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Abstract
We systematically study the first- and second-order band topologies, which are tied to the pseudospin and valley degree of freedoms (DOFs), in honeycomb-kagome photonic crystals (HKPCs). We first demonstrate the quantum spin Hall phase as the first-order pseudospin-induced topology in HKPCs by observing the partial pseudospin-momentum locked edge states. By employing the topological crystalline index, we also discover the multiple corner states emerging in the hexagon-shaped supercell as the manifestation of the second-order pseudospin-induced topology in HKPCs. Next, by gapping the Dirac points, a lower band gap associated with the valley DOF emerges, in which the valley-momentum locked edge states are observed as the first-order valley-induced topology. Such HKPCs without inversion symmetry are proved to be Wannier-type second-order topological insulators, which manifested with valley-selective corner states. Additionally, we also discuss the symmetry breaking effect on pseudospin-momentum locked edge states. Our work realizes both pseudospin-induced and valley-induced topologies in a higher-order manner and thus provides more flexibility in manipulating electromagnetic waves, which may find potential applications in topological routings.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Photonic crystals (PCs) are periodic optical structures consisting of dielectric media, in which the electromagnetic wave behaves in a similar way as the electron wave in crystalline materials [1,2]. Since the dielectric media can be manufactured with different materials and formed in any periodic arrays, the PCs provide a versatile platform for manipulating light as well as realizing various functional devices [3]. Recently, the extensive studies of the topological phases in PCs give birth to topological PCs [4–7]. The first topological photonics is a photonic analog of quantum Hall state [8–10], which was demonstrated in the gyromagnetic PCs in the microwave regime [11]. However, the use of magneto-optic material hinders its further application due to the weak magneto-optical effect in the optical frequency.
To overcome this shortage, topological photonics using all-dielectric PCs attract much attention. These include photonic Floquet topological insulators [12–15], photonic quantum spin Hall insulators [16–23], photonic valley Hall insulators [24–33], and higher-order topological insulators [34–44]. By taking the advantage of the crystalline symmetry of the all-dielectric PCs, the pseudospin and valley degree of freedoms could be synthesized (or introduced), leading to the realization of the photonic quantum spin Hall insulators and valley Hall insulators. These topological insulators featured with unique topological edge states, which enable topological wave transmission and hence greatly extend the flexibility in manipulating electromagnetic waves. Very recently, the concept of higher-order topological insulators has been introduced in all-dielectric PCs, with a signature of the robust lower-dimensional corner or hinge states emerging in the gap. Interestingly, it is recognized that the interplay between higher-order topology and pseudospin and valley DOFs would result in pseudospin-dependent higher-order topological states [38–41]. These photonic higher-order topological states may find potential applications in the wave location, such as topological optical switches, energy splitters, topological lasings [45,46], and light rainbow trappings [47].
Although topological PCs based on pseudospin or valley DOFs have been extensively studied, most of them focus on the first- (or second-) order topology based on a single DOF. The discussion of the first- or/and second-order topologies with two DOFs is scarce. It is reported that the valley and pseudospin edge states as the manifestation of the first-order topology can coexist in a single system [48–50], while the second-order topology in these systems remains unexplored. In this work, we systematically study the first- and second-order band topologies, which are tied to the pseudospin and valley DOFs in HKPCs. By tuning the geometric parameters, a complete band gap hosts quantum spin Hall phase is demonstrated, in which the edge states exhibit partial pseudospin-momentum correlation and are gapped due to the reduced symmetry at the edges. These gapped edge states further stabilize the topological corner states emerging in the band gap. When the symmetry is reduced from $C_{6v}$ to $C_3$, a lower band gap emerges due to the gapping out of Dirac points associated with the valley DOF. Such a gap supports both valley-polarized edge states at domain wall interfaces and valley-selective corner states at triangular-/hexagon-shaped supercells. We also discuss the effect of symmetry breaking on pseudospin-momentum locked edge states. We enclose our work by presenting a potential application of frequency-dependent topological routings.
2. Honeycomb-kagome photonic crystals
As shown in Fig. 1(a), the HKPC forms a composite lattice that consists of both honeycomb and kagome sublattices with the same lattice vectors $\vec {a}_1$, $\vec {a}_2$ and lattice constant $a$. The adopted primitive cell (the side length is denoted as $L$) includes two sets of all-dielectric rods with relative permittivity $\epsilon =8.9$. One set (indicated by red circles with radius $R_h$) is arrayed in the honeycomb sublattice, whereas the other set (indicated by the blue circles with radius $R_k$) is arrayed in the kagome sublattice. To control the spatial symmetry of the HKPC, we introduce another geometric parameter $D$, which describes the displacement of the all-dielectric rods at the kagome sites away from their centers. For simplicity, we denote that $D>0 (D<0)$ when kagome sites move along (opposite to) the direction indicated by the arrows. Apparently, the primitive cell hosts $C_{6v}$ symmetry when $D=0$, while reduce into $C_{3}$ symmetry when $D \neq 0$ [see the dashed circle].
Fig. 1. (a) Schematic of HKPCs arranged in a hybrid lattice that consisting of honeycomb and kagome sublattices with same lattice vectors of $\vec {a}_1$ and $\vec {a}_2$. Right panel: the adopted primitive cell. (b) Left panel: the band structure of HKPC with $R_h=R_k=0.08a$ and $D=0$. Inset: the first Brillouin zone. Right panel: the eigenstates of the $d$-doublets, $p$-doublets, and $f$ singlet at $\Gamma$ point. (c) The frequency evolution of the $p,d,f$ modes versus the geometric ratio $R_k/R_h$ with $R_h=0.08a$. The purple area refers to the complete band gap. (d) The frequency evolution of the first two bands at $K (K^\prime )$ points versus geometric ratio $D/L$ with $R_k=R_h=0.08a$. The colored areas refer to the complete band gaps with different valley Hall phases.
To start, we consider the HKPC with $R_k=R_h=0.08a$ and $D=0$. Throughout this work, we use $\frac {c}{a}$ as the frequency unit ($c$ is the speed of light in vacuum) and only consider the transverse magnetic (TM) modes, i.e., the magnetic field in the $xy$-plane and out-of-plane $E_z$ are finite while components else are zero. All the numerical simulations of the PCs are performed using the module "electromagnetic waves, frequency domain" of the commercial software COMSOL Multiphysics. As shown in Fig. 1(b), the band structure hosts a Dirac point between the first and second bands at $K$ point protected by $C_{3v}$ symmetry. In addition, by carefully checking the eigenstates at $\Gamma$ point, we find a pair of dipole modes (also known as $p$-doublets) appear at the higher frequency, whereas a pair of quadrupole modes (also known as $d$-doublets) and a hexapole mode (also known as $f$-singlet) appear at lower frequency [see the left panel of Fig. 1(b)]. Since these modes originate from the two-dimensional irreducible representation of the $C_{6v}$ symmetry and therefore, cannot be removed as long as the symmetry remains unchanged. Specifically, the degenerate dipole (quadrupole) modes can form a new representation with finite orbital angular momentums (OAMs), as we will elaborately discuss in the Sec. 3.
To unveil the possible topological phases transition in the HKPCs, we first study the frequency evolution of the $p,d,f$ modes at $\Gamma$ point versus the geometric ratio $R_k/R_h$ [see Fig. 1(c)]. It is seen that the $d$-doublets have lower frequencies than that of $p$-doublets, which is a hallmark of the quantum spin Hall phase [51]. Note that a complete band gap indicated by the purple region emerges after passing the critical value $R_k/R_h=1.04$. We remark that such a band gap hosts quantum spin Hall phase in spite of the existence of $f$-singlet band between $d$- and $p$-doublets bands [18]. In addition, Fig. 1(d) displays the frequency evolution of the lowest two bands at $K$ and $K^\prime$ points versus geometric ratio $D/L$. When the kagome sites move away from its center, i.e., $D \neq 0$, the Dirac point is gapped and forms two complete band gaps (indicated by the colored regions), which are associated with valley DOF. Evidently, the frequency order of the lowest two bands at $K$ and $K^\prime$ points, which are locked to the phase distributions of electric fields with specific chirality (see details in Sec. 4.), is inverted after passing the Dirac point. As a signal of valley Hall phase transition, such a band inversion can be explained by a massive Dirac equation, which gives
where $\sigma _i (i=x,y,z)$ are Pauli matrices acting on the valley subspace, $v_D$ is the group velocity, $(k_x,k_y)$ is the momentum deviation from the $K$ point. Consistent with the above band inversion picture, the sign of the effective mass $m$ characterizes two different valley phases separated by the Dirac point, as we will elaborately discuss in Sec. 4.3. First- and second-order pseudospin-induced topologies in the HKPCs
We first focus on pseudospin-induced topology in the HKPCs. As an illustration, a typical HKPC with $R_k=0.14a, R_h=0.04a$, and $D=0$ (termed as HKPC0), which the band structure associated with the electric field $E_z$ distributions at $\Gamma$ point are displayed in Fig. 2(a). It is seen that a pair of dipole modes (i.e., $p_x$ and $p_y$ modes) appear at the higher frequency, whereas a pair of quadrupole modes (i.e., $d_{xy}$ and $d_{x^2-y^2}$ modes) appear at a lower frequency, which can form a new representation with finite OAMs, i.e., $d_\pm =d_{x^2-y^2} \pm id_{xy}$ and $p_{\pm }=p_x \pm i p_y$. Note that these photonic OAMs play a role of the pseudospin DOF, which enables an analog with the Bernevig-Hughes-Zhang model for the quantum spin Hall insulators within the framework of the $k\cdot p$ theory around the $\Gamma$ point [18,51]. Regardless that a trivial band emerges in the gap, we remark the topological nature of the band gap remains unchanged. Since the crystalline symmetry plays a vital role on the formation of the topological band gap, we utilize the symmetry-eigenvalues at high-symmetry points (HSPs) of the Brillouin zone to characterize the band topology [52]. Figure 2(b) displays the phase profiles of the electric field patterns at $\Gamma, M, K$, from which gives the rotation eigenvalues as well as the topological crystalline index. In parallel, a honeycomb photonic crystal (HPC) with $R_h=0.13a$ hosts a common band gap with that of HKPC0 is designed [see the band structure in Fig. 2(c)], of which the trivial topological property can be demonstrated via the phase profiles of the electric field patterns at $\Gamma, M, K$ [see Fig. 2(d)].
Fig. 2. (a) Photonic band structures of HKPC0 with $R_k=0.14a, R_h=0.04a$ and $D=0$. (b) The phase profiles of the electric fields, $arg(E_z)$, of the photonic Bloch functions below the band gap at the HSPs for HKPC0. The rotation eigenvalues are also labeled. (c) Photonic band structures of HPC with $R_h=0.13a$. (d) same with (b) except for HPC.
According to the bulk-edge correspondence, topological edge states can be formed at the interface between two PCs with different topologies. To this end, a supercell consisting of HKPC0s and HPCs is employed to calculate the topological edge states. As can be seen from the simulated band structure in Fig. 3(a), a pair of topological edge states featured a tiny gap crossing almost the whole band gap. Since the boundary configuration cannot preserve the $C_6$ symmetry, the gap opening in the edge states are inevitable. Remarkably, such an edge gap opening induces mixing between the edge states with opposite pseudospins. As a consequence, the edge states become partially spin-momentum locked: away from the time-reversal invariant momenta (i.e., $k_x=0, \frac {\pi }{a}$), the pseudospin polarization is still prominent, while close to the time-reversal invariant momenta the pseudospin polarization is suppressed. At time-reversal invariant momenta, the pseudospin polarization strictly vanishes due to time-reversal symmetry. To elucidate the pseudospin DOF, we present the distribution of the Poynting vectors as well as the phase distribution of the electric field patterns at two marked points of the edge states. As shown in Fig. 3(b), away from the time-reversal invariant wave vectors (e.g., $k_x = \pm 0.55\frac {\pi }{a}$), the finite angular momenta are manifested in the phase vortices of the photonic edge wave functions as well the winding of the Poynting vectors [see green arrows]. Specifically, the direction of energy flow depends on the sign of the photonic OAMs, implying the unique pseudospin-momentum locking behavior.
Fig. 3. (a) Photonic band structures of the edge boundary between HKPC0 ad HPC for the zigzag edge along the $x$ direction. Right panel: the schematic of the ribbon-shaped supercell consisting of HKPC0 and HPC. (b) Phase distribution and amplitudes of the electric field patterns for the edge states $A$ with pseudospin up and the edge state $B$ with pseudospin down. (c,d) The simulation of the OAM-selective excitation of the edge states at a frequency of $0.55\frac {c}{a}$, where the details of the point source with specific phase winding are illustrated in the inset. (e,f) Suppressed OAM-selectively in the excitation of the edge states simulated at (e) $0.51\frac {c}{a}$ and (f) $0.586\frac {c}{a}$.
To demonstrate the partial pseudospin-momentum correlation, we use a source with OAM to excite the pseudospin-polarized edge states. The source is composed of three-point sources with a phase delay $\pm \frac {2\pi }{3}$ between them. Depending on whether the phase winding is clockwise or anti-clockwise, electromagnetic waves with opposite angular momentum can be excited. Although the OAM of the edge states is not quantized, in the case without pseudospin mixing, the source with OAM $l=1(l=-1)$ can excite only the edge states with positive (negative) OAM. As shown in Figs. 3(c) and 3(d), a three-point source, of which the center approximately coincides with the center of the phase vortices, is placed in the zigzag boundary. Note that the phase winding of the three-point sources is also seen in the insets. Obviously, these two types of edge states propagate in opposite directions, leading to OAM-selective excitation and unidirectional edge state propagation, which have been shown experimentally in Ref. [17]. In contrast, at a higher or lower frequency, e.g., $0.51\frac {c}{a}$ or $0.586\frac {c}{a}$, such an OAM-selective excitation phenomenon disappear, which is mainly due to the pseudospin mixing in the edge states [see Figs. 3(e) and 3(f)].
To reveal the second-order topology in the HKPC0, we utilize the symmetry eigenvalues of the photonic Bloch functions at the HSPs to deduce the bulk topology. For an HSP denoted by the symbol $\Pi$, the $C_n$-rotation eigenvalue of the Bloch functions can only be $\Pi _p^{(n)}=e^{2\pi i (p-1)/n}$ with $p=1,\ldots,n$. The HSPs of the Brillouin zone for 2D hexagonal lattice are $\Gamma, M, K$. The full set of the rotation eigenvalues of the HSPs is 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 characterize the band topology can be obtained by using the following quantities:
where $\#\Pi _p^{(n)}$ and $\#\Gamma _p^{(n)}$ are the numbers of bands below the band gap with eigenvalues $\Pi _p^{(n)}$ and $\Gamma _p^{(n)}$, respectively, with $\Pi =M, K$. In this scheme, the symmetry eigenvalues at the $\Gamma$ point are taken as the reference to get rid of the redundance. Any nonzero $\left [\Pi \right ]_n$ indicates a topological band gap that is adiabatically disconnected from the trivial insulators. For the $C_6$ symmetric PhC, the topological indices $\xi$ can be written as follows:From the phase distribution of the electric field patterns at the HSPs below the band gap [see Fig. 2(b,d)], one can directly obtain the symmetry eigenvalues. The topological index are $\xi =(0,0)$ for HPC, and $\xi = (-2,0)$ for HKPC0.
According to Ref. [52], the bulk topological invariant is connected to the bulk-induced corner topological index, $Q_c$, as
From the bulk topological indices $\xi =([M_1^{(2)}],[K_1^{(3)}])$, we find that for HPC, the corner indec is $Q_c=0$, whereas for the HKPC0, the corner index is $Q_c=\frac {1}{2}$. Such a nontrivial corner index difference indicates the emergence of the topological corner states at the corner boundaries between the HPCs and HKPC0s.
To visualize the key manifestation of the second-order pseudospin-induced topology in the HKPC, we show the eigenstates spectrum of a hexagonal-shaped large structure consisting of HKPC0 and HPC. The structure, as depicted in Fig. 4(a), has both edge and corner boundaries where the edge and corner states reside, respectively. The eigenstates thus include the bulk, edge, and corner states. Interestingly, Fig. 4(b) indicates there are totally three sets of corner states, one of which emerges in the common spectral gap of the edge and bulk, whereas the other two sets emerge in the gap between bulk and upper edge states. In contrast to the bulk-edge correspondence, We remark that the corner index difference between HKPC0 and HPC does not tell the exact number of the corner states. For each set of corner states, it is predicted that they would interact and hybridize with each other due to the finite-size effect. Indeed, the zoom-in corner spectrum shows the evident frequency splitting of the set of corner states with the lowest frequency, while the other two sets of corner states do not exhibit evident frequency splitting. We further plot the summation of the intensity of the electric field pattern of the six states for each set of corner states in Figs. 4(c-e). It is seen the electric field localized at the corner exhibits different patterns. More differences between these three sets of corner states can be explored by testing the robustness against defects or disorders, which are out of our scope.
Fig. 4. (a) Schematic of hexagon-shaped corner structures, which the HKPC0 is surrounded by HPC. (b) The eigen spectrum of the hexagonal-shaped large structure. The bulk, edge, and corner states are indicated by gray, blue, and red, respectively. (c-e) The intensity of the electric field pattern of the corner states which are arranged in increasing frequency order.
4. First- and second-order valley-induced topologies in the HKPCs
Next, we introduce the valley DOF in HKPC by setting $D\neq 0$. For convenience, we term HKPC with $D=\frac {L}{10}$ and $D=-\frac {L}{10}$ as HKPC1 and HKPC2, respectively. Figures 5(a) and 5(b) present the band structures of HKPC1 and HKPC2 and their phase profiles of $E_z$ at $K$ valleys. For HKPC1 (HKPC2), it is seen that the phase profile of $E_z$ at $K$ point at the first band decreases anticlockwise (clockwise) by $2\pi$ and thus exhibits intrinsic circular-polarized OAMs. Note that they featured with opposite circular-polarized OAM due to inversion symmetry breaking. On the other hand, the frequency order of the valley states, of which the difference is proportional to the Dirac mass term in Eq. (1), flips for HKPC1 and HKPC2, indicating a typical band inversion associated with the topological phase transition.
Fig. 5. (a,b) The photonic band structure for (a) HKPC1 ($R_K=0.14a, R_H=0.04a, D=L/10$) with the phase distribution of the eigenstates of the lowest two bands at the $K$ point, and (b) HKPC2 ($R_K=0.14a, R_H=0.04a, D=-L/10$) associated with the phase distribution of the eigenstates of the lowest two bands at the $K$ point. (c,d) The calculated Berry phase as a function of $k_1$ is presented, respectively, for (c) the HKPC1, and (d) the HKPC2. Inset: the Wannier center configurations and the adopted rhombic Brillouin zone in the calculation of the bulk polarization.
To explore the second-order topology physics in HKPC1 and HKPC2, we also calculate bulk polarization which is associated with the Wannier centers, namely, the center of the maximally localized Wannier function [40,53]. In a 2D system, the bulk polarization is defined in terms of the Berry phase vector potential as
where $\alpha$ denotes the two directions of the triangular lattice vectors and running over the bands below the considered band gap. The associated Wannier centers are pinned to $(P_1,P_2)$. We then implement the numerical calculation of the bulk polarization via the Wilson-loop approach [54] as where $L$ denotes the projection length of the Brillouin zone along the direction. Figures 5(c) and 5(d) present the $\theta _{\alpha,k_1}$ as a function of $k_1$ for HKPC1 and HKPC2, respectively. Thanks to the restriction of $C_3$ symmetry, the Berry phase along loop $k_1$ for a fixed $k_2$ is equal to that along $k_2$ for a fixed $k_1$. Using Eq. (6) the bulk polarization for HKPC1 and HKPC2 are $(\frac {1}{3},\frac {1}{3})$ and $(-\frac {1}{3},-\frac {1}{3})$. Accordingly, the Wannier centers configuration indicated by red and blue dots are shown in the insets of Fig. 5(c) and 5(d).Given the bulk-edge correspondence, the nontrivial polarization difference across domains manifests in the emergence of topological edge states, which appear at the interface between HKPC1 and HKPC2. From the viewpoint of valley Hall physics, across the interface, the massive Dirac Hamiltonian in Eq. (1) produces a nontrivial Berry curvature as
in the first band, which can be integrated into a topological number $C_K=sgn(m)/2$. Therefore, for an interface separating two PCs with distinct valley Hall phases, the difference in the topological number across the interface is quantized ($|\Delta C_K|=1$), which predicts a valley-polarized edge state emerges in the interface, acting as the first-order valley-induced topology. Figure 6(a) displays the eigen spectrum of a supercell formed by HKPC1 and HKPC2, where two valley-polarized edge modes emerge within the topological band gap as expected. As shown in Fig. 6(b), the phase vortices of the photonic edge wave functions exhibit clockwise (anticlockwise) winding around $K(K^\prime )$ valley, indicating the finite angular momentum induced by symmetry breaking.Fig. 6. (a) The dispersion of the edge boundary between HKPC1 and HKPC2 for the zigzag edge along the $x$ direction. Right panel: schematic of the ribbon-shaped supercell. (b) Phase and amplitude of the electric field pattern for edge states $A$ with clockwise phase winding and edge state $B$ with anticlockwise phase winding. The frequency is $0.34\frac {c}{a}$. The Poynting vector is denoted by the green arrows. (c,d) The simulated electric field pattern at $0.346c/a$ when the point source with (c) positive and (d) negative OAM is applied, illustrating the robustness of the valley-dependent edge states.
To demonstrate the unique valley-polarized edge states, we implement a numerical simulation in a $z$-bend consisting of HKPC1 and HKPC2 via point source excitation. As shown in Figs. 6(c) and 6(d), a three-point source, of which the center approximately coincides with the center of the phase vortices, is placed in the zigzag boundary. The phase winding of the three-point sources is also seen in the insets. When a point source with $l=1$ OAM is excited, the waves propagate unidirectionally along the sharp bend without any backscattering. In contrast, when a point source with $l=-1$ OAM is excited, the wave propagates along the opposite direction [see Fig. 6(d)].
To unveil the valley-induced second-order topology in HKPC1 and HKPC2, we construct two types of supercells with triangular (with zigzag edges) and hexagonal (with both zigzag and armchair edges) configurations, as depicted in Figs. 7(a-d). In each case, we label the Wannier center configurations for the boundaries, where the Wannier centers in the bulk cells are hidden for clarification. In Fig. 7(a), it is seen that zigzag edges cut through Wannier centers and the corner terminates exactly at the Wannier center, which suggests the existence of both edge and corner states. To visualize it, we present the eigen spectrum of the triangular-shaped supercell in Fig. 7(e), in which HKPC2 are surrounded by HKPC1. As is expected, there exists both edge and corner states within the band gap, where the summation of the intensity of the electric field patterns of the corner states are shown in the inset. Next, we consider the corner structure made of HKPC1 [see Fig. 7(b)]. The Wannier center configurations indicate the existence of edge states and the absence of corner states. Indeed, the eigen spectrum of the triangular-shaped supercell [see inset of Fig. 7(f)] gives only edge states but finds no signs of corner states. Interestingly, the valley selective is further manifested in the hexagonal structures in Figs. 7(c) and 7(d), where the corner states surprisingly only emerge at three (out of six) corners. When the HKPC2 are surrounded by HKPC1, the corner states emerge at the upper left, upper right, and bottom corners, while exchanging the HKPC1 and HKPC2, the corner emerges at the top, lower left, and low right corners, exhibiting the interesting selectivity. The eigen spectrum associated with the field patterns in Figs. 7(f) and 7(h) are consistent with the Wannier center description.
Fig. 7. (a-d) Schematic of (a,b) triangular-shaped and (c,d) hexagon-shaped corner structures, wherein (a,c), HKPC2 is surrounded by HKPC1 (not shown here), and (b,d) HKPC1 is surrounded by HKPC2 (not shown here). Note the Wannier center configurations are also labeled. (e-h) The eigen spectrum of the corresponding corner structures in (a-d). The emergence of the corner states are well explained by the Wannier center configurations.
5. Excitation of the pseudospin-like and valley Hall edge states
Since both pseudospin-induced and valley-induced band topologies exhibit pseudospin-momentum locked edge states, a natural question is whether these topological effects can coexist in a single system. From the symmetry consideration, the valley Hall topology originates from the inversion symmetry breaking, while pseudospin-dependent edge states require for the inversion symmetry. Hence, strictly speaking, it is impossible to simultaneously have these two topological phases in a single system. However, it is still interesting to explore the effect of symmetry breaking on the pseudospin-dependent edge states.
To this end, we study the evolution of pseudospin-dependent edge states in Fig. 3 versus the geometric parameter $D$. The calculated eigen spectrum the supercells with $D=L/30, L/10, L/6$ are displayed in Figs. 8(a-c), respectively. Note that the edge states localized at the upper (lower) interface are indicated by dashed (solid) lines. Obviously, the degeneracies of the pseudospin-dependent edge states are lifted because the inversion symmetry breaking leads to distinct geometric configurations between upper and lower interfaces. It is also seen that the edge gap is becoming larger accompanying the increase of $D$. For simplicity, we only focus on the edge states at the lower interface. By checking the distribution of the Poynting vectors as well as the phase distribution of the electric field patterns at two typical edge states $A$ and $B$ in Figs. 8(d) and 8(e), it is recognized that the key feature of the pseudospin-momentum locking is still applied when $D=L/30$ and $D=L/10$. Nevertheless, the larger $D=L/10$ makes the upper branch of edge states dispersionless and thus freezing the propagation of the electromagnetic wave. By further increasing $D$ to $L/6$, as shown in Fig. 8(f), the direction of the Poynting vectors of the phase distribution of the electric field pattern of state $B$ flips, which is opposite to that in Fig. 3(b). Hence, the edge states no longer exhibit pseudospin-momentum locked effect when $D=L/6$.
Fig. 8. (a-c) The evolution of the pseudospin-dependent edge states with (a) $D=L/30$, (b) $D=L/10$, (c) $D=L/6$. Note the adopted supercells are the same with Fig. 3 except for the geometric parameter $D$. (d-f) Phase and amplitude of the electric pattern for edge states $A$ and $B$ that are labeled in (a-c), respectively.
From the viewpoint of application, the coexisting of the valley and pseudospin-like edge states revealed above could be useful for designing frequency-dependent topological routings [49,50]. As depicted in Fig. 9(a), we construct a four-port routing system consisting of two routing channels, which would be selectively activated by either pseudospin or valley DOF. When an excitation source, of which the frequency is within the higher-frequency gap, is placed at the interface between HKPC1 and HPC, it is expected the wave would propagate from port 2 to port 4. In contrast, an excitation source, of which the frequency is within the lower-frequency gap, placed at the interface between HKPC1 and HKPC2 would excite a wave propagating from port 1 to port 3. The simulated real-space distributions of the electric field within the two band gaps are depicted in Figs. 9(b) and 9(c). At the frequency of $0.52\frac {c}{a}$, the pseudospin-polarized edge state is successfully excited by the chiral source. As shown in Fig. 9(b), the field is well confined at the interface and propagates only along the direction correlating with the chirality of the source without being backscattered by the sharp bend. Meanwhile, the valley-polarized edge state is also excited by a chiral source with the frequency of $0.33\frac {c}{a}$ [see Fig. 9(c)]. Owning to the small band gap, there exists some intervalley scattering at the turning place.
Fig. 9. Schematic of a four-channel system for wave routing that consisting HPC, HKPC1 and HKPC2. The valley channel is indicated by red lines, whereas the spin channel is indicated by green lines. (b,c) The stimulated electric field pattern of (b) pseudospin-polarized edge states the excited by a source with OAM at the frequency of $0.52\frac {c}{a}$, and (c) valley-polarized edge states that excited by a source with OAM at the frequency of $0.33\frac {c}{a}$.
6. Conclusion
In conclusion, We systematically study the first- and second-order topologies, which are tied to the pseudospin and valley DOFs in HKPCs.By tuning the geometric parameters, we demonstrate that the HKPCs with $C_{6v}$ symmetry host quantum spin Hall phase as the first-order pseudospin-induced topology by presenting the pseudospin-dependent edge states. As the manifestation of the second-order pseudospin-induced topology in HKPCs, we also discover multiple corner states in the hexagon-shaped supercell by employing the topological crystalline index. When the symmetry of HKPC is reduced from $C_{6v}$ to $C_3$, a lower band gap emerges due to the gapping out of Dirac points and exhibits valley Hall topology, in which both the valley-polarized edge states as the first-order and valley-selective corner states as the second-order valley-induced topologies are observed. Meanwhile, the inversion symmetry breaking effect on the pseudospin-momentum locked edge states are also discussed. Our work combines the pseudospin and valley DOFs and provides more flexibility in manipulating electromagnetic waves, which may find potential applications in topological routings.
Funding
National Natural Science Foundation of China (11904060, 62171215).
Acknowledgments
Hai-Xiao Wang thanks Prof. Jian-Hua Jiang for the fruitful discussions and the support from the project of all-English course construction for graduate students in Guangxi Normal University.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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