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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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This is the Accepted Manuscript version of an article accepted for publication in Nanotechnology.  IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it.  The Version of Record is available online at https://dx.doi.org/10.1088/1361-6528/ad2483[Creative Commons BY-NC-ND Attribution-NonCommercial-NoDerivs 4.0 International](https://creativecommons.org/licenses/by-nc-nd/4.0/)

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[Topological electronic states in holey graphyne](https://mdr.nims.go.jp/datasets/c3bb5205-8b4a-4884-85db-3b88c603ea46)

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Topological electronic states in holey graphyneYong-Cheng Jiang,1, 2 Toshikaze Kariyado,1 and Xiao Hu1, 2, ∗1Research Center for Materials Nanoarchitectonics (MANA),National Institute for Materials Science (NIMS), Tsukuba 305-0044, Japan2Graduate School of Science and Technology,University of Tsukuba, Tsukuba 305-8571, JapanAbstractWe unveil that the holey graphyne (HGY), a two-dimensional carbon allotrope where benzenerings are connected by two C C bonds fabricated recently in a bottom-up way, exhibits topo-logical electronic states. Using first-principles calculations and Wannier tight-binding modeling,we discover a higher-order topological invariant associated with C2 symmetry of the material, andshow that the resultant corner modes appear in nanoflakes matching to the structure of precursorreported previously, which are ready for direct experimental observations. In addition, we find thata band inversion between emergent g-like and h-like orbitals gives rise to a nontrivial topology char-acterized by Z2 invariant protected by an energy gap as large as 0.52 eV, manifesting helical edgestates mimicking those in the prominent quantum spin Hall effect, which can be accessed experi-mentally after hydrogenation in HGY. We hope these findings trigger interests towards exploringthe topological electronic states in HGY and related future electronics applications.∗ HU.Xiao@nims.go.jp11. INTRODUCTIONHaldane was the first to notice that a typical topological state, i.e., quantum Hall effect,can be realized without Landau levels induced by an external magnetic field. He developeda tight-binding (TB) model on honeycomb lattice with complex next-nearest-neighbor hop-pings which break time-reversal symmetry (TRS) but introduce no net external magneticfield, generating quantum anomalous Hall effect characterized by a Chern number [1, 2]. Al-though this toy model was thought to be unrealistic at the time of the original proposal, soonafter the discovery of graphene [3] it was pointed out by Kane and Mele that if spin-orbitcoupling (SOC) is taken into account, graphene can be regarded as two copies of Haldanemodel, one for up spin and the other for down spin, manifesting the TRS-preserved quantumspin Hall effect (QSHE) characterized by a Z2 invariant [4, 5] (see also Ref. 6). However,the predicted QSHE is extremely hard to observe experimentally in graphene because ofits very weak SOC. In order to resolve this issue, honeycomb systems with heavy elementshave been investigated, aiming at having stronger SOC. Recently large-gap QSH insulatorshave been realized, in Bismuthene [7], monolayer 1T’-WTe2 [8], stanene [9] and ultrathinNa3Bi [10] for instance. For monolayer 1T’-WTe2 with an energy gap ∼55 meV [11], thequantized edge conductance was observed in a transport channel of 100 nm up to 100 K [8];while for the other materials, although energy gaps range from 0.3 eV to 0.8 eV, evidencefor nontrivial topology is still limited in showing the local density of states (LDOS) for edgestates, partially due to their small sample sizes below 40 nm. Despite all the noticeable pro-gresses, further investigations are needed on materials possessing sizable energy gaps andamenable to fabrication into structures for device applications.Besides utilizing strong SOC, deforming honeycomb lattice while preserving C6v crys-talline symmetry is also able to open a sizable gap in the Dirac dispersion associated withgraphene structure, where the band inversion between bands with p- and d-like charac-ters generates nontrivial topology [12–14]. This idea has been experimentally verified firstin photonic crystals [15–19] due to their comparably easy fabrication, and later in elec-tronic systems using molecular graphene [20], which is constructed by aligning carbon-monoxide molecules regularly on the Cu(111) surface in terms of the scanning tunnelingmicroscopy (STM) technique [21, 22]. Such a superstructuring with C6v symmetry can alsogenerate corner modes originating from higher-order topology (HOT) [23], which has been2observed in various metamaterials [24–29] and electronic artificial lattices [30]. However, del-icate deformations at the angstrom scale in real electronic materials are extremely difficult.While it was proposed that graphene with nanohole arrays [31] might reduce the difficulty,so far the top-down approach has not succeeded in fabricating nanohole arrays with desiredpatterns and scales. As the alternative approach, bottom-up synthesis methods have beentried, such as graphene nanoribbon [32], nanoporous graphene [33] and graphyne [34–36] withatomic precision, which yield interesting electronic [37–43] and mechanical properties [44].Although topological corner modes have been predicted to exist in carbon allotropes suchas graphyne [45] and graphdiyne [46], experimental observations have so far been limited.Here, we unveil topological electronic states in the holey graphyne (HGY) fabricatedrecently in a bottom-up way [47] using first-principles calculations and Wannier TB model-ing [48]. HGY is a two-dimensional (2D) carbon allotrope where benzene rings are connectedby two C C bonds (figure 1(a)). A nontrivial higher-order topological invariant is ob-tained from the C2 symmetry of the material, and the resultant corner modes appear innanoflakes with the edge morphology matching to the structure of precursor [47], which areready for direct experimental observations. In addition, we find that intriguing molecularorbitals emerge in the hexagonal unit cell, and specially a band inversion between g-like andh-like modes gives rise to a nontrivial topology characterized by the Z2 topological invariant.In order to observe the helical edge states similar to those in QSHE, we propose in-planehydrogenation in HGY to absorb the px,y orbitals of carbon atoms in the pristine material,which opens a global topological energy gap at the appropriate energy and makes the helicaledge states with opposite orbital angular momenta (OAM) observable in experiments.2. METHODS2.1. First-principles calculationsFirst-principles calculations are performed within the density-functional-theory (DFT)scheme using the Vienna Ab initio Simulation Package [49], where the projector augmented-wave method [50], the Perdew-Burke-Ernzerhof type generalized gradient approximation [51]for the exchange-correlation potential and a plane-wave basis set with a cutoff energy of520 eV are adopted. A 11 × 11 × 1 Γ-centered k-point mesh is used for both structure3relaxations and self-consistent calculations. The structure relaxations are performed untilthe Hellmann-Feynman forces acting on ions are smaller than 10−4 eV/Å and the energytolerances are below 10−6 eV/atom. Graphyne sheets are separated by a vacuum layer of2 nm, ensuring that interlayer couplings are negligible. The data post-processing is doneusing VASPKIT [52].2.2. Wannier TB modelingMaximally localized Wannier functions (MLWF) are obtained from data in first-principlescalculations using WANNIER90 [53]. For HGY, the Wannier localization procedures areperformed by projecting the eigenstates onto pz orbitals on each atom and a superpositionof px and py orbitals on atoms C’ (figure 1(a)). The outer energy window is set as Eout ∈[−7 eV, 12 eV] to include all p orbitals, and the frozen energy window is set as Efroz ∈[−5.3 eV, 2.4 eV].For hydrogenated HGY (HHGY), the projection orbitals are pz orbitals on each carbonatom and bonding sp orbitals at the centers of C–H bonds. The outer energy window is thesame as the case for HGY, while the frozen energy window starts from −4.5 eV to avoidother orbitals.In the calculation of the HOT index γ (equation 4) for HGY, a pair of twist phases θ areintroduced in a supercell consisting of 3×3 rhombic unit cells with periodic boundary condi-tions applied in two in-plane directions. For the ribbon structure of HHGY, the calculationis performed on a supercell composed of 20 hexagonal unit cells.3. RESULTS AND DISCUSSIONS3.1. Nontrivial higher-order topology and corner modes in HGYBy employing DFT calculations, we obtain the stable real-space structure of HGY witha lattice constant of a0 = 10.85 Å (see bonding information in Supplementary Note S1) andthe band structure as depicted in figures 1(a) and (b), respectively, in good agreement withthe results in previous works [47, 54]. The structure in figure 1(a) can be regarded as anetwork of hexagons (benzene rings) and octagons, and there are two types of carbon sites,one belongs to the hexagon (shared with the octagon) and the other only belongs to the4M Γ K M-6-5-4-3-2-1012Energy(eV)𝑝!𝑝",$Γ MK(b)M Γ K M-6-5-4-3-2-1012Energy(eV)𝑝!𝑝∥𝜃𝜃C’C𝑎!𝑝!𝑝∥𝑥𝑦𝑧𝑧𝑥𝑦(d)(c)(a)Figure 1. Holey graphyne (HGY) and its band structures. (a) Stable structure of HGY withlattice constant a0 obtained by DFT calculations. Carbon atoms are divided into two types, Cand C’, where the former belongs to the benzene rings and the latter only belongs to the octagon.The rhombic unit cell corresponds to the precursor in the bottom-up synthesis as discovered inRef. [47], whereas the hexagon denotes the highest symmetric unit cell. A pair of twist phases θare introduced in the two bonds where the precursor is terminated for the calculation on the Berryphase γ for HOTI. (b) DFT band structure and the Brillouin zone. The Fermi energy is set tozero, and the eigenstates are projected to pz and px,y orbitals. (c) pz-like and p∥-like maximallylocalized Wannier functions obtained from the localization procedure, where the p∥ orbital is thesuperposition of px and py orbitals. (d) Wannier-interpolated band structure obtained from thesubspace selected by projecting onto pz orbitals on each atom and p∥ orbitals on atoms C’.octagon. We name the former and the latter C and C’, respectively. For the band structurein figure 1(b), we project the eigenstates onto the pz and px,y orbitals of carbon atoms, andfind that occupations are either zero or unity, indicating no hybridization between them dueto the mirror symmetry of the 2D material. The pz orbital is also found orthogonal to thes orbital, and thus no hybridization occurs between these two orbitals. Therefore, the pz5bands can be treated independently of other bands.From the C2 symmetry of the precursor octagon (see figure 1(a)), we surmise a HOT inthe HGY. In order to explore the possibility, we first employ Wannier localization procedureto obtain the Wannier TB Hamiltonian. The MLWF shown in figure 1(c) indicate that pzorbitals exist on each atom and a superposition of px and py orbitals exist on atoms C’,which is named p∥ because it forms the π bond parallel to the plane of the material. TheWannier-interpolated band structure is displayed in figure 1(d), which reproduces the DFTband structure shown in figure 1(b).With the obtained Wannier TB Hamiltonian, we calculate the Berry phase γ for HOTinsulator (HOTI) [55–57] associated with a pair of twist phases θ introduced on the twobonds of HGY as depicted in figure 1(a). The Hamiltonian H(θ) can be separated into twoparts asH(θ) = h0(θ) + h1, (1)h0(θ) = −t∑⟨mn⟩(eiθc†mcn + e−iθc†ncm), (2)h1 = −∑mntmn(c†mcn + c†ncm), (3)where h0(θ) with nearest-neighbor hopping energy t = 3.0 eV obtained by MLWF is for thepart with twist phases and h1 is for the rest of the system including long-range hoppings.Note that since the phase twist is applied only on the selected bonds, this phase twist cannotbe gauged out. The Berry phase for HOTI is defined asγ = −i∫ 2π0dθ⟨Ψ(θ)|∂θ|Ψ(θ)⟩ (mod 2π) (4)with |Ψ(θ)⟩ being the ground state of H(θ). Since the system has the C2 symmetry at thecenter of octagon, γ remains the same when θ → −θ, i.e. γ = −i∫ −2π0dθ⟨Ψ(θ)|∂θ|Ψ(θ)⟩ =−i∫ 02πdθ⟨Ψ(θ)|∂θ|Ψ(θ)⟩ (mod 2π), where we shift the integration by 2π since H(θ) = H(θ+2π). Knowing that the integration over a close loop θ = 0 → 2π → 0 is 0 modulo 2π, γ is aZ2 index being either 0 or π. We find outγ = π at the924filling of pz bands, (5)which corresponds to the filling upto the gap at −2.5 eV. Our calculation indicates clearlya nontrivial HOT in the HGY.6●●●●●●● ● ●● ●●● ●●●● pz● s+px,y150 155 160-2.2-2.0-1.8-1.6-1.4State numberEnergy(eV)1 2●●●●●●● ● ●●●●● ●●●● pz● s+px,y150 155 160-2.2-2.0-1.8-1.6-1.4State numberEnergy(eV)1 2(a)(b)1 21 2Figure 2. Topological corner modes in nanoflakes of HGY. (a) (left panel) Energy spectrum forthe straight-type nanoflake with three rhombic unit cells and boundary morphology matching tothe precursor. (right panel) Wave functions of the corner states 1 and 2. C, Br and H atoms arecolored in gray, brown and pink, respectively. (b) Same as (a) except for the boomerang-type.The nontrivial HOT will manifest corner modes in flake structures within the topologicalenergy gap of pz bands, which ranges from −2.5 eV to −1.8 eV. However, as can be seenin figure 1(b), this energy gap is covered by the px,y bands, which hampers the observationof corner modes if the corner modes and the bulk states of px,y orbitals are energeticallyclose. In order to observe topological corner states arising from pz orbital experimentally,the nanoflake should be small enough to have the bulk spectrum discretized, which makesit possible to distinguish the corner states from bulk states energetically.Here we perform DFT calculations on two minimal nanoflakes consisting of three rhombicunit cells with boundary morphology exactly matching the precursor [47]. In the left panelof figure 2(a) we display the energy spectrum for the straight-type nanoflake. As shown inthe right two panels of figure 2(a), the two states labeled by 1 and 2 formed by pz orbitalare corner states (see the other states of pz orbital in Supplementary Note S2). The wavefunctions of the corner states possess even and odd C2 parity with respect to the center of thenanoflake, respectively, corresponding to the bonding and antibonding states. The smallestenergy difference between the antibonding corner state 2 and other states is 40 meV, which7is large enough for the STM technique to detect the corner states by measuring differentialconductance (dI/dV ) maps [40]. The bonding corner state 1 in figure 2(a) might be difficultto detect due to the small energy gap. Similarly, for the boomerang-type nanoflake, therealso exist two corner states as shown in figure 2(b), and the antibonding corner state 2 canbe detected by the STM. The antibonding corner states 2 in figure 2 have equal distributionsof wave functions at two corners due to the C2 symmetry and/or mirror symmetry of thenanoflake. When the Fermi energy is shifted to the energy of antibonding corner states 2, afractional charge of e/2 is expected to appear at each corner.3.2. Nontrivial first-order topology in HGYWe also investigate first-order topology which can be characterized by parity index (C2eigenvalue with respective to the center of hexagonal unit cell shown in figure 1(a)) [58]. Forthis purpose, we evaluate the parity index by counting the numbers of parity-even states atboth Γ and M points (N+Γ/M) for all the valence bands below the global bandgap at −2.4 eVand find (N+Γ , N+M) = (21, 21), which apparently suggests that the material is topologicallytrivial. However, it turns out that the topology of HGY is much richer than it looks at thefirst glance.Since the pz bands are independent of other bands as discussed in the previous section, wecan extract the parity index for the pz bands which gives (N+Γ , N+M) = (3, 5). This imbalanceof parity index indicates the presence of nontrivial topology. Meanwhile, the parity indexfor the other bands, namely s and px,y bands, is (N+Γ , N+M) = (18, 16). By checking theparity index in detail, we find out the imbalance originated from the three bands of px,yorbitals around E = −3 eV, which give (N+Γ , N+M) = (3, 1). Therefore, HGY hosts two setsof topological bands, one from the pz orbital and the other from the px,y orbitals. Namely,these two orthogonal sets of bands exhibit nontrivial topology, showcasing rich topologicalcharacteristics in HGY.Associated with the imbalance of parity index, we expect that the material manifeststopological edge states within the energy gap of pz bands around −2 eV (see figure 1(b)).Unfortunately, as can be seen in figure 1(d), these edge states are covered in energy by thebulk bands of p∥ orbital although orthogonal to each other. In order to solve this issue,we consider in-plane hydrogenation on HGY, where hydrogen atoms are attached to carbon8atoms C’. The stable structure of HHGY with a lattice constant a′0 = 11.02 Å obtained byDFT calculations is illustrated in figure 3(a) (see bond lengths and angles in SupplementaryNote S1). The DFT band structure is shown in figure 3(b), where a global energy gap of0.52 eV is opened at the energy of −2 eV compared to the band structure of pristine HGYshown figure 1(b). Similar to the situation in HGY, the pz bands in HHGY are orthogonal toother bands because of the mirror symmetry with respect to the horizontal plane. Therefore,we perform the Wannierization procedure by projecting the eigenstates onto pz orbitals oneach carbon atom, and the interpolated band structure is shown in figure 3(c), reproducingthe pz bands obtained by DFT calculations shown in figure 3(b).We display the projected density of states (PDOS) for HHGY in figure 3(d), where thepx,y orbitals of carbon atoms and the s orbital of hydrogen atoms exist below the bandgapat −2 eV. This is in sharp contrast to the situation in HGY where the px,y orbitals fill upalmost all the bandgap of pz orbital as displayed in figure 1(b). The in-plane hydrogenationleads to the sp-hybridization between the px,y orbitals of carbon atoms and the s orbital ofhydrogen atoms, lowering the energy of the original p∥ orbital. Therefore, the hydrogenationeffectively kills the original p∥ orbital and results in an energy gap as large as 0.52 eV.In order to double-check our strategy, we re-examine the topology for the pz bands us-ing parity index, and find imbalance by (N+Γ , N+M) = (3, 5) for the states below the en-ergy gap, indicating the presence of nontrivial topology as expected (see SupplementaryNote S3). We can also check the parity index for all states including other orbitals, whichgives (N+Γ , N+M) = (24, 26), meaning that the nontrivial topology originates purely from thepz bands as discussed above.Moreover, in figures 3(b) and (c) we notice that the unbalanced parity index is inducedby a band inversion between g- and h-like modes around the bandgap at the Γ point. Asillustrated in figures 3(e) and (f), the eigenstates g-, h- and i-like modes are named bycounting the number of nodes along the perimeter of hexagonal unit cell, i.e., 8, 10, and12, with even, odd and even parity, respectively. Note that if the green hexagonal cell infigure 3(a) is isolated and the hoppings between pz orbitals are uniform, the classification bythe number of nodes is exact and the eigenenergy should increase with the number of nodes.However, in figure 3(b), g-like mode comes above h-like mode, signaling a band inversion.For the two isolated valence pz bands with h-like mode with odd parity at the Γ point andparity-even states at the M point, we evaluate the Wilson loop [2, 59] and find a phase9M Γ K M-6-5-4-3-2-1012Energy(eV) ghi++−−+M Γ−+++−g1 g2ih2h1g1 g2h1 h2i(e) (f)(d) C(pz)C(px,y)H-6 -5 -4 -3 -2 -1 0 1 2051015Energy (eV)PDOS(states/eV)(c)(a)𝑎!"M Γ K M-6-5-4-3-2-1012Energy(eV) ghi++−−+−++Γ+−M(b)Figure 3. Hydrogenated holey graphyne (HHGY) and its band structures. (a) Stable structure ofHHGY with lattice constant a′0 obtained by DFT calculations, where carbon and hydrogen atomsare colored in black and cyan, respectively. The unit cell with the highest symmetry is denotedby a hexagon. (b) Band structures obtained by DFT calculations, with eigenstates projected topz orbitals and the bandgap shown by a region highlighted in blue. The g-, i- and h-like modesare labeled explicitly, and the parity of the eigenstates at Γ and M points for the correspondingfive bands are denoted by plus and minus signs. (c) Same as (b) except for Wannier interpolationwhich is performed by projecting onto pz orbitals on each carbon atom. (d) Projected density ofstates (PDOS) obtained by DFT calculations, with the bandgap shown by a region highlighted inblue. (e) Wave functions of g-, h- and i-like modes at the Γ point obtained by DFT calculations,with red/blue color denoting the plus/minus value. Notation: g1 ≡ gx4−6x2y2+y4 , g2 ≡ gxy(x2−y2),h1 ≡ hx(x4−10x2y2+5y4), h2 ≡ hy(5x4−10x2y2+y4), i ≡ ix6−15x4y2+15x2y4−y6 . (f) Same as (e) except forWannier interpolation, with amplitude represented by size of dots.winding of 2π, which indicates a nontrivial band topology (see Supplementary Note S4).Coming back to the full argument with the actual crystalline symmetry, doubly degener-ate g- and h-like modes are the 2D irreducible representations of the C6v symmetry in thematerial, while the singlet i-like mode is the 1D irreducible representation. With the doubledegeneracy, we can construct pseudospin states using g- and h-like modes:|g±⟩ =1√2(|g1⟩ ± i|g2⟩) , |h±⟩ =1√2(|h1⟩ ± i|h2⟩) . (6)10We regard |g+⟩ and |h+⟩ as pseudospin-up states, since their phases of wave functions in-crease +8π and +10π counterclockwisely along the perimeter of hexagonal unit cell, whichcorrespond to the states with OAM +4ℏ and +5ℏ, respectively, noting that the operator ofOAM is −iℏ ∂∂ϕ. The time-reversal counterparts |g−⟩ and |h−⟩ are regarded as pseudospin-down states with OAM −4ℏ and −5ℏ, respectively. Again, this argument is exact in the caseof an isolated cell with a uniform hopping. In this system where there is only C6v symmetry,the eigenstates are classified by OAM up to mod 6ℏ, and ±4ℏ and ±5ℏ correspond to ∓2ℏand ∓ℏ, respectively.3.3. Helical edge states in a ribbon of HHGYThe nontrivial topology characterized by the Z2 topological index will manifest topolog-ical edge states. As can be seen in figure 4(a) obtained by the Wannier TB calculations onthe ribbon structure with a molecule-zigzag edge morphology [14] where the hexagonal unitcells remain intact, a pair of topological edge states carrying opposite pseudospins appearwithin the DFT bulk bandgap (see Supplementary Notes S5 for details). The LDOS of edgestates are shown in figure 4(b) (for detailed understanding see Supplementary Notes S6),and the wave functions including phases with comparison to the bulk pseudospin states |±⟩(= (|g±⟩ ± i|h±⟩)/√2) are shown in figure 4(c).It should be noted that, in the previous work [43] the topological control means real-space geometrical control, namely changing neck widths between holes in holey graphene;no topology in momentum space has been touched. In the present work, the topologyis related to Berry phase defined by Bloch wave functions in momentum space, and weshow successfully that this can be engineered by real-space control, such as arrangement ofprecursors and/or edge morphology of ribbon.A direct comparison between figures 2(a) and 4(a) is difficult, since they stand for twodifferent systems with energy shift: figure 2(a) for small nanoflakes without hydrogenation,whereas figure 4(a) for a ribbon with hydrogenation. The edge morphology exhibiting thetopological edge states in this work cannot be achieved by the available precursor directly,and thus trials on edge treatments are required, such as out-of-plane hydrogenation.111423𝐸!,#$%𝑘 (1/ 3𝑎&' )(b)−1 1𝑖−𝑖12|+⟩|−⟩ 34(c)31𝛼1-10(a)Figure 4. Topological edge states in a ribbon of HHGY. (a) Energy dispersion of a ribbon of HHGYobtained by Wannier TB calculations on pz orbitals. The region highlighted in blue represents theDFT bulk bandgap, where the edge states are colored by their pseudospin polarizations α (seeSupplementary Note S5). (b) LDOS |ψi|2 for the edge state 1 (upper panel) and 3 (lower panel)as labeled in (a), with the outmost unit cell highlighted by a dashed hexagon. (c) Comparisonbetween the bulk pseudospin states |±⟩ and the wave functions with spinful parts in the outmostunit cell of the four edge states 1 to 4, with amplitude and phase denoted by size and color of dots,respectively.4. CONCLUSIONSIn summary, we uncover topological electronic states in holey graphyne, through DFTcalculations and Wannier TB modeling. Firstly, a nontrivial Z2 higher-order topologicalinvariant is obtained arising from the C2 symmetry of the material. We display explicitly theminimal nanoflakes hosting the corresponding corner states, which can be fabricated usingthe same precursor in the experiment [47], and the corner modes are ready to be observed12by measuring differential conductance maps using STM. While so far the observation oftopological corner modes in electronic systems has been limited to artificial lattices madeby positioning carbon-monoxide molecules on Cu(111) surface [30], holey graphyne couldbe the non-artificial material where topological electronic corner modes are experimentallyobserved. Secondly, by introducing an in-plane hydrogenation, holey graphyne opens a largeenergy gap of 0.52 eV at −2 eV, with a nontrivial topology arising from the band inversionbetween g- and h-like modes and characterized by a Z2 invariant. Topological edge statescarrying opposite OAMs propagate in opposite directions, indicating potential applicationsin orbitronic devices with low energy loss workable at room temperature. Our work pointsout that holey graphyne serves as an ideal platform to observe topological electronic cornermodes and edge states with OAMs protected by a large energy gap, opening a new avenue forfurther investigation and exploitation of topology in 2D carbon allotropes for both scientificunderstanding and practical applications.ACKNOWLEDGEMENTThis work is supported by CREST, JST (Core Research for Evolutionary Science andTechnology, Japan Science and Technology Agency) (Grant Number JPMJCR18T4).DATA AVAILABILITY STATEMENTAll the data that support the findings of this study are available from the correspondingauthor upon reasonable request, following the policy of JST.[1] F. D. M. Haldane, Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the ”Parity Anomaly”, Phys. Rev. Lett. 61, 2015 (1988).[2] H. Weng, R. Yu, X. Hu, X. Dai, and Z. Fang, Quantum anomalous Hall effect and relatedtopological electronic states, Adv. Phys. 64, 227 (2015).[3] A. H. 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