## Abstract

Synchronized rotation of unit cells in a periodic structure provides a novel design perspective for manipulation of band topology. We then design a two-dimensional version of higher-order topological insulator (HOTI) by such rotation in a triangular photonic lattice with ${\mathcal{C}}_{3}$ symmetry. This HOTI supports the hallmark zero-dimensional corner states and, simultaneously, the one-dimensional edge states. We also find that our photonic corner states carry chiral orbital angular momenta locked by valleys, whose wave functions are featured by the phase vortex (singularity) positioned at the maximal Wyckoff points. Moreover, when excited by a fired source with various frequencies, the valley topological states of both one-dimensional edges and zero-dimensional corners emerge simultaneously. Extendable to higher or synthetic dimensions, our paper provides access to a chiral vortex platform for HOTI realizations in the terahertz photonic system.

© 2022 Chinese Laser Press

## 1. INTRODUCTION

With further development of photonic crystals (PhCs) armored by topological understanding for condensed matter, people have found miscellaneous photonic counterparts of topological phases [1–5]. Among them, the topological edge states generated on the interface between different topological phases promise superior features, such as robustly smooth transmission, backscattering suppression, and defect immunity despite rather strong perturbation of the local boundary. Following quantum Hall phases [2], more intricate topological phases, such as quantum spin Hall phases [3,5] and quantum valley Hall phases [4], are also invented in the context of analog PhC systems, the two of which, respectively, exploit the dichroism freedom by redefining pseudospin/valley concepts in classical wave setups. Such configurable symmetrical lattices, furthermore, provide easy access to topological crystalline insulators (TCIs), for example, those with synchronous rotation giving rise to high tunability in practical realization [6]. Specifically for a ${\mathcal{C}}_{3}$ kagome lattice of broken inversion symmetry, distinct valley states will emerge in the first Brillouin zone (FBZ) and produce their Berry curvature of opposite values [4,7,8]. Such a valleytronics concept calls for bulk valley states locked to their chiralities, which are possible to couple into and out of communication devices, such as valley filters and valley sources, respectively [4,7–10].

Nevertheless, a concept of corner states from higher-order topology that is one further dimension lower than the edges in a two-dimensional (2D) setup [11–15] has added new bricks to the premise for topological information devices. Among the class of higher-order phases, one type of topology is measured by the fractional bulk polarization (or the position of Wannier centers) [15]. For instance, 0D corner states, other than the 1D edge ones, emerge in the second-order TCIs, whose spatial positions are associated with Wannier centers determined from the polarization value [16–19].

Peculiar to the classical analog for topological quantum physics, the spatial vortex, i.e., the wave function with an undefined phase in certain spatial locations, remains less explored in the context of topological photonics despite its mechanical power to manipulate macroparticles. The vortex flow of electromagnetic waves, also defined as the orbital angular momentum (OAM) of light, may open up new avenues to exert optical torques to matters in a noninvasive manner. In this paper, we will reveal such a possibility by designing a valley higher-order topological insulator (HOTI) in a triangular lattice with ${\mathcal{C}}_{3}$ symmetry, which is fueled by synchronous rotation of each unit cell. By observing the phase of electric fields near $K$ and ${K}^{\prime}$ points, we recognize a valley selection feature discussed previously [4]. We also find that the synchronous rotation mechanism of unit cells induces a band inversion at valleys, which leads to a topological phase transition in our photonic system. This topological transition can be characterized by the extended 2D bulk polarization related to the Zak phase [15,20,21]. In the electric field of the valley HOTI, pointwise corner states are predicted by the 2D bulk polarization. Furthermore, not only does our proposed HOTI have a vortex edge state locked to one of the dichroic valleys [22,23], but also it supports a topologically corner state. Using chiral point sources of different frequencies, our simulations verify that the electromagnetic waves shape into high-quality corner states and robust edge states. Our idea can be extended to higher or synthetic dimensions, which contributes to an experimentally feasible platform for HOTI in the photonic vortex system [4,8,9,24–26].

## 2. THEORY AND MODEL

We propose a 2D PhC in a triangular lattice with ${\mathcal{C}}_{3v}$ symmetry, the unit cell of which is composed of six identical pure dielectric cylinders embedded in air as shown in the left panel of Fig. 1(a). Additionally, the maximal Wyckoff points in the unit cell are represented by labels $o$, $p$, and $q$ in real space. The dielectric permittivity is ${\epsilon}_{d}=7.5$, ${a}_{0}$ is the lattice constant, and ${\mathbf{a}}_{1}$ and ${\mathbf{a}}_{2}$ are the lattice vectors with cylinder diameter $d=0.2{a}_{0}$, the lattice constants ${a}_{0}=50\text{\hspace{0.17em}}\mathrm{\mu m}$, and ${a}_{0}/R=3.5$. The synchronous rotation angle of the dielectric cylinders in the unit cell is represented by $\theta $, shown in the right panel of Fig. 1(a) with counterclockwise rotation as the positive direction of rotation whose maximum rotation angle is 60°.

In this paper, a finite element method is used to calculate the PhC dispersion and to solve for the related electric fields. In a ${\mathcal{C}}_{3}$-symmetric lattice, the photonic FBZ contains a pair of $K$ and ${K}^{\prime}$ points in its vertices, which are named valley points [27,28] as shown in the Fig. 1(c) inset. Here, the valley states at $K$ and ${K}^{\prime}$, connected by time-reversal (TR) symmetry [29,30], are both linearly dispersed, which are, hence, named Dirac points [31]. We only focus on the eigenstates near these two valley points and refer valleys $K$ and ${K}^{\prime}$ to Dirac points throughout our whole paper to be succinct. *De facto*, there are more valley points with Dirac dispersion [cf. Fig. 6(b) in Appendix A]. Considering the transverse magnetic (TM) mode for simplicity, the band degeneracy at two Dirac points in Fig. 1(c) is levitated away from linear dispersion by rotating the dielectric cylinders in every unit, which are shown in Figs. 1(b) and 1(d). For the complete band diagram, see Appendix A. To be specific, when rotated away from the original lattice $(\theta =0\xb0)$ in Fig. 1(c), the Dirac degeneracy is levitated to open a bandgap near the Dirac points. We define the lower- and higher-frequency states at $K$ (${K}^{\prime}$) as represented by ${K}_{1}\text{\hspace{0.17em}}({K}_{1}^{\prime})$ and ${K}_{2}\text{\hspace{0.17em}}({K}_{2}^{\prime})$, respectively, in Figs. 1(b) and 1(d). When the unit cells are rotated counterclockwise $\theta =30\xb0$, two pairs of valley states are presented as insets in Fig. 1(d). In addition, these valley states in gap occupy chirality in the sense of circularly polarized OAM, which is manifested by the phase distribution of ${E}_{z}$, i.e., $\mathrm{arg}({E}_{z})$ [10,32]. For the $K$ valley, the phases of ${K}_{1}$ and ${K}_{2}$ have opposite vortex chirality at the positions of $p$ and $q$, respectively [denoted as ${p}_{1}$ and ${q}_{1}$ for $\theta =-30\xb0$ and ${p}_{2}$ and ${q}_{2}$ for $\theta =+30\xb0$ shown in the insets in Figs. 1(b) and 1(d)] and vice versa for the ${K}^{\prime}$ valley. With the opposite signs of $\theta $, the frequency orders of the valleys corresponding to $p$ and $q$ positions are reverse as shown in Fig. 1(e), indicating a typical band inversion that leads to a topological phase transition [5]. The crayon and orange shadings in Fig. 1(e) mark out the complete bandgap width of the system during synchronous rotation of unit cells.

Let us focus on the properties of the $K$-valley state, i.e., ${K}_{1}$ and ${K}_{2}$ for its lower and higher band in frequency, respectively, whereas the counterparts for the ${K}^{\prime}$ valley can be deduced by TR symmetry (cf. Appendix B) [17,29,33]. We find that the photonic valley states are chiral in the sense of phase singularity, which can be readily seen from the electric fields in Fig. 2 where the top and bottom panels display the phase and amplitude distributions, respectively. In the positions of maximal Wyckoff [13] $q$ and $p$, the electric amplitudes ${E}_{z}$ vanish, and, thus, the phases become singular for the chiral valley states [34]. Note that in our PhC unit of ${\mathcal{C}}_{3}$ symmetry, it has three maximal Wyckoff positions: $o$ at the center of the unit cell, and $q$ and $p$ at the vertices of it [cf. Fig. 1(a) and Appendix D]. The electric fields above reveal a typical feature of the vortex field, aligning in flow directions defined by time-averaged Poynting vectors $\mathbf{S}=\mathrm{Re}[\mathbf{E}\times {\mathbf{H}}^{*}]/2$ [35], which are represented by the arrows of the lower panels in Fig. 2. Therefore, we can control the chirality of the valley vortex by choosing the source chirality. Other than such valley-chirality locking, we also note that the ${K}_{1}$ state in ${E}_{z}$ field distribution in Fig. 2(a) actually supports a whole circle of zero amplitude and singularity, and that in Fig. 2(b) a Y-type singularity curve, other than discrete singularity points. Recently, it has been suggested that the HOTI state can be evaluated by integrating the Berry connection in the FBZ, which is actually the Zak phase along the wave-vector direction [17,20,21,33]. The 2D Zak phase is connected to the fractional polarization through ${\theta}_{i}=2\pi {P}_{i}$ for $i=1,2$ where its Zak phase or polarization is completely determined by the bulk property. In the 2D system, the bulk polarization is defined in terms of the Berry connection as [36]

Therefore, our HOTI supports, thus, defined corner states in the bandgap, which appear at the maximal Wyckoff positions of the unit cell. Generally, in ${C}_{n}$-symmetric lattices, given a choice of unit cell, there exist special high-symmetry points (HSP) in the unit cell, which are called the maximal Wyckoff position (cf. Appendix D). As in Eq. (1), the nontrivial second-order topology and emergence of the valley-selective corner states are theoretically characterized by the nontrivial bulk polarizations and the associated Wannier centers. Here, the Wannier center refers to the center of the maximally localized Wannier function and for nontrivial polarization insulators, the Wannier center is located at the same position with the maximal Wyckoff position in the unit cell [12,13].

## 3. NUMERICAL RESULTS AND DISCUSSION

To investigate the concept of valley-selective HOTI, we construct nanodisks made of two types of triangular lattices with distinct polarizations. When $\theta \in (-30\xb0,-10\xb0)$, the eigenspectra of our nanodisk are shown in Fig. 4(a). The two colored curves indicate the eigenfrequency functions with $\theta $ for the two types of vertices [up-corner I (U-I) and up-corner II (U-II) for shorthand, respectively]. Here, we refer to the PhC with $\theta =-30\xb0$ as the up-triangular PhC (UPC) and $\theta =30\xb0$ as the down-triangular PhC (DPC). A schematic for our simulation is shown in the left panel of Fig. 4(b) where the UPC is surrounded by the DPC to interface a zigzag edge mode. The eigenfrequencies of a bulk-edge corner in the UPC structure are shown in the right panel of Fig. 4(b) where the U-I and U-II corner states both are triply degenerate. In the insets of Figs. 4(b) and 4(e), Wannier centers are colored at the corners of the UPC structure. As the electric field shows in Fig. 4(c), Wannier center representation, illustrated by the red dots $q$ in the UPC structure, reveals the valley selectivity of U-I corner states. Additionally, the blue dots $q$ in the UPC structure reveal the valley selectivity of the U-II corner state. When $\theta \in (10\xb0,30\xb0)$ the D-I and D-II corner states, respectively, appear below and above the edge state as shown in Fig. 4(d). From the eigenfrequency distribution of the DPC structure, we find that D-I and D-II corner states each also have three degenerate corner states, and the Wannier center configurations (in red dots) of the corner UPC structure are shown in the right panel of Fig. 4(e); whereas the electric field in Fig. 4(f) shows, Wannier centers (cf. $p$, $q$ points in the picture) of the UPC and DPC are both fired by corner states. We speculate that in the DPC case [cf. the left panel of Fig. 4(e)] the zigzag boundary appears to disrupt and, hence, the corner states of the two models are excited mixedly at the same time. Moreover, the amplitude of the D-I corner electric field ($f=5.8301\text{\hspace{0.17em}}\mathrm{THz}$) is higher than that of the D-II corner electric field ($f=6.4264\text{\hspace{0.17em}}\mathrm{THz}$). For a further view of valley-selective corner states, we construct two kinds of hexagonal nanodisks forming armchair edges. Along some position of the armchair edge, the corner state of a polarization model can be excited separately, whose position is determined by its polarization value of the unit cell (cf. Appendix E). We remark that the valley selectivity behaves globally, which should apply beyond the UPC and the DPC cases here.

Now, we set up full-wave simulation to verify the corner and edge states above in one where the valley dependence of OAM chirality can be exploited to achieve unidirectional excitation of valley chiral states. In Fig. 5, we consider chiral line sources (in blue pentagrams where we choose an LCP OAM source) with a chiral phase, which are fired near the bottom of our PhC with three zigzag boundaries. By switching the source frequency, we can directly control the appearance of edge and corner states as shown in Figs. 5(a) and 5(b). In the superunit where the UPC is surrounded by the DPC, U-I and U-II corner states are, respectively, excited at frequencies $f=5.9656\text{\hspace{0.17em}}\mathrm{THz}$ and $f=6.0309\text{\hspace{0.17em}}\mathrm{THz}$ at the same frequencies as in Fig. 4(c). We note that corner states rely more sensitively on frequency parameters than edge ones do. Since the corner state transmits with loss, the electric amplitude of the corner state near the source remains higher than the further one. We choose frequency $f=6.1700\text{\hspace{0.17em}}\mathrm{THz}$ to fire the edge states, and our simulation shows that electromagnetic waves propagate smoothly along the interface even around sharp corners. It will promise new methods for streering electromagnetic waves along arbitrarily cornered pathways (cf. Appendix G). In the nanodisk where the DPC is surrounded by the UPC, D-I and D-II corner states are excited at $f=5.8301\text{\hspace{0.17em}}\mathrm{THz}$ and $f=6.4264\text{\hspace{0.17em}}\mathrm{THz}$, respectively. In addition, the edge states are excited at $f=6.1400\text{\hspace{0.17em}}\mathrm{THz}$. Our results then show that corner states can be selectively excited by tuning the source frequency in addition to valley selection.

## 4. CONCLUSION

To summarize, we numerically realized a valley-type second-order topology due to unit-cell rotation characterized by the nontrivial bulk polarization. Specifically, the corner states are found to be valley dependent and, therefore, enable flexible manipulation on the wave localization. Thus, topological switches by valley selection of the corner states were numerically demonstrated in our paper. Our valley HOTI and the valley-selective corner states provide preliminary understanding on the interplay between the higher-order topology and the valley degree of freedom, which may find potential applications in valleytronics for future information carriers, such as waveguides, couplers, and topological circuit switches in the terahertz regime [24,30,37–42].

## APPENDICES

Appendices A–G below are for the details of the complete band diagram, symmetry operation, polarization theoretical analysis, maximal Wyckoff positions, corner state distribution along armchair interfaces, and transport efficiency of the edge state along two boundary types.

## APPENDIX A: COMPLETE BAND DIAGRAM

The nontrivial bandgaps were distributed in the range of 5.5–6.5 THz, i.e., seventh band, which is what we focused on in the main text. We then showed the complete band diagram of the PhCs for three rotation angles. When $\theta =-30\xb0$ and $\theta =30\xb0$, four complete bandgaps were produced. The green regions indicated the nontrival bandgap, and the brown regions indicated the trivial bandgap as shown in Figs. 6(a) and 6(c). At $\theta =0\xb0$, bands 2, 3, 7, and 8 were closed in Fig. 6(b). Four Dirac points (phase-transition points) appeared at $K$ and ${K}^{\prime}$, and the topological nontrivial bandgap disappeared. In the main text, we analyzed that bands 7 and 8 reversed in frequency order during rotation, and we noted that no edge states appeared between bands 1 and 2 by rotation of the unit cells. Here, we focused on analyzing the properties of $K$ and ${K}^{\prime}$ points in bands 2 and 3. At $\theta =-30\xb0$ and $\theta =30\xb0$, we found the $K$-valley states had opposite vortex chirality, and vice versa for the ${K}^{\prime}$ valley shown in Fig. 6(d). With the opposite signs of $\theta $, the frequency orders of the valleys corresponding to $p$ and $q$ states were reversed, indicating a typical band inversion that led to a topological phase transition. Additionally, there was no complete bandgap in bands 2 and 3, which, although, might still host topological edge states (not shown here). In the main text, we determined through the polarization value that the bandgap between bands 7 and 8 was nontrivial. Furthermore we numerically calculated the Berry curvature of band 7 according to its eigenstates and found the opposite sign of Berry curvature in different valleys as shown in Fig. 6(e). The Berry connection of the 7th band can be defined as [32]

## APPENDIX B: TR SYMMETRY AND ROTATION SYMMETRY

We first reviewed in concept TR symmetry (TRS), then rotation symmetry, and finally the interplay of the two of them. Using these constraints, we then constructed the complete set of invariants for ${C}_{3}$-symmetry insulators. Insulators in this class have TRS with a Bloch Hamiltonian satisfying [15]

Here, $\mathrm{\Theta}=K$ is the TR operator, which consists only of complex conjugation $K$. The operator obeys ${\mathrm{\Theta}}^{2}=1$. Acting on an energy eigenstate, we haveInvariant points under rotation with ${C}_{3}$ symmetry, in ${C}_{3}$-symmetric TCIs, there are only three threefold HSPs: $K,{K}^{\prime}$, and $\mathrm{\Gamma}$. These points are shown in Fig. 7(b) for all the crystalline symmetries.

Finally, we inspect the interplay between TRS and rotation symmetry. The two operators commute

Thus, on one hand, we have## APPENDIX C: QUANTIZATION OF POLARIZATION

In this appendix, we review the quantization of polarization due to ${C}_{n}$ symmetry [15]. We denote the lattice vectors in real space as ${\mathbf{a}}_{1},{\mathbf{a}}_{2}$ and the corresponding reciprocal lattice vectors in $K$ space as ${\mathbf{b}}_{1},{\mathbf{b}}_{2}$. The reciprocal lattice vectors satisfy

Without loss of generality, we choose our lattice vectors and reciprocal lattice vectors for each symmetry to be those shown in Fig. 7. The conventional modern definition of polarization per unit cell in 2D crystals is

Since ${p}_{1},{p}_{2}$ are defined as mod $e$, the constraints from the above equations imply that with ${C}_{3}$ symmetry, ${p}_{1},{p}_{2}$ are quantized to be $0,e/3,2e/3$, and the difference of the two polarization components ${p}_{1}-{p}_{2}$ is a multiple of the integer charge ${n}_{2}$. Therefore, the two polarization components are the same,

## APPENDIX D: UNIT CELLS AND MAXIMAL WYCKOFF POSITIONS

In ${C}_{n}$-symmetric lattices, given a choice of unit cell, there were special high-symmetry points within the unit cell, called maximal Wyckoff positions, that were invariant under rotations (about the center of the unit cell) up to lattice translations. Let us take the ${C}_{3}$ symmetric lattice as an example (Fig. 7). In Figs. 7(a) and 7(c), we have three maximal Wyckoff positions: the ${C}_{3}$-symmetric point $o$ at the center of the unit cell and the ${C}_{3}$-invariant points $p$, $q$ at the corners of the unit cell [15].

## APPENDIX E: SELECTION OF VALLEY HOTI CORNER STATES

Cutting the ${C}_{3}$ lattice structure in different directions, at least, two types of boundaries can be formed: the zigzag and the armchair types. In the eigenmodes, the UPC and DPC are bounded by each other to create edges [4]. To investigate the valley-locking property in our valley HOTI, we constructed two types of boundary structures with zigzag and armchair edges. The valley-selection property of HOTI in the zigzag boundary was shown in the main text where the U-II corner states were induced by long-range interactions of the unit cell [25] as shown in Fig. 4(d).

The valley selectivity was further manifested in the armchair edges where the corner states surprisingly only emerged at three (out of six) corners as shown in Figs. 8(a), 8(c), 8(d), 8(e), 8(f), and 8(g). In the UPC case, the eigenfrequencies of U-I and U-II corner states were represented by blue and red dots, and the brown dots indicated that there were corner states at the edge of the model. Observing the electric field of the U-corner states, it was found that the corners appeared in the blue (red) positions [cf. the inset of Fig. 8(a)] as shown in Figs. 8(b) and 8(c). It is worth noting that the corner state only appeared at the Wannier center with a polarization of $\mathbf{P}=(-1/3,-1/3)$ as shown in Figs. 8(c) and 8(d). In the DPC case, we found the frequencies of D-I and D-II corner states were the same as that of U-I and U-II corner states. It was also found that the corner state of the polarization of $\mathbf{P}=(1/3,1/3)$ structure was excited alone. Whereas the D-I corner states only appeared in the blue positions [cf. inset of Fig. 8(e)], the D-II corner states only appeared in the red positions as shown in Figs. 8(g) and 8(h). In short, in the armchair-type model, the corner state of a polarization model was excited separately, whose position was determined by its polarization value of the unit cell.

## APPENDIX F: THE BAND STRUCTURES OF NANORIBBON SUPERCELLS

In this appendix, we calculate the projected band of the ribbon surpercells for our valley photonic crystal (VPC). Consider four types of supercells, comprising the zigzag and armchair interfaces between the UPC and the DPC. From calculation, we know that in the trivial bandgap between bands 1 and 2, rotation of unit cells cannot induce topological edge states. Here, we focus on the edge states of the nontrivial bandgap between band 7 and band 8 [cf. Figs. 6(a)–6(c)]. In the nontrivial bandgap, the black solid line indicated the dispersion curve of the edge state, and the gray area was marked according to the frequency intervals of the corner states as shown in Fig. 9. Note that, in Figs. 9(a) and 9(b), frequency ranges of the zigzag edge states and corner states match the eigenspectra of Figs. 4(b) and 4(e) in the main text. A similar rule of valley selection for corner states applies as in Figs. 9(c) and 9(d). Because the two armchair-type interfaces have the same shape, the edge state transmission is also the same [26].

## APPENDIX G: TRANSPORT OF VALLEY EDGE STATES

In order to further verify the edge-transmission features of our design, we test five boundary types for waveguides and use the chiral OAM source in Figs. 10(a)–10(e). It was shown that the zigzag and the armchair boundaries were able to host scatteringless states. Hereby, we define transmission efficiency in Figs. 10(a), 10(b), and 10(d) as

## Funding

Young Scientists Fund (NSFC11804087, NSFC11704106); National Natural Science Foundation of China (NSFC12047501, NSFC12074108, NSFC41974195); Fundamental Research Funds for the Central Universities (CCNU19TS073, CCNU20GF004); Hongque Innovation Center (HQ202104001); Hubei University (030-090105, A201508); Science and Technology Department of Hubei Province (2018CFB148).

## Acknowledgment

R.Z., H.L., Y.W., Z.L. and Z.Y. thank the Central China Normal University. Y.L. and D.-H.X. thank the Chutian Scholars Program in Hubei Province, the Hubei Key Laboratory of Ferro- and Piezoelectric Materials and Devices (Hubei University), and the Institute of Physics Carers’ Funds (IOP, U.K.). R.Z. and D.-H.X. proposed the idea. R.Z. performed the calculation, produced all the figures, and wrote the paper draft. H.L. and Y.L. led the project and revised the whole paper thoroughly. R.Z., Y.L. and D.-H.X. put inputs together from all other coauthors in the paper revision.

## 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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