Topological phases and phase transitions have been extensively studied in electronic [1,2], photonic [3], and acoustic [4,5] systems in the past decades. Recently, a new class of topological insulators, called higher-order topological insulators (HOTIs) that are characterized by higher-order bulk-boundary (e.g., bulk-corner or bulk-hinge) correspondence, were discovered [6–36]. HOTIs set up examples with multidimensional topological physics going beyond the bulk-edge correspondence in conventional topological insulators and semimetals and thus attract growing attention. Prototype HOTIs include quadrupole and octupole topological insulators [6–16,37,38], 3D HOTIs in electronic systems with topological hinge states [17–20], and HOTIs with quantized Wannier centers [21–34,39–41]. Among these prototype HOTIs, the breathing kagome lattice is regarded as an excellent platform to study higher-order topological phases and phase transitions. It was first proposed in Ref. [21], and subsequently experimentally realized in acoustic [22,23] and photonic [24,25] systems. In the breathing kagome lattice, the higher-order topology is characterized by the quantized bulk polarization (or the position of the Wannier center). When there is a mismatch between the Wannier center and the lattice site, the breathing kagome lattice becomes a higher-order topological phase and exhibits gapped edge states and in-gap corner states. On the contrary, the breathing kagome lattice becomes a topological trivial phase when the Wannier center overlaps with the lattice site. Despite extensive studies on HOTIs based on the breathing kagome lattice, most studies only distinguish the higher-order topological phases from the trivial phases. As a result, the distinctions between two higher-order topological phases and phase transitions have not yet been revealed.

Here, we study multiple higher-order topological phases and phase transitions in $C_3$ symmetric 2D photonic crystals (PhCs). We show that by moving the dielectric rods continuously, the $C_3$ symmetric PhCs can switch between triangle and kagome lattice configurations, leading to rich higher-order topological phases and phase transitions. Accompanying such phase transitions, the corner and edge states emerge or disappear, while the corner charge changes between 0 and 1/3. The topological indices for various phases are deduced from the symmetry indicators that are closely related to the fractional corner charge [42]. We also discuss the physical meaning of the fractional corner charge in the photonic context. The richness of the higher-order topological phases and their evolutions provide intriguing photonic phenomena and potential applications in topological photonics that can be readily realized in genuine materials.

We study 2D hexagonal PhCs of $C_3$ rotation symmetry, as illustrated in Fig. 1. The lattice vectors are denoted as ${\overrightarrow{a}}_1=(a,0)$ and ${\overrightarrow{a}}_2=(\frac{a}{2},\frac{\sqrt{3}a}{2})$, where $a$ is the lattice constant. The side length of the unit cell is denoted as $l=a/\sqrt{3}$. Throughout this study, we consider the $C_3$ symmetric PhC to consist of three identical dielectric rods with permittivity $ϵ=15$ and radii $r=0.1a$. The simplest configuration is the triangular lattice with a dielectric rod at the center of each unit cell [Fig. 1(a)], which can be regarded as a special case where three identical dielectric rods overlap with each other. By moving the three dielectric rods along the three symmetry lines, as indicated by the arrows in Fig. 1(b), the PhC undergoes a continuous geometry transformation that includes three triangular lattice configurations (denoted as triangular I, II, and III) and three kagome lattice configurations (denoted as kagome I, II, and III), as shown in Figs. 1(b)–1(f). The configurations between these six special cases are the breathing kagome lattices. The whole cycle of the continuous deformation encompasses $d$ from 0 to $3l$, as shown in Fig. 1. The kagome lattices are characterized by $d=(n+\frac{1}{2})l$ with $n=\mathrm{0,1,2}$, while the triangular lattices are characterized by $d=nl$ with $n=\mathrm{0,1,2}$.

Intuitively, as the dielectric rods move, the Wannier center changes. We consider the band gap between the first and the second photonic bands; therefore, there is only one Wannier center in the unit cell that can locate at the center ($\alpha$) or the corner of the unit cell ($\beta$ or $\gamma$), as shown in Fig. 1(g). The Wannier center positions of the triangular lattices with different configurations are discussed in detail in Appendix A. Unlike the positions of the dielectric rods, the Wannier center is constrained by the crystalline symmetry and thus cannot continuously change. The change of the Wannier center is nonadiabatic, which must be achieved by the closing and reopening of the band gap, as shown in detail below.

We first provide the photonic band structures for nine prototype cases in Figs. 2(a)–2(c), where we use $c/a$ as the frequency unit ($c$ is the speed of light in vacuum). Throughout this work, we focus on the low-lying photonic bands due to the transverse magnetic (TM) harmonic modes. All the numerical simulations are carried out using the finite element numerical solver COMSOL Multiphysics. The photonic bands for the triangular I, II, and III configurations are shown in Fig. 2(a), which indicates that the three triangular configurations have an identical band structure. This phenomenon happens because the three triangular configurations differ only by a partial lattice translation, while the structure of the 2D array of the dielectric rods remains the same. However, the symmetry representations of the photonic bands are distinct for the three triangular configurations. Hence, their topological properties are different. In particular, the location of the Wannier center is distinct for the three triangular configurations, as revealed below.

Similarly, the photonic band structures for the kagome I, II, and III configurations are identical because they can be related to each other by partial lattice translations, as shown in Fig. 2(b). Such translations change the location of the Wannier center as well as the symmetry representations of the Bloch bands and their topological properties.

Furthermore, as shown in Fig. 2(d), the photonic band structure is identical, if two configurations differ by an integer time of $l$ in the geometry parameter $d$. Since the lattice periodicity for the translation along the black arrows in Fig. 1(b) is 3l, there are three different configurations with the same photonic band structure, where the geometry parameter $d$ differs by an integer time of $l$. The translation of $l$ along the black arrows in Fig. 1(b) shifts the unit cell center to the unit cell corner without changing the pattern of the 2D array of the dielectric rods. The photonic band structure is insensitive to such global shifts. Therefore, configurations that differ by an integer time of $l$ in the geometry parameter $d$ have the same photonic band structure. To further demonstrate such a periodicity of the photonic band structure, we present the photonic bands for three breathing kagome configurations with $d=0.25l$, 1.25l, and $d=2.25l$ in Fig. 2(c). It is seen that their photonic band structures are identical to each other.

The evolution of the first two photonic bands at the $K$ point [i.e., $\mathbf{k}=(\frac{4\pi }{3},0)$] with the geometry parameter $d$ is summarized in Fig. 2(d). There are three topologically distinct photonic band gaps (regions painted by different colors) that are characterized by three different locations of the Wannier center, as indicated in Fig. 2(d). The band gap between the first two bands closes and reopens during the change of the parameter $d$. We find that the band gap closes at the kagome I, II, and III configurations where $d=(n+\frac{1}{2})l$ with $n=\mathrm{0,1,2}$, separately. These three d separate the whole region $d\in [\mathrm{0,}3l]$ (bearing in mind that the parameter $d$ is modulo $3l$, since $3l$ corresponds to a lattice periodic translation) into three topologically distinct phases: $d\in (-0.5l,0.5l)$, where the Wannier center is at $\alpha$; $d\in (0.5l,1.5l)$, where the Wannier center is at $\beta$; and $d\in (1.5l,2.5l)$, where the Wannier center is at $\gamma$.

The symmetry representation of the first photonic band at the $K$ point is depicted by the phase profile of the electric field $E_z$, which is shown in Fig. 2(d) for several d. We find that the $C_3$ symmetry eigenvalue does not change within the same phase. Upon the topological phase transitions (i.e., the band gap closing and reopening), the symmetry eigenvalue changes abruptly. We use the symmetry indicators to characterize the bulk band topology. Following Ref. [42], the topological crystalline index can be expressed by the full set of the $C_3$ eigenvalues at the high-symmetry points (HSPs). For an HSP denoted by the symbol $\mathrm{\Pi }$, the $C_3$ eigenvalues can only be ${\mathrm{\Pi }}_n=e^{i2\pi (n-1)/3}$ with $n=\mathrm{1,2,3}$. Here, the HSPs include the $\mathrm{\Gamma }$, $K$, and $K^{\prime }$ points. The full set of $C_3$ eigenvalues at the HSPs are redundant due to the time-reversal symmetry and the conservation of the number of bands below the band gap. The minimum set of indices that describe the band topology is given by [42] $${[K]}_n=\#K_n-\#{\mathrm{\Gamma }}_n,n=\mathrm{1,2},$$(1)where $\#K_n$ ($\#{\mathrm{\Gamma }}_n$) is the number of bands below the band gap with the $C_3$ symmetry eigenvalue $K_n$ (${\mathrm{\Gamma }}_n$) at the $K$ ($\mathrm{\Gamma }$) point. In this scheme, the $\mathrm{\Gamma }$ point is taken as the reference point to get rid of the redundance. For the trivial atomic insulators (i.e., the band gap formed by uncoupled atoms), all the HSPs have exactly the same symmetry eigenvalues. Therefore, the trivial atomic insulators have $[{\mathrm{\Pi }}_n]=0$ for all the HSPs. In contrast, any nonzero $[{\mathrm{\Pi }}_n]$ indicates a topological band gap that is adiabatically disconnected from the trivial atomic insulator.

For the $C_3$ symmetric PhCs, the topological indices can be written in a compact form as $$\chi =[[K_1],[K_2]].$$(2)We find that for all d, $\#{\mathrm{\Gamma }}_1=1$ and $\#{\mathrm{\Gamma }}_2=0$. Furthermore, from the $C_3$ eigenvalue at the $K$ point, as indicated in Fig. 2(d), we find that the topological indices for the three parameter regions are: $\chi =[\mathrm{0,0}]$ for $d\in (-0.5l,0.5l)$; $\chi =[-\mathrm{1,1}]$ for $d\in (0.5l,1.5l)$; and $\chi =[-\mathrm{1,0}]$ for $d\in (1.5l,2.5l)$.

Both the phase with $\chi =[-\mathrm{1,1}]$ and that with $\chi =[-\mathrm{1,0}]$ are higher-order topological phases that host gapped edge states and in-gap corner states. In contrast, the phase with $\chi =[\mathrm{0,0}]$ is the trivial phase. To demonstrate the higher-order topology, we construct a large triangular supercell that is schematically shown in Fig. 3(a). In the supercell, the inside is the phase that we study, while the outside is the trivial band gap phase with $d=0.25l$. The side length of the supercell is $10a$, while the inside structure has a side length of $4a$. The whole structure is surrounded by the PEC (i.e., perfect electric conductor) boundary condition that is physically a hard-wall boundary for photons.

We study the evolution of the edge and corner states when the parameter $d$ of the inside PhC structure goes from 0 to $3l$. Several prototype geometries are shown in Fig. 3(a). The results are presented systematically in Fig. 3(b). Figure 3(c) gives the electric field $|E_z|$ distributions of the eigenstates. Throughout this paper, the electric field patterns of the corner states are given by the superposition of $|E_z|$ on the three degenerate corner states. From the figure, it is seen that the edge and corner states emerge only in the region with $0.5l<d<2.5l$ (i.e., the two higher-order topological phases). In particular, in the region with $0.5l<d<1.5l$, two types of edge states emerge, as revealed previously in Ref. [25]: type-I corner states [denoted by the purple curve in Fig. 3(b), and two examples (“A” and “C”) are shown in Fig. 3(c)] due to the nearest neighbor couplings; and type-II corner states [denoted as the blue curve in Fig. 3(b), and two examples (“B” and “D”) are shown in Fig. 3(c)] due to long-range couplings. As the common band gap between the inside and outside structures becomes smaller, at $d=l$, the type-II corner states are much less localized and become similar-looking edge states [see “D” in Fig. 3(c)]. In contrast, the type-I corner states remain well-localized and distinguishable from the edge states [denoted in green in Fig. 3(b)] and bulk states [denoted in gray in Fig. 3(b)]. This indicates that the type-I corner states are due to the bulk topology, while the type-II corner states may originate from long-range couplings between the adjacent edge states.

The region with $1.5l<d<2.5l$ has not yet been studied in the literature. We find that in this region the type-II corner states are hardly seen. Meanwhile, there are two sets of type-I corner states. Each set has three degenerate corner states. One set has a frequency higher than the edge states, while the other set has a frequency lower than the edge states. In both sets, the wave functions of the corner states are well-localized around the corners, distinguishable from the edge states (the green band) and the bulk states (the gray bands). Examples of the corner and edge wave functions at $d=1.9l$ and $d=2.25l$ are shown in Fig. 3(c). In some cases, for instance, $d=2.25l$, type-II corner states can be found. However, the wave functions are not well-localized at the corners.

In addition, we consider another type of supercell by exchanging the outer and inner PhCs of the above supercell. As shown in Fig. 4(a), the trivial PhCs with phase $\chi =[\mathrm{0,0}]$ are surrounded by the PhCs with adjustable parameter $d$. The side length of the supercell is $10a$, while inside structure has a side length of $4a$.

We then study the evolution of the edge and corner states when the parameter $d$ of the outside PhC structure goes from $0$ to $3l$. Four prototypes of supercells are shown in Fig. 4(a). The eigenfrequencies of the photons as functions of the geometry parameter $d$ are displayed in Fig. 4(b). As expected, the edge and corner states only emerge in the region with $0.5l<d<2.5l$ (i.e., the two higher-order topological phases). Figure 4(c) further gives the electric field $|E_z|$ distributions of the eigenstates. From the figures, it is seen that there are both type-I corner states [denoted by the purple curve] and type-II corner states [denoted by the blue curves] emerging in the region with $0.5l<d<1.5l$. Examples of type-I (“C” and “F”) and type-II corner states (“A,” “B,” “D,” and “E”) are shown in Fig. 4(c). The fundamental difference between the field patterns of the type-I corner states and that of the type-II corner states also indicates that they have distinct origin, as revealed previously in Ref. [25].

For the region with $1.5l<d<2.5l$, it is seen that two corner modes split off from the edge states continuum. The corresponding field patterns (“G,” “I,” and “J”) in the Fig. 4(c) resemble those of the topological edge states, but exponentially decay away from the corners, indicating that they are type-II corner states. Interestingly, we also find that there only exist type-II corner states while the type-I corner states are absent in this region.

We now explore the corner and edge states in another type of supercell. We design the supercell in such a way that the inner structure is a PhC with parameter $d$ while the outer structure is the PhC with parameter $3l-d$; that is, we consider the edge and corner boundaries between complementary PhC structures. Such a supercell architecture will induce intriguing edge and corner boundaries [e.g.,  various zigzag edge boundaries, as depicted schematically in Fig. 5(a)]. The evolutions of the bulk, edge, and corner states are systematically summarized in Figs. 5(b) and 5(c).

In this type of supercell, the edge and corner states emerge only in the two topological regions, $0.5l<d<1.5l$ and $1.5l<d<2.5l$, as shown in Fig. 5(b). In the region with $0.5l<d<1.5l$, the inner PhC has the topological index $\chi =[-\mathrm{1,1}]$, while the outer PhC has the topological index $\chi =[-\mathrm{1,0}]$. In the other region with $1.5l<d<2.5l$, the inner PhC has $\chi =[-\mathrm{1,0}]$ and the outer PhC has $\chi =[-\mathrm{1,1}]$. The emergence of the edge and corner states, which has not yet been discovered in the literature, reveals that higher-order topological phenomena can appear at the boundaries between two topologically distinct higher-order phases. This is consistent with the topological band theory [42,43] and the Wannier center picture.

From Fig. 5(b), the bulk band gap closing is clearly seen at the phase transition points, $d=0.5l$, 1.5l, and 2.5l. Interestingly, type-II corner states can be found only in the higher-order phase with $\chi =[-\mathrm{1,1}]$ (i.e., the phase studied in Ref. [25]; here $0.5l<d<1.5l$), but not in the higher-order phase with $\chi =[-\mathrm{1,0}]$ (i.e., $1.5l<d<2.5l$). This finding can be demonstrated by the tight-binding approach, which is discussed in detail in Appendix B. For all cases in the region $0.5l<d<2.5l$, the edge states are clearly visible [Fig. 5(c)]. The bandwidth of the edge states is considerably larger in this type of supercell compared to the supercell studied in Fig. 3. As a consequence, the corner states are less localized, particularly in the region with $1.5l<d<2.5l$ where the corner states live in the small band gap between the edge and bulk states.

We now show that the higher-order band topology can also be manifested in the fractional corner charge. Even though we are considering photonic bands and photonic states in this work, it is possible to define an analog of “charge” through the local density of states (LDOS), ${\rho }_e(\mathbf{r},E)$. In electronic systems, the charge contributed by the filling of the valence bands in the $j$-th unit cell is given by $$Q_{j,e}=e{\int }^{E_{\mathrm{gap}}}\mathrm{d}E{\int }_j\mathrm{d}\mathbf{r}{\rho }_e(\mathbf{r},E).$$(3)Here, $e$ is the charge of an electron, $E_{\mathrm{gap}}$ is an energy in the topological band gap that is below the eigenenergies of the edge and corner states. In the above equation, the integration over the position $\mathbf{r}$ is defined within the $j$-th unit cell. The filling of all the valence bands below the topological band gap contributes a fractional charge in the corner region. It was predicted [42] that the fractional corner charge $e{Q}_c$ is completely determined by the topological indices of the bulk bands as $$Q_c=-\frac{1}{3}([K_1]+[K_2])\mathrm{mod}1.$$(4)The actual size of the corner region depends on the specific model. However, one can often choose one or a few unit cells around the corner boundary to converge the fractional corner charge.

We then check the theoretical prediction in photonic system by calculating the quantity $$Q_j={\int }_0^{f_{\mathrm{gap}}}\mathrm{d}f{\int }_j\mathrm{d}\mathbf{r}{\rho }_p(\mathbf{r},f),$$(5)which is the analog of the “charge” in the $j$-th unit cell in the photonic system. Here, we omitted the elementary charge $e$, which does not have a physical meaning in photonics. The integration over the frequency is from 0 to a frequency in the band gap $f_{\mathrm{gap}}$ that is below the eigenfrequencies of the edge and corner states. The photonic LDOS is calculated through the following spectral decomposition of all the photonic eigenstates of the valence bulk bands: $${\rho }_p(\mathbf{r},f)=\sum _n\frac{\mathrm{\Gamma }}{\pi }\frac{ϵ(\mathbf{r})|E_z^{(n)}(\mathbf{r}{)|}^2}{{(f-f_n)}^2+{\mathrm{\Gamma }}^2}.$$(6)Here, $n$ labels the photonic eigenstates of the valence bulk bands, and $\mathrm{\Gamma }\to 0$ is a sufficiently small number that converges the calculation. $E_z^{(n)}(\mathbf{r})$ is the scaled electric field distribution of the $n$-th photonic eigenstate that satisfies the normalization condition, $$1=\int \mathrm{d}\mathbf{r}ϵ(\mathbf{r})|E_z^{(n)}(\mathbf{r}{)|}^2,$$(7)where the integral of $\mathbf{r}$ is over the whole photonic system, and $ϵ(\mathbf{r})$ is the position-dependent relative permittivity.

The photonic “charge” defined above does have a physical meaning. It represents the number of the photonic modes contributed from the $j$-th unit cell from the valence bulk bands. We calculate the photonic “charge” for each unit cell and present the results in Fig. 6 for various configurations.

For all four cases considered in Fig. 6, the calculated charge for the bulk unit cells is close to 1. This is consistent with the fact that there is only one band below the band gap [i.e., each unit cell contributes a single charge (mode) to the bulk band]. Figures 6(a) and 6(c) show that for both $d=0.25l$ and $d=2.75l$ (i.e., $\chi =[\mathrm{0,0}]$), the fractional corner charge is zero, since the corner unit cell has a charge very close to 1. Figure 6(b) shows that for $d=0.75l$ (i.e., $\chi =[-\mathrm{1,1}]$), the fractional corner charge is $Q_c=0$, which is indicated by that the corner unit cell has a charge very close to 0. In this case, despite that the band gap carries higher-order topology and the resultant corner states, the corner charge vanishes, which is consistent with the theoretical prediction given in Eq. (4). The origin of the corner states in this case can be explained in detail in the previous study [25]. Figure 6(d) shows that for $d=2.25l$ (i.e., $\chi =[-\mathrm{1,0}]$), the fractional corner charge is $Q_c=1/3$ which is manifested by the fact that the corner unit cell has a charge very close to $1/3$. These photonic charges can be measured through the classical or quantum versions of the Purcell effect, as indicated by Ref. [44].

In conclusion, we demonstrate that rich higher-order topological phases and multiple phase transitions can be obtained in $C_3$ symmetric PhCs by tuning a single geometry parameter $d$. These higher-order topological phases yield intriguing multidimensional topological phenomena where the corner and edge states can be tuned in versatile ways. Our study shows that continuously configurable dielectric PhCs [45] can be useful in generating topological photonic circuits with tunable edge and corner states. The emergent fractional photonic charge indicates that photonic systems can be powerful in revealing the fundamental properties of topological bands.

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