Flat bands with fragile topology through superlattice engineering on single-layer graphene
Anastasiia Skurativska, Stepan S. Tsirkin, Fabian D Natterer, Titus Neupert, Mark H Fischer
FFlat bands with fragile topology through superlattice engineering on single-layer graphene
Anastasiia Skurativska, Stepan S. Tsirkin, Fabian D Natterer, Titus Neupert, and Mark H Fischer
Department of Physics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland (Dated: January 22, 2021)‘Magic’-angle twisted bilayer graphene has received a lot of interest due to its flat bands with potentially non-trivial topology that lead to intricate correlated phases. A spectrum with flat bands, however, does not requirea twist between multiple sheets of van der Waals materials, but rather can be realized with the applicationof an appropriate periodic potential. Here, we propose the imposition of a tailored periodic potential onto asingle graphene layer through local perturbations that could be created via lithography or adatom manipulation,which also results in an energy spectrum featuring flat bands. Our first-principle calculations for an appropriatedecoration of graphene with adatoms indeed show the presence of flat bands in the spectrum. Furthermore, wereveal the topological nature of the flat bands through a symmetry-indicator analysis. This non-trivial topologymanifests itself in corner-localized states with a filling anomaly as we show using a tight-binding model. Ourproposal of a single decorated graphene sheet provides a new versatile route to study correlated phases intopologically non-trivial, flat band structures.
Engineering the desired functionality of a material throughnanostructuring has proven a powerful approach that is partic-ularly well suited for two-dimensional (2D) materials. Of-ten, the goal of such engineering involves shifting spectralweight to the Fermi level or increasing the coupling of theelectrons to other degrees of freedom, such as light for im-proved optoelectronic properties or phonons for supercon-ductivity . Nanostructuring can be achieved in a variety ofways: standard cleanroom techniques, such as electron beamlithography , photolithography, or focused ion beam lithogra-phy , allow to realize patterns of a few nanometers in size.Moir´e engineering, in other words using the potential land-scape from a Moir´e lattice that emerges due to a finite twistangle between two 2D lattices, has attracted a lot of atten-tion in recent years. This attention stems in large part fromthe discovery of extremely flat bands for certain, very smalltwist angles, referred to as ‘magic’ angles, in twisted bilayergraphene (TBG) . Ideally, these bands concentrate spec-tral weight around the Fermi level, are separated by a gapfrom other bands in the spectrum, and potentially possess non-trivial topology with intriguing implications for the many-body ground states . Moreover, as a result of the flatness ofthese bands the electron-electron coupling becomes the dom-inant interaction. Consequently, this system shows correlatedinsulator states at integer fillings and superconductivity inbetween . Finally, even quantum anomalous Hall phases,driven by the strong interactions, have been observed .More specifically, for a discrete set of angles, the Moir´epattern creates a commensurate hexagonal supercell and theelectronic structure can be described again in terms of Blochstates. The superstructure significantly affects the tunneling ofelectrons between the two graphene layers and the hybridiza-tion of electronic states near the two graphene Dirac-conesresults in a decrease of the Fermi velocity at the charge neu-trality point and the formation of nearly flat bands in the spec-trum. The twist angle in bilayer graphene directly controls thesize of the Moir´e supercell and thus acts as the tuning param-eter for the band flattening.Motivated by magic-angle TBG, many more systems wereproposed, where a small twist angle between stacked layers,such as other multilayer graphene heterostructures or transi- tion metal dichalcogenides, is used to create novel correlatedphases. Furthermore, Moir´e engineering on a single Diraccone of a topological-insulator surface state was recently dis-cussed . Besides the restriction on the types of periodicpotentials that can be realized with Moir´e pattern, however,producing samples with predefined twist angle and sufficienthomogeneity is a further intricate experimental challenge .There is thus an ongoing search for other systems with elec-tronic properties similar to the ones of magic-angle TBG ,but with more control over the design. Such systems providenovel platforms to study the physics of correlated electrons intopologically non-trivial bands.In this work, we propose an alternative approach for cre-ating topologically non-trivial flat bands in a single graphenesheet by the application of a periodic potential. In particular,using first-principle calculations, we investigate a single layerof graphene decorated with a periodic, C -symmetric distribu-tion of adatoms, see Fig. 1a. While such an approach allowsfor high control over the applied potential through an adatomsuperlattice via atom manipulation using scanning tunnellingmicroscopy , our principle is amenable also to artificialgraphene or engineered lattices , and via nanofabrica-tion to graphene .Within our first-principles calculations, we indeed find flatbands separated by gaps from other bands in the spectrum ofthe system. Employing the recently introduced framework oftopological quantum chemistry , we further reveal the frag-ile topological nature of these bands. Bands with this typeof topology stand in between strong topological and trivialphases, based on the topological robustness against additionof trivial degrees of freedom. Specifically, the main charac-teristic of bands with fragile topology is that they can be triv-ialised by addition of bands permitting an atomic limit belowthe Fermi level .Systems with fragile topology protected by n -fold rotationsymmetry often feature a filling anomaly: In open boundaryconditions, the system possesses n degenerate in-gap states,which are only partially occupied at charge neutrality . Thisimplies a degeneracy of the many-body ground state in thethermodynamic limit protected by rotation symmetry. We il-lustrate this bulk-boundary correspondence employing a tight- a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n d. v v a a x y a. x y b.c. y k x k MKΓ s M s K K KΓ s Γ y k x k k x k x k y k y Density of States W- W- W- W- C- total sd z d xy / x − y d xz / yz p z Γ Γ
K M E − E F [ e V ] Γ Γ Γ Γ Γ Γ M M Γ Γ M FIG. 1.
System setup and electronic structure. ( a ) The unit cell consisting of a graphene monolayer with adatoms placed in the center ofthe unit cell as indicated by the red circle. The lattice vectors ( v , v ) are related to the lattice vectors of pure graphene ( a , a ) by Eq. (1)with ( n, m ) = ( − , . ( b ) The Brillouin zone (BZ) of monolayer graphene shown in red and the reduced BZ of our setup depicted in black.The K and K (cid:48) points of graphene map to the Γ point in the reduced BZ of the superstructure. ( c ) For comparison, the BZ of TBG (in black)tilted with respect to the two BZs of graphene twisted by a small angle (in blue and red). ( d ) The DFT spectrum of graphene with tungstenadatoms. Irreducible representations of the bands at high-symmetry points are indicated in the figure. Right side shows the density of statesprojected onto the adatom and carbon p z orbitals, respectively. binding calculation of a C -symmetric flake, where the fillinganomaly manifests itself in an excess charge accumulation inthe corners of the flake, suitable for experimental discovery. MODEL SYSTEM AND FLAT BANDS
Conceptually, the electronic structure of both TBG and ourapproach can be understood starting from that of a single layerof graphene. The band structure is characterized by two Diraccones around the K and K (cid:48) points in the Brillouin zone (BZ).For TBG with discrete commensurate twist angles, the result-ing periodic superstructure allows for a momentum-space de-scription in a reduced BZ. As shown in Fig. 1c, this reducedBZ can be geometrically constructed directly in momentumspace, by twisting the two original BZs resulting in a reducedBZ with new K and K (cid:48) points stemming from the K pointsfrom the two individual layers, K and K . For small twistangles, the tunneling between the two graphene layers thathybridizes the bands around K and K becomes comparableto the band width of the respective bands in the reduced BZ,resulting in flat bands over the whole reduced BZs. Note thatthere are two time-reversal related BZs, one from the K andone from the K (cid:48) points of the individual graphene layers. Forthe small angles required for flat bands, the resulting unit cellin real space contains thousands of carbon atoms.Inspired by the construction in TBG, we build in the follow-ing flat bands starting again from a single graphene sheet, butenlarging the unit cell using nanostructuring. We consider asuperlattice potential arising from adatoms placed in the hol-low sites (H) of the graphene lattice, meaning in the center ofthe C hexagon, with a periodicity described by the superlatticevectors v and v , see Fig. 1a. In consequence and contrastto the TBG case, we therefore only have a single original K and K (cid:48) point. We choose the superlattice vectors in such away, that both K and K (cid:48) are mapped to the Γ point of the re- duced BZ. Crucially, this allows for a strong hybridization ofthe original with the adatom bands. The lattice vectors leadingto such a configuration are given by v = n a + (3 m + n ) a , (1)where n, m ∈ Z , a and a are the lattice vectors forgraphene, and v is related to v by a 60-degrees rotation.For concreteness, we use in the following ( n, m ) = ( − , .The BZ resulting from this construction is shown in Fig. 1b.As a guiding principle, we choose non-magnetic transition-metal adatoms, which, as we will discuss below, genericallylead to flat bands that are topologically non-trivial due to their d orbitals. We further require the candidate adatoms to be suf-ficiently stable at the H site . As a figure of merit, we con-sider the ratio of spectral gap to the closest C-based bands andthe band width. Finally focusing on situations, where the flatbands are separated from other bands related to the adatoms,these guiding principles result in a set of most promisingadatoms: W, Ta, and Ru. Figure 1d shows the band structurethat results from the above construction with tungsten adatomsobtained using first-principles calculations (see Methods fordetails). For these calculations, the graphene lattice is ori-ented in the x - y plane and the tungsten atoms are relaxed totheir equilibrium position in z direction over the H site. Forcompleteness, Tab. I summarizes the band width and gaps tothe nearest C-based bands of tungsten and all other consideredadatoms (see Methods for their corresponding DFT spectra). Element W Ir Cr Mo Rh Ta Ru Re Os NbWidth [eV] .
26 0 .
56 0 .
21 0 . .
34 0 .
26 0 .
26 0 . x y Γ Γ MK x y + - + - - b.a.c. d. y x y x B a n d e n e r g y k − π πN uc = 61 FIG. 2.
Tight-binding calculations for open geometries ( a ) Thegeometry of C -symmetric flake with the number of the unit cells N uc = 61 . ( b ) Band structure of the tight-binding model (in blue)with flat bands highlighted in yellow and C -symmetry eigenvaluesat Γ and M points indicating the topological nature of the bands. Thespectrum for the flake (in red) shows in-gap states around the energyindicated by the green line. ( c ) The unit cell of the ribbon geometry.( d ) The spectrum for ribbon geometry (in blue) shows a small gap.In the flake geometry, there are six additional states that lie in thisgap (in red). In the following we focus on the flat bands highlightedin Fig. 1d, assuming that the chemical potential through thegraphene channel can be directly controlled by back gating .The orbital projected density of states on the right side ofFig. 1d shows that these flat bands are originating from thehybridization of the C p z orbitals of graphene and the d xz,yz orbitals of tungsten. As we will discuss in the following,this is crucial for the topological properties and needs to betaken into account when constructing a minimal tight-bindingmodel. TOPOLOGICAL PROPERTIES
To examine the topological properties of the flat bands con-structed by the superlattice, we employ a symmetry analysiswithin the context of topological quantum chemistry . Thisanalysis requires the transformation properties of the wavefunctions that make up the flat bands at the Γ and the M points. While the decorated graphene lattice retains the C symmetry of graphene, all mirror symmetries are broken bythe adatom arrangement, such that, including translations, oursystem reduces to P space-group symmetry. For the situ-ation shown in Fig. 1d, the flat bands highlighted with yel-low transform as Γ and Γ at the Γ point and M at the M point. In particular, the C eigenvalues associated withthese irreducible representations at the Γ and M points are α C (Γ) = +1 and α C ( M ) = − . These eigenvalues can beunderstood by inspecting the orbital content of the bands inFig. 1d: At the Γ point, the bands stem from p z orbitals, while at the M point, the bands originate from d xz and d yz orbitals.Such a combination of irreducible representations cannot arisefrom an atomic limit of exponentially-localized Wannier func-tions , implying that the bands are indeed topological. Fol-lowing the terminology introduced in this context, these bandsdo not form an elementary band representation (EBR). The p z - d xz / d yz hybridization is thus the crucial ingredient for theformation of bands with non-trivial topology.While the flat bands cannot be adiabatically connected toan atomic limit, they can be written as a difference betweentwo EBRs (with integer coefficients), which indicates the frag-ile nature of their topology . In particular, the bands canbe expressed as the difference FT = AL − AL , whereAL = [Γ ⊕ Γ Γ , M ⊕ M ] and AL = [Γ ⊕ M ] aretwo sets of band representations forming an EBR. This fea-ture distinguishes the current case from a strong topologicalphase, which cannot be trivialised by adding trivial degrees offreedom.Unlike strong topological phases, which are characterizedby topological edge states due to the bulk-boundary corre-spondence, bands with fragile topology protected by C n -symmetry can exhibit a filling anomaly. This topologicalfeature describes the situation, when a mismatch exists be-tween the number of electrons required to simultaneously sat-isfy charge neutrality, a unique ground state in open boundaryconditions, and the crystalline symmetry. In the spectrum, n degenerate states associated with the filling anomaly appear inthe gap with n/ states occupied for charge neutrality. In ourcase, adding one more electron will lead to a quantized excesscharge of e/ in each corner of the flake .To investigate the appearance of this type of bulk-boundarycorrespondence associated with the fragile topology of the flatbands, we introduce a tight-binding model starting from the C p z and the transition-metal d orbitals (see Methods). Such amodel allows for the simulation of any open geometry, suchas the ones shown in Fig. 2. Before investigating this finite L DO S b.c. E a. x y unit cells along the blue/red lines N uc = 331 N uc FIG. 3.
Corner-localized in-gap states ( a ) The spatial distributionof the absolute values squared of the eigenvectors corresponding tothe in-gap states for C -symmetric graphene flake shown in Fig. 2awith N uc = 331 . ( b ) Line profile of the local density of states atthe bound state, integrated along the blue and red lines given in a .Dashed red line indicates exponential fit. ( c ) Finite-size scaling of thein-gap states showing their degeneracy in the thermodynamic limit.An arrow indicates the states with spatial distribution given in a . system further, we note that for appropriate parameters, thetight-binding model indeed yields flat bands with the correctirreducible representations as discussed above and shown inFig. 2.The example of the C -symmetric flake in Figure 2a leadsto the spectrum in panel b with the spectrum of the transla-tionally invariant system added for comparison. Indeed, wefind in-gap states, though, more states than the six anticipatedfrom fragile topology. We can understand the origin of theseadditional states considering a ribbon geometry as shown inFig. 2c. As can be seen in Fig. 2d, most of the in-gap statesare associated with edge states which are, however, not com-pletely gapless. These states are connected to an only weakly-broken mirror symmetry perpendicular to the open directionin the ribbon geometry. In the hybridization gap of these edgestates, we find six in-gap states for the flake geometry, whichwe attribute to the fragile topology of the system.Figure 3a shows the contribution of these six in-gap statesto the local density of states of the flake geometry. Their dom-inant spectral weight is localized at the corners of the flake,as further emphasized in Fig. 3b, which shows the unit-cell-averaged weight of the wave functions of the states in the gapalong the edge. Relevant to their experimental discovery is anexponentially decaying LDOS towards the center of the struc-ture with a characteristic length of . /N uc , which wouldallow for a distinction with respect to other edge-state obser-vations. Finally, while the six states associated with the fillinganomaly are not degenerate in a finite geometry, a finite sizescaling, Fig. 3c, shows that they indeed become degenerate inthe thermodynamic limit. As such, panel c in Fig. 3 serves asa useful guide for how the degeneracy of the corner-localizedgap states evolves in the gradual buildup of such a structure. DISCUSSION
The system we propose here is conceptually simple, yet fea-tures intriguing topological properties. We demonstrated howthe fragile topology manifests itself through a filling anomaly,which can be mapped with a local scanning probe. In ad-dition to the finite geometries that are required to probe thecorner-localized states, our approach allows for more designfreedom. In particular, while we focused here on a superstruc-ture with P symmetry, any subgroup of the graphene spacegroup P /mmm can be realized by choosing the appropriatesuperlattice vectors. Furthermore, defects in the lattice, whichare a distinct way of probing topological bands, can be readilyimplemented by the deliberate addition or removal of atoms.Our ideas of engineering topologically non-trivial flat bandsthrough nanostructuring go beyond the periodic decoration ofgraphene with adatoms. A further promising route towardstheir realization can be based on artificial graphene, for exam-ple using scanning-tunneling-microscopic methods to arrangeCO molecules on a Cu(111) surface . Finally, in view ofthe required nanometer periodicity, we expect that graphenesheets could even be engineered by lithography techniques. ACKNOWLEDGEMENTS
A.S., S.S.T. and T.N. were supported by funding fromthe European Research Council (ERC) under the Euro-pean Union’s Horizon 2020 research and innovation program(ERC-StG-Neupert-757867-PARATOP). A.S. was also sup-ported by Forschungskredit of the University of Zurich, grantNo. FK-20-101. S.S.T. and T.N. were additionally sup-ported by NCRR Marvel. S.S.T. also acknowledges supportfrom the Swiss National Science Foundation (grant number:PP00P2 176877). F.D.N. thanks SNSF (PP00P2-176866) andONR (N00014-20-1-2352) for generous support. S. Yan, X. Zhu, J. Dong, Y. Ding, and S. Xiao, Nanophotonics ,74 (2020). M. P. Allan, M. H. Fischer, O. Ostojic, and A. Andringa, SciPostPhys. , 010 (2017). A. E. Grigorescu and C. W. Hagen, Nanotechnology , 292001(2009). C. Genet and T. W. Ebbesen, Nature , 39 (2007). R. Bistritzer and A. H. MacDonald, Proceedings of the NationalAcademy of Sciences , 12233 (2011). Y. Cao, V. Fatemi, A. Demir, S. Fang, S. L. Tomarken, J. Y. Luo,J. D. Sanchez-Yamagishi, K. Watanabe, T. Taniguchi, E. Kaxiras,R. C. Ashoori, and P. Jarillo-Herrero, Nature , 80 (2018). Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras,and P. Jarillo-Herrero, Nature , 43 (2018). L. Zou, H. C. Po, A. Vishwanath, and T. Senthil, Phys. Rev. B ,085435 (2018). Z.-D. Song, B. Lian, N. Regnault, and B. A. Bernevig, arXiv e-prints , arXiv:2009.11872 (2020). X. Lu, P. Stepanov, W. Yang, M. Xie, and D. K. Efetov, Nature ,653 (2019). Y. Wang, J. Herzog-Arbeitman, G. W. Burg, J. Zhu, K. Watanabe,T. Taniguchi, A. H. MacDonald, B. A. Bernevig, and E. Tutuc, arXiv e-prints , arXiv:2101.03621 (2021). A. T. Pierce, Y. Xie, J. M. Park, E. Khalaf, S. H. Lee, Y. Cao,D. E. Parker, P. R. Forrester, S. Chen, K. Watanabe, T. Taniguchi,A. Vishwanath, P. Jarillo-Herrero, and A. Yacoby, arXiv e-prints, arXiv:2101.04123 (2021). M. Serlin, C. L. Tschirhart, H. Polshyn, Y. Zhang, J. Zhu,K. Watanabe, T. Taniguchi, L. Balents, and A. F. Young, Science , 900 (2020). J. Cano, S. Fang, J. H. Pixley, and J. H. Wilson, arXiv e-prints ,arXiv:2010.09726 (2020). T. Wang, N. F. Q. Yuan, and L. Fu, arXiv e-prints ,arXiv:2010.09753 (2020). A. Uri, S. Grover, Y. Cao, J. A. Crosse, K. Bagani, D. Rodan-Legrain, Y. Myasoedov, K. Watanabe, T. Taniguchi, P. Moon,M. Koshino, P. Jarillo-Herrero, and E. Zeldov, Nature , 47(2020). T. Benschop, T. A. de Jong, P. Stepanov, X. Lu, V. Stalman, S. J.van der Molen, D. K. Efetov, and M. P. Allan, arXiv e-prints ,arXiv:2008.13766 (2020). J. M. Lee, C. Geng, J. W. Park, M. Oshikawa, S.-S. Lee, H. W.Yeom, and G. Y. Cho, Phys. Rev. Lett. , 137002 (2020). V. W. Brar, R. Decker, H.-M. Solowan, Y. Wang, L. Maserati,K. T. Chan, H. Lee, C¸ . O. Girit, A. Zettl, S. G. Louie, M. L. Cohen,and M. F. Crommie, Nature Physics , 43 (2011). Y. Wang, D. Wong, A. V. Shytov, V. W. Brar, S. Choi, Q. Wu, H.-Z. Tsai, W. Regan, A. Zettl, R. K. Kawakami, S. G. Louie, L. S.Levitov, and M. F. Crommie, Science , 734 (2013). J. Wyrick, F. D. Natterer, Y. Zhao, K. Watanabe, T. Taniguchi,W. G. Cullen, N. B. Zhitenev, and J. A. Stroscio, ACS Nano ,10698 (2016). K. K. Gomes, W. Mar, W. Ko, F. Guinea, and H. C. Manoharan,Nature , 306 (2012). A. H. Robert Drost, Teemu Ojanen and P. Liljeroth, NaturePhysics , 668 (2017). L. Yan and P. Liljeroth, Advances in Physics: X , 1651672(2019). A. A. Khajetoorians, D. Wegner, A. F. Otte, and I. Swart, NatureReviews Physics , 703 (2019). O. Dyck, S. Kim, S. V. Kalinin, and S. Jesse, Applied PhysicsLetters , 113104 (2017). B. Bradlyn, L. Elcoro, J. Cano, M. G. Vergniory, Z. Wang,C. Felser, M. I. Aroyo, and B. A. Bernevig, Nature , 298 (2017). H. C. Po, A. Vishwanath, and H. Watanabe, Nature Communica-tion , 50 (2017). H. C. Po, H. Watanabe, and A. Vishwanath, Phys. Rev. Lett. ,126402 (2018). W. A. Benalcazar, T. Li, and T. L. Hughes, Phys. Rev. B ,245151 (2019). K. Nakada and A. Ishii,
DFT Calculation for Adatom Adsorptionon Graphene (IntechOpen, 2011). K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang,S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science ,666 (2004). M. Aroyo, J. Perez-Mato, D. Orobengoa, E. Tasci, G. De La Flor,and A. Kirov, Bulgarian Chemical Communications , 183(2011). M. I. Aroyo, A. Kirov, C. Capillas, J. M. Perez-Mato, andH. Wondratschek, Acta Crystallographica Section A , 115(2006). M. I. Aroyo, J. M. Perez-Mato, C. Capillas, E. Kroumova,S. Ivantchev, G. Madariaga, A. Kirov, and H. Wondratschek,Zeitschrift f¨ur Kristallographie - Crystalline Materials , 15(2006). B. Bradlyn, Z. Wang, J. Cano, and B. A. Bernevig, Phys. Rev. B , 045140 (2019). G. Kresse and J. Furthm¨uller, Phys. Rev. B , 11169 (1996). P. E. Bl¨ochl, Phys. Rev. B , 17953 (1994). G. Kresse and D. Joubert, Phys. Rev. B , 1758 (1999). J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. ,3865 (1996). M. Iraola, J. L. Ma˜nes, B. Bradlyn, T. Neupert, M. G. Vergniory,and S. S. Tsirkin, arXiv e-prints , arXiv:2009.01764 (2020). L. Elcoro, B. Bradlyn, Z. Wang, M. G. Vergniory, J. Cano,C. Felser, B. A. Bernevig, D. Orobengoa, G. de la Flor, and M. I.Aroyo, Journal of Applied Crystallography , 1457 (2017). METHODS1. Tight-binding model
To study low-energy properties of the described system, webuild a minimal tight-binding model. For ( n, m ) = ( − , ,we have to include 42 p z orbitals and 4 d orbitals of thetransition-metal adatom in the unit cell, which in a nearest-neighbour approximation give rise to the following tight-binding model: H = − t (cid:88) (cid:104) i,j (cid:105) (cid:16) c † i c j + h.c. (cid:17) + (cid:88) α, (cid:104) i,j (cid:105) (cid:16) ˜ t α,j d † αi c j + h.c. (cid:17) , (2)where c ( † ) j , d ( † ) αj correspond to creation and annihilation of p z and d orbitals respectively with α ∈ { xy, x − y , xz, yz } and the hopping parameters ˜ t α,j , which depend on the typeof d orbital and can be simplified by taking into account localsymmetries of the lattice. d x − y − γ /2− γ /2 − γ /2− γ /2 γγ d xz δ / 3 δ / 3− δ / 3− δ / 3−2 δ / 3 2 δ / 3 d yz δ − δδ − δ d xy γ /23 γ /2− 3 γ /2 − 3 γ /2 FIG. 4. The values of the hopping parameters between four different d -orbitals of adatom shown in the center of each hexagon and thenearest p z -orbitals of graphene. Locally around the position of the adatom, the unit cell isinvariant under mirror symmetry M x and C -rotation repre-sented by C = R xzyz R xyx − y , (3)where R xzyz = (cid:18) cos π sin π − sin π cos π (cid:19) , R xyx − y = (cid:18) cos ( π ) − sin ( π ) 2 cos π sin π π sin π cos ( π ) − sin ( π ) (cid:19) . and M x = − − . (4)The part of the Hamiltonian corresponding to the hoppingswithin the central hexagon including four d orbitals in the cen-ter and six nearest p z orbitals surrounding them is given by thematrix H ˜ t = t if i, j = { , } , ˜ t α,j if i = { , } , j = { , } and vice versa, if i, j = { , } , (5)with unknown hopping parameters ˜ t α,j , which can be ex-pressed in terms of only two parameters γ and δ using the fol-lowing set of equations provided by the symmetry constraints C H ˜ t C − = H ˜ t M x H ˜ t M x − = H ˜ t . (6)The results are summarised in Fig. 4.
2. First-principles calculations
We performed first-principles calculations within thedensity functional theory (DFT) framework using theVASP package , employing the projector-augmented wavemethod and the Perdew-Burke-Ernzerhof generalized-gradient approximation (GGA-PBE) for the exchange-correlation energy. We employ the following scheme consist-ing of two steps: In a first step, we perform self-consistentDFT calculations on a × × grid in momentum spacewith fixed in-plane atomic positions and allow for the latticerelaxation in the out-of plane direction. In this way, we findthe distance between the graphene lattice and adatom whichminimises the total energy and we get the correspondingcharge-density profile. In the second step, we use this chargedensity to perform non-self-consistent calculations along the Γ − K − M − Γ path in momentum space to obtain the bandstructure plots. Note that the calculations do not include spin-orbit coupling. The irreducible representations were deter-mined using the IrRep code and given in the notation ofthe Bilbao Crystallographic Server .Performing these calculations for different chemical ele-ments we find a set of elements, which give rise to flat bandswith fragile topology (see Fig.5). In Tab. I we summarise theresults of DFT calculations for a set of adatoms, which show similar behaviour of the band structure to what has been pre-sented in the main text. Γ Γ
K M E − E F [ e V ] Γ Γ Γ Γ Γ Γ M M Γ Γ
K M E − E F [ e V ] Γ Γ Γ Γ Γ Γ M M Γ Γ
K M E − E F [ e V ] Γ Γ Γ Γ M M Γ Γ Γ Γ
K M E − E F [ e V ] Γ Γ Γ Γ M M Γ Γ Γ Γ
K M E − E F [ e V ] Γ Γ Γ Γ Γ Γ M M Γ Γ
K M E − E F [ e V ] Γ Γ Γ Γ Γ Γ M M Γ Γ
K M E − E F [ e V ] Γ Γ Γ Γ Γ Γ M M Γ Γ
K M E − E F [ e V ] Γ Γ Γ Γ Γ Γ M M Γ Γ