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[Yong-Cheng Jiang](https://orcid.org/0000-0001-9824-0907), [Toshikaze Kariyado](https://orcid.org/0000-0002-3746-6803), [Xiao Hu](https://orcid.org/0000-0001-6880-402X)

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[Higher-Order Topology in Honeycomb Lattice with Y-Kekulé Distortions](https://mdr.nims.go.jp/datasets/2885e521-d57a-4631-8734-08456de65824)

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Higher-Order Topology in Honeycomb Lattice with Y-KekuléDistortionsYong-Cheng Jiang,1, 2 Toshikaze Kariyado,1 and Xiao Hu1, 2, ∗1Research Center for Materials Nanoarchitectonics,National Institute for Materials Science, Tsukuba 305-0044, Japan2Graduate School of Science and Technology,University of Tsukuba, Tsukuba 305-8571, Japan(Dated: August 2, 2024)AbstractWe investigate higher-order topological states in honeycomb lattice with Y-Kekulé distortionsthat preserve C6v crystalline symmetry. The gapped states in expanded and shrunken distortionsare adiabatically connected to isolated hexamers and Y-shaped tetramer states, respectively, wherethe former possesses nontrivial higher-order topology characterized by a Z6 invariant. Topologicalcorner states exist in a flake structure with expanded distortion where the hexamers are brokenat the corners. Our work reveals that honeycomb lattice with Y-Kekulé distortions serves as apromising platform to study higher-order topological states.1Introduction.—Topological insulators1–4 are the materials that, while insulating in bulk,have conducting boundary states robust against nonmagnetic disorders due to the protectionby time-reversal symmetry. Because of this property, they have attracted much attention fortheir potential applications in low-energy loss spintronic devices. The topological insulatorin two dimensions (2D), also known as quantum spin Hall insulator (QSHI), was first theo-retically proposed in graphene5 and then experimentally realized in HgTe quantum well6,7.The topology of QSHI is captured by a Z2 invariant8 protected by time-reversal symmetry,where spin-orbit coupling plays an essential role. It is found that topological states can alsoarise from crystalline symmetries9–11 even without spin-orbit coupling, where orbital degreesof freedom play a role similar to spin. It has been proposed that deforming honeycomb lat-tice in a C6v symmetric way, namely with O-Kekulé distortions, can open an energy gap inthe double Dirac dispersions with nontrivial topology induced by a band inversion betweenp- and d-like modes11–14. In a simple tight-binding picture, the distortions are capturedby hoppings with two different strengths, and the band inversion is induced by changingthe ratio of the hopping amplitude. This idea, which relies purely on crystalline symmetryrather than spin-orbit coupling, has been experimentally verified in spinless systems suchas photonic crystals15–20, acoustic systems21, and electronic artificial lattices22–24, throughobserving topological edge states.With crystalline symmetries, the concept of topological insulator has been further ex-tended to higher-order topology associated with boundary states of co-dimension largerthan one25–38, for example, 0D corner states in 2D materials, which arise from the mis-match between the number of electrons for charge neutrality and that preserving crystalsymmetry26,28. Corner states induced by nontrivial higher-order topology have been ob-served in a flake structure of honeycomb photonic crystal with O-Kekulé distortion29. Onthe other hand, Y-Kekulé distortions with C3v symmetry have been experimentally observedin graphene recently, which was induced by the vacancies of copper substrate39. This pro-vides a novel platform for exploring electronic states associated with higher-order topologyin real materials.In this Letter, using a tight-binding model, we unveil that higher-order topological statescan be achieved in honeycomb lattice by introducing Y-Kekulé distortions with the C6vcrystalline symmetry preserved where half of the Y-shaped distortions point upward andthe other half point downward. As in the case of the model with O-Kekulé distortions, Y-2M Γ K M-4-2024Energy[t 0](a) (b)(c) (d)𝑡! = 0.5𝑡" 𝑡! = 2𝑡"𝑡"𝑡!𝑡" 𝑡!M Γ K M-2-1012Energy[t 0]ΓMK𝐸#,!𝐸#,%FIG. 1. (Color online) (a) Schematic picture for honeycomb lattice with expanded Y-Kekulédistortion and global C6v symmetry. t0 and t1 are the nearest-neighbor hopping energies, and thehighest-symmetric unit cell is shown by a hexagon (blue) including eight sites. The hexamer insidethe unit cell with Hamiltonian (2) is picked up for introducing local twist phases. (b) Same as (a)except for shrunken distortion. (c) Band structure of (a) with t1 = 0.5t0. For energy gaps belowthe Fermi level, we denote the lower one by Eg,1 and the upper one by Eg,2. Inset: Brillouin zone.(d) Band structure of (b) with t1 = 2t0.Kelulé distortions are effectively captured by introducing two types of hoppings. By tuningthe distortion (or, the ratio of the hoppings), there appear gapped states in both expandedand shrunken Y-Kekulé distortions, which are adiabatically connected to the set of isolatedhexamers and the set of Y-shaped tetramers, respectively. Especially, we reveal a nontrivialhigher-order topology in the expanded distortion, which is characterized by a Z6 Berry phasewith respect to the local twist of the Hamiltonian. We show that the corresponding cornerstates can be observed in a flake structure where the hexamers are broken at the boundary.Model.—We consider honeycomb lattice with expanded and shrunken Y-Kekulé distor-tions as shown in Figs. 1(a) and 1(b), respectively. With these distortions, the lattice can beseen as a network of hexamers (black bonds) and tetramers (red bonds). Both the expandedand shrunken structures respect C6v crystalline symmetry, with half of Y-shaped distortionspointing upward/downward.In order to describe the electronic states in this system, we consider a tight-binding model3for pz orbitals with nearest-neighbor (NN) hopping, which gives the HamiltonianH = −t0∑〈i,j〉c†icj − t1∑〈i′,j′〉c†i′cj′ , (1)where t0 and t1 are the NN hopping energies within hexamers (black bonds) and tetramers(red bonds), respectively, in Figs. 1(a) and 1(b). Because we focus on pz-orbitals, this modelis an eight-band model coming from eight sites in a unit cell.We show the typical band structures for the expanded (t1 = 0.5t0) and shrunken (t1 = 2t0)Y-Kekulé distortions in Figs. 1(c) and 1(d), respectively. Both band structures are symmet-ric with respect to E = 0 due to the sublattice symmetry. For the expanded distortionin Fig. 1(c), there are two energy gaps below the Fermi level (E = 0), one between thefirst (counting from low energy) and the second bands and one between the third and thefourth bands, which we denote as Eg,1 and Eg,2, respectively. For the shrunken distortion inFig. 1(d), there is only one energy gap below the Fermi level, which is between the secondand the third bands.In Fig. 2(a) we show the band gaps Eg,1 and Eg,2 as functions of t1/t0 for the expandeddistortion. The energy gap Eg,1 remains open for 0 ≤ t1/t0 . 0.85. For 0.85 . t1/t0 < 1,the system becomes metallic around E = −2t0 with a negative indirect bandgap betweenthe energy at the K point in the first band and the one at the M point in the second band.The energy gap Eg,2 remains open for 0 ≤ t1/t0 < 1. At t0 = t1, both energy gaps close,since the system is nothing but the pristine honeycomb lattice without distortion. For theshrunken distortion, a global energy gap is opened when t1/t0 & 1.63.Higher-order topology.—We characterize the higher-order topology in our system using aBerry phase defined with respect to the local twist in Hamiltonian40–43. We first assume asufficiently large supercell of our system, and pick up one of the hexamers (see Fig. 1(a)),for which the Hamiltonian is given byh0 = −t06∑j=1c†j+1cj + H.c., (2)with NN-hopping terms within the hexamer. The Hamiltonian for the rest part of the systemis H − h0. Now we introduce local twist phases in the hexamer as shown in the inset ofFig. 2(b), rendering Hamiltonian (2) ash0(Θ) = −t06∑j=1eiθj+1c†j+1cj + H.c., (3)4Eg,1Eg,20 0.2 0.4 0.6 0.8 101/62/63/64/65/61t1/t0γ/2π(a)(b)θ1θ2θ3θ4θ5 θ6Eg,1Eg,20 0.2 0.4 0.6 0.8 100.20.40.60.81t1/t0Energygap[t 0]FIG. 2. (Color online) (a) Energy gap as a function of t1/t0 for expanded Y-Kekulé distortion forEg,1 (lower curve) and Eg,2 (upper curve), respectively. (b) Z6 Berry phase γ as a function of t1/t0for Eg,1 (lower line) and Eg,2 (upper line), respectively. Inset: Twist phases for definition of Berryphase.with Θ = (θ1, θ2, θ3, θ4, θ5) and θ6 = −∑5i=1 θi. Hamiltonian (1) for the whole system isthen rewritten asH(Θ) = h0(Θ) + (H − h0). (4)Denoting the ground state by |Ψ(Θ)〉, the Berry phase is defined asγ =∫LjdΘA(Θ) (mod 2π), (5)where A(Θ) = −i〈Ψ(Θ)|∂Θ|Ψ(Θ)〉 is the Berry connection. Here Lj = Ej−1 → G → Ej is apath in the Θ parameter space with Ej = 2πêj and G = 15∑5j=1Ej, where êj (j = 1, · · · , 5)are the unit vectors and E0 = E6 = 0. Due to the C6 symmetry with respect to the centerof the chosen hexamer which cycles θ1 to θ6 as shown in the inset of Fig. 2(b) (equivalentlyin the parameter space the C6 symmetry with respect to the central point G cycles E1 toE6), the summation of γ in paths L1 to L6 is 0 modulo 2π. Therefore, the C6 symmetryguarantees a quantization ofγ = 2πn6, n = 1, · · · , 5, (6)5(c)(a) (b)(d) (e)1 2 30 10 20 30 40-2.0-1.5-1.0-0.5State numberEnergy[t 0]123𝑡! = 0.5𝑡"corner stateFIG. 3. (Color online) (a) Flake structure with expanded Y-Kekulé distortion. (b) Energy spectrumof (a) with t1 = 0.5t0, where triplet states of 1 to 3 (nine states in total) are corner states. The lowerand upper shaded regions represent the bulk energy gaps Eg,1 and Eg,2, respectively. (c)–(e) Wavefunctions of corner states 1 to 3 with +1 C3 eigenvalue with respect to the center of flake. Eachcorner possesses fractional charge of e/3.which indicates that the Berry phase is a Z6 index.In Fig. 2(b) we show the Berry phases as functions of t1/t0 for the expanded distortioncalculated with a supercell including totally 7× 7 unit cells where periodic boundary condi-tions are applied. For energy gap Eg,1 (Eg,2), the Berry phase remains quantized to π/3 (π)as long as the global energy gap does not close. This is intimately related to the fact thatthe ground state is adiabatically connected to the isolated hexamer cluster with t1 = 0 at1/6 (1/2) filling where the Wannier center is localized at the center of hexamer42,44.For the shrunken distortion, the gapped state between the second and third bands isadiabatically connected to the isolated Y-shaped tetramer cluster with t0 = 0 at 1/4 filling.The state is topologically trivial since the Wanneir center for the shrunken distortion islocated at the central lattice site of the tetramer, in contrast to the expanded distortionthat the Wannier center does not coincide with any lattice sites.Topological corner states.—Associated with the higher-order topological index, in-gapstates are expected to appear when the hexamers are broken at the boundary42. Here weconsider a flake structure where the hexagons are broken at the boundary, as shown inFig. 3(a). The energy spectrum with t1 = 0.5t0 is shown in Fig. 3(b). There appear twotriplets of in-gap states in energy gap Eg,1 and one triplet of in-gap states in energy gap Eg,2.6Due to the C3v symmetry of the flake, three corners are equivalent and give rise to the threedegenerate states (with small degeneracy lifting due to finite-size effect). In each triplet wepick up the state with +1 C3 eigenvalue, labeled as 1 to 3 in Fig. 3(b), and plot their wavefunctions in Figs. 3(c)–3(e), which show a good localization at the corners. Corner states 1and 2 mainly localize at the two boomerang-like trimers, which are fragments of hexamer,with t0 hopping. The two trimers sit on the two sides of the mirror plane at the corner,and state 1 exhibits even mirror symmetry whereas state 2 shows odd mirror symmetry. Incontrast, corner state 3 mainly localizes at the Y-shaped tetramer with t1 hopping, and thuscarries a higher energy. Since the tetramer sits at the mirror plane, only states with evenmirror symmetry are allowed. The wave functions in Figs. 3(c)–(e) show that the electroniccharge is distributed equally at three corners, resulting in fractional charge of e/3.Since the Wannier centers in our system are located at the centers of hexamers andthe boundary of nanoflake cuts the hexamers, there is a mismatch between the numberof electrons for charge neutrality and that preserving crystal symmetry. This mismatchgenerates corner states, even when they are shifted energetically into bulk bands. However,detecting fractional charges would be very challenging if the corner states fall into bulkbands.Discussion.—In the flake structure with the expanded distortion shown in Fig. 3(a),there also appear edge states between corner states 1 and 2, as shown in Fig. 3(b). Theseedge states originate from weak first-order topology45, indicated by the imbalance of parityindex29,46 between the Γ point and two of the three distinct M points for the rhombic unitcell matching the corner of the flake. The imbalance of parity index is one, resulting in onenearly dispersionless edge state.The energy difference between corner states and edge states in Fig. 3(b) are larger than0.04t0, corresponding to 0.1 eV in graphene (t0 = 2.7 eV), which is large enough for scanningtunneling microscope to distinguish the corner states and the edge states. Moreover, itis possible to diminish the edge states within energy gap Eg,1 by making the central twohexamers complete out of four hexamers along each edge of the nanoflake in Fig. 3(a), whichleaves the Wannier centers associated with edge states unbroken along the edges13.Conclusions.—We study a honeycomb structure with Y-Kekulé distortions that maintainsthe C6v symmetry, and uncover its higher-order topology using a tight-binding model withnearest-neighbor hopping. For the expanded distortion where hopping energy t0 within7hexamers is stronger than hopping energy t1 within Y-shaped tetramers, two energy gapsare found below the Fermi level. Both energy gaps exhibit nontrivial higher-order topologyas characterized by finite Berry phases, which are defined with respect to a local twist inone hexamer. The Berry phase is quantized to a multiple of 2π/6 due to the C6 symmetry,and remains unchanged as long as the energy gap does not close. The corresponding cornerstates can be observed in a flake where the hexamers are broken at the boundary.Our model can be realized not only in real materials39, but also in electronic artificiallattices, photonic crystals47 and acoustic systems. Even when next-nearest-neighbor hoppingthat breaks sublattice symmetry is taken into account, the topology remains unaffectedprovided that the energy gap does not close. As elucidated in Ref. 43, the nontrivial higher-order topology in our model characterized by the finite Berry phase in Eq. (5) survives evenin the presence of many-body interactions as long as the energy gap remains open. 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