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Abstract
Recently, higher-order topological insulators have been investigated as a novel topological phase of matter that obey an extended topological bulk–boundary correspondence principle. In this paper, we study the influence of BNN interaction on photonic higher-order corner states. We find both next-nearest-neighbor (NNN) hopping and perfect electric conductor (PEC) boundaries can solely result in two kinds of corner states which are quite different from the traditional “zero-energy” state. To demonstrate this intuitively, we design a novel all-dielectric structure that can effectively shield the influence of NNN couplings while remain the effect of PEC boundaries, so that we can distinguish the contributions from NNN hopping and PEC boundaries. In addition, we also investigate the total contribution on corner states when NNN couplings and PEC boundaries coexist, and some interesting features are revealed. These findings may expand our understanding of the high-order corner modes in a more general framework.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Photonic crystal (PC) is a spatially periodic nanostructure characterized by bandgaps, which provides an innovative platform to realize flourish functional devices [1–4]. Recently, topological insulators based on PCs have attracted enormous attention for their ability to mold the fascinating wave transport properties [5–9]. Topological phases enable protected edge states, offering immunity against scattering from disorder and imperfections [7,8,10,11]. These robust properties are of great promise in the application fields including quantum computers, highly efficient lasers and information transfer devices [12–15]. In contrast to intensive studies on topologically protected waveguides [7–11], the explorations of nanoscale stationary resonators in topological photonics are relatively lacking. However, the idea of recently introduced higher-order topological insulators (HOTIs) provides new opportunity for this [16–19]. For the 2-dimensional (2D) case, a second-order topological insulator has gapped 1D edge states but gapless 0D corner states [17–19]. The arising of these lower-dimensional topological corner states can either originate from the quantization of quadrupole moments such as the topological quadrupole insulators [20,21], or stem from the quantization of the dipole moments [17,18,22] such as the higher-order topological (HOT) states in a 2D breathing kagome lattice [23,24]. In this way, HOTIs can naturally introduce stationary cavity modes into the topological systems.
So far, most of the theoretical studies on HOTIs with corner states only take the nearest-neigbour (NN) coupling into account, and ignore the influence of the interactions beyond NN (BNN) coupling. This idealized case can be well described by generalized tight-binding Su-Schrieffer-Heeger (SSH) model, which predicts the well-known zero-energy corner states tightly localized at one of the sublattices [16–18]. However, for most of the proposed cases (e.g., a topological system composed of all-dielectric rods), the BNN couplings inevitably exist in the systems. If we ignore it, we might miss some important effects on corner states [25,26]. In this paper, we investigate the influence of BNN interactions on photonic higher-order corner states. We find both next-nearest-neighbour (NNN) hopping and perfect electric conductor (PEC) boundaries can solely result in two kinds of corner states (one with odd symmetry while another with even symmetry) that are quite different from the traditional “zero-energy” state. Although the odd symmetric corner state mentioned-above has also been reported by Li et al. [27], the even symmetric one has not been observed yet, no matter numerically or experimentally, since it strongly interacts with the continuum of the bulk states and hybridizes with the continuum in the reported system [27]. Here, we successfully separate these new corner states from the bulk and edge continuum, and visualize them directly. More importantly, considering that NNN hopping always exist in the common kagome lattices and other all-dielectric lattices [25–28], we present a novel method that can effectively shield the influence of NNN hopping, which makes it possible to investigate the independent contribution from other BNN interaction. To the best of our knowledge, it is for the first time to eliminate NNN hopping in an all-dielectric kagome system. In this way, we give an intuitive demonstration that either NNN hopping or PEC boundaries can independently induce the formation of these new kinds of corner states. Finally, we also investigate the total contribution on corner states when NNN couplings and PEC boundaries coexist, and some interesting features are revealed. These findings may expand our understanding of the high-order corner modes in a more general framework.
2. Model and theory
We start by considering a 2D PC structure as sketched in Fig. 1(a), which represents an array of silicon rods with the permittivity of ε = 11.4 in the air to form a kagome lattice. Each rod has a radius of r0 = 0.15a, where a = 1 μm is the lattice constant. There are three identical dielectric rods in each primitive cell with C3 rotational symmetry. The intra-unit-cell hopping (t1) and inter-unit-cell hopping (t2) between the neighboring rods are denoted by green and red lines, respectively. The relative strength of t1 and t2 can be readily tuned by changing the distance between the three rods inside the unit cell (as characterized by L in Fig. 1(a) inset), while keeping the size of the unit cell fixed.
Fig. 1. (a) Schematics of the 2D kagome lattice. The green (red) lines denote NN hopping of strength t1 (t2). P and Q represent the NNN hopping among different sublattice and 3rd NN hopping among same sublattice, respectively. (b-d) Band structures for $L = 0.3a/\sqrt 3 $, $L = 0.5a/\sqrt 3 $, and $L = 0.7a/\sqrt 3 $, which represent a shrunken, normal, and expanded lattice, respectively. The insets in Figs. 1(b) and (d) are the Ez field patterns of the unit cell; the symbols “+” and “-” denote different parities of the first and second bands at the K point.
Figures 1(b-d) show the evolution of band gaps for transverse magnetic (TM) modes by employing the finite-element method (FEM) to make the first-principles calculations. When $L = 0.5a/\sqrt 3 $, the PC is a conventional 2D triangular lattice. In this case, the intra-cell coupling t1 and the inter-cell coupling t2 are equal, and the band structure exhibits a Dirac-like degeneracy at the K point, as shown in Fig. 1(c). However, when $L = 0.3a/\sqrt 3 $(which corresponds to a “shrunken” lattice) and $L = 0.7a/\sqrt 3 $(which corresponds to an “expanded” lattice), t1 and t2 are no longer equal. Although the intra-cell rotational symmetry is preserved, the rotational symmetry of the entire structure is broken. Therefore, the Dirac-like point at point K no longer exist, inducing a photonic band gap, which also means the occurrence of a topological phase transition associated with a band-inversion process, as shown in Figs. 1(b) and (d). The insets in Figs. 1(b) and (d) are the Ez field patterns of the unit cell for the “shrunken” and “expanded” cases, respectively. One can see that the parities for the first and the second bands have changed as we change $L = 0.3a/\sqrt 3 $ to $L = 0.7a/\sqrt 3 $, which implies the bands have been inversed.
Topological phase transition can be characterized by 2D bulk polarization [29,30], expressed as:
Fig. 2. Variation of the bulk polarization when the lattice change from a shrunken one to an expanded one. The solid (hollow) circles denote the calculated P1 (P2) based on FEM, and the dashed line is theoretically calculated results based on Eqs. (1) and (2).
We now establish a tight-binding model to investigate the influence of NN and NNN hopping on HOT corner states. For the all-dielectric kagome lattice discussed here, the BNN interactions mainly include the NNN hopping among different sublattice, and 3rd NN hopping among same sublattice, as illustrated by P and Q in Fig. 1(a). Considering the influence of 3rd NN hopping is relatively weaker since it is shielded by a medium rod, in the following theoretical analysis we can temporarily neglect 3rd NN hopping for simplicity, as well as the other higher-order hopping. We suppose the hopping coefficient decreases exponentially with distance l, and has the form (the accuracy of this supposition has been testified by comparing the theoretical predictions with the simulation results, as shown below):
where lij is the center distance between the ith and jth sites; t0 and α are two system constants and can be calculated via theoretically fitting the simulation results, e.g., the data in Fig. 2. In this way, t0 and α are calculated to be 2.811 and 3.446 μm-1 in our system, respectively, and very good agreements are achieved between the theoretical curve (the dashed line in Fig. 2) and FEM simulation results (scatted dots).The Hamiltonian of the kagome lattice sketched in Fig. 1(a) is then given by
where $c_i^\dagger $ and cj are the creation operators and annihilation operators in the ith and jth sites, respectively; tij is the corresponding hopping strength, and tij = t1, t2, P when lij = l1, l2, l3, respectively, as illustrated in Fig. 1(a). The ground frequency (energy) for each site (i.e., the diagonal terms in H) is set to be zero.To study how the NN hopping and NNN hopping affect HOT corner states, we change the relative strength of t1, t2 and P gradually by varying l1 (note that l2 = a - l1, and ${l_3} = \sqrt {l_1^2 + l_2^2 + {l_1}{l_2}} $). By solving the eigenvalue equation $H|\psi \rangle = E|\psi \rangle $ for a finite kagome lattice consisting of 210 unit cells, we obtain the evolution spectrum of eigenenergy as a function of l1/l2, as shown by Fig. 3(a).
Fig. 3. Tight binding calculations with NN hopping and NNN hopping. (a) and (b) Spectrum of eigenenergy as a function of l1/l2 when (a) with NNN hopping and (b) without NNN hopping (P = 0). (c-f) Calculated eigenmode field profiles of corner states and edge state at l1/l2 = 2 (the points marked by 1-4 in Fig. 3(a)). (c) N-type corner state. (d) Edge state. (e) Odd B-type corner state. (f) Even B-type corner state.
Figures 3(a) clearly reveals three kinds of corner states. The first kind (denoted by red curve) emerges at midgap with a flat spectral position just between the edge states and the bulk bands, while the other two kinds (denoted by green curve) evolve from the edge states. Figures 3(c)-(f) show the corresponding eigenmode field ($|\psi \rangle $) profiles of corner states and edge state at l1/l2 = 2 (the points marked by 1-4 in Fig. 3(a)). We can see the first kind of corner state (marked by “1”) does localize at only one of the sublattices (the corner site), and is pinned to the “zero energy”. Obviously, this familiar corner state originates from NN coupling, thereby we refer to it as “NN-coupling-induced” (hereafter called “N-type”) corner state. Compared with N-type corner state, the other two kinds (marked by “2, 4”) exhibit some unique features: the field profile no longer localize at the corner site, but distributes along the two sides of each corner within short distance. Further, seen form Figs. 3(e) and (f), these new kinds of corner states have different eigenmode-field symmetry: the one at higher frequency (marked by “2”) exhibits odd symmetry (Fig. 3(e)), while the one at lower frequency (marked by “4”) exhibits even symmetry (Fig. 3(f)). They evolve from the edge states, but all decay exponentially away from the corners and show good localization at the three corners, showing significant difference from the edge states (Fig. 3(d)). Since these new corner states cannot be predicted from the tight-binding model with only NN coupling, the BNN interactions (beyond NN coupling) should be responsible for them, therefore we refer to them as “BNN-coupling-induced” (hereafter called “B-type”) corner states. Thus, we can refer to the one with odd symmetry as “odd B-type”, and the one with even symmetry “even B-type”.
To further theoretically demonstrate the separation of the B-type corner states from edge states is really caused by NNN hopping in above case, we then intensively set P = 0 in Eq. (3), and observe what will happen. The result is shown in Fig. 3(b). We see both odd and even B-type corner states do disappear, and there is not any separation from the edge state any longer. Therefore, NNN hopping is responsible for the generation of B-type corner states.
3. Numerical analysis
Based on the above tight-binding model, we first consider a hybrid structure where a topologically nontrivial super cell of 136 unit cells is surrounded by a topologically trivial PC, as shown in Fig. 4. The angle at each corner is 60°. By using first-principles calculations, we obtain the evolution spectrum of eigenfrequency in Fig. 4(a) when we gradually increase the distance L between the three rods inside the unit cell to tune the relative strength of inter- and intra-unit-cell hopping. One can see a topological phase transition occurs when L is longer than $0.\textrm{517}a/\sqrt 3$(the transition line is denoted by a dashed vertical line), which is consistent with the nonzero bulk polarization as depicted in Fig. 2. Figure 4(b) shows the eigenmodes of non-trivial lattice at frequencies between 0.22 and 0.38 (2πc/a) when L = $0.\textrm{7}a/\sqrt 3$.
Fig. 4. N-type and B-type Corner states and edge states of a hybrid structure with a kagome lattice surrounded by a topologically trivial PC. (a) Evolution spectrum of eigenfrequency. (b) Calculated eigenmodes at $L = 0.7a/\sqrt 3$. (c-f) The simulated electric field (polarized in the z direction) distribution of the (c) N-type corner state at 0.3108, (d) edge state at 0.2850, (e) odd B-type corner state at 0.2888, and (f) even B-type corner state at 0.2561 (all in units of 2πc/a).
As expected, seen from Figs. 4(a) and (b), three kinds of triply-degenerate corner states appear: except for the N-type corner state with flat spectrum (the red line or dots), the other two B-type corner states (the green lines or dots) really split off from the edge states. Their eigenmode field profiles in shown in Figs. 4(c-f). Figure 4(c) clearly indicates that the N-type corner state tightly localizes at the corner site, and is pinned to the “zero energy” (the ground frequency). While in Figs. 4(e) and (f), the two B-type corner states distribute along the two sides of each corner with opposite symmetries. Obviously, these unique localization features of the B-type states are quite different from the edge states shown in Fig. 4(d). By comparing Figs. 3 and 4, one can see very good agreements are achieved between the theoretical predictions and first-principle FEM simulation results.
To demonstrate the B-type corner states really originate from BNN interactions, a natural idea is, if we “eliminate” the NNN and 3rd NN hopping among the kagome lattice, will the B-type corner states disappear?
To do so, we design a novel PC structure composed of three kinds of rods with same size but different resonant frequencies (denoted by red, green, and blue rods, respectively), and the rods are arranged in a specified order, as shown in Fig. 5 (a). The three kinds of rods are intensively set detune, so that the NNN hopping, as well as the 3rd NN hopping, will be significantly restrained, but the NN hopping t1 will be preserved via on-resonance effect. Although another NN hopping t2 will be also weakened, it barely influences the formation of either the N-type or B-type corner states. Therefore, we can employ this system to verify if the BNN interactions play a decisive role on the formation of B-type corner states.
Fig. 5. (a) Sketch of a detuned kagome lattice. Here the rods’ permittivity ε = 11, 12, and 13 are taken for the red, green, and blue rods, respectively. The dashed polygon at the top right of (a) denote a unit cell with C3 rotational symmetry. (b) Band structures for uniform ($L = 0.\textrm{5}a/\sqrt 3 $) (left panel), shrunken ($L = 0.3a/\sqrt 3 $) and expanded ($L = 0.\textrm{7}a/\sqrt 3$) lattices (both at right panel). (c) Evolution spectrum of eigenfrequency for a hybrid structure with a detuned kagome lattice surrounded by a topologically trivial PC. One can see that the B-type corner states indeed disappear, and only N-type corner states exist. (d) Eigenfrequencies taken from (c) at $L = 0.7a/\sqrt 3$. The insets show a N-type corner state at 0.3128 and a hybrid state at 0.2562 (all in units of 2πc/a).
As an example for illustration, here the rods’ permittivity ε = 11, 12, and 13 are taken for the red, green, and blue rods, respectively. The unit cell contains sublattices in three colors with C3 rotational symmetry, as depicted at the top right of Fig. 5(a) (inside the dashed polygon). The band structures for shrunken ($L = 0.3a/\sqrt 3 $), expanded ($L = 0.\textrm{7}a/\sqrt 3$), and uniform ($L = 0.\textrm{5}a/\sqrt 3 $) lattices are presented in Fig. 5(b). Similar to Figs. 1(b-d), when the lattice is distorted from “uniform” case to a “shrunken” or “expanded” one, topological phase transition occurs and leads to the opening up of a full photonic bandgap at the K point. If we cut a large triangle from the lattice and let the boundaries passing through the downward-pointing triangles (the three black dashed long lines depicted in Fig. 5(a)), HOT corner states will occur, as shown below.
For better comparison with the case based on common kagome lattice shown in Fig. 4, we consider a similar hybrid structure where a topologically nontrivial lattice is surrounded by a topologically trivial PC. Except for the permittivity of the rods (i.e., resonant frequencies) within the topological region, the other parameters are the same as that used in Fig. 4.
From Figs. 5(c) and (d), we see the triply degenerate N-type corner state still exists, with a nearly-flat band and an ideally location at only one corner site. However, as expected, the B-type corner states indeed disappear, since they now merge into the edge states and cannot split from the continuum, no matter how we “expand” the lattice via changing L, just as what is predicted by Fig. 3(b) when set P = 0 in the tight-binding theory. By comparing the two cases shown in Figs. 4 and 5, we can draw a conclusion that the NNN hopping among the kagome lattice does play a critical role on the formation of the B-type HOT corner states.
Another BNN interaction is from PEC boundaries. PEC boundaries have been widely employed in topological photonics [32–34], due to its perfect reflectivity, which is similar to a topological trivial forbidden band in a full frequency range. However, to the best of our knowledge, whether or not the PEC boundaries can solely result in the B-type HOT corner states in a kagome lattice has not been reported yet. Actually, even if the PEC boundaries do have this function, it is very difficult to distinguish the contribution is from NNN hopping or from PEC boundaries or both, since NNN hopping always exist in the common kagome lattices. But now, the detuned kagome lattice presented above makes it possible for us to investigate the independent contribution of PEC boundaries, since the NNN hopping has now been significantly restrained.
Figure 6 exhibits the simulation results of the N-type and B-type corner states when the detuned kagome lattice in Fig. 5 is surrounded by PEC boundaries, instead of topologically trivial PC. One can see that under the influence of the interactions with PEC boundaries, both odd and even B-type corner states all split from the edge states (Fig. 6(a)), and are all triply-degenerated (Fig. 6(b)), with field profiles (Figs. 6(c) and (d)) almost the same with that in Figs. 4(e) and (f), except that the field images are some asymmetric due to the permittivity difference of the rods along the two sides of each corner. This clearly implies that the PEC boundaries can solely result in the B-type HOT corner states in a similar way as NNN coupling in a kagome lattice, since the radiated electromagnetic waves (imaginary) from arbitrary site near the boundary will now have more opportunity to couple into the non-neighbouring sites via the perfect reflectivity of the PEC boundaries. This is in some sense equivalent to the coupling between the sites near the boundary and their “virtual images” (the PEC boundary is an ideal mirror), leading to the split of the edge state and B-type corner states are therefore created.
Fig. 6. Detuned kagome lattice with PEC boundaries. (a) Evolution spectrum of eigenfrequency for different L. (b) Calculated eigenmodes taken from (a) at $L = 0.7a/\sqrt 3$. (c) Even B-type corner state at 0.2592 (2πc/a). (d) odd B-type corner state at 0.3035 (2πc/a). The field images are some asymmetric due to the permittivity difference of the rods along the two sides of each corner.
On the other hand, owing to the interaction between the corner sites and the PEC boundaries, the eigenfrequencies of the N-type corner states are lifted into the bulk continuum (see the red line in Fig. 6(a)), and become normal resonance states because the chiral symmetry is now broken. The breaking of the chiral symmetry is attributed to the interactions from PEC boundary, rather than NNN hopping, since the latter is now significantly restrained in this detuned kagome lattice. Actually, even if the NNN hopping is included, it will not break the chiral symmetry, as has been confirmed both theoretically and numerically in Figs. 3 and 4: the N-type corner states are kept triply degenerate and pinned to the “zero energy”. In contrast, under the influence of the PEC boundary, the spectral position of the N-type corner states is not flat any longer. This clearly indicates that the chiral symmetry is broken, since the inhomogeneous spectral shift of the resonance of each rod (note that the interaction with PEC boundary is position-dependent) makes the trace of the Hamiltonian of the system no longer be zero. As the result, the precondition for the generalized chiral symmetry is then broken [23].
Next, we discuss the robustness of the above B-type corner states to the perturbations. The disorder is introduced through randomly varying the radius of each rod, namely, ri = r0 (1 + δi), where i is the site index, r0 is the original radius, and δi is a random variation distributed from -δ0 to δ0, with δ0 denoting perturbation strength. As an example, we calculate the eigenfrequencies of the above lattice with PEC boundaries when δ0 = 5%, as shown in Fig. 7. We see the odd and even B-type corner states still survive, but their eigenfrequencies are shifted comparing to the original frequency (i.e., 0.2592 and 0.3035) in the defect-free case, and the triple degeneracy is broken due to the perturbation. The insets in Fig. 7 further show the field patterns of the B-type corner states, which clearly indicate that these B-type corner states are robust against disorders.
Fig. 7. Robustness of the B-type corner states in a detuned kagome lattice with PEC boundaries. The dots are calculated eigenfrequencies when disorder is introduced to all sites through randomly varying the radius of each rod. The insets are the Even B-type corner state at 0.2655 (2πc/a) and odd B-type corner state at 0.3104 (2πc/a), respectively.
Finally, we investigate what will happen when NNN coupling and PEC boundaries coexist in a kagome lattice. We then select a similar topological system as that shown in Fig. 6, but this time the permittivity of all rods are set to the same (ε = 11.4, just as that in Fig. 4). Therefore, the NNN hopping is revived, and will work side by side with the PEC boundaries.
The combination of these two BNN interactions yields some interesting features. From Fig. 8, we find that the eigenfrequencies of the B-type corner states split from the edge states more obviously, compared with the cases when NNN hopping (Fig. 4(b)) and PEC boundaries (Fig. 7) exist alone. Besides, the N-type corner states are immersed into the bulk continuum due to the spectral blue shift caused by their interaction with the metal, and become normal resonance states just like that in Fig. 6(a). The field images of the B-type corner states shown in Fig. 8 are quite symmetric, since the permittivity of the rods along the two sides of each corner are now the same.
Fig. 8. Investigation of the case when NNN coupling and PEC boundaries all exist. The dots are the calculated eigenfrequencies for a normal kagome lattice with PEC boundaries. The insets are the Even B-type corner state at 0.2664 (2πc/a) and odd B-type corner state at 0.3146 (2πc/a), respectively.
4. Conclusion
In summary, we have demonstrated that both NNN hopping and PEC boundaries can solely result in HOT B-type corner state, which has unique local field distribution that is quite different from the traditional N-type corner state. To distinguish the contributions from NNN hopping and PEC boundaries, we design a detuned kagome lattice that can effectively shield the influence of NNN couplings while remain the effect of PEC boundaries, in all-dielectric fashion. The approach of restraining the influence of NNN hopping can be readily generalized to other topological systems, such as square lattices and bipartite lattices [17,18,25,26]. In addition, we also investigate the total contribution on corner states when NNN couplings and PEC boundaries coexist, which will further strengthen the separation of the B-type corner states from the edge state and improve the robustness against perturbations. The fact of the B-type corner states evolving from the edge states makes the coupling between these new corner states is more easily to be tuned in contrast to the conventional “zero-energy” N-type corner state. Therefore, these B-type corner states can be employed to realize some new optical functionalities, e.g., topologically-protected electromagnetically induced transparency (EIT), light delay, etc. These results may expand our understanding of the HOT corner modes in a more general framework, and find applications in the fields of quantum computing, highly efficient lasers and on-chip optical signal processing.
Funding
National Natural Science Foundation of China (11774098, 11304099); Natural Science Foundation of Guangdong Province (2017A030313016, S2013040015639); Guangzhou Municipal Science and Technology Project (202002030500).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper may be obtained from the authors upon reasonable request.
References
1. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University, 1995).
2. M. Soljačić and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. 3(4), 211–219 (2004). [CrossRef]
3. H.-G. Park, S.-H. Kim, S.-H. Kwon, Y.-G. Ju, J.-K. Yang, J.-H. Baek, S.-B. Kim, and Y.-H. Lee, “Electrically-driven single-cell hexapole mode photonic crystal laser,” Science 305(5689), 1444–1447 (2004). [CrossRef]
4. T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2(8), 465–473 (2008). [CrossRef]
5. J.-W. Dong, X.-D. Chen, H. Zhu, Y. Wang, and X. Zhang, “Valley photonic crystals for control of spin and topology,” Nat. Mater. 16(3), 298–302 (2017). [CrossRef]
6. 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]
7. F. Alpeggiani and L. Kuipers, “Topological edge states in bichromatic photonic crystals,” Optica 6(1), 96–110 (2019). [CrossRef]
8. Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljačić, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100(1), 013905 (2008). [CrossRef]
9. M. I. Shalaev, W. Walasik, and N. M. Litchinitser, “Optically tunable topological photonic crystal,” Optica 6(7), 839–845 (2019). [CrossRef]
10. N. Stroev and N. G. Berloff, “Managing the flow of liquid light,” Phys. Rev. B 102(20), 201114 (2020). [CrossRef]
11. T. L. van den Berg, M. R. Calvo, and D. Bercioux, “Living on the edge: Topology, electrostatics, and disorder,” Phys. Rev. B 2(1), 013171 (2020). [CrossRef]
12. A. Lubatsch and R. Frank, “Evolution of Floquet topological quantum states in driven semiconductors,” Eur. Phys. J. B 92(9), 215 (2019). [CrossRef]
13. W. Yang, G. Tian, Y. Zhang, F. Xue, D. Zheng, L. Zhang, Y. Wang, C. Chen, Z. Fan, Z. Hou, D. Chen, J. Gao, M. Zeng, M. Qin, L.-Q. Chen, X. Gao, and J.-M. Liu, “Quasi-one-dimensional metallic conduction channels in exotic ferroelectric topological defects,” Nat. Commun. 12(1), 1306 (2021). [CrossRef]
14. Z. Zhang, X. Qiao, B. Midya, K. Liu, J. Sun, T. Wu, W. Liu, R. Agarwal, J. M. Jornet, S. Longhi, N. M. Litchinitser, and L. Feng, “Tunable topological charge vortex microlaser,” Science 368(6492), 760–763 (2020). [CrossRef]
15. A. Lubatsch and R. Frank, “Behavior of floquet topological quantum states in optically driven semiconductors,” Symmetry 11(10), 1246 (2019). [CrossRef]
16. A. E. Hassan, F. K. Kunst, A. Moritz, G. Andler, E. J. Bergholtz, and M. Bourennane, “Corner states of light in photonic waveguides,” Nat. Photonics 13(10), 697–700 (2019). [CrossRef]
17. B.-Y. Xie, G.-X. Su, H.-F. Wang, H. Su, X.-P. Shen, P. Zhan, M.-H. Lu, Z.-L. Wang, and Y.-F. Chen, “Visualization of higher-order topological insulating phases in two-dimensional dielectric photonic crystals,” Phys. Rev. Lett. 122(23), 233903 (2019). [CrossRef]
18. X.-D. Chen, W.-M. Deng, F.-L. Shi, F.-L. Zhao, M. Chen, and J.-W. Dong, “Direct observation of corner states in second-order topological photonic crystal slabs,” Phys. Rev. Lett. 122(23), 233902 (2019). [CrossRef]
19. M. Ezawa, “Higher-order topological insulators and semimetals on the breathing kagome and pyrochlore lattices,” Phys. Rev. Lett. 120(2), 026801 (2018). [CrossRef]
20. C. W. Peterson, W. A. Benalcazar, T. L. Hughes, and G. Bahl, “A quantized microwave quadrupole insulator with topologically protected corner states,” Nature 555(7696), 346–350 (2018). [CrossRef]
21. M. Serra-Garcia, V. Peri, R. Susstrunk, O. R. Bilal, T. Larsen, L. G. Villanueva, and S. D. Huber, “Observation of a phononic quadrupole topological insulator,” Nature 555(7696), 342–345 (2018). [CrossRef]
22. Y. Ota, F. Liu, R. Katsumi, K. Watanabe, K. Wakabayashi, Y. Arakawa, and S. Iwamoto, “Photonic crystal nanocavity based on a topological corner state,” Optica 6(6), 786–789 (2019). [CrossRef]
23. X. Ni, M. Weiner, A. Alù, and A. B. Khanikaev, “Observation of higher-order topological acoustic states protected by generalized chiral symmetry,” Nat. Mater. 18(2), 113–120 (2019). [CrossRef]
24. M. Proctor, M. B. de Paz, D. Bercioux, A. García-Etxarri, and P. A. Huidobro, “Higher-order topology in plasmonic Kagome lattices,” Appl. Phys. Lett. 118(9), 091105 (2021). [CrossRef]
25. C. Poli, H. Schomerus, M. Bellec, U. Kuhl, and F. Mortessagne, “Partial chiral symmetry-breaking as a route to spectrally isolated topological defect states in two-dimensional artificial materials,” 2D Mater. 4(2), 025008 (2017). [CrossRef]
26. D. Leykam, S. Mittal, M. Hafezi, and Y. D. Chong, “Reconfigurable topological phases in next-nearest-neighbor coupled resonator lattices,” Phys. Rev. Lett. 121(2), 023901 (2018). [CrossRef]
27. M. Li, D. Zhirihin, M. Gorlach, X. Ni, D. Filonov, A. Slobozhanyuk, A. Alù, and A. B. Khanikaev, “Higher-order topological states in photonic kagome crystals with long-range interactions,” Nat. Photonics 14(2), 89–94 (2020). [CrossRef]
28. A. Cerjan, M. Juergensen, W. A. Benalcazar, S. Mukherjee, and M. C. Rechtsman, “Observation of a higher-order topological bound state in the continuum,” Phys. Rev. Lett. 125(21), 213901 (2020). [CrossRef]
29. S. King and Vanderbilt, “Theory of polarization of crystalline solids,” Phys. Rev. B 47(3), 1651–1654 (1993). [CrossRef]
30. R. Resta, “Macroscopic polarization in crystalline dielectrics: the geometric phase approach,” Rev. Mod. Phys. 66(3), 899–915 (1994). [CrossRef]
31. H. Xue, Y. Yang, F. Gao, Y. Chong, and B. Zhang, “Acoustic higher-order topological insulator on a kagome lattice,” Nat. Mater. 18(2), 108–112 (2019). [CrossRef]
32. W.-J. Chen, S.-J. Jiang, X.-D. Chen, B. Zhu, L. Zhou, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5(1), 5782 (2014). [CrossRef]
33. H.-C. Chan and G.-Y. Guo, “Tuning topological phase transitions in hexagonal photonic lattices made of triangular rods,” Phys. Rev. B 97(4), 045422 (2018). [CrossRef]
34. Y. Li, Y. Yu, F. Liu, B. Zhang, and G. Shvets, “Topology-controlled photonic cavity based on the near-conservation of the valley degree of freedom,” Phys. Rev. Lett. 125(21), 213902 (2020). [CrossRef]
