brillouin zone boundary .

To sum up, in a symmetric system satisfying with two co-prime numbers (an odd integer) and (an even integer), two distinct eigenstates should be connected to each other at the Brillouin zone boundary . Namely, the eigenstate with the eigenvalue should be smoothly connected to the other eigenstate with the eigenvalue at the Brillouin zone boundary in the following way,

(46)

where we have used the fact that is an odd integer and is well-defined modulo . It is interesting to note that the eigenvalue at the zone boundary becomes

(47)

Namely, the eigenvalue is simply given by at the zone boundary. This additional factor gives rise to the following relations between the screw rotation and the inversion at the zone boundary ,

(48)

which can be easily derived from Eq. (35) and (36). Hence and anticommute when the Bloch state is used as a basis for the representation. From this, we obtain the following general principle to create a stable Dirac semimetal with a single Dirac point. Namely,

  • The Dirac point should be located at a TRIM at the Brillouin zone boundary () where the screw rotation symmetry anti-commutes with the inversion symmetry.

Figure 6: (a) An example of the band connection required by the screw rotation in Eq. (IV.3.1). For each band, the eigenvalue of the screw rotation is marked in the figure. (b,c) Possible band structure protected by a two-fold screw rotation with a Dirac point at and at , respectively.

iv.3 Applications

In the following, we examine the possible Dirac semimetals with a single Dirac point by considering various screw rotation symmetries explicitly.

iv.3.1 Two-fold screw rotation

A two-fold screw rotation symmetry has the following two eigenvalues

(49)

where

(50)

for and

(51)

Now we prepare two bands and with an eigenvalue and , respectively, and construct a band structure with a Dirac point. Here the crucial point is that the band () should be smoothly connected to the other band () at the Brillouin zone boundary () to satisfy Eq. (50) and (IV.3.1) as shown in Fig. 6. This naturally gives rise to a band structure with a single band crossing point at a TRIM. Considering the or symmetry, there are two possible band structures having a single band crossing point as shown in Fig. 6 (b) and (c). In each case, the band crossing point locates at a TRIM either at (Fig. 6 (b)) or at (Fig. 6 (c)). However, let us note that, due to the symmetry, the state with the eigenvalue () should be locally degenerate with the other state with the eigenvalue