## Abstract

We formulate a degenerate perturbation theory for the vector electromagnetic field of periodic structures and apply it to the problem of the creation of Dirac cones in the Brillouin-zone center by accidental degeneracy of two modes. We derive a necessary condition by which we can easily select candidates of mode combinations that enable the creation of the Dirac cone. We analyze the structure of a matrix that determines the first-order correction to eigen frequencies by examining its transformation by symmetry operations. Thus, we can obtain the analytical solution of dispersion curves in the vicinity of the zone center and can judge the presence of the Dirac cone. All these findings clearly show that the presence or absence of the Dirac cone in the zone center is solely determined by the spatial symmetry of the two modes.

© 2012 Optical Society of America

## 1. Introduction

Photonic Dirac cones, or the linear dispersion relation around certain points in the Brillouin zone, have been attracting considerable interest during the last five years [1–12]. Haldane et al. [1, 2] pointed out the presence of photonic Dirac cones on the Brillouin-zone boundary of two-dimensional triangular-lattice photonic crystals due to their structural symmetry and discussed unidirectional propagation of surface modes caused by time-reversal symmetry breaking. Ochiai et al. extended the discussion to honeycomb-lattice photonic crystals [3]. Zhang proposed optical simulation of *Zitterbewegung*, or trembling motion, in particle physics by propagating an optical pulse of the Dirac point frequency [4]. A pseudo-diffusive transmission was found by Sepkhanov et al. [5] and numerically demonstrated by Diem et al. [6].

Recently, Huang et al. reported that Dirac cones can also be created in the Brillouin-zone center of two-dimensional dielectric photonic crystals by accidental degeneracy of two modes [7]. They showed for square- and triangular-lattice photonic crystals that combinations of a non-degenerate mode and a doubly degenerate mode yield a Dirac cone together with a quadratic dispersion surface. Because the Dirac point in the Brillouin-zone center is equivalent to a zero effective refractive index [7], it has much potential for various applications like scatter-free waveguides [13] and lenses of arbitrary shapes [14].

On the other hand, we showed by tight-binding approximation and group theory that Dirac cones can also be created in the Brillouin-zone center of metamaterials, which are characterized by well-defined electromagnetic resonant states localized in their unit structures, by accidental degeneracy of two modes [8–12]. First, we analyzed one-dimensional regular arrays of metallic unit cells to show that one-dimensional photonic Dirac cones can be created by accidental degeneracy of *A*_{1} and *B*_{1} modes and that of *A*_{2} and *B*_{2} modes [8]. We also showed that a one-dimensional Dirac cone can be created by the *controlled symmetry reduction* of doubly degenerate modes of two-dimensional square-lattice metamaterials [9]. In addition, we proved that the combination of *A*_{1} and *E* modes of square-lattice metamaterials and the combination of *A*_{1g} and *T*_{1u} modes of simple-cubic-lattice metamaterials create an isotropic Dirac cone with quadratic dispersion surfaces [10]. We further proved that the combination of *E*_{1} and *E*_{2} modes of triangular-lattice metamaterials yields two-dimensional double Dirac cones, or a pair of identical Dirac cones [11].

Quite recently, we systematically examined the relation between the symmetry of modes and the shapes of dispersion curves generated by accidental degeneracy for both dielectric photonic crystals and metamaterials with localized electromagnetic resonant states by numerical photonic-band calculation and tight-binding approximation, respectively [12]. We found that the two calculations gave the same results, which strongly suggested the presence of universality of mode symmetries that enabled the creation of photonic Dirac cones irrespective of the details of the sample structure, irrespective of the presence or absence of localized resonant states, and irrespective of the approximation used in the calculation.

In this paper, anticipating that the presence or absence of photonic Dirac cones is solely determined by the combination of mode symmetries, we develop a new theory that does not depend on the details of the sample structure, but depends only on the spatial symmetry of the accidentally degenerate modes.

This paper is organized as follows. In Section 2, a degenerate perturbation theory is formulated for the vector electromagnetic field of periodic systems and it is applied to the problem of the creation of Dirac cones in the Brillouin-zone center by accidental degeneracy of two modes. A necessary condition is derived by which we can easily select candidates of mode combinations that enable the creation of the Dirac cone. In Section 3 to Section 6, this method is applied to four periodic systems, that is, the one-dimensional lattice of *C*_{2v} symmetry, the square lattice of *C*_{4v} symmetry, the triangular lattice of *C*_{6v} symmetry, and the simple-cubic lattice of *O _{h}* symmetry. The condition for obtaining Dirac cones and/or double Dirac cones is clarified for each case. A summary of the present study is given in Section 7.

## 2. Theory

The eigen equation for the magnetic field of periodic systems is given by

*n*and

**k**denote the band index and a wave vector in the first Brillouin zone, respectively,

*ε*is the periodic dielectric constant, and operator ℒ is defined by the first equality. We assume that the magnetic permeability of the system is equal to that of free space, since we do not deal with magnetic materials. Then the

**k**dependent eigenvalue is given by where

*c*is the speed of light in free space and

*ω*

_{kn}denotes the eigen angular frequency.

*ε*should be real for

*ω*

_{kn}to be real. According to Bloch’s theorem, the eigen function,

**H**

_{kn}, is a product of an exponential factor and a vector field with the lattice-translation symmetry:

**a**is the elementary translation vector.

When we assume that *ε*(**r**) is real and independent of frequency and impose a periodic boundary condition to make our problem well-defined, ℒ is a Hermitian operator in the Hilbert space of complex vector fields, for which the inner product of two vector fields is defined as

*V*is the volume on which the periodic boundary condition is imposed [15]. We normalize the eigen functions as

**u**

_{kn}is an eigen function of operator ℒ

**defined by**

_{k}**u**

_{kn}|

*n*= 1, 2,…} is a complete set. We normalize it as

*V*

_{0}denotes the volume of the unit cell. Thus, in particular, for

**k**= 0, is an orthonormal complete set. Therefore, we can express any eigen function

**u**

_{kl}of operator ℒ

**by a linear combination of eigen functions {**

_{k}**u**

_{0n}} of operator ℒ

_{0}. Thus, for small

**k**in the vicinity of the Γ point, we can calculate

*λ*

_{kl}perturbatively using {

**u**

_{0n}} as a basis set. Because we are interested only in the presence or absence of terms linear in

**k**, we neglect the quadratic term of the perturbation operator:

We assume according to the situation of our problem that {**u**_{0l}| *l* = 1, 2, … ,*M*} are degenerate and denote their eigenvalue by
${\lambda}_{0}\left(={\omega}_{0}^{2}/{c}^{2}\right)$. By the degenerate perturbation theory, the first-order solution for **u**_{kl} (*l* = 1, 2, … ,*M*) is obtained by diagonalizing the matrix whose *ij* (1 ≤ *i*, *j* ≤ *M*) element is given by

**k**. Thus, our problem on the creation of photonic Dirac cones is reduced to examining whether the eigenvalues of matrix ${\text{C}}_{\mathbf{k}}=\left({C}_{ij}^{(\mathbf{k})}\right)$ are non-zero.

For this purpose, we examine C** _{k}** using the spatial symmetry of {

**u**

_{0}

*}. We assume that the periodic structure that we deal with is invariant by symmetry operations of point group 𝒢. We denote the symmetry operations and their matrix representations by ℛ and R, respectively. First, we should note that*

_{l}**P**

*is defined as*

_{ij}*S*

_{0}denotes the surface of

*V*

_{0}and the first integral on the right-hand side is the surface integral of the normal component of the integrand. This surface integral is equal to zero because of the periodicity of

*ε*and

**u**. By the same transformation as in Eq. (15), we finally obtain

Next, by introducing five pairs of ℛ^{−1}ℛ (= identity operator) to the definition of **P*** _{ij}*, we obtain

**r**′ = R

**r**and by definition,

*ε*is invariant by ℛ and that R does not change the size of the volume element, since it is an orthogonal matrix.

So, unless the product of **k**, **u**_{0i}, and **u**_{0j} contains a term invariant for all ℛ ∈ 𝒢, **k** · **P*** _{ij}*, and consequently,
${C}_{ij}^{(\mathbf{k})}$ vanish. In this case, the linear term is absent in the dispersion curve, so Dirac cones do not exist. Therefore, the presence of an invariant term in the above-mentioned product is a distinct necessary condition for the Dirac cone.

Because both **u**_{0i} and **u**_{0j} are eigen functions of ℒ_{0}, which commutes with ^{∀}ℛ ∈ 𝒢, they are irreducible representations of group 𝒢, so their transformation by ℛ is well known. The transformation of vector **k** by ℛ can also be found easily. Thus, we can select the candidates of combinations of mode symmetries that enable the creation of Dirac cones by using this necessary condition. The exact shapes of dispersion curves can also be clarified by examining the structure of C** _{k}**. Some examples will be given in next four sections.

## 3. One-dimensional lattice of *C*_{2v} symmetry

We start with the one-dimensional lattice of *C*_{2v} symmetry. An example of such structures was previously analyzed by tight-binding approximation in Ref. [8]. There are four one-dimensional representations (*A*_{1}, *A*_{2}, *B*_{1}, *B*_{2}) for the *C*_{2v} point group [16]. As can be verified easily, vector **k**, which should be regarded as a one-dimensional vector (or scalar), has the *B*_{1} symmetry. We can examine by the well-known reduction procedure [16] whether the product of **k**, **u**_{0}* _{i}*, and

**u**

_{0j}contains an invariant term, or the totally symmetric

*A*

_{1}representation. As is summarized in Table 1, { ${C}_{ij}^{(\mathbf{k})}$} is non-zero only for the combinations of (

*A*

_{1},

*B*

_{1}) modes and (

*A*

_{2},

*B*

_{2}) modes.

By examining all symmetry transformations ℛ ∈ *C*_{2v}, we can prove that matrix C** _{k}** has the following form for the combination of the

*A*

_{1}(

**u**

_{01}) and

*B*

_{1}(

**u**

_{02}) modes.

**e**is a unit vector in the direction of the one-dimensional alignment of unit cells. Then, the secular equation to determine the first-order correction, Δ

*λ*, to the eigenvalue of Eq. (1) is given by

*A*

_{1}mode and a

*B*

_{1}mode by tight-binding approximation based on the resonant states of unit cells of periodic metamaterials. For the combination of the

*A*

_{2}and

*B*

_{2}modes, we have the same relations as Eq. (24), so there is also a Dirac cone.

Let us make a remark here. If two modes have the same symmetry, they are mixed and their dispersion curves repel each other when their eigen frequencies are close. So, accidental degeneracy does not take place in this case. Nevertheless, this case is included in Table 1 and other tables in the following sections to show that their dispersion curves are quadratic for small **k**.

## 4. Square lattice of *C*_{4v} symmetry

Next, we examine the two-dimensional square lattice of *C*_{4v} symmetry. An example of such structures was previously analyzed by tight-binding approximation in Ref. [10]. There are four one-dimensional representations (*A*_{1}, *A*_{2}, *B*_{1}, *B*_{2}) and one two-dimensional representation (*E*) [16]. We can verify easily that vector **k**, which should be regarded as a two-dimensional vector for this case, has the *E* symmetry. The nature of
${C}_{ij}^{(\mathbf{k})}$ is summarized in Table 2. It is non-zero for the combinations of an *A*_{1}, *A*_{2}, *B*_{1}, or *B*_{2} mode and an *E* mode.

Let us first examine the case of the combination of an *E* mode and an *A*_{1} mode. From group theory [16], we can assume without loss of generality that one of the two eigen functions of the *E* mode is transformed like the *x* coordinate and the other like the *y* coordinate by the symmetry operation ℛ ∈ *C*_{4}* _{v}*. We denote the first and second functions by

**u**

_{01}and

**u**

_{02}, respectively. The eigen function of the

*A*

_{1}mode is denoted by

**u**

_{03}. By examining all transformation ℛ ∈

*C*

_{4v}, we can prove that matrix C

**has the following form:**

_{k}For the combinations of (*E*, *A*_{2}), (*E*, *B*_{1}), and (*E*, *B*_{2}) modes, we can prove that C** _{k}** has the following forms, respectively:

## 5. Triangular lattice of *C*_{6v} symmetry

In this section, we examine the two-dimensional triangular lattice of *C*_{6v} symmetry. There are four one-dimensional representations (*A*_{1}, *A*_{2}, *B*_{1}, *B*_{2}) and two two-dimensional representations (*E*_{1}, *E*_{2}) [16]. We can verify easily that vector **k** has the *E*_{1} symmetry. The nature of
${C}_{ij}^{(\mathbf{k})}$ is summarized in Table 3.

Let us examine the shapes of the dispersion curves for the cases of non-zero {
${C}_{ij}^{(\mathbf{k})}$}. For the combination of an *E*_{1} mode and an *E*_{2} mode, we can prove by examining all transformations ℛ ∈ *C*_{6v} that matrix C** _{k}** has the following form:

*E*

_{1}mode (

**u**

_{01},

**u**

_{02}) are transformed like

*x*and

*y*and those of the

*E*

_{2}mode (

**u**

_{03},

**u**

_{04}) are transformed like 2

*xy*and

*x*

^{2}−

*y*

^{2}[16]. The secular equation for the first-order correction can be solved easily and its solutions are given by

For the (*E*_{1}, *A*_{1}) and (*E*_{2}, *B*_{2}) combinations, matrix C** _{k}** has the following form:

*E*

_{1},

*A*

_{2}) and (

*E*

_{2},

*B*

_{1}) combinations,

## 6. Simple-cubic lattice of *O*_{h} symmetry

_{h}

Finally, we examine the simple-cubic lattice of the *O _{h}* symmetry. There are four one-dimensional representations (

*A*

_{1g},

*A*

_{1u},

*A*

_{2g},

*A*

_{2u}), two two-dimensional representations (

*E*,

_{g}*E*), and four three-dimensional representations (

_{u}*T*

_{1g},

*T*

_{1u},

*T*

_{2g},

*T*

_{2u}) [16]. We can verify that vector

**k**has the

*T*

_{1u}symmetry. The nature of ${C}_{ij}^{(\mathbf{k})}$ is summarized in Table 4.

Let us start by examining the shapes of the dispersion curves for the combination of a *T*_{1u} mode and an *A*_{1g} mode [10]. From group theory, we can assume without loss of generality that three eigen functions of the *T*_{1u} mode are transformed like the *x*, *y*, and *z* coordinates [16], which we denote by **u**_{01}, **u**_{02}, and **u**_{03}, respectively. The eigen function of the *A*_{1g} mode is denoted by **u**_{04}. We can prove by examining all transformations ℛ ∈ *O _{h}* that matrix C

**has the following form:**

_{k}**is sufficiently sparse, we can easily diagonalize it to obtain the first-order correction to the eigenvalues. The result is**

_{k}*T*

_{1g},

*A*

_{1u}), (

*T*

_{2u},

*A*

_{2g}), and (

*T*

_{2g},

*A*

_{2u}), we can also prove that C

**has the same form as in Eq. (38). So, all these combinations yield an isotropic Dirac cone and two quadratic dispersion surfaces.**

_{k}Next, we examine the combination of a *T*_{1u} mode (**u**_{01}, **u**_{02}, **u**_{03}) and an *E _{g}* mode (

**u**

_{04},

**u**

_{05}). We can assume that two eigen functions of the

*E*mode are transformed like 2

_{g}*z*

^{2}−

*x*

^{2}−

*y*

^{2}and $\sqrt{3}\left({x}^{2}-{y}^{2}\right)$ [16]. We can prove easily that matrix C

**has the following form:**

_{k}*k*,

*θ*,

*ϕ*),

*F*(

*θ*,

*ϕ*) ≠ 0. When

*F*(

*θ*,

*ϕ*) = 0, which happens for

**k**in the (1,0,0) direction and its equivalent directions, two of the four

*k*-linear eigenvalues vanish, so the corresponding dispersion curves are quadratic in

*k*. For the combination of (

*T*

_{1g},

*E*), we can prove that C

_{u}**has the same structure as Eq. (40), so it also yields two anisotropic Dirac cones and a quadratic dispersion surface.**

_{k}For the combinations of (*T*_{2u},*E _{g}*) and (

*T*

_{2g},

*E*), C

_{u}**has the following form:**

_{k}**leads to exactly the same secular equation as Eq. (41). So, the combinations of (**

_{k}*T*

_{2u},

*E*) and (

_{g}*T*

_{2g},

*E*) also yield two anisotropic Dirac cones and a quadratic dispersion surface.

_{u}Finally, let us examine the case of two triply degenerate modes. For the combination of a *T*_{1g} mode (**u**_{01}, **u**_{02}, **u**_{03}) and a *T*_{1u} mode (**u**_{04}, **u**_{05}, **u**_{06}), C** _{k}** has the following structure:

*T*

_{2g}and

*T*

_{2u}modes has C

**of the same structure, so this combination also yields double Dirac cones and two quadratic dispersion surfaces.**

_{k}For the combinations of (*T*_{1g}, *T*_{2u}) and (*T*_{2g}, *T*_{1u}), C** _{k}** has the following form:

*l*= 1, 2, 3. So, six solutions of Eq. (48) are

To check these analytical results, we performed photonic-band calculations by the plane-wave expansion method [15]. We assumed a regular simple-cubic lattice of dielectric spheres with a dielectric constant of 12.6 (GaAs). By changing the radius of the spheres, we found several cases of accidental degeneracy among the lowest 20 dispersion curves, three of which are shown in Fig. 1, which agree with the results of analytical calculations listed in Table 4. The rest of the cases also agreed with Table 4.

Let us make six remarks. First, in this paper, we formulated a degenerate perturbation theory for the vector electromagnetic field of periodic structures and applied it to the problem of the creation of Dirac cones by accidental degeneracy of two modes. We obtained an efficient necessary condition (non-zero C** _{k}**) by which we can select candidates of mode combinations that enable the creation of single and double Dirac cones. In addition, by examining the transformation of C

**by symmetry operations for four periodic systems of different spatial symmetries, we could determine the structure of C**

_{k}**and obtain the shapes of dispersion curves in the vicinity of the zone center by purely analytical calculations. All these findings clearly show that the presence or absence of the Dirac cone by accidental degeneracy does not depend on the details of the periodic structure but is solely determined by the spatial symmetry of the two modes.**

_{k}Second, the present theory can of course be applied to periodic structures of other spatial symmetries. We can examine C** _{k}** and obtain the shapes of dispersion curves in quite a similar manner. We should also note that if the periodic structure does not have a spatial symmetry, any two modes are mixed and repel each other when their eigen frequencies are close, so they do not create Dirac cones.

Third, the secular equations obtained by the present theory are generally much simpler than those obtained by tight-binding approximation in Refs. [8], [10], [11], and [12] due to the perturbative nature of the calculations. As a result, we could easily solve equations up to the sixth degree to obtain the first-order corrections to the eigen values.

Fourth, we assumed that the dielectric constant of the periodic structure does not depend on frequency in order to make our problem well-defined. When we have to deal with dispersive materials with frequency-dependent dielectric constants, we can use the value at the degenerate frequency. Thus, we can obtain qualitatively correct solutions in the vicinity of the degenerate frequency.

Fifth, although the numerical confirmation was performed for photonic crystals because of the ease of calculation, the present theory is equally applicable to periodic metamaterials with well-defined resonant states localized in unit cells.

Finally, the present theory may be regarded as an extension of the **k** · **p** theory in semiconductor physics [17] to the vector electromagnetic field. In the usual **k** · **p** theory, the first-order term is absent and an effective mass is derived by the second-order perturbation, so the energy bands are quadratic in the Brillouin-zone center. However, in the present theory, the first-order term can be non-zero because of the accidental degeneracy of two modes, which resulted in the creation of Dirac cones.

## 7. Conclusion

We formulated a degenerate perturbation theory for the vector electromagnetic field of periodic structures like photonic crystals and metamaterials and applied it to the problem of the creation of Dirac cones in the center of the Brillouin zone by accidental degeneracy of two modes. We derived an efficient necessary condition for the Dirac cone by which we can easily select candidates of mode combinations that enable the creation of the Dirac cones. We can analyze the structure of a matrix (C** _{k}**) that determines the first-order correction to eigen frequencies by examining its transformation by symmetry operations. Thus, we can obtain the analytical solutions of the dispersion curves in the vicinity of the zone center and judge the presence of the Dirac cone.

This method was applied to four periodic systems, that is, the one-dimensional lattice of *C*_{2v} symmetry, the square lattice of *C*_{4v} symmetry, the triangular lattice of *C*_{6v} symmetry, and the simple-cubic lattice of *O _{h}* symmetry. We succeeded in examining all possible combinations of mode symmetries and obtaining the analytical solution of dispersion curves for each case. Thus, we fully clarified the conditions required to obtain Dirac cones and double Dirac cones in these four systems. These results are consistent with our previous calculations for periodic metamaterials by tight-binding approximation and for photonic crystals by plane-wave expansion.

All these findings clearly show that the presence or absence of the Dirac cone in the zone center due to accidental degeneracy of two modes does not depend on the details of the periodic structure that we analyze but is solely determined by the spatial symmetry of the two modes. Thus, we succeeded in proving the universality of mode symmetries in creating photonic Dirac cones, which was anticipated in our recent study [12].

## Acknowledgments

This study was supported by a Grant-in-Aid for Scientific Research on Innovative Areas from the Japanese Ministry of Education, Culture, Sports and Technology (Grant number 22109007).

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