## Abstract

A simple core-shell two-dimensional photonic crystal is studied where the triangular lattice symmetry and the *C*_{6} point group symmetry give rich physics in accidental touching points of photonic bands. We systematically evaluate different types of accidental nodal points at the Brillouin zone center for transverse-magnetic harmonic modes when the geometry and permittivity of the core-shell material are continuously tuned. The accidental nodal points can have different dispersions and topological properties (i.e., Berry phases). These accidental nodal points can be the critical states lying between a topological phase and a normal phase of the photonic crystal. They are thus very important for the study of topological photonic states. We show that, without breaking time-reversal symmetry, by tuning the geometry of the core-shell material, a phase transition into the photonic quantum spin Hall insulator can be achieved. Here the “spin” is defined as the orbital angular momentum of a photon. We study the topological phase transition as well as the properties of the edge and bulk states and their application potentials in optics.

© 2016 Optical Society of America

## 1. Introduction

Since Haldane and Raghu’s seminal work [1] which brought the quantum Hall effect [2] to the realm of photonics, the field of topological photonics attracted significant research interest. The study of topological properties of photonic bands [3–9], polarization states [10,11], as well as their unique physical consequences and applications motivated the field. Recent developments of the field extend to time-reversal symmetric all-dielectric photonic crystals (PhCs) which are easier to fabricate (especially for visible-frequency applications) compared to time-reversal symmetry broken systems [12–17]. In time-reversal symmetric all-dielectric PhCs, because the Kramers theorem does not guarantee double degeneracy for photons, additional symmetry is needed to induce topological effects. These symmetry-protected topological photonic states enrich the study and application values of Berry phases in photonic bands [18]. Many novel properties are to be explored for fundamental physics and applications. Examples include the study of photonic graphene [19], Dirac-like points with zero effective refractive index [20,21], etc.

In spite of much research efforts in the literature, the design and understanding of topological photonic bands remain a challenging task. The existing designs are accidental: there is no clue or guiding principles on how to realize photonic bands with nontrivial topology. One may use Dirac cones as mother states of topologically nontrivial photonic band gap. However, the emergence of Dirac cone is also accidental unless for a few deterministic cases such as in honeycomb lattice photonic crystals where the sub-lattice symmetry dictates Dirac dispersion at the *K* and *K′* points [1,14].

Here we present a systematic study on accidental nodal points in photonic bands in 2D triangle PhCs with core-shell dielectric structures. We show that some of those nodal points are Dirac (and Dirac-like) points. Their dispersion around the Γ point can be easily understood via the *k⃗* · *P⃗* theory with the assistance of the picture that the Mie resonances can be regarded as atomic orbits for photonic bands [22]. Those atomic orbits can be of *s*, *p*, *d* and *f* nature and have well-defined parities at the Γ point. The couplings between photonic bands are readily determined by their parity differences. In our core-shell structures, by tuning the inner and outer radii of the core-shell cylinder, we are able to flip the order of the *p* and *d* photonic bands, which leads to a phase transition from topologically trivial photonic bands to quantum spin Hall insulator of light: a photonic band gap with nontrivial *Z*_{2} topology. The properties of the bulk and edge states of the *Z*_{2} topological photonic band gap are studied in details, where essential difference between electronic and photonic systems are emphasized. Differing from realization of *Z*_{2} topological photonic band gap using deterministic Dirac cone in honeycomb lattice [14], here the topological phase transition demonstrates rich physics as revealed by our phase diagram and *k⃗* · *P⃗* analysis. The core-shell triangle photonic crystal structure is simple and mechanically stable. In addition, it is compatible with colloidal self-assembled structure [23]. Our study shows how topological phases of photons can be realized in simple 2D dielectric PhCs and various phases can be induced by tuning the geometry of the dielectric materials, offering guidelines for future studies.

## 2. 2D triangle lattice photonic crystals

Among common 2D PhCs, triangle lattice PhCs have the highest point group symmetry [24]. As will be shown below the 2D triangle PhCs can host several kinds of topological nodal points. These nodal points are the mother states of different types of photonic topological insulators. In the triangular lattice, the Γ point has two kinds of 2D irreducible representations in the *C*_{6} symmetry, which are denoted as the *E*_{1} and *E*_{2} representations, respectively [21, 24]. The associated Bloch waves are referred as the *E*_{1} and *E*_{2} modes, respectively. Each representation has two Bloch waves, which can be reorganized into a pair of modes connected by time-reversal operation. Here the relevant modes are the *p*_{+} = *p _{x}* +

*ip*and

_{y}*p*

_{−}=

*p*−

_{x}*ip*modes, which are denoted as (pseudo-) spin-up and spin-down states of the

_{y}*p*bands [14]. Similarly, the

*d*

_{+}=

*d*

_{x2−y2}+

*id*and

_{xy}*d*

_{−}=

*d*

_{x2−y2}−

*id*modes are denoted as the (pseudo-) spin-up and spin-down states for the

_{xy}*d*bands [14].

The 2D core-shell dielectric structure in triangular lattice is shown in Fig. 1(a). Its outer and inner radii are denoted as *R*_{1} and *R*_{2}, respectively. The dielectric constant of the core-shell material and the background material are *ε*_{1} and *ε*_{2}, respectively. The lattice constant is set as *a*_{1} = *a*_{2} ≡ 1. We define the inverse structure by the exchange of *ε*_{1} and *ε*_{2}. The inverse structure is hence a core-shell air cylinder in dielectric background. Our architecture is one of the simplest 2D PhCs that can support rich physics of topological nodal points, photonic *Z*_{2} topological insulators, and topological phase transitions.

By numerically solving the Maxwell equations for transverse magnetic (TM) harmonic modes, of which the electric field is along the core-shell cylinder (i.e., the *z* direction). Figure 1(b) shows a typical band structure of the PhC. The band structure is calculated by COMSOL based on the finite-element method. In this band structure, the eigen-modes at the Γ point exhibit orbital symmetry of the *s*, *p*, *d* and *f* waves from low frequency to high frequency. These modes correspond to the *A*_{1}, *E*_{1}, *E*_{2} and *B*_{1} modes of the *C*_{6} point group symmetry [24]. The labeling of bands with *s*, *p*, *d* and *f* modes is only effective at the Γ point which possesses the *C*_{6} symmetry. Away from the Γ point these modes couple with each other and their order in frequency may change [14]. Nevertheless, the labels clearly reveal the accidental band degeneracy and band inversion at the Γ point. Besides, the *s*, *p*, *d* and *f* modes carry physical meanings. The photonic bands can be viewed as derived from transfer (hopping) of local Mie resonances of the core-shell structures between adjacent unit cells (except the lowest photonic band which becomes plane wave in the low frequency limit). Such a tight-binding understanding of the photonic bands has successfully connected the photonic bands with the Mie resonances [22]. Thus the Mie resonances can be regarded as the “atomic orbits” for photonic bands. This picture is useful in our design of photonic *Z*_{2} topological insulators and topological nodes.

The *s* and *f* (*A*_{1} and *B*_{1}) modes are singlet states, while the *E*_{1} and *E*_{2} modes are doubly degenerate, according to the *C*_{6} symmetry [21]. By continuously changing the radii *R*_{1} and *R*_{2} or the dielectric constant, different bands cross each other, leading to accidental degeneracy’s at the Γ point. The accidental degeneracy usually results in linear or quadratic dispersion. The scale invariance of the Maxwell equations dictates that the independent variables in our system are the inner and outer radii (divided by the lattice constant) as well as the ratio of the two dielectric constants *ε*_{1}/*ε*_{2}. The properties of the photonic bands for *R*_{2} = 0 are well-studied in the literature [25], where no topological nodal point or topological band structure is found.

## 3. Phase diagram

We first discuss the eigen-frequency of the *p* and *d* bands at the Γ point. We use the following dimensionless quantity to characterize the relative *p*-*d* band-gap size at the Γ point,

*ω*and

_{p}*ω*are the egien-frequency’s of the

_{d}*p*and

*d*modes at the Γ point, respectively.

By numerically calculating the photonic band structure, we obtain in the *R*_{1}–*R*_{2} parameter space with *ε*_{1} = 1 and *ε*_{2} = 12 (see Fig. 2). The figure clearly demonstrates the *p*-*d* band inversion which can result in photonic analog of the *Z*_{2} topological insulator. The properties of this photonic *Z*_{2} topological states will be discussed in details below. We incorporate in the upper triangle of Fig. 2 the phase diagram for the reversed structure, where the core-shell cylinder of air induces the photonic energy bands. The diagonal line is the homogeneous limit, which can be regarded as no dielectric (for the normal structure) or no air (for the inverse structure). The influence of *ε*_{1}/*ε*_{2} on the phase diagram will be discussed below.

The color scheme in Fig. 2 is as follows: the red region has *p* bands lower (in frequency) than the *d* bands at the Γ point, while the blue region has *p* bands higher than the *d* bands (i.e., *p*-*d* band inversion), at the Γ point. The blue region thus represents the *Z*_{2} topological phase (*Z*_{2} = 1), while the red region represents the normal phase (topologically trivial, *Z*_{2} = 0). The parity inversion at the Γ point for the *p*-*d* reversed photonic bands dictate the *Z*_{2} topology [26]. At the boundary between the topological phase and the normal phase, the double Dirac-cone dispersion emerges at the Γ point. The properties of the photonic *Z*_{2} topological insulator will be discussed in details in next section.

In addition, we use the blue (red) curve in Fig. 2 to denote the crossing between the *s* and *p* bands (the *f* and *d* bands) in the phase diagram. These two situations lead to Dirac-like cones at the Γ point. The crossing between the *s* and *d* bands (the *f* and *p* bands) are denoted as the green and purple curves in the phase diagram. These crossings lead to quadratic band touching at the Γ point. All these accidental band degeneracy’s and the properties of the nodal point will be discussed in details below.

Other important information revealed by Fig. 2 is the geometric conditions for the emergence of various nodal points in the photonic spectrum. These nodal points are important for several reasons. First, they are the mother states of topologically nontrivial states of photons. An example has been demonstrated above for the relation between the double Dirac cone and the photonic *Z*_{2} topological insulator. The Berry phase of each band for a loop *L*_{Γ} enclosing the Γ point
${\oint}_{{L}_{\mathrm{\Gamma}}}\u3008{\overrightarrow{\mathrm{\Psi}}}_{n\overrightarrow{k}}\left|i{\mathbf{\nabla}}_{\overrightarrow{k}}\right|{\overrightarrow{\mathrm{\Psi}}}_{n\overrightarrow{k}}\u3009$ (here |Ψ⃗* _{nk⃗}*〉 denotes the vectorial photonic Bloch wavefunction), is closely related to the formation of the photonic topological states [18,27]. Usually, topological nodal points are the critical states between normal band gaps and topological insulators. By introducing time-reversal/inversion symmetry breaking perturbations, the double-Dirac-cone state can become a photonic quantum anomalous Hall insulator (

*Z*topological insulator), a

*Z*

_{2}topological insulator, or a trivial photonic band gap material, depending on the specific gap opening perturbations [28]. Our study includes the triangle PhCs with dielectric rods as a special limit of

*R*

_{2}=0. From the phase diagram, it is clear that the

*p*-

*d*band inversion cannot be attained using dielectric rods with

*R*

_{2}=0. Thus the dielectric rod PhCs cannot support double Dirac cone. They can only support Dirac-like cone due to

*s*-

*p*degeneracy or quadratic band touching due to

*s*-

*d*degeneracy.

We plot in Fig. 3 the band structures of different kinds of accidental degeneracy’s marked by the black points in Fig. 2. In Figs. 3(a) and 3(b), the linear dispersion due to the accidental degeneracy of the singlet *s* mode and the doubly degenerate *p* modes are plotted, leading to the Dirac-like cone. The Dirac-like cone is associated with a cone and a flat band intersecting at the Dirac point. Its effective Hamiltonian is *ℋ* = *v*_{0}*k⃗* · *S⃗* where *S⃗* is the pseudo-spin 1 consisting of three modes *s*, *p _{x}*, and

*p*[20].

_{y}*v*

_{0}is the group velocity around the Dirac point. The upper, lower and flat bands have pseudo-spin along the wave vector direction as 1, −1, and 0, respectively.

The Dirac-like cone can emerge in PhCs with *R*_{2} = 0 (cylinder rods) as well as for finite *R*_{2} (hollow cylinders). The photonic dispersions for these two situations are illustrated in Figs. 3(a) and 3(b), respectively. Another Dirac-like cone dispersion is shown in Fig. 3(c) due to accidental degeneracy of the singlet *f* mode and the doubly degenerate *d* modes. Linear, conical dispersion appears for the *s*-*p* and *d*-*f* degeneracy’s because the parity of the *s* and *p* (*d* and *f*) modes are different. According to the *k⃗* · *P⃗* theory [29,30], the linear in *k* coupling between two photonic bands originates from the *P* matrix element which is finite only between bands with different parities. For bands with the same parity, such coupling is quadratic [29, 30]. The quadratic dispersion around the touching point of the *s* and *d* bands is shown in Fig. 3(d), while the quadratic band touching around the *f* -*p* degeneracy is shown in Fig. 3(e).

The accidental degeneracy of the *p* doublets and the *d* doublets [Fig. 3(f)] carries nontrivial topological properties which are the main focus of this work. Since the parity of the *p* modes and the *d* modes are different, the dispersion around the accidental degeneracy is conical. The two fold degeneracy’s of the *p* and *d* bands result in a double Dirac cone that resembles the dispersion of Dirac’s famous equation for electron and positron with vanishing mass in 2D systems [14,21].

We now exploit the *k⃗* · *P⃗* theory to derive the effective Hamiltonian for the photonic bands near the nodal points. The Maxwell equations for the TM modes can be written as

*n*is the band index and

*h⃗*(

_{n,k⃗}*r⃗*) is the Bloch function of the magnetic field of photon. The Bloch function is normalized as ${\int}_{u.c.}d\overrightarrow{r}{\overrightarrow{h}}_{{n}^{\prime},\overrightarrow{k}}^{*}(\overrightarrow{r})\cdot {\overrightarrow{h}}_{n,\overrightarrow{k}}(\overrightarrow{r})={\delta}_{n{n}^{\prime}}$ with

*u.c.*denoting the unit cell (i.e., integration in a unit cell). The Hermitian operator $\mathbf{\nabla}\times \frac{1}{\epsilon (\overrightarrow{r})}\mathbf{\nabla}\times $ can be regarded as the photonic Hamiltonian. Expanding the Bloch function

*h⃗*(

_{n,k⃗}*r⃗*) in the basis of the Bloch wavefunctions at the Γ point,

*h⃗*

_{n,0}(

*r⃗*), one can establish a

*k⃗*·

*P⃗*Hamiltonian,

*ω*

_{n,0}is the eigen-frequency of the

*n*band at the Γ point. The matrix element of

^{th}*P⃗*is given by

*P⃗*is nonzero only when the

*n*and

*n′*bands are of different parity. Using the above

*k⃗*·

*P⃗*theory, to the linear order in

*k⃗*, the effective Hamiltonian of the

*p*and

*d*bands is written in the basis of (

*p*

_{+},

*p*

_{−},

*d*

_{+},

*d*

_{−})

*as,*

^{T}*k*

_{±}=

*k*±

_{x}*ik*, and

_{y}*A*is the coupling coefficient. The double Dirac-cone appears at the situations with

*p*-

*d*degeneracy,

*ω*=

_{p}*ω*≡

_{d}*ω*

_{0}. The group velocity for the double Dirac-cone dispersion at Γ point is then $\pm \frac{\left|A\right|{c}^{2}}{2{\omega}_{0}}$ (positive group velocity for bands above the Dirac point, negative group velocity for bands below). The

*p*bands behave as the valence band and the

*d*bands behave as the conduction band in our PhCs. Note that the coupling between the

*p*and

*d*bands are within the same pseudo-spin, i.e., between

*p*

_{+}and

*d*

_{+}, or between

*p*

_{−}and

*d*

_{−}. The Berry phase for a loop circulating the Dirac point is ±

*π*for spin-up/down bands above the Dirac point. The total Berry phase is zero, in accordance with time-reversal symmetry.

The physics described by Eq. (5) resembles that of the quantum spin Hall effect in electronic systems [18]. The *p*-*d* inversion at the Γ point leads to the formation of photonic *Z*_{2} topological insulators which have helical edge states. This phenomenon is first discovered in [14]. The phase transition from normal photonic band gaps with trivial topology to the photonic *Z*_{2} topological insulator takes place at the black line in Fig. 2, where the double Dirac cone emerges. The key information in Fig. 2 is the appearance of two regions support photonic topological insulators (the two blue regions). This takes place for hollow dielectric cylinders with large outer and inner radii, or for hollow air cylinders with small inner radius. We remark that although Δ*ω _{pd}* can be quite large, ∼ 40%, for both normal and reversed structures, the complete photonic band gap is considerably smaller, particularly for the inverse structure. The photonic

*Z*

_{2}topological insulators have helical edge states which can enable unprecedented manipulation of light flow. For example, light propagation can be controlled by the orbital angular momentum. We shall discuss the properties of the edge states below.

It is natural to ask how the phase diagram changes when the permittivity ratio *ε*_{1}/*ε*_{2} is tuned. In most PhCs the photonic band gap increases with the permittivity ratio *ε*_{1}/*ε*_{2} [25]. Here we find that, quite interestingly, the phase boundary between the *Z*_{2} topological insulator and the normal photonic band gap changes negligibly for a *very broad range* of the permittivity ratio. This interesting property is demonstrated in Fig. 4 where we examine the phase boundary along the *R*_{2} axis for different permittivity ratio *ε*_{1}/*ε*_{2} for fixed outer radius *R*_{1}. The critical value of *R*_{2} where the double Dirac cone emerges is insensitive to the permittivity ratio *ε*_{1}/*ε*_{2} in a very wide range. Although the calculation is done for *R*_{1} = 0.45*a*_{1}, the observed behavior holds true for other values of the outer radius *R*_{1}. In fact, we have chosen the *R*_{1} in the calculation such that the dependence of the critical value of *R*_{2} on the permittivity ratio *ε*_{1}/*ε*_{2} is the strongest. The regions with negative value of ln(*ε*_{1}/*ε*_{2}) in Fig. 4 stands for the inverse structure PhC. The horizontal axis represents the case with *ε*_{1}/*ε*_{2} = 1, i.e., the homogenous limit. In the homogeneous limit the photonic band gaps vanish and all the photonic bands become plane waves. Thus they cannot be associated with the *s*, *p*, *d*, and *f* symmetries. The homogeneous limit is a singular limit for the discussion the topological nodal points and *p*-*d* inversion. This explains the discontinuity of, e.g., the *s*-*p* band crossing curve for the normal and inverse structure in the phase diagram upon crossing the homogeneous limit. Nonetheless, the *p*-*d* band crossing seems to undergo a “continuous transition” from the normal structure to the inverse structure. The possible physical scenario in approaching the homogeneous limit is that the plane-wave component continuously increase to 100%, while the *s*, *p*, *d*, *f* wave (the local Mie resonances of the hollow cylinder) components gradually vanish in approaching the homogeneous limit.

## 4. Edge states of photonic *Z*_{2} topological insulators

We now discuss the properties of the edge states of the photonic *Z*_{2} topological insulators. First, let us consider the *G* and *H* points in Fig. 2. The photonic bands *p* and *d* are normally ordered (trivial band topology) at the *G* point, whereas they are reversely ordered (*Z*_{2} band topology) at the *H* point. There is a common complete band gap marked by the cyan region in Figs. 5(a) and 5(b). When these two PhCs are put together, there is topology-induced edge states. The *Z*_{2} topology is protected by the pseudo-time-reversal symmetry, of which the operation is defined as *𝒯 _{p}* =

*iσ̂*

_{y}*K*(

*K*is the complex conjugation operator). Here

*σ̂*is an operator acting on the pseudo-spin space for both the

_{y}*p*and

*d*bands which can be written as a combination of the

*C*

_{6}symmetry operations (see [14]) to ensure that ${\mathcal{T}}_{p}^{2}=-1$. The

*Z*

_{2}band topology is protected by the double degeneracy at time-reversal invariant wavevectors induced by ${\mathcal{T}}_{p}^{2}=-1$ [26]. We remark that such anti-unitary operator can always be constructed whenever the double degeneracy of the

*p*and

*d*bands are kept [31]. Therefore, to simulate the

*Z*

_{2}band topology in PhCs, it is crucial to have two doubly degenerate bands with opposite parity (e.g., the

*p*and

*d*bands in our case).

The *C*_{6} symmetry breaks down at the boundary between the two different PhCs. Hence the boundary introduces perturbations that gap the helical edge states. The size of the gap depends on the strength of the perturbation. However, since the bulk topology of the two PhCs are distinctive and well-defined, the emergence of edge states is deterministic, although they may *not* be gapless. This is the essential feature for all bosonic analog of the *Z*_{2} topological insulators [32]. A typical calculation of the edge states using 16 unit cells of each type of PhCs (32 unit cells in total) via COMSOL MultiPhysics is presented in Fig. 5(c). The helical edge states are gapped at *k _{x}* = 0. Away from the

*k*= 0 point the helical feature of the edge states is clearly demonstrated in Figs. 5(d) and 5(e). The edge state at the

_{x}*B*point is mostly spin-down as recognized from the real space distribution of the Poynting vector. Similarly the

*A*point is mostly spin-up and it has positive group velocity, while the

*B*point has negative group velocity. Hence the helical character of the edge states is preserved away from the

*k*= 0 point.

_{x}From symmetry considerations, the boundaries break the pseudo-time-reversal symmetry but still keep the genuine time-reversal symmetry. The genuine time-reversal operator is *𝒯* = *σ̂ _{x}K̂* (since |↑〉 = |

*p*+

_{x}*ip*〉 and |↓〉 = |

_{y}*p*−

_{x}*ip*〉 for

_{y}*p*bands, while ↑〉 = |

*d*

_{x2−y2}+

*id*〉 and |↓〉 = |

_{xy}*d*

_{x2−y2}−

*id*〉 for the

_{xy}*d*bands). The spin-operators that are even under time-reversal transformation are

*σ̂*and

_{x}*σ̂*. Thus there can be two types of “mass terms” that gap the edge states. The general form of the edge Hamiltonian that obeys the (genuine) time-reversal symmetry is

_{y}*ℋ*=

_{edge}*vk*+

_{x}σ̂_{z}*m*+

_{x}σ̂_{x}*m*where

_{y}σ̂_{y}*v*is the group velocity at the Γ point,

*m*and

_{x}*m*are two real quantities. The magnitude of the two masses,

_{y}*m*and

_{x}*m*, depend on the specific geometry of the boundary. The energy gap at

_{y}*k*= 0 is $2\sqrt{{m}_{x}^{2}+{m}_{y}^{2}}$.

_{x}The edge thus has a single branch of massive Dirac spectrum with Chern number ±1/2 (sign depends on the mass terms). This extraordinary property can not be realized in a normal waveguide and is unique to the edge states of the *Z*_{2} topological photonic band gap materials. This is a realization of “parity anomaly” in quantum field theory for bosonic particles. If the signs of the mass terms can be modulated, there can be edge solitons emerging at the boundaries [33].

In situations where the two PhCs do not share a common photonic band gap. We can change the *ε*_{1} or *ε*_{2} of the structure to get the common photonic band gap. Since the *p*-*d* band inversion is insensitive to the permittivity (as shown in Fig. 4), such variation of the permittivity does not change the topological properties of the two PhCs. For example, if the PhC with normal band structure has *R*_{1} = 0.45*a*_{1}, *R*_{2} = 0.2*a*_{1}, *ε*_{1}=9, and *ε*_{2}=1 [see in Fig. 6(a)], while the PhC with nontrivial *Z*_{2} topology has *R*_{1} = 0.45*a*_{1}, *R*_{2} = 0.3*a*_{1}, *ε*_{1}=12, and *ε*_{2}=1 [see in Fig. 6(b)]. The two PhCs have a common band gap marked by the cyan ribbon. When those two PhCs are put together the topology-induced edge states are found [see in Fig. 6(c)].

## 5. Application potentials

One of the applications of the topological nodes, such as Dirac and Dirac-like points, is that they can be used as effective medium with unconventional refractive index. It was first found in [20] that PhCs with a Dirac-like point can be used as effective medium with zero refractive index when the frequency of light is at the Dirac point [20]. We now show that for the double Dirac cone, the effective refractive index of the PhC can be positive, zero, or negative, depending on the frequency of the light. By matching the frequency and wave vector parallel to the boundary, we find, in agreement with [34], that the effective refractive index is frequency dependent,

*θ*

_{1}and

*θ*

_{2}are the angle of incidence and refraction, respectively.

*ω*

_{0}is the frequency of the Dirac point. The above equation demonstrates that the effective refractive index can be tuned via the frequency. Both positive and negative refractive indices can be achieved, as indicated in Fig. 7. We emphasize that the effective refractive index here are in the range −1 <

*n*(

*ω*) < 1, which are unattainable for natural (lossless) materials. Therefore, the PhC with double Dirac cone can serve as a particular type of lossless metamaterial with unprecedented ability of manipulating light.

The edge states can be viewed as unconventional waveguides that correlates the direction of light flow and its group velocity with the angular momentum of light (as demonstrated in Fig. 5). In addition, when the frequency of the light is in the bulk photonic bands, the propagation of light are influenced by the Berry phase in the bulk bands due to the pseudo-spin-orbit coupling, which leads to the spin Hall effect of light [35,36]. The spin-dependent Berry phase can lead to anomalous velocity and shift in a scattering/reflection set-up [36]. Due to the time-reversal symmetry, these anomalous velocity and shift are opposite for opposite pseudo-spins. These properties can be exploited as angular-momentum selective transmission and filtering for advanced photonic circuits.

## 6. Conclusion and discussions

Using a simple architecture of core-shell triangle PhC with *C*_{6} point group symmetry we have systematically studied the evolution of the nodal points in photonic energy bands for various geometry and (isotropic) permittivity parameters. We show that such a simple PhC can support *Z*_{2} photonic topological insulators. The edge states of the *Z*_{2} topological insulators, unlike for electronic systems, are not ensured to be gapless. Nevertheless, they show properties similar to the helical edge states in electronic systems, such as spin-dependent propagation directions. We also give a full phase diagram for the topological nodal points, Dirac-like cones and double Dirac cones, for various geometric parameters. These topological nodal points are proximate to topologically nontrivial photonic band gaps. The physical origin of the topological nodal points as well as their properties such as Berry phase, (pseudo-)spin-orbit coupling are revealed. Dirac cones at *k⃗* = 0 can be exploited to tune the effective refractive index to the range −1 < *n*(*ω*) < 1 which are unattainable in natural, lossless materials. These properties, together with the Berry phases of photon, offer great potential for future advanced photonics.

## Funding

Soochow University; National Natural Science Foundation of China (NSFC) for Excellent Young Scientists (grant no. 61322504).

## Acknowledgments

We thank support from the faculty start-up funding of Soochow University. J.H.J thanks Sajeev John, Xiao Hu, and ZhiHong Hang for helpful discussions.

## References and links

**1. **F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. **100**(1), 013904 (2008). [CrossRef] [PubMed]

**2. **K. v. Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance,” Phys. Rev. Lett. **45**(6), 494–497 (1980). [CrossRef]

**3. **Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature (London) **461**(7265), 772–775 (2009). [CrossRef]

**4. **A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. **12**(3), 233–239 (2013). [CrossRef]

**5. **M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature (London) **496**(7444), 196–200 (2013); [CrossRef]

**6. **L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Weyl points and line nodes in gyroid photonic crystals,” Nat. Photon. **7**(4), 294–299 (2013). [CrossRef]

**7. **L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photon. **8**(11), 821–829 (2014). [CrossRef]

**8. **M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photon. **7**(12), 1001–1005 (2013). [CrossRef]

**9. **L. Lu, Z. Wang, D. Ye, L. Ran, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Experimental observation of Weyl points,” Science **349**(6248), 622–624 (2015). [CrossRef] [PubMed]

**10. **N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. **13**(2), 139–150 (2014). [CrossRef] [PubMed]

**11. **B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljačić, “Topological nature of optical bound states in the continuum,” Phys. Rev. Lett. **113**(25), 257401 (2014). [CrossRef]

**12. **W.-J. Chen, S.-J. Jiang, X.-D. Chen, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. **5**, 5782 (2014). [CrossRef] [PubMed]

**13. **W.-Y. He and C. T. Chan, “The emergence of Dirac points in photonic crystals with mirror symmetry,” Sci. Rep. **5**, 8186 (2014). [CrossRef]

**14. **L. H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. **114**(22), 223901 (2015). [CrossRef] [PubMed]

**15. **H.-X. Wang, L. Xu, H.-Y. Chen, and J.-H. Jiang, “Three-dimensional photonic Dirac points stabilized by point group symmetry,” Phys. Rev. B **93**, 235155 (2016). [CrossRef]

**16. **W.-J. Chen, M. Xiao, and C. T. Chan, “Experimental observation of robust surface states on photonic crystals possessing single and double Weyl points,” arXiv:1512.04681.

**17. **L. Lu, C. Fang, L. Fu, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Symmetry-protected topological photonic crystal in three dimensions,” Nat. Phys. **12**(4), 337–340 (2016). [CrossRef]

**18. **X.-L. Qi and S.-C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. **83**(4), 1057–1110 (2011). [CrossRef]

**19. **R. A. Sepkhanov, Ya. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A **75**(6), 063813 (2007). [CrossRef]

**20. **X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. **10**(8), 582–586 (2011). [CrossRef] [PubMed]

**21. **K. Sakoda, “Dirac cone in two- and three-dimensional metamaterials,” Opt. Express **20**(4), 3898–3917 (2012). [CrossRef] [PubMed]

**22. **E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, “Tight-binding parametrization for photonic band gap materials,” Phys. Rev. Lett. **81**(7), 1405–1408 (1998) [CrossRef]

**23. **K. P. Velikov, A. Moroz, and A. van Blaaderen, “Photonic crystals of core-shell colloidal particles,” Appl. Phys. Lett. **80**(1), 49–51 (2002). [CrossRef]

**24. **K. Sakoda, *Optical Properties of Photonic Crystals*, 2nd ed. (Springer, 2005).

**25. **J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, *Photonic Crystals: Molding the Flow of Light* (Princeton University, 2011).

**26. **L. Fu and C. L. Kane, “Topological insulators with inversion symmetry,” Phys. Rev. B **76**(4), 045302 (2007). [CrossRef]

**27. **J.-H. Jiang and S. Wu, “Non-Abelian topological superconductors from topological semimetals and related systems under the superconducting proximity effect,” J. Phys.: Condens. Matter **25**(5), 055701 (2013).

**28. **T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B **80**(15), 155103 (2009). [CrossRef]

**29. **Y. Li, Y. Wu, X. Chen, and J. Mei., “Selection rule for Dirac-like points in two-dimensional dielectric photonic crystals,” Opt. Express **21**(6), 7699–7711 (2013). [CrossRef] [PubMed]

**30. **J. Mei, Y. Wu, C. T. Chan, and Z.-Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals,” Phys. Rev. B **86**(3), 035141 (2012). [CrossRef]

**31. **J.-M. Hou and W. Chen, “Accidental degeneracy and hidden antiunitary symmetry protection: a novel Weyl semimetal phase as an example,” arXiv:1507.02024.

**32. **C. L. Kane and T. C. Lubensky, “Topological boundary modes in isostatic lattices,” Nat. Phys. **10**(1), 39–45 (2014). [CrossRef]

**33. **D.-H. Lee, G.-M. Zhang, and T. Xiang, “Edge solitons of topological insulators and fractionalized quasiparticles in two dimensions,” Phys. Rev. Lett. **99**(19), 196805 (2007). [CrossRef]

**34. **L. Wang, S.-K. Jian, and H. Yao, “Topological photonic crystal with equifrequency Weyl points,” arXiv:1511.09282.

**35. **M. Barkeshli and X.-L. Qi, “Topological response theory of doped topological insulators,” Phys. Rev. Lett. **107**(20), 206602 (2011). [CrossRef] [PubMed]

**36. **M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. **93**(8), 083901 (2004). [CrossRef] [PubMed]