ChiralspinliquidsintriangularlatticeSU(N)fermionicMottinsulatorswithartificialgaugefields PierreNataf,1 Miklo´sLajko´,2 AlexanderWietek,3 KarloPenc,4,5 Fre´de´ricMila,1 andAndreasM.La¨uchli3 1InstituteofTheoreticalPhysics,EcolePolytechniqueFe´de´raledeLausanne(EPFL),CH-1015Lausanne,Switzerland 2Institute for Solid State Physics, University of Tokyo, Kashiwa 277-8581, Japan 3Institut fu¨r Theoretische Physik, Universita¨t Innsbruck, A-6020 Innsbruck, Austria 4Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, Hungarian Academy of Sciences, H-1525 Budapest, P.O.B. 49, Hungary 5DepartmentofPhysics,BudapestUniversityofTechnologyandEconomics,1111Budapest,Hungary (Dated:January6,2016) 6 Weshowthat,inthepresenceofaπ/2artificialgaugefieldperplaquette,Mottinsulatingphasesofultra- 1 coldfermionswithSU(N)symmetryandoneparticlepersitegenericallypossessanextendedchiralphasewith 0 intrinsictopologicalordercharacterizedbyamultipletofN low-lyingsingletexcitationsforperiodicboundary 2 conditions,andbychiraledgestatesdescribedbytheSU(N)1Wess-Zumino-Novikov-Wittenconformalfield theoryforopenboundaryconditions.ThishasbeenachievedbyextensiveexactdiagonalizationsforNbetween n 3and9,andbyapartonconstructionbasedonasetofN Gutzwillerprojectedfermionicwave-functionswith a fluxπ/N pertriangularplaquette.Experimentalimplicationsarebrieflydiscussed. J 5 PACSnumbers:67.85.-d,71.10.Fd,75.10.Jm,02.70.-c ] s a The search for unconventional quantum states of matter istheSU(N)HubbardHamiltonian g in realistic models of strongly correlated systems has been - N t an extremely active field of research over the last 25 years. (cid:88)(cid:88) (cid:88) n H =−t (eφijc† c +H.c.)+U n n (1) Mott insulating phases in which charge degrees of freedom i,α j,α i,α iβ a u aregappedhavebeenarguedtopotentiallyhostseveralfami- (cid:104)i,j(cid:105)α=1 i,α<β q liesofquantumspinliquidsrangingfromResonatingValence where the phases φ are chosen in a such a way that the t. Bond Z2 quantum spin liquids [1–3] to U(1) algebraic spin (gauge-invariant) fluixj through each triangular plaquette is a liquids [4–6] and chiral spin liquids [7–14]. The topological m equal to π/2. Then, at a filling of one particle per site, propertiesofthesephaseshaveattractedalotofattentiondue and for large enough U/t, the effective model is an SU(N) - to their potential impact on the implementation of quantum d Heisenbergmodelwithlocalspinsinthefundamentalrepre- n computers[15]. sentationofSU(N)endowedwithrealpairwisepermutations o and purely imaginary three-site permutations defined by the c [ Coldatomsopennewperspectivesinthatrespect.Inpartic- Hamiltonian[24,25] ular,alkalinerareearthsallowtorealizeSU(N)Mottphases (cid:88) (cid:88) 1 H =J P +K (iP +h.c.) (2) with N as large as 10 [16–19], and if a chiral phase can be ij 3 ijk v 8 stabilized,itslow-energytheoryisexpectedtobetheSU(N) (cid:104)i,j(cid:105) (i,j,k) 5 levelk =1Chern-Simonstheory. Thefirstproposalofachi- wherethesumover(i,j,k)runsoveralltriangularplaquettes, 9 ralphaseinthiscontextgoesbacktotheworkofHermeleetal andP andP arecircularpermutationoperators. Tosec- 0 ij ijk [20,21],whoshowedthatamean-fieldapproachleadstothe 0 ondorder,theamplitudeofthepairwisepermutationissimply stabilizationofchiralphasesonthesquarelatticeinthelimit . givenbyJ = 2t2/U,whilethe3-sitepermutationappearsat 1 oflargeN andlargenumberofparticlespersitemwithN/m 0 third order in perturbation theory with K3 = 6t3/U2 [26]. integer and ≥ 5. The same mean-field applied to SU(6) on 6 In the following, we will discuss the properties of the model the honeycomb lattice with one particle per site has also led 1 (2) as a function of J and K using the parametrization 3 : to the prediction of a chiral state, with a competing plaque- v J = cosθ and K3 = sinθ. We will discuss the experimen- tte state very close in energy [22]. More recently Ref. [23] i talprospectsofrealizingthisHamiltoniantowardstheendof X suggested the stabilization of SU(N) chiral spin liquids on the manuscript. It is interesting to note that parent Hamilto- r thesquarelatticeusingstaticsyntheticgaugefields,basedon a nians for SU(N) chiral spin liquids have been proposed re- aslave-rotormean-fieldapproach. Inallthesescases,there- cently[27,28]. Theyinclude both thetwo-sitepermutations sultscallforfurtherinvestigationwithmethodsthatgobeyond and the imaginary part of the cyclic three-site permutations, mean-fieldtheory. butalsoinadditiontherealpartofthecyclicthree-sitepermu- tations, whichweomit. Therangeofthetermsintheparent In this Letter, we show that the ground state of the Mott Hamiltonians are however not restricted to nearest neighbor phase of N-color fermions on the triangular lattice with one ortheelementarytriangularplaquetteonly,buttheamplitudes particle per site is a SU(N) chiral spin liquid in a large pa- dependinapower-lawfashiononthedistancesamongthetwo rameterrangeifthesystemissubjecttoastaticartificialgauge orthreespins. Whiletherearesomestructuralsimilarities,it fieldwithfluxπ/2pertriangularplaquette. Thestartingpoint isnotobviousthatthespatiallycompactHamiltonian(2)fea- 2 -0.5 1 4 4 SU(3), Ns=21(a) (b) (c) (d) SU(4), Ns=20 SU(5), Ns=25 SU(6), Ns=24 0.9 SU(N) SU(7), Ns=21 -1 SU(8), Ns=24 3 CSL 3 SU(9), Ns=27 E/Ns-1.5 E / EVMCED00..78 SU(N) ∆GS2 Ns=21, SU(3) Ns=24, SUNN(4ss==)2270,, SSUU((34)) 2∆singlet CSL 1 Ns=25, SU(5) SU(N) 1 0.6 Ns=24, SU(8) CSL -2 Ns=24, SU(6) 0 0.1 0.2 0.3 0.4 0.5 0.50 0.1 0.2 0.3 0.4 0.5 00Ns=21,0 S.1U(7) 0.N2s=27, S0U.(39) 0.4 0.5 0 0.1 0.2 0.3 0.4 0.50 θ/π θ/π θ / π θ / π FIG.1. Panel(a):GroundstateenergypersiteasafunctionofθforvariousN andN . Opensymbols(fulllines)denoteED(VMC)results. s (b): QualityoftheVMCwavefunctionasmeasuredbytheratioE /E . (c): EnergysplittingamongtheexpectedN singletstates VMC ED formingthegroundspacemanifoldofaSU(N)chiralspinliquid.(d)Energygapfromthegroundstatetothefirstexcitedsingletstatewhich isnotpartoftheexpectedgroundspacemanifold. turesCSLphases. ItisthegoalofthisLettertoprovidecom- that the small and large θ regimes for all considered N are pellingnumericalevidence,basedonlarge-scaleExactDiag- mostlikelyotherphases,whiletheintermediateregioncould onalizations(ED)andGutzwillerprojectedpartonwavefunc- harbourchiralspinliquids. tions,thattheaboveHeisenbergHamiltonianindeedfeatures SU(N)chiralspinliquidsareintrinsicallytopologicallyor- extendedregionsofSU(N)CSLsforallvaluesofN = 3to dered: Theyexhibitanon-trivialgroundstatedegeneracyon 9consideredhere. thetorus[21]andfractionalexcitations. Thegroundstatede- Exact diagonalizations – We start by investigating finite generacy on the torus is expected to be N for these partic- periodictriangularlatticeclustersasafunctionofθ forvari- ular states with N different abelian anyons [20, 21]. In our ous values of N. We focus on the range θ ∈ [0,π/2] in the numerical simulations, we can detect this degeneracy by in- following. θ > π/2 is likely to be dominated by ferromag- vestigating the low-energy spectrum on samples with a total netism, while θ < 0 yields the time-reversed, but otherwise numberoflatticesitesNs thatisanintegermultipleofN. In identical physics as −θ. For small values of N = 3,4 we Fig. 1(c) we display the energy spread ∆GS of these N ex- usedthestandardEDapproachemployingallthespacegroup pected ground states for different N as a function of θ. As symmetries,whileonlyconsideringtheindividualcolorcon- a general trend we observe that the splitting reduces signifi- servation, corresponding to an abelian subgroup of SU(N). cantly as we increase N. On the other hand several samples ForallotherN arecentlydevelopedEDapproachbytwoof still show a substantial splitting. Naively one would expect the authors [29], exploiting the SU(N) symmetry at the ex- asimpleexponentialsuppressionofthesplittingwithsystem pense ofspatial symmetries, iscurrently the only wayto ad- size,howeverintherelatedcontextoffractionalCherninsula- dress these systems within ED. Depending on N, the largest torsamoresubtledependenceofthegroundspacesplittingon systemsizesN rangefrom21to27latticesites. the actual shape of the clusters has been observed and ratio- s nalized[30]. Wethinkthatsimilarconsiderationsapplyhere InFig.1(a)weplottheEDresultsfortheenergypersiteof aswell. thegroundstateasafunctionofθforallconsideredN (open symbols). WhilethecurvesforN (cid:46) 5lookrathersmoothat Finallywealsomeasurethegap∆singlet fromtheabsolute first sight, it is visible that the energy per site displays kinks groundstatetothefirstsingletlevelthatisnotpartoftheex- aroundθ/π ∼0.05−0.1andatθ/π ∼0.35−0.4forN =6 pectedgroundstatemanifold. Thisisameasurefortheexci- to9. Forcomparisonweplottheenergyexpectationvalueof tationgapinthegappedchiralspinliquidstates. InFig.1(d), parameter-free Gutzwiller projected chiral spin liquid model one observes an approximate dome-shaped behaviour of this wavefunctionsforallvaluesofN (fulllines).Wewilldiscuss gapforallN,andfurthermorethisgapseemstodependonly thepropertiesofthesewavefunctionsinamoment. Interest- weakly on N. The approximate region in θ where the N- ingly, these model wave functions have very competitive en- fold ground state degeneracy splitting is small compared to ergies,especiallyintheθ regionslightlyabovethefirstkink. the excitation gap (for large N) is indicated as a shaded re- ForaquantitativecomparisonweshowinFig.1(b)theratioof gioninallthepanels,andindicatesaroughstabilityregionfor thevariationalenergydividedbytheEDgroundstateenergy. the SU(N) chiral spin liquids on the triangular lattice. One It is impressive that for N beyond 3 the best ratio exceeds shouldnotehoweverthatthepreciseextentofthechiralspin 0.98forthesystemsizesconsidered. Sothepicturesofaris liquidsforsmallN isanopenquestionatthispoint. 3 10 x 3.5 atri m 1 3 p a erl 0.1 S2.5 v G o Eigenvalues of 0.00.000.000111 SSSSSSSUUUUUUU(((((((3456789))))))),,,,,,, NNNNNNNsssssss=======8818891440446008 E - E 01..5521 three-sublattice flavor order S CUS(3L) 1e-05 1 2 3 4 5 6 7 8 9 10 0 index 0 0.1 0.2 0.3 0.4 0.5 θ / π FIG.2. VMCgroundspacedegeneracy: orderedsequenceofeigen- valuesoftheoverlapmatrixofGutzwillerprojectedwavefunctions FIG. 3. Summed squared overlaps of the VMC model wave func- with 30 different values of threaded flux. The overlap matrix has tionswithEDeigenstatesforN =3andN =12.Thebluecrossed s preciselyN largeeigenvaluesforanSU(N)chiralspinliquid. denoteEDeigenstates,whilethediameterofthefilledredcirclesde- notesthetotalsquaredoverlaponthoseeigenstates. Inthebestcase thesummedoverlapsonthelowestthreeEDeigenstates(degeneracy 1+2)accountforover90%ofthetotalweight. Variational parton approach – An appealing way to de- scribetheSU(N)chiralspinliquidsistouseaparton-based mean field approach [20, 21, 31–36], complemented with a at the mean-field level leads to a robust rank-N overlap ma- Gutzwiller projection. The idea is to fractionalize the ele- trix,thereforecorroboratingtheexpectationofanN-foldde- mentary spin degree of freedom into fermionic spinons (par- generate ground state manifold in the thermodynamic limit tons) with N flavors. For an exact description a dynamical alsoattheVMClevel. gauge field needs to enforce the physical constraint of one Since the variational energies for SU(3) turned out not to fermion per site. At the mean-field level however it is suffi- be very competitive, as shown in Fig. 1(a)/(b), we explicitly cienttospecifythebandstructureandfillingofthefermionic calculated the overlaps of individual ED eigenstates of the spinons. In the SU(N) chiral spin liquids of interest here, Hamiltonian (2) with the three orthogonal Gutzwiller wave thespinonbandstructureconsistsofN bands,wherethelow- functionsobtainedonthesamesystemsize. InFig.3weplot est band is completely filled for all N flavors and separated the summed squared overlap of all three wave functions (di- by a gap from the other bands. In addition this band is re- ameteroffilledcircles)withtheEDeigenstates(crosses)asa quiredtohaveChernnumber±1.Forthetriangularlatticewe functionofθ.HereweconsideraN =12sitesystem,where s use a Hofstadter-type tight-binding Hamiltonian with a uni- themomentaofthethreeEDgroundstatesinthechiralspin form flux of π/N per triangular plaquette [37], fulfilling the liquid phase are at the zone center (one) and at the corners requirementsonthebandstructure. Thismean-fieldstatecan of the Brillouin zone (twofold degenerate). Around θ = 0 nowbeturnedintoavalidspinwavefunctionbytheapplica- the SU(3) triangular lattice Heisenberg model is in a three- tionofanexactGutzwillerprojection,enforcingthepresence sublatticeflavororderedstate[41,42],howeverintheregion of exactly one fermionic spinon per site. Such a wave func- around θ/π ∼ 0.25, the three lowest ED eigenstates indeed tioncanbehandledbyVariationalMonteCarlo(VMC)tech- have sizeable overlap with the VMC model wave functions, niques,andinparticularonecaneasilycalculatetheenergyof therebyunderliningthepresenceofanSU(3)chiralspinliq- the Hamiltonian (2) on rather large lattices. The VMC ener- uidforsufficientlylargevaluesofθalsoforN =3. giesdisplayedinFig.1(a),(b)havebeenobtainedthisway. Edge states – Another hallmark of chiral topological ThenextquestionishowtheVMCapproachisabletoac- phasesisthepresenceofchiraledgemodesintheenergyspec- countforthenon-trivialgroundstatedegeneracyonthetorus. trumofsystemswithaboundary. Ithasbeenunderstoodthat Itturnsoutthatbythreadingfluxthroughthenon-contractible thecharacteristicenergylevelstructureoftheedgeexcitations loopsaroundthetorus,oneisabletospananN-dimensional asafunctionofthemomentumalongtheboundaryservesas subspace of Gutzwiller projected wave-functions, with al- a fingerprint of the type of topological order realised in the most identical local properties on finite lattices. From the bulk [43]. The SU(N) CSLs considered here are expected viewpoint of topological order this corresponds to a charge to exhibit a chiral edge energy spectrum described by the pumping procedure, where one cycles through the N differ- SU(N) Wess-Zumino-Novikov-Witten(WZNW)conformal 1 entgroundstatesbythreadingdifferentanyonicfluxthrough fieldtheory(CFT)[21].ThisisthesameCFTthatgovernsthe the interior of the torus. These concepts have recently been low-energyspectrumofwell-studiedone-dimensionalcritical exploredinthecontextofSU(2)CSLonseverallattices[38– SU(N)spinchains[27,28,44,45]. 40]. We have checked in Fig. 2 that the subspace of wave Inordertotestthishypothesisnumerically, onehastode- functionsspannedbyusing30differentboundaryconditions sign a setup where one can detect the edge modes in a clean 4 SU(3) SU(4) SU(5) SU(6) SU(7) SU(8) Ns=7 Ns=13 Ns=15 NN=s=1199 168 C6 C6 C7 sC6 x2 224 x2 56 15 3 20 21 168 4 490 45 6 3 20 224 56 3 6 5 120 735 2800 6 36 50 21 168 56 20 4 45 47505 6 218604 224 2281 56 550044 4 5 84 84 2281 504 1008 3 4 5 6 21 56 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 l-l l-l l-l l-l l-l l-l 0 0 0 0 0 0 FIG. 4. Edge states in SU(N) chiral spin liquids: the leftmost panel displays the N = 19 sites triangular cluster with open boundary s conditionsused.Inthevariousotherpanelsweexhibitthelowenergyspectrumasafunctionoftheangularmomentumaroundthecentralsite (l denotesthegroundstateangularmomentum). Thechiraledgestatesareclearlyvisible,withacharacteristicSU(N)multipletstructure, 0 whichcorrespondstoaparticularsectorofachiralSU(N) Wess-Zumino-Novikov-Wittenconformalfieldtheory.Theanalyticalpredictions 1 areindicatedbythedimensionsoftheSU(N)multipletsandcanbefoundinTab.Iofthesupplementarymaterial. way. Startingfromatorusanaturalwaywouldbetocutthe summarytheanalysisofthestructureoftheedgeexcitations torus open into a cylinder. This geometry however has two performed here confirms the SU(N) WZNW CFT predic- 1 independent,counter-propagatingedges,makingacleananal- tionsandthusstrengthensthecaseforabelianSU(N)chiral ysisdifficult,giventhesystemsizesN accessibletoED.We spinliquidsinthemodelHamiltonian(2). s therefore choose to emulate a disk geometry by considering Experimental considerations – With the recent realiza- thespecificN =19sitetriangularlatticewithopenboundary s tion of the Mott-crossover regime in 3D optical lattices with conditions depicted in the left panel of Fig. 4. Such a lattice fermionicYtterbiumatoms[46,47]theprospectforthereal- might actually be built in future ultracold atom experiments izationofstronglycorrelatedSU(N)quantummagnetismis withopticallatticesandatightconfiningpotential. Thissam- becoming bright. Our proposal for triangular lattices builds plestillhasasixfoldrotationaxisaboutthecentralsite,yield- oningredientswhichhavebeendemonstratedseparately: the inganangularmomentumquantumnumberwhichweuseto possibility to realize Mott insulators in optical lattices, and plottheenergyspectrum.Theenergyspectrumofthedischas to create static artificial gauge fields in an optical lattice (for no topological ground state degeneracy, but features gapless alkaline atoms) [48, 49]. Beside, working with the triangu- edge modes whichtypically propagate only inone direction. lar lattice is a big advantage because the 3-site permutation Theprecisemultipletstructureoftheedgemodesdependson term is the first and only term to appear to third order per- theanyonicsector. Inoursetupthissectorcanbesimplyla- turbation theory starting from the Hubbard model with one beled as a = (N mod N). In Tab. I of the supplementary s particlepersite,bycontrasttoe.g. thesquareandhoneycomb materialwehavecompiledtheSU(N) WZNWCFTpredic- 1 lattice, where they appear at order 4 and 6 respectively, and tions for the different irreducible representations of SU(N) are not the first corrections. The chiral phase typically ap- which appear at a given excitation energy, here qualitatively pears for θ (cid:39) 0.3, which, using the perturbation expressions labeledbytheexcessangularmomentuml−l0.Intheremain- ofJ =2t2/U andK =6t3/U2,correspondstot/U (cid:39)0.1. 3 ingpanelsofFig.4wedisplaytheactualEDenergyspectrum This might besmall enough to be still inthe Mott insulating of the Hamiltonian (2) for a fixed value of θ/π = 0.25 for phase, and to ensure that higher order corrections are negli- N =3upto8asafunctionoftheangularmomentuml−l . 0 gible. Infuturestudiesonemightalsorelaxtheπ/2fluxper For all N one can clearly identify a branch of chiral excita- plaquette condition, and explore the extent of the expected tions propagating to the right. The analytical predictions are stabilityregionoftheSU(N)CSLphases. indicated by the dimensions of the SU(N) irreducible rep- Several interesting questions need to be addressed in fu- resentations. For all N the numerical data for the first three turework. Forexample,isitpossibletodirectlyengineerthe sectors(l−l =0,1,2)isinfullagreementwiththeanalyti- 0 required three site exchange terms in Hamiltonian (2) using calpredictions.Thesplittingbetweenthemultipletsatagiven sophisticatedquantumopticsschemes? Thereishopethatthe valueoflisexpectedtovanishasN grows,andthespectrum s current activity on lattice gauge-theory implementations will should become linear with a certain edge state velocity. In bringtechniquestoaddressthisquestion. 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[26] H.-H.Lai,Phys.Rev.B87,205131(2013). 6 NinSU(N) N mod N l=0 l=1 l=2 s 2 1 (2) ⊕ (4) 3 1 (3) ⊕ (¯6) 2× ⊕ ⊕ (15) 4 3 (¯4) ⊕ (20) 2× ⊕2× ⊕ (3¯6) 5 4 (¯5) ⊕ (45) 2× ⊕2× ⊕ (50)⊕ (7¯0) 6 1 (6) ⊕ (8¯4) 2× ⊕2× ⊕ (120)⊕ (21¯0) 7 5 (2¯1) ⊕ (2¯8)⊕ (224) 3× ⊕ ⊕2× ⊕ (490)⊕ (73¯5) 8 3 (56) ⊕ (168)⊕ (5¯04) 3× ⊕2× ⊕2× ⊕ (10¯08)⊕ (2800) 9 1 (9) ⊕ (3¯15) 2× ⊕2× ⊕ (396)⊕ (27¯00) TABLEI. ThethreefirstangularmomentumsectorsofthechiraledgemodeoftheN =19dropletforSU(N) .Thenumber(N mod N) s 1 s selectstheanyonicsector(primaryfieldoftheCFT)whichconsequentlydeterminestheedgespectrum. 7 SUPPLEMENTARYMATERIAL variantcombinationofN −1LuttingerliquidCFTs(thusthe SU(N) WZNWCFTcentralchargec=N −1). 1 SU(N) WZWNpredictionsforthechiraledgestates 1 InTab.IweexplicitlylisttheexpectedSU(N)irreducible SU(N) countingrule OEISidentifier representationswiththeirmultiplicityforthefirstthreeexci- SU(2) 1,1,2,3,... A000041 tationlevelsl = 0(primaryfield)andl = 1,2(firsttwode- SU(3) 1,2,5,10,... A000712 scendantlevels)ofachiralSU(N) WZNWconformalfield 1 SU(4) 1,3,9,22,... A000716 theory. TheprimaryfieldforeachN isdictatedbytheopen SU(5) 1,4,14,40,... A023003 boundaryclusterssizeN =19viathelength(N mod N) s s SU(6) 1,5,20,65,... A023004 of the single-column young diagram at l = 0. We have de- SU(7) 1,6,27,98,... A023005 rived these results using a successive SU(N) coupling se- SU(8) 1,7,35,140,... A023006 quence with the adjoint representation starting from the irre- duciblerepresentationatl=0andsubsequentthenull-vector SU(9) 1,8,44,192,... A023007 elimination based on the SU(N) counting rule restrictions SU(10) 1,9,54,255,... A023008 listed in Tab. II. This simplified procedure uses the fact that TABLEII.SU(N)countingrulesusedtoderivetheresultsinTab.I. the SU(N) CFT can also be seen as particular SU(N) in- 1