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Photo-inducedSU(3)topologicalmaterialofspinlessfermions Sayonee Ray, Ananya Ghatak and Tanmoy Das Department of Physics, Indian Institute of Science, Bangalore-560012, India. (Dated:January24,2017) GenerationoftopologicalphasesofmatterwithSU(3)symmetryinacondensedmattersetupischallenging duetothelackofanintrinsicthree-foldchiralityofquasiparticles.Weuncovertwosalientingredientsrequired toexpressathree-componentlatticeHamiltonianinaSU(3)formatwithnon-trivialtopologicalinvariant. We findthatallthreeSU(3)componentsmustbeentangledviaagaugefield,withoppositechiralitybetweenanytwo components,andtheremustbebandinversionsbetweenallthreecomponentsinagiveneigenstate.Forspinless particles,weshowthatsuchchiralstatescanbeobtainedinatripartitelatticewiththreeinequivalentlatticesites 7 1 inwhichtheBlochphaseassociatedwiththenearestneighborhoppingactsask-spacegaugefield.Thesecond 0 andamorecrucialcriterionisthattheremustalsobeanodd-parityZeeman-liketerm,i.e.sin(k)σztermwhere 2 σz is the third Pauli matrix defined in any two components of the SU(3) basis. Solving the electron-photon interactionterminaperiodicpotentialwithamodifiedtight-bindingmodel,weshowthatsuchatermcanbe n engineeredwithsite-selectivephotonpolarization. Suchsiteselectivepolarizationcanbeobtainedinmultiple a ways, such as using Sisyphus cooling technique, polarizer plates, etc. With the k-resolved Berry curvature J formalism,wedelineatetherelationshipbetweentheSU(3)chirality,bandinversion,andk-spacemonopoles, 3 governingfiniteChernnumberwithoutbreakingthetime-reversalsymmetry.Thetopologicalphaseisaffirmed 2 byedgestatecalculation,obeyingthebulk-boundarycorrespondence. ] l PACSnumbers:67.85.–d,03.75.–b,37.10.Gh,71.70.Ej l a h - I. INTRODUCTION Chernnumber(topologicalinvariants). s e Encouraged by the tremendous success of material re- m The discovery of quantum Hall effect in 19801 has ush- alization and engineering of the SU(2) based topological t. ered in a new era of quantum states of non-interacting elec- materials,4,9,12 we explore the possibility of designing a lat- a trons,distinguishedbynon-trivialtopologicalinvariant. Sub- ticemodelwithSU(3)symmetry. SU(3)flavorsymmetrywas m sequently, Haldane’s proposal2 of quantum Hall effect with originally proposed in the quantum chromodynamics (QCD) - synthetic gauge field in a honeycomb lattice gave an impor- field for systems with three basis vectors (namely, colors) d n tant clue on how a specific lattice structure can give rise to which can be expanded in the basis of eight 3 × 3 Gell- o non-trivialbandtopologies. Therearevariousmechanismby Mann matrices λˆ.13–15 The non-Abelian theory of the SU(3) c whichthemomentum-spacemappingoftheHamiltonianwith groupspredictsseveralelementaryexcitationssuchasquark, [ specificcrystalsymmetry,orbitalsymmetry,and/orspin-orbit and gluons. Prediction of emergent SU(3) symmetry in con- 1 coupling (SOC) results in an irreducible SU(2) representa- densed matter systems is rather limited,16,17 and no physical v tionoftheDiracorWeylHamiltonian,garneringalargeclass systemor opticallatticeis realized todatewith suchproper- 9 of Dirac and Weyl materials.3–10 In bipartite lattices, such as ties. In a recent work, it is shown that a SOC generated in a 1 in the so-called Su-Schrieffer-Heeger model,3 or in honey- square optical lattice in the presence of a spatially homoge- 3 comb lattice,7 the hopping between inequivalent lattice sites neousSU(3)gaugefieldcangiverisetonon-trivialtopologi- 6 carries a net Bloch phase eik·r. This acts as a momentum- calcharacteristics16.Thiscanbeexperimentallyrealizedwith 0 . space gauge field, in analogy with the Wilson loop formula ultra-cold atoms having internal spin degrees of freedom or 1 foramagneticfieldwhich,inspecialcases,isassociatedwith anysuchthreecomponentHamiltonian.Anexplicittopology- 0 an integer phase winding number − a non-trivial topologi- engineeringschemeinathreebandmodelhasbeendiscussed 7 calinvariant. Again,electronhoppingbetweenevenandodd in another recent work17, where the generation of arbitrary 1 : parity orbitals (such as s- and p-orbitals) often results in a Chernnumberisbasedontheequivalenceofthetopological v net hopping ∝ eik·r −e−ik·r = 2isink·r, giving a linear numberofagivenbandonthemonopolecharge, andcanbe i X orbital-momentumlocking.11 Asorbitaltextureinversionoc- extendedtohigherChernnumberderivatives. Thefeasibility r cursatdiscretek-points,thesepointsactasmonopolesinthe of the materials realization of the corresponding models has a momentum-space, giving rise to Dirac or Weyl cones. Fur- notbeendiscussedinthesepapers. thermore,SOCprovidesacommonorigintoavarietyofquan- tum Hall and topological classes of materials.4–6 In all these The basic principles for SU(3) symmetric TI follows that examples, the spinor of the SU(2) representation (or its gen- oftheSU(2)counterpart. Amomentumrepresentationofthe eration to a SU(2N) version, where N is integer) comprises SU(3) Hamiltonian incipiently demands the entanglement of oftwochiralspeciesoriginatingfromthemomentumlocking a three components quantum state via some sort of intrinsic withtwosublattices,ortwoorbitals,orspin-1/2particles. As gauge field. We start with a generalized form of the SU(3) a chiral object forms an orbit, surrounding an external mag- HamiltonianHˆ(k)=b(k)·λˆ,whereb(k)istheeightcom- neticfield,ormomentumspaceBerrycurvature,itproducesa ponents vector made of electron hoppings in a lattice. The non-zero flux, which is quantified by the winding number or momentum-space magnetic field (Berry curvature) can thus 2 beexpressedin(2+1)dimension,16 3𝑡𝑥13𝑒−𝑖𝑘𝑥𝑎1 𝑡𝑥12𝑒𝑖𝑘𝑥𝑎 2 𝑡𝑥23𝑒𝑖𝑘𝑥𝑎 3 1 1 Ω (k)= (cid:15)µνρb ∂ b ∂ b , (1) mn 2|b|3 µ m ν n ρ wnehnetrseoifndthiceesb-mve,cntor=. k(cid:15)xµ,νρkyis, athneduµs,uνal, Lρegviiv-Ceitvhietactoemnspoor-. 𝑡𝑦𝑒𝑖𝑘𝑦𝑏 a b Eq.(1)impliesthatchiralorbitsorvorticesareformedinall threebands,withtheircenterslocatedat|b(k∗)| = 0,where theBerrycurvatureΩobtainssingularity. ThefluxofΩ(k∗) 𝑡𝑦𝑒−𝑖𝑘𝑦𝑏 through the first Brillouin zone for each band is quantized, andisquantifiedbytheassociatedChernnumber. Therefore, theChernnumberessentiallydictatesthenumberoforbitcen- ters, while its sign corresponds to the direction of associated -𝜖𝑦Ƹ +𝜖𝑦Ƹ 𝜖𝑦Ƹ =0 -𝜖𝑦Ƹ +𝜖𝑦Ƹ phase winding or chirality. Without any external magnetic field,thetotalChernnumberforallbandsmustvanish. This FIG.1.(Coloronline)Schematicdrawingoftheproposedsetup.The impliesthatthechiralityofoneoftheSU(3)componentmust differentcolouredspheresdenotethreedifferentbasisoftheHamil- be opposite to the other two components, and also the asso- tonian, whichcanbethreedifferentatomicspeciesororbitals, ψ , 1 ciatedChernnumber(ork−spacemonopoles), say, C must ψ and ψ . E denotes the direction of the vector potential of the 1 2 3 beequaltothetotalChernnumberfromtheothertwochiral electromagnetic(EM)wave,whichcausesthedifferentPeierlsphase states,i.e.C =C +C .InthecaseofC =0,theothertwo couplingforthedifferentorbitals. Thehoppingamplitudesaremul- 1 2 3 1 bands give C2 = −C3 which mimics the quantum spin-Hall tiples of tx and ty , and are different for the different components systemforspinfulsystems.5,6 (orbitals).Also,b=2ainthepresentmodel. Theabove-mentionedfeaturescanalsobeunderstoodsim- plybythecorrespondingbandtopology. Thecenteroforbits atk∗-pointsarethosediscretepointswherebanddegeneracy occurs. Therefore,thenumberofvorticesdictatethenumber matterinteractionwithsite-selectivephotonpolarization. We ofbandinversionsintwodimensions(2D).InSU(2)topolog- showwithamodifiedtight-bindingmodelinatripartitelattice icalsystems,theoddnumberofbandinversions(atthetime- thatthedipoleinteractionterm∝k·A,whereAisthevector reversal momenta, if this symmetry is present) between two potentialofthephoton,naturallyleadstoa±sin(βk·(cid:15)ˆ)term basiscomponentsgivesafiniteChernnumberorZ2invariant. (where(cid:15)îstheunitvectoralongthepolarizationdirection,and ForSU(3)systems, thebandinversionmusthappenbetween β isatunableparameter). Thenbytuningthedirectionofthe all three bands, or at least, the band with the highest Chern polarizationparallelororthogonaltok(orwithothermethods number(C1)mustundergoinversionwithboththeothertwo asdiscussedlater),onecanselectivelygenerate,reverse,and bands.Theothertwobandsdonotnecessarilyundergoaband destroy this term in different sublattices. This way, we can inversion between them unless their Chern numbers are also generateodd-parityZeemantermbyusingoppositepolariza- different. ThisisanimportantdistinctionoftheSU(3)frame- tion in lattice sites 1 and 2, while it is destroyed in the 3rd work. Another unique requirement of the SU(3) topological site,asshowninFig.1. ThisgeneratesourSU(3)topological stateisthatherenotonlyagaugefieldisrequiredtobepresent system,whichwecharacterizebythedetailedanalysisofthe intheoff-diagonaltermoftheHamiltonian(asinSU(2)case), Berrycurvature,Chernnumber,andedgestates. Wealsodis- but an odd parity Zeeman-like term, i.e. sinkσz term must cuss two realizable setup to obtain site-selective polarization also be present between any two basis. Such an odd parity withexistingtools. onsite Zeeman-like term does not arise naturally from Bloch phaseorfromconventionalSOC. The rest of the paper is arranged as follows. In Sec. II, A focal point of our work is to construct a lattice model wedescribethegeneralcharacteristicsthattheSU(3)Hamil- which inherits the required suitable gauge fields, and odd- tonian should possess to obtain non-zero Chern number. In parity Zeeman term, and thus intrinsically performs as topo- Sec. IIB we discuss the setup. Derivation of the odd- logical material distinguished by finite Chern number. We parity Zeeman term with site-selective EM fields is given in present a model Hamiltonian of a spinless SU(3) topologi- Sec.IIB1. Thetight-bindingmodelforatriparticlelatticeis cal system in a tripartite lattice with three inequivalent lat- discussedinSec.IIB2.Designprinciplesforsuchlatticewith ticesites. Eachsublatticeissittingindistinct1Dchains,and Sisyphuscoolingtechniqueorwithpolarizerplatearegivenin they are coupled by nearest-neighbor quantum tunneling, as Sec.IIB3.InSec.IIIweelaborateonthegeometricalmethod shown in Fig. 1 . This would naturally give an uncompen- for calculating the Berry curvature and Chern number. We satedBlochphaseforallthreeinter-sitehoppingsas: eik.r12, detail the calculation of edge states using the strip geometry eik.r23, while the third one is naturally reversed to e−ik.r13, approach in Sec. IV. In Sec. V we discuss the robustness of wherer aredistancesbetweeniandj sublattice. Thisnat- thespinlessSU(3)topologicalphasestospinfulperturbations. ij urally enforces opposite chirality between any two band in a WealsodiscussedapossiblemodelofengineeringSU(3)by givenparameterspace. Finally,forthegenerationoftheodd suitablycombiningSU(2)andU(1)species.Weendthepaper parity‘onsite’matrix-element,weproposetoutilizethelight- withdiscussionsandconclusionsinSec.VI. 3 II. MODEL its sign can be simultaneously reversed by using antiparallel photon polarization. We notice that all three diagonal terms A. GeneralcharacteristicsofSU(3)TIHamiltonians are taken to depend only on ky which is consistent with the setup drawn in Fig. 1. By comparing Eqs. (4) and (3), we obtain WedesignaSU(3)HamiltonianwithoneyeonfiniteChern number in a bottom-up approach. A generic Hamiltonian, a(k)= ξ1+ξ2+ξ3 = 1(cid:0)−3cosαk +2(m +m )sinβk (cid:1), obeyingtheSU(3)decomposition,canbewrittenas, 3 6 y 1 3 y ξ −ξ 1 Hˆ(k)=a(k)Î3+b(k)·λˆ, (2) b3(k)= 1 2 2 = 4(−7cosαky+2m1sinβky), 2ξ −ξ −ξ whereÎ3isa3×3identitymatrix,λîaretheSU(3)generators b8(k)= 3 √1 2 2 3 (Gell-Mannmatrices),anda(k),b (k)arethecorresponding i 1 coefficients. TheexplicitmatrixformoftheHamiltonian(the = √ (3cosαk +2(m −2m )sinβk ), (5) y 1 3 y k-dependenciesinaandb areimplied)is, 4 3 i a+b3+ √b8 b1−ib2 b4−ib5 Looking at the Hamiltonian in Eq. (3), we notice that a(k) 3 givesaoverallshifttoallthebandsandthusdoesnotplayany Hˆ(k)= b1+ib2 a−b3+ √b83 b6−ib7. (3) specificroleonthetopology. b3(k)givesananisotropicZee- b4+ib5 b6+ib7 a− 2√b8 man splitting between 1st and 2nd basis in the Hamiltonian, 3 whileb (k)givesasimilarsplittingofthe3rdbasisfromthe 8 We work with a three component spinor Ψ† = othertwoones. Itiseasytoseethatthebandinversionalong k (cid:16)ψ1†(k),ψ2†(k),ψ3†(k)(cid:17)T, where ψi(k) are the basis rep- tthheekeiygednirveacltuioesnaisredprirvoepnorbtyiobn3alantodbb38atenrdmbs8.,Awnedsaelesoth,astinthcee resenting different orbitals, or sublattices, and so on (but we bands become anisotropic between ±k . On the other hand, y do not consider spin here). Each b term requires special alongthek directiontheyaresymmetric,sincetheeigenval- i x treatmentsuchthatoppositechirality,andodd-parityZeeman uesdependontheabsolutevalueoftheotherb terms. This m termcanbesimultaneouslyachievedinsuchawaythatBerry asymmetryalsoreflectsintheBerrycurvaturemapsshownin curvaturesingularitiesatdiscretek-pointscanbeattained. Fig.3. 1. Diagonalterms 2. Off-diagonalterms We start with the diagonal terms of Eq. (3). We denote Nextweconsiderthethreeoff-diagonaltermswhichfollow the three onsite, intra-basis, dispersions as ξi(k) where i = ageneralformbν(k)±ibσ(k),whereν =1,4,6,σ =2,5,7. 1, 2, 3.Ingeneraltight-bindingHamiltonians,diagonalterms In condensed matter systems, such a complex term usually compriseofcosinefunctionsofmomentum,andchemicalpo- has two origins: (1) Rashba- or Dresselhaus-type spin-orbit tential. The sine term of the Bloch phase is associated with coupling(SOC),(2)Blochphasefromnearestneighborelec- imaginary‘i’whichcannotappearinthediagonaltermforit tron’s hopping. (1) Rashba and Dressenhaus SOC yields to be a Hermitian one. It drops out even in centrosymmetric b (k) = α sink , and b (k) = α sink (where α is ν R x σ R y R latticesasthesamehoppingonbothpositiveandnegativedi- the SOC strength). SOC is however difficult to achieve for rectionsareadded. WiththeanalysisofBerrycurvatureand allthreeSU(3)spinsinbothcondensedmatterandopticallat- SU(3) symmetry, we recognize that an essential requirement ticesetups.Moreimportantly,wefindthatthecomputationof for non-zero Chern number in this case is that the diagonal ChernnumberwithSOCintheoff-diagonaltermsoftengives termsξ1andξ3mustcontain±sin(ky)terms,whichisequiv- zeroChernnumber.Therefore,wefocusonthepossibility(2). alent to having odd-parity Zeeman term. Without specifying Assigning b (k) = t cosk , and b (k) = t sink (where ν x x σ x x the origin at this point, we start with a combination of three t is a parameter which can be different for different ν and x diagonaltermsina1Dlattice: σ),weseethatthistermsimplifiesto∼ t exp(ik ). Thisis x x justaBlochphaseassociatedwiththenearest-neighborhop- ξ (k)=t cos(αk )+m sin(βk )−µ, (4) i i y i y pingbetweendifferentsublattices(sinceitappearsintheoff- diagonaltermintheHamiltonian). Wealsofindthatforfinite wheret ,m aretheexpansionparameters,andµisthechem- i i Chern number, the Bloch phase must be reversed in at least icalpotential. αandβ arearbitraryparametersdependingon oneoftheoff-diagonalterms,comparedtotheothertwo. thecrystalstructureandlatticeconstants. FiniteChernnum- ber arises for a set of parameters as t = −2t , t = 3t , 1 y 2 2 y t = −t , and m = −m , and m = 0. The cosine terms 3 y 1 3 2 arisefromthenearestneighborhoppingalongthey-direction. 3. FullHamiltonian InSec.IIB1below,wediscusshowtoobtainm sink term i y withthehelpoflight-matterinteractioninwhichwefindthat Basedontheconstraintsforbothdiagonalandoff-diagonal m dependsonbotht aswellasthevectorpotentialA. Thus terms, we now seek a minimal model for the realization of i i 4 SU(3)Cherninsulatorinthespinlessbasis: cannowderivethetight-bindingdispersionas H(k)=−−ξt1txx(eke−iyki)kxx −−12ξt2txx(keeiy−k)xikx −−ξ21t3xt(exk−eyii)kkxx , (6) ξ(k)==(cid:104)N1ηk(cid:48)n(cid:88)|H,n(cid:48)(cid:48)|eηik(cid:48)k(cid:105)·(Rn−Rn(cid:48))(cid:90) dr(cid:104)u(cid:48)n(cid:48)|H(cid:48)|u(cid:48)n(cid:105) 1 (cid:88) (cid:90) wheretxisthenearestneighbortight-bindinghoppingparam- = N eik·(Rn−Rn(cid:48))ei(φn−φn(cid:48)) dr(cid:104)un(cid:48)|H|un(cid:105) eter between different basis. Without loosing generality, we n,n(cid:48) settx = ty=1. Thisgivesalleightcomponentsofthebvec- = (cid:88)tnn(cid:48)eik·(Rn−Rn(cid:48))ei(φn−φn(cid:48)). (8) tortobe: n,n(cid:48) (cid:2) 1 Heret = 1 (cid:82) dr(cid:104)u |H|u (cid:105)istheTBhoppingamplitude b(k)= −coskx,−sinkx,4(−7cosαky+2m1sinβky), betweennn(cid:48)nanNdn(cid:48) sitesnw(cid:48) ithountthevectorpotential. Wehere 1 1 restrict ourselves to the nearest neighbor hopping, i.e., n(cid:48) = −cosk ,sink ,− cosk ,− sink , x x 2 x 2 x n±1. Letthelatticeconstantalongthey-directionbeb. By √1 (3cosαk +2(m +2m )sinβk )(cid:3). setting tn(n±1) = ty, and ±φ = φn − φn±1 = ±(cid:126)eAb = 4 3 y 1 3 y (cid:126)eAbyˆ·(cid:15)ˆ,weobtain, (7) (cid:104) (cid:105) ε(k)=t ei(kyb+φ)+e−i(kyb+φ) . y =2t [cos(k b)cosφ−sin(k b)sinφ]. (9) y y y B. Setup We absorb cosφ in to the TB term as t (φ) = 2t (0)cosφ, y y and define m(φ) = −2t sinφ. Then we see that Eq. (9) is y Next we discuss how to obtain such a Hamiltonian with the same as Eq. (4). From this definition, it is easy to see realistic crystal structure and orbital symmetry. The phase thatasthedirectionofpolarizationisreversed,φ→−φ,and dependent off-diagonal term e±ikx, and the Zeeman term thusm→−m,t →t whiletheperpendicularpolarization y y sin(βky)σztermcanbesimultaneouslyobtainedinatripartite yieldsm(φ=0)=0,andty remainsthesame. latticebyapplyinglinearlypolarizedlightoneachsublattices. Attheendofthissection,wediscusshowtodesignsuchalat- tice. 2. Tripartitelattice For the SU(2) case, the phase dependent hopping term ∼ texp(ik ) is obtained in bipartite lattice (c.f. Su-Schrieffer- 1. Tight-binding(TB)modelforelectron-photoncouplinginduced x sin(βk )term Heeger model in 1D,3 or honeycomb lattice7 in 2D) or for y hopping between even and odd-parity orbitals.11 Similarly, for the SU(3) case, we need the same term for all three off- Themotivationfortheoriginofsinβk termcanbedrawn y diagonal terms. Therefore, we propose a tripartite lattice as from the fact that the dipole interaction between an electron depictedinFig.1. Alsonotethat,thecomplexhoppingterm with momentum p = (cid:126)k and an EM wave with potential onlyincludesk terms,implyingthatdifferentbasiselements A = A(cid:15)ˆ((cid:15)îs the light polarization) is H = −ep·A = x int m should be aligned along the x-direction only. Therefore, we −em(cid:126)Ak · (cid:15)ˆ. We choose a linearly polarized light with its considerthreechainsofdifferentspecieswhichareconnected polarization oriented along the y-direction. We take a sin- viaquantumtunnelinginbothdirections.Weassumeperiodic gleelectronHamiltonianundertheperiodicpotentialU(r)of boundaryconditionsalongbothdirections. the lattice as H = p2 +U(r). The corresponding Bloch The nearest neighbor hoppings along the x−direction be- 2m∗ twoatavlefnuunmctbioernoisf ηuknit=ce√ll1Ns, (cid:80)u n(re)iki·sRtnhuenW(ra)n,nwiehrersetaNte aist tthhee tdwepeeenndbeansciese1ik→x, w2h,ialendth2at→for31g→ive3thisees−amikex m(woemseenttuthme ntiathl Asit,ethloecHataemdialttonRiann. beIncnotmheespHres(cid:48)en=ce(opf−evAe)c2to+r pUot(ern)-. caonrdret2xs3po=nd−in21gtxh)o.pTphinisgraemveprlsiatuldoefsBaloscth1x2ph=as−estexr,vte1xs3th=ep−utrx- 2m∗ poseofchiralityinversionalongthisdirection.Cautionshould For the EM wave, the spatial dependence of A can be ne- betakenwhenthenext-nearesthoppingtermbecomesturned glected, and thus, the translational symmetry of the lattice remains the same. Therefore, the new Bloch wavecfunc- on, whose k-dependence is given by −2t2ycoskye±ikx, re- tion simply changes to η(cid:48) = √1 (cid:80) eik·Rnu(cid:48) (r), where spectively. SuchtermadiabaticallydestroystheintegerChern k N n n number. Therefore, to avoid it we propose to increase the u(cid:48)n(r) = un(r)ei(cid:126)e(cid:82)RrnA·dl = un(r)eiφn(r). φn(r) is called inter-atomic distance between adjacent chains to be as large the Peierls phase at r acquired by the charged particle in as possible so that the hopping term t → 0. This setup 2y traversing from the nth lattice site. It can be shown that simultaneouslyproducesacosk termfortheintra-basisdis- x H(cid:48)|u(cid:48)n(r)(cid:105) = eiφn(r)H|un(r)(cid:105). Using these ingredients, we persionsinEq.(4)duetonearestneighborhopping. 5 (a) tripartitelattice,wecanconstructthedesiredsite-selectivepo- 𝜖Ƹ σ+ 𝜖𝑦Ƹ 𝑦 σ− larizationasfollows. 1 2 3 1 2 3 In Fig. 2(a), we demonstrate the setup. We take two 𝜖𝑧Ƹ −𝜖𝑧Ƹ 𝜖𝑧Ƹ counter-propagating, oppositely oriented circularly polarized −𝜖𝑦Ƹ −𝜖𝑦Ƹ (σ±) EM fields ((cid:15)ˆy + i(cid:15)ˆz)eiq.x and ((cid:15)ˆy − i(cid:15)ˆz)e−iq.x to trap atoms along the x-direction. q = 2π/λ is the light’s z λ λ 𝑎 λ 𝑎 y 2=𝑎 4=2 4=2 wavevector and λ is its corresponding wavelength (the fre- quencydependenceofthefielddonotmakeanycontribution Polarizer plates x to our analysis and thus not discussed). The resultant field (b) 2[(cid:15)ˆycos(qx) − (cid:15)ˆzsin(qx)] has a spatially dependent polar- 𝜖𝑦Ƹ 𝜖𝑦Ƹ ization. Atx=0,itstartsoffwithalinearpolarizationalong 1 2 3 1 2 the(cid:15)ˆy-direction, atx = λ/4thepolarizationisalongthe(cid:15)ˆz- direction,andatx=λ/2itisrotatedalongthe−(cid:15)ˆ -direction. y InFig.1,weassumesites1and3haveoppositepolarization −𝜖𝑧Ƹ −𝜖𝑦Ƹ −𝜖𝑧Ƹ atiloonn.gTthheereyf-odriere,cttoioinm,pwlehmileentsitthee2Sihsaysphourtshotegcohnnailqupeo,lasritieza1- 𝑎 sitsatx = 0,whilesite3residesatx = a = λ/2. Thenthe distance of site 2 from 1 and 3 is a/2 to gain the orthogonal polarization ((cid:15)ˆ ). Since the system is confined in 2D, it has FIG.2.(Coloronline)Schematicdiagramofapossibleexperimental z set-upstorealizeasite-selectivepolarization. (a)Sisyphuscooling no momentum along the z-direction and thus no dipole term techniquegivespolarizationgradientina1Dlattice,whencounter- arisesforsite2. Withthisatomicposition,thecorresponding propagating circularly polarized waves σ± are used. This creates Blochphasesforhopping1 → 2and2 → 3iseikx/2,while a linear polarization that rotates in space (at x = 0 polarization is thatfor1 → 3,ande−ikx. ThisgivestheChernnumber(see along(cid:15)ˆy,atx=λ/4andλ/2polarizationalong(cid:15)ˆzand−(cid:15)ˆy,respec- calculationsinSec.IIIas(−2,4,−2). tively). Atomsaretrappedatthesex-valuestocreatethenecessary Discussions of the advantage and limitation of using this oddparityZeemanterm. b)AlinearlypolarizedEMwavealong(cid:15)ˆ y technique are in order. In this specific Sisyphus technique, isincidentontheatomtrappedinthe1statomicchain. Apolarizer the resulting EM field has only linear polarization along the plate,placedbetweenthe1stand2ndwires,rotatesthepolarization vectorbyπ/2.TheEMwaveinthe2ndwireisthenpolarizedalong chains where the atoms are placed18. In the ground state, (cid:15)ˆz. Anotherpolarizerplate,betweenthewireswiththe2ndandthe atomshavehyperfinelevelsgs = ±1/2. Forthistwostates, 3rd atom, rotates the incident polarization vector (cid:15)ˆ by π/2 again, thelightshiftcausedbytheinteractionbetweentheatomsand z and the polarization vector along the 3rd wire is along −(cid:15)ˆy. Two EM field is exactly equal, and also it does not vary with the quarter-wavepolarizeroronehalf-wavepolarizerplatesareneedto directionofpropagation(herex-direction). Thisisagreatad- beplacedbetweenthe3rdand4thwire(siteindex1asperiodicityis vantage to our setup where we do not have to deal with the imposed)torotatethepolarizationvectorfrom−(cid:15)ˆ to(cid:15)ˆ .Thisgen- y y light-shiftsplittingofthehyperfinelevelscomingfromthein- eratesthedesiredpolarizationgradientinourlatticegridstructure. teraction with the EM field. However, this technique has a limitation. Thelowesttemperaturethatcanbeattainedbythis coolingissetbytherecoilenergy(cid:126)2q2/2m,whichistheki- 3. Designprinciples neticenergyanatomgainsafterabsorbingaphoton. Inmost alkaliatoms,thistemperatureisbelow1µK.Atsuchlowtem- For all the above terms, no constraint arose about the peratures,theatomicdeBrogliewavelengthbecomescompa- specific parity (or orbital symmetry) of each basis. There- rabletothecoolinglaserwavelength(thoughstillshorterthan fore, coupled chain structure can be engineered with ultra- isrequiredfortheBECphasetransition)andhencetotheex- cold fermionic or bosonic atoms in optical lattice setup, or tent of the potential wells. It is therefore no longer possible withquasi-1Dquantumwiresofelectronswithlithographyor tolocalizetheatomicwavepacketinthepotentialwells,even pulselaserdepositionmethod. However,forthegenerationof if they were deeper than the photon recoil energy. However, site-selective sinβk term with light-matter interaction, spe- thislimitationhasbeensuccessfullyovercome,andthereare y cificstructureortuningisrequired. Wesuggesttwopractical numerous usage of this technique in the literature (see e.g. experimentalsetupsfortheengineeringofthisphenomenon. Refs.20,21). Ofcourse,thepossibilitiesareabundance. (b) Using polarizer plates: Another simpler setup can (a)SisyphusCooling:-Sisyhuscoolingtechniqueprovides be made using polarizer plates between consecutive atomic aspatialmodulationofthestaticpolarization.18,19Inthistech- chains,whichrotatesthepolarizationoftheincidentemfield nique, two counter-propagating laser beams with orthogonal by π/2, as illustrated in Fig. 2(b). This can be repeated pe- polarization are used to create standing wave with a polar- riodically along the x-direction (propagation direction of the izationgradientthatalternatesbetweencircularand/orlinear incidentEMfield)toachievethedesiredpolarizationreversal polarizationwithrightandlefthandedness.Atomstrappedby alongeachchain. Westartwithalinearlypolarizedlight((cid:15)ˆ ), y thelaser canacquiredifferentpolarization accordingtotheir incidentonsite1.Aquarter-wavepolarizerbetweensite1and positionswithrespecttothewavelengthofthelasers. Bytun- 2, rotates the polarization to (cid:15)ˆ , having no dipole interaction z ingthelaserwavelengthwithrespecttolatticeconstantofthe here.Againanotherquarter-wavepolarizerbetweensite2and 6 3,rotatesthepolarizationalongthe−(cid:15)ˆ . Toachieveperiodic y boundaryconditions,thenextsite4(whichshouldbeequalto site1),weneedthepolarizationtobealong(cid:15)ˆ ,whichcanbe y achievedbyeither2quarter-waveoronehalf-wavepolarizer. (a) Thisconfigurationisrepeatedalongthex-direction. Polarizer plates,likehalf-waveandquarter-waveplates,areassociated withintensitylosses, whichcanbeencounteredbyreplacing them with birefringent filter plates. Polarizer plates may at- (b) (c) tenuatethehoppingamplitudesbetweentheatoms. However, since the value of the Chern number does not depend on the hoppingamplitudes,thesystemwillremaintopologicallyin- variantaslongasthetunnelingisfinite. (c) Other possible condensed matter setups: Few other possibleengineeringprinciplescanbeenvisionedusingcon- densed matter setups. A site-selective electric field grid can becreatedin1Dlattices.(i)Alternatively,itisshownrecently thathigh-energyphotonscanbeusedtoselectivelyphotonize the valence electrons of hydrogen chloride by using the ro- FIG.3.(Coloronline)Bandsdemonstratingtheenergydispersionfor tational dependence of the photonization profiles.22 (ii) One theproposedmodelareshown.(a)Surfaceplotwithbothkx(−πto can also pursue a possibility of using ferroelectric substrate π)andky(−π2 to π2),(b)Keepingky =0,dispersionplotalongkx, from−π toπ, and(c)Keepingk = 0, dispersionplotalongk , in which due to in-plane inversion symmetry breaking, elec- x y from−π to π. Allquantitiesaremeasuredinunitst = t = 1, tric field polarization in different layer or chain can induce 2 2 √ x y and|m |=|m |= 3t . site-selective polarization via proximity effect to the top lat- 1 3 y tice of our interests. (iii) We can also use orbital-selective chains. Recallingthatthesink termisabsentinthesecond y diagonalterm(Eq.(4)),wecanthinkofanorbitalsymmetry notnecessarilyhavetobealongthek -andk -directions). In x y for the second basis which is orthogonal to the direction of thisspiritwecandefineabandinversionstrengthviaoccupa- light’spolarization. Correspondingchoicesoforbitalsarepx- tionnumberorthe‘orbitalweight’(hereorbitalreferstothe or dxz orbitals for which Hint = 0 with polarization along SU(3)components)ofthebandateachk-points. Anotherin- the y-direction. For the other two orbitals, we can consider terestingfeatureofthemonopoleisthatitrepresentsasaddle combinationssuchass-andpy orbitals,etc. pointintheorbitalweight(seeFig.4),inthesensethatifan orbitalcharacterobtainsamaximumalongthek direction,it x obtainsaminimuminthek direction. Therightmostcolumn y III. BANDTOPOLOGYANDBERRYCURVATURE ofFig.4referstothek-resolvedBerrycurvature. Weseethat atallthek∗-pointswhereΩ(k∗)divergesinagivenband,the correspondingorbitalweightprofileexhibitsasaddlepoint. In Fig. 3, we plot the band structure in the mom√entum- space for the parameter values of m = −m = 3t and Let us define a quantity κν = 1 3 (cid:104) (cid:105) kl α = β = 2, and tx = ty. The electronic structure consists sgn |γν(k )|2−|γν (k )|2 , where γν is the eigenfunction i l j(cid:54)=i l i of three well separated bands, with only Dirac-like nodes at oftheHamiltonianinEq.(6)correspondingtoνth bandand variousdiscretenon-highsymmetrick-points(seeFig.3(a)). ith SU(3)spinorcomponent. Intheνthband,iftheith orbital Therefore, a topological invariant can be separately assigned obtains a maximum (and jth orbital then obtains a minima) foreachband. However,projectingtheorbitalcharacteronto along, say k -direction, then κν = +1. On the other hand, each band, we observe that substantial exchange of orbital itsminimumxwouldcorrespondksxtoκν =−1. Therefore,the characteroccursineachband. Wevisualizethethreeorbital kx locusofthemonopoleisdefinedbythediscretepointswhich characters(indifferentrow)forthreedifferentbands(indif- satisfy κν κν = −1. This is of course an indication of the ferentcolumn)intheentire2Dk-spaceinFig.4. kx ky Asdiscussedintheintroductionsection, bandinversionis location of finite Berry curvature. A singularity in Ω(k) is an important criterion for both the SU(2) and SU(3) topo- obtainedwherethebandsarefullydegenerate. logical classes. In time-reversal invariant SU(2) topological For the higher energy band, among the six visible saddle- classes, bands are only required to be inverted at the time- points, three of them reside in the −k region. They give y reversalinvariantk-points. Thismakesiteasiertodefinethe threenegativespikesintheBerrycurvatures,asshowninthe band inversion strength simply by defining the band gap be- correspondingrightmostcolumnofFig.2. (Thetwoextreme tweenthetwobandsatthetime-reversalinvariantk-points.4,23 peaksoccurringatthezoneboundaryarerelatedbyreciprocal Such simple definition becomes difficult to implement for lattice vectors). This gives the corresponding Chern number (cid:80) SU(3) materials. On the other hand, we recognize that the C = Ω (k)(wherenstandsforbands)tobe-3. Inthis n k n Berrycurvatureacquiresspikeatthediscretebanddegenerate band, inversion occurs between orbitals 1 and 3 across the k-points(k-spacemonopoles)acrosswhichbandsareinverted three saddle points. In the lowest band, the Berry curvature intwoorthogonaldirections(thetwoorthogonaldirectionsdo peaks occur in the corresponding +k side due to the band y 7 Basis 1 Basis 2 Basis 3 Berry curvatures 1 d n a B 2 d n a B 3 d n a B FIG.4. (Coloronline)Firstthreecolumns: Theorbitalweightsofthebasisstatesψ (j = 1,2,3)areshownforeachenergybandsE ,E j 1 2 andE respectively. Thearrowinthefigurecorrespondingtothethirdcolumnshowsthepositionofoneminimawhichiscorrespondingto 3 thegaplesspointofoneoftheedgestatesinthesystem. Fourthcolumn: TheberrycurvatureΩ (i = 1,2,3)correspondingtoeachenergy i bandisshowninthefourthcolumn.Theberrycurvaturesshowasharppeakatthecorrespondinggaplesspointsoftheedgestates,indicative ofabandinversionandatopologicalphasetransitionattherespectivek-points. inversions between orbitals 2 and 3. The Chern number of this band comes out to be the same as -3. The middle band shows band inversions between all three orbitals at the same locations in both ±k sides, with positive Berry curvatures y andthusobtainsaChernnumberof+6. ItcanalsobeshownthattheChernnumberdoesnotdepend ifthestrengthofthePierels’phaseisdifferentinthedifferent orbitals. Aslongasthesinβk termshaveoppositesignsin y the diagonal terms (i.e. mimicking a Zeeman-like term), the modelistopologicallynon-trivial. Changingtheoff-diagonal elements,sayfromeikxtoe2ikx,wefindhigherChernnumber (−4,8,−4). IntheAppendixbelow,wediscussseveralother parametersetswhereChernnumbercanbetunedbydifferent valuesofα,β,andothertermsintheHamiltonian. (cid:1) (cid:1) IV. CALCULATIONOFEDGESTATES FIG.5. (Coloronline)Edges√tatestructureforN = 30,withα = β = 2and|m | = |m | = 3t . Also,t = t = 1. Thereare 1 3 y x y sixgaplesspointswithinthefirstBrillouinzone,asindicatedbythe Non-trivial topological character can be observed chernnumbersbeing{−3,6,−3}. from the edge state dispersion and non-local electrical measurements.4–6 We study the characteristics of the edge state parallel to the y-direction for the above parameter set 8 whichgivesChernnumbers(−3,6,−3). Therefore,wesolve V. EXTENSIONTOSPINFULCASE theHamiltonianinEq. 6withperiodicboundaryconditions in the y-direction, and open boundary condition in the In our model, SU(3) symmetry is obtained for spinless x-directionwithafinitesizelatticeofN atoms. Considering fermions, in which spin of the particles is a dummy vari- Φ (k )astheWannierstatelocalizedontheith atomwecan i y able. This Hamiltonian respects the spin-rotational sym- nowexpandtheHamiltonianinaN ×N matrixas metry. Therefore, as long as this symmetry is held (in the absence of spin orbit coupling or magnetic moment), (cid:88)(cid:104) H(k)=− Φ†(k )B (k )Φ (k ) the topological invariant remains the same for all values of i y i,i y i y i spin in a given system, that means for both fermionic and +Φ†(k )A Φ (k )(cid:105)+h.c., (10) bosonic systems. We now discuss how the result changes i y i,i+1 i+1 y when the spin rotational symmetry is broken. For generality, we assume the atoms/electrons have a spin value S, which splits into 2S+1 multiplets once the rotational sym- −2t 0 0  metry is lifted. In this case, our starting SU(3) spinor y where, B(ky)=cosαky 0 32ty 0  has the dimension of 3(2S+1). For illustration we take 0 0 −t S = 1/2, while the obtained conclusions below remain the y +sinβky−2m013−m3 m1+03m3 00  , (11) Ψsa†kme=fo(cid:16)rψan1†↑y(kot)h,eψr2†↑v(aklu)e,sψ3†o↑f(kS).,ψIn1†↓t(hki)s,ψca2†s↓e(,kt)h,eψ3†s↓p(ikno)(cid:17)rTis, 0 0 m1−2m3 inwhichtheHamiltonianbecomesa6×6matrix.TheHamil- 3 toniancanbesplitintotwo3×3diagonalblocksasH ,and ↑↑ and, H ,andoffdiagonalblocksH ,andH† . Intheabsenceof ↓↓ ↑↓ ↑↓ thespin-flipterms,i.e.,whenH = 0atallmomentum,the ↑↓   0 0 −1 Hamiltonian breaks into a block diagonal one. As the time- A=tx−1 0 0  . (12) reversalsymmetryisbrokenbyintroducinganexchangeterm 0 −1/2 0 E ,theblockdiagonaltermsbecomeH =H±E I , z ↑↑/↓↓ z 3×3 where H is given in Eq. (6) above. Note that Chern num- EigenvaluesoftheEq.(10)areplottedinFig.5withthesame ber does not depend on an overall energy shift (a(k) term parameter set. The essence of the topological edge state is in Eq. 2). Therefore, the system remains topologically non- that it must adiabatically connect to the bulk states which is trivial. If the exchange energy Ez is increased beyond the clearlyobservedinthepresentcase. Owingtothemaximum band gap (determined by b3 and b8 terms), it can introduce Chern number of 6, there are 6 edge states also (red lines). newbandinversion. Dependingonthenatureofthebandin- Edge states with opposite dispersion, connecting to different version, the system can obtain either higher or lower Chern bulk bands with opposite sign of the Chern number, meet at number(atopologicalphasetransition). discretek∗-pointswheretheBerrycurvatureobtainedsingu- In what follows, as long as a simple exchange field is laritiesinFig.4. Consistently, therearetotalof6suchband presenttosplitthespinstates,suchasmagneticfieldorIsing touchingpointsfortheedgestates. likeferromagneticstate, thesystemmaintainsitstopological We see that B term is diagonal in this basis which gives property. We have also checked that when the exchange en- the dispersion along the k direction. These states become ergy is taken to be orbital dependent (breaking parity), the y gapped by A. However as the number of lattice site is in- system continues to obtain the same Chern number untill a creased, the gap at the edge state vanishes at the ky-points new band inversion occurs. When the spin-flip term H↑↓ is whereBerryphaseacquiresdivergence.Therefore,expanding introducedforthecaseofspin-orbitcouplingorantiferromag- theedgestatenearthesepoints,wefindthatthreeeigenstates netism,spinisnolongeragoodquantumnumber.Inthiscase, uptolinear-in-k as(substituting|m |=|m |=m) Chern number cannot be defined for each spin state of band. y 1 3 Therefore,therewillbeatopologicalphasetransitionintoei- (cid:18) (cid:19) theratrivialcase,ortoanothertopologicalclass,suchasZ m 2 E (k )=−t 2+ k family for fermions. We discuss one such possibility using 1 y y t y y Isingspin-orbitcoupling. 3 E (k )= t Recently, thereisanumberoftransitionalmetaldichalco- 2 y 2 y genide materials synthesized in isolated monolayers which (cid:18) (cid:19) m havebuckledhoneycomblatticestructure. Thebuckledstruc- E (k )=−t 1− k . (13) 3 y y t y ture looses inversion symmetry in all three directions. This y causesalargeout-of-planespinpolarization, similartoIsing We notice that the second term represents a localized bound spin which then becomes locked to the only one momentum state (soliton), while the other two bands are linear with k componentviaSOC.ThecorrespondingSOCcomponent,as y inthelow-energyregion. AsAtermisturnedon,thesethree knownbyIsingSOC,isexpressedbyH = α d (k)σ , ISOC I z z statesspitintosixstates,inaccordancewiththehigherChern whereα istheSOCstrengthandz-axisischosenastheeasy I numberinthebulkstate. axis of the Ising spins. For a buckled honeycomb lattice, it 9 is shown that the momentum dep√endent term comes out to condensedmattersetup. Herewemainlyfocusedondesign- be42 d (k) = sin(k ) − 2cos( 3/2k )sin(k /2). This ing such term through electromagnetic wave by generalizing z y x y term essentially gives an odd-parity (along k ) Zeeman-like the tight-binding Hamiltonian with an applied vector poten- y term. To go from a SU(2) system to SU(3), one requires tial. This gives a spinless SU(3) topological material. We another spinless (or singlet) U(1) basis. One possible way proposed two realistic techniques, using either the Sisyphus to obtain such a Hamiltonian is to introduce a spin-singlet coolingtechniqueorpolarizerplatestoobtainthedesiredsite- Kondo impurity as new basis in each unit cell. Solving a selectivepolarization. Kondolatticemodel, onecanobtainthereqiredBlochphase Robustness of the spinless SU(3) topological phase when eikx for the hopping between SU(2) species to the Kondo spin isintroduced isalso discussed. Weshowed thatas long impurity. Finally, the Rashba-SOC between the SU(2) basis asthereisonlyamagneticexchangetermpresentwithoutany providesthechiralitybetweenthem. Inthisway,aneffective spin-flip term, the topological invariant is robust upto a new SU(3)topologicalHamiltoniancanbeconstructedwhichcan bandinversion. Astheexchangeenergysurpassthebandgap, bewrittenintermsofλˆ-matrices. Thecorrespondingresults a topological phase transition occurs, reducing or increasing willbepublishedelsewhere39. theChernnumberbyanintegervalue. Whenaspin-flipterm is introduced (such as spin orbit coupling or antiferromag- netic like term), spin is no longer a good quantum number, and Chern number can no longer be defined for each spin or band. Soourformalismdoesnotholdanymore. VI. DISCUSSIONANDCONCLUSION Thepresenceofedgestatescanbedetectedbyusingnon- local electrical measurements43,44, where a current applied The paper delivers the following messages. We showed betweentwoprobescreatesanetcurrentalongtheedge. This that for the SU(3) topological phase, two of the basis com- current is then measured by a pair of voltage probes which ponents must contain counter-propagating chiral phase. The isplacedawayfromthebulkcurrentpathtoavoiddissipation. third component can take any of the chirality and/or may be chiral-free. ThissituationisanalogoustoaSU(2)topological insulatorinwhichchiralityinversionisalsoanimportantcri- terion.OnewaytoobtainchiralityinversioninSU(2)systems ACKNOWLEDGMENTS is through reversing the SOC, either intrinsically in layered materials,4 or in engineered heterostructures in which adjacent layers are assumed to have opposite SOC,12,24 or even WeacknowledgeKallolSenforvaluablediscussionsduring the course of the work. AG acknowledges the financial sup- throughFermisurfacenestingbetweenspin-orbitcoupledob- jectswhichinspecialcasescanrenderchiralityreversal.25 portfromScienceandEngineeringResearchBoard(SERB), DepartmentofScience&Technology(DST),Govt. ofIndia We engineered the complex phase dependent off-diagonal fortheNationalPostDoctoralFellowship. TDacknowledges terms in a tripartite lattice through uncompensated Bloch acknowledges the financial support from the same board un- phase. The chirality inversion is naturally obtained between derStartUpresearchGrant(YoungScientist). the sublattices 1 & 3, which is reversed from that between 1 &2and2&3. Themulti-channelset-upofone-dimensional atomic chains suggested in our model can be visualized as AppendixA:OtherformsofHamiltonians anarrayofquantumwires,hostingdifferenttypesoforbitals, beingsubjectedtoalinearlypolarizedstaticelectromagnetic Sofar,wehadconsideredaspecificformofthemostgen- field. Quantum wires are being studied extensively to de- scribevariedtopologicalphenomenatheoretically26–29. Frac- eral Hamiltonian given in Eq. (3). This model Eq. (6) is re- alizedinatripartitelatticewithsite-selectiveelectromagnetic tionaltopologicalphasesarebeingstudiedinweaklycoupled quantum wires, in both two and three dimensions.30–33 Peri- field. Withrespecttothestructureofourmodel,theHamilto- niancanberewritteninamoregeneralizedformbyexpress- odically driven systems can also show interesting non-trivial topological effects in spinless systems 34–38 and can also be ingtheoff-diagonaltermsas, extendedtoSU(3)systems. Furthermore,Fermisurfacenest- H =t12e−ikx ;H =t13e−ikx ;H =t23e−ikx(.A1) inginquantumwireswithSOCcanalsoleadtochiralityin- 12 x 13 x 23 x versionbetweendifferentsublatticesthroughtheformationof Where the t provide the inter-basis hopping strengths ij an exotic spin-orbit density wave, as predicted40 and subse- between nearest neighbour sites. The diagonal terms are quentlyrealized41inPb-,andBi-basedquantumwires.Quan- kept in the same form as in Eq. (4) with various choice of tumwirescanbegrownonelectricfieldgridoronferroelec- t and m . Here we discuss various other combinations of i i tric subtracts which can be tailored to obtain chirality inver- the diagonal and off-diagonal terms which give finite Chern sionintheoff-diagonalanddiagonalterms. number,someofwhichmayrequiredifferentlatticestructure ThesecondcriterionfortheSU(3)topologicalHamiltonian thanthetriparticlelatticediscussedinthemaintext. Itshould is unique to this system. Here, two of the diagonal terms be noted that the following list is not necessarily exhaustive, in the Hamiltonian must contain an odd parity term, such as and more combinations can be derived based on the basic a sinusoidal function of the momentum. This poses an im- principles deduced in the main text. In all combinations, portant bottleneck to engineer SU(3) topological phase in a the Hamiltonian must respect the SU(3) symmetry and is 10 representedbytheeightGell-Mannmatrices. 1. With t = m = 0 and t = t /2, as in point 3 of 2 2 3 y CaseI,replacet12 =−t ,t13 =2t cos2k andt23 = √x x x x x x CaseI:α=β =1 −t sin(k − 3), keeping the rest of the coefficients x y √ sameasinpoint1inCaseI.ThisHamiltoniangivesthe 1. Witht1 =t3 =−ty,t2 =2ty,m1 =−m3 =− 3ty, Chernnumbers(4,0,−4). m = 0, t12 = t13 = t23 = −t , this Hamiltonian 2 x x x x providesintegerChernnumberset(−3,6,−3). 2. Sameas(1)butwitht1x2 = −txcosky. ThisHamilto- 2. The almost similar configuration as in point√1 in Case nianprovidesChernnumbers(−1,2,−1). II,onlychangingt12ast12 =±t (cosk + 3sink ) x x x y y andusingthesamesign(either+or−)fortij i.ewith 3. Witht = m = 0i.e. ξ (k) = 0,andt = t /2with x√ 2 2 2 3 y t13 = ±t cosk and t23 = ±t sin(k − 3) the restofthecoefficientsasin2. 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