Topological insulators, which host gapped bulk states and robust gapless edge states [1–4], brought new concepts and ideas in photonics in the past decade. Topologically protected photonic edge states can serve as robust waveguides, which have been demonstrated to be useful in integrated photonics [5–9], information transport [10–17], quantum photonics [18,19], lasing [20–28], and exciton-polariton devices [29–31]. The ability to visualize the wavefunctions of the bulk and edge photonic states as well as the controllability of photonic systems makes them a highly desirable platform for the study of topological phenomena.

Recently, it was predicted [30–39] and observed [40–68] that topological boundary states can emerge not only on the boundaries with n−1 dimensions, but also on the boundaries with n−2 and lower dimensions of n-dimensional (nD) topological insulators. Such exotic topological insulators are termed as higher-order topological insulators (HOTIs) [30–39]. For instance, a two-dimensional (2D) HOTI hosts 1D edge states at the edge boundaries as well as 0D corner states at the corner boundaries [30–39]. The underlying mechanism is that, due to the intricate role of the crystalline symmetry, the 1D edge states become gapped and hence can be regarded as emergent 1D insulators. At the corner boundaries between the edge boundaries, the 0D corner states emerge in the common band gap of the edge and bulk states due to edge or bulk band topology [30–68]. Such topologically protected multidimensional boundary states beyond the bulk-edge correspondence introduce new degrees of freedom in the design of photonic states for wave-guiding, trapping, and manipulation, which may lead to potential applications in integrated photonics, quantum photonics, and high-performance lasing [57–71]. However, such a discipline is still at its infant stage and yet to be fully developed. In particular, hexagonal photonic crystals (PhCs) with the six-fold rotation (C6) crystalline symmetry are well-known for hosting topological edge states [11,12,15,18,72–75] and can also support higher-order band topology [43,65]. Such PhCs, when made of all dielectric materials, host promising applications in advanced photonic and quantum technology [18,74,75].

In this work, we propose an all-dielectric hexagonal PhC that exhibits a topological band gap as the photonic analog of an unprecedented topological crystalline insulator. The unique band topology, as protected by the C6 crystalline symmetry, is manifested first by the coexisting gapped edge states and in-gap corner states, indicating a photonic HOTI. From topological band theory, the photonic HOTI hosts a fractional corner charge of Qc=2/3. We demonstrate, using the first-principle simulations, that such a fractional charge can be verified through the concept of local spectral charge as an analog of the local electric charge due to a band filling anomaly in electronic systems. Physically, the spectral charge measures how many photonic modes exist in a local area in a given frequency range. Exploiting such a concept, we further show that disclinations, topological defects that disrupt the crystalline rotation symmetry, can induce a fractional spectral charge Qdis=2/3 and trap localized photonic states around the disclination core. We emphasize that the all-dielectric PhC proposed here hosts what we believe, to the best of our knowledge, is an unprecedented higher-order band topology that is particularly reflected by the unique topological indices and the fractional spectral charge of 2/3. In comparison, previous studies focused on cases with 1/2 fractional spectral charges [76]. First-principle calculations give consistent results with the bulk-disclination correspondence picture predicted by the topological band theory. The unique topological phenomena found in this work can be readily observed in the state-of-the-art photonic experiments and may offer potential applications in topological quantum photonics and topological lasing.

This paper has seven sections. Section 2 focuses on the design of the PhC and its bulk topological indices. Section 3 studies the topological edge and corner states as well as the fractional corner charge. Section 4 explores the manifestation of the bulk-disclination correspondence in the PhC. Section 5 demonstrates the disclination in the trivial PhC. Section 6 shows the robustness of the disclination states. Section 7 gives the conclusions and outlooks.

The photonic band gaps discussed here are analogs of topological crystalline insulators protected by the C6 crystalline symmetry. In this context, the topological indices of the band gap can be deduced from the symmetry indicators of the bands below the gap. Using the theory in Ref. [39], the topological indices are given by χ(6)=(χM,χK).(1)

Here, the symmetry indicators are χM=#M1(2)−#Γ1(2) and χK=#K1(3)−#Γ1(3), respectively. The symbol #Πl(n) denotes the number of bands with the Cn rotation eigenvalue ei2π(l−1)/n (l=1,…,n) below the band gap at a high-symmetry point Π=Γ,M,K. The symbol χM (χK) thus stands for the change of the symmetry representations between the M (K) and the Γ points. For instance, χM represents the parity inversion between the Γ and M points, as generalized parity-inversion [1] (or generalized Fu–Kane [2]) indices. According to the symmetry eigenvalues at the high symmetry points [as illustrated in Figs. 1(b) and 1(c)], we find that our PhC has χ(6)=(0,−2), whereas Wu–Hu’s PhC (also studied recently in Ref. [76]) has χ(6)=(−2,0). Moreover, Wu–Hu’s PhC has a nontrivial second Stiefel–Whitney number ν=1 [i.e., 4m+2 (where m is an integer) odd-parity Bloch states at all time-reversal invariant momenta (the Γ point and the three M points)], whereas our PhC here has a trivial second Stiefel–Whitney number ν=0 (i.e., 4m odd-parity Bloch states at all time-reversal invariant momenta), according to Ref. [77].

Geometrically, the key difference between our PhC and Wu–Hu’s PhC is that in our PhC, the dielectric rods are aligned along the line from the corners to the center of the unit cell [see the brown dashed line in the inset of Fig. 1(a)]. In comparison, in Wu–Hu’s PhC the dielectric rods are aligned from the edge centers to the center of the unit cell, as shown by the green dotted line in the inset of Fig. 1(a).

The evolution of the lowest six photonic bands with the geometry parameter d (i.e., the distance between the rod center and the unit cell center), is presented in Figs. 1(d) and 1(e). From Fig. 1(e), one can see that only the band gap between the second and the third bands is topological, whereas the other band gaps are all trivial (i.e., these band gaps have trivial indices χM=χK=0). We find that the topological band gap is finite only when d>0.418a.

Although the results in this paper do not depend on the lattice constant (since the Maxwell’s equations are scale-invariant), most of the calculations here focus on the situation with a=25mm and d=0.47a (unless specified as other values), while the relative dielectric constant and the diameter of the rods are εd=24 and D=4mm, respectively. At microwave frequencies, such dielectric rods can be realized using zirconia ceramics. For these parameters, we find a bulk band gap ranging from 4.0 GHz to 5.15 GHz, yielding a large photonic band gap. Moreover, we find that for the same geometry parameters specified above, when the relative permittivity εd varies, the topological photonic band gap survives for permittivity larger than 9 [see Figs. 1(f) and 1(g)]. In the near-infrared regime, silicon has a permittivity as large as 13, which is sufficiently large to induce the topological photonic band gap proposed in this work.

In addition to the edge and corner states, topological band theory predicts that in finite systems, a fractional electric charge appears at the corner boundary due to the filling anomaly of the occupied bulk bands in electronic systems [39]. The fractional part of the corner charge, eQc, is connected to the bulk topological indices through [39] Qc=(14χM+16χK)mod1.(2)

The equation above states that the fractional corner charge is determined by the bulk band topology. In electronic systems, the fractional corner charge is directly related to the local electric charge. However, photons are neutral particles that lack such a property. Nevertheless, it has been shown that through the concept of “spectral charge,” the fractional corner charge can still be measured in bosonic systems [78].

To verify the fractional corner charge in photonic systems, we calculate the spectral charge for each unit cell in the structure illustrated in Fig. 3(c). Specifically, the spectral charge for the p-th unit cell in the structure is defined as Qp=∫0fgapdf∫p-thU.C.dr→ρ(f,r→),(3)where fgap is a frequency in the common band gap of the bulk and edge states, and ρ(f,r→) is the local density of states (LDOS) of photons (f is the frequency and r→ is the position vector). The integration over the coordinates is performed for the region of the p-th unit cell. The equation above is an analog of the electric charge for the p-th unit cell by band filling up to the Fermi energy in the bulk band gap in electronic systems. For topologically trivial insulators, for each unit cell Qp is equal to the number of bands below the band gap [39], which is consistent with the picture that each unit cell contributes Qp modes to form the bulk bands below the trivial band gap. However, for topological crystalline insulators, fractional spectral charges can appear due to band filling anomaly [39]. We remark that the applications of the PEC boundary are important here for the evaluation of the fractional mode charge. Particularly, the PEC boundary confines the photons in a finite supercell and makes the system closed and Hermitian; therefore, the boundary preserves the quantization of the fractional charges.

We calculate the photonic LDOS and obtain the spectral charges for all the unit cells. The photonic spectral charge for each unit cell can be obtained by integrating the photonic LDOS ρ(f,r→) up to a frequency in the band gap, fgap, as indicated by Eq. (3). The LDOS is calculated through the photonic eigenstates. If we denote the wavefunction of the i-th photonic eigenstate as ψi and the frequency of the eigenstate as fi, then the LDOS can be written as ρ(f,r→)=∑iΓπ[Γ2+(f−fi)2]|ψi(r→)|2.(4)

Here, Γ is a parameter used to model the Lorentz broadening of the eigenstates. In our calculation, Γ is set to be sufficiently small to converge the calculation. The normalized photonic wavefunction is given by |ψi(r→)|2=ε(r→)|Ez,i(r→)|2,(5)where ε(r→) is the relative permittivity and Ez,i(r→) is the rescaled electric field of the i-th photonic eigenstate that satisfies ∫dr→ε(r→)|Ez,i(r→)|2=1. The spectral charge is then given by Eq. (3), where the integration over the coordinates is performed for the region of the p-th unit cell.

Based on the calculation method above, we find that in the bulk region (gray), the spectral charge is nearly 2 for each unit cell in Fig. 3(b). In comparison, in the edge region (blue), the spectral charge is close to 1 for each unit cell, while in the corner region (red), the spectral charge is close to 2/3.

The above spectral charges can be understood through the Wannier centers. In our photonic system, there are only bulk states below the frequency fgap=4GHz. These bulk modes are pictorially represented by the Wannier centers away from the edge and corner boundaries. In each bulk unit cell, there are six Wannier centers locating at the corners of the unit cell. Each Wannier center is shared by three neighboring unit cells, thus contributing 1/3 spectral charge to one of these unit cells. Therefore, each bulk unit cell has a spectral charge of 2 (i.e., there are two photonic modes in each bulk unit cell). These two modes interact with the modes in other unit cells and form the two bulk Bloch bands below the band gap.

In comparison, there are three Wannier centers in an edge unit cell. The other three Wannier centers in the edge unit cell are obstructed by the edge boundary and become edge states. For a corner unit cell, there are only two Wannier centers associated with the bulk, while the other four Wannier centers are obstructed by the boundary and become edge and corner states. As a consequence, an edge unit cell has a spectral charge 1, while a corner unit cell has a spectral charge 2/3.

Our first-principle calculations give a spectral charge 1.07 for an edge unit cell that is close to the theoretical value 1. On the other hand, each corner unit cell has a spectral charge 0.75 that is not far away from the theoretical value 2/3. In general, because PhCs are in the continuum limit, the PEC boundaries may slightly affect the specific mode charges. The corner unit cells have three facets with the PEC boundaries and thus are more affected by such effects, compared to the edge unit cells that have two facets with the PEC boundaries. Note that the charge has to be fractionalized to integer times of 1/3 or 1/2, not to other fractional values, according to Eqs. (2) and (6). Therefore, the small deviations of the numerical value of the corner (and disclination) charge from the theoretical value do not cause a problem in identifying the correct fractional corner or disclination charges. The slight deviation of the spectral charges calculated using the first-principle methods from the theoretical spectral charges essentially originates from the fact that photons in PhCs do not strictly follow the tight-binding theory. Nevertheless, the fractional corner charge is still revealed approximately from the first-principle calculations.

To visualize the bulk-disclination correspondence in our PhC, we calculate the eigenstates and the LDOS of photons for the disclination structure in Fig. 4(a). The photonic spectrum [Fig. 4(b)] shows that there are five disclination states in the bulk band gap, in addition to the edge states. By integrating the photonic LDOS up to a frequency in the band gap, fgap=4GHz, we calculate the spectral charges for all the unit cells. Due to the five-fold rotation symmetry of the disclination structure, we present the spectral charges only for part of the disclination [see Fig. 4(a)]. Again, each bulk unit cell has a spectral charge close to 2, while each edge unit cell has a spectral charge approximately 1. The disclination unit cells have spectral charges of 1.4, which is close to the theoretical value of 4/3.

We calculate the spectral charge for each unit cell by integrating the photonic LDOS from zero up to a frequency fgap′=3.2GHz in the concerned photonic band gap, as shown in Fig. 5(b). The results in Fig. 5(a) indicate that all the spectral charges are close to 1. There is no signature of charge fractionalization in the disclination structure. Note that the charge has to be fractionalized to integer times of 1/3 or 1/2, not to other fractional values, according to the bulk-disclination correspondence [i.e., Eq. (6)]. Moreover, there is no disclination state. All the eigenstates are bulk states.

In this section, we study the robustness of the disclination states against two kinds of disorder: (i) defects that preserve the five-fold rotation symmetry of the disclination (i.e., at the center of the disclination core); and (ii) defects that break such rotation symmetry (i.e., away from the center of the disclination core). The defect is realized by inserting a dielectric rod with a radius of 2 mm but with varying relative permittivity. To facilitate the discussions, we remove the PEC boundary in the disclination core. Such a setup creates a hollow core in the disclination. As a consequence, a defect photonic mode emerges at the disclination core in the trivial PhC. In the meanwhile, there are still five disclination states localized around the disclination core.

We propose a hexagonal PhC with unconventional higher-order topology. In addition to the conventional spectral features of HOTIs (i.e., gapped edge states and in-gap corner states), the unique band topology here gives rise to a fractional charge of 2/3 at the corner boundaries, which is confirmed by the first-principle calculations through the concept of spectral charges. The spectral charges measure the number of photonic modes within a local area (e.g.,  a unit cell) for all the bulk states below the band gap. We also show that the bulk-disclination correspondence leads to a fractional spectral charge of 1/3 at the disclination core. Besides, we find that there are five localized states bound to the disclination, which is robust against disorders when compared to the conventional defect mode in PhCs. In contrast, the above phenomena disappear in trivial photonic band gaps.

The localized states bound to disclinations can be used as photonic cavity modes that are robust against disorders. Such robust subwavelength cavities are useful in integrated photonic systems as well as for quantum photonics. In addition, these cavity modes can also be exploited for lasing, as demonstrated in recent works. The fractional charges at the corners and disclination cores can be used to control the LDOS of the bulk photonic states. Our work may inspire future studies on similar phenomena and their applications in photonic and optoelectronic systems.

Acknowledgment. J.-H. Jiang is supported by the Jiangsu specially appointed professor funding, and a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).