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Synthetic gauge fields stabilize a chiral spin liquid phase Gang Chen,1,∗ Kaden R. A. Hazzard,2 Ana Maria Rey,3,4 and Michael Hermele1,4 1Department of Physics, University of Colorado-Boulder, Boulder, Colorado 80309-0440, USA 2Department of Physics, Rice University, Houston, Texas 77005, USA 3JILA and Department of Physics, University of Colorado-Boulder, NIST, Boulder, Colorado 80309-0440, USA 4Center for Theory of Quantum Matter, University of Colorado, Boulder, Colorado 80309, USA (Dated: January 26, 2015) WecalculatethephasediagramoftheSU(N)Hubbardmodeldescribingfermionicalkalineearth atoms in a square optical lattice with on-average one atom per site, using a slave-rotor mean-field 5 approximation. WefindthatthechiralspinliquidpredictedforN ≥5andlargeinteractionspasses 1 through a fractionalized state with a spinon Fermi surface as interactions are decreased before 0 transitioning to a weakly interacting metal. We also show that by adding an artificial uniform 2 magneticfieldwithfluxperplaquette2π/N,thechiralspinliquidbecomesthegroundstateforall n N ≥ 3 at large interactions, persists to weaker interactions, and its spin gap increases, suggesting a that the spin liquid physics will persist to higher temperatures. We discuss potential methods to J realize the artificial gauge fields and detect the predicted phases. 6 1 Introduction.—Theexperimentalrealizationofatopo- Severaltheoryworkshaveaddressedquestionsrelatedto ] logically ordered phase of matter other than the frac- the expected SU(N) magnetic phases in the strongly in- s a tionalquantumHalleffectthatoccursintwo-dimensional teracting limit [5, 6, 27–43]. g electron gases is a major goal in both condensed matter Inparallel,otherultracoldatomexperimentshavereal- - t andatomicphysics. Phaseswithintrinsictopologicalor- izedsyntheticgaugefields[44–51]. Intheseexperiments, n der [1] are of fundamental interest, as they exist outside the atoms behave as if they were charged particles in ex- a u of the standard symmetry-breaking framework for clas- ternal electromagnetic fields despite their neutrality. Al- q sifying phases of matter and display exotic phenomena though many schemes in principle can create the gauge . such as fractionalized excitations and edge states that field that we study in this paper, we focus on methods t a are robust to local perturbations [2]; in some cases these utilizing laser-induced tunneling [52–54]. m phases have been predicted to be useful for topological AEA in optical lattices with synthetic gauge fields.— - quantum computation [3, 4]. Ultracold atomic systems AEAinasufficientlydeepopticallatticearedescribedby d n are uniquely tunable and clean systems that offer a plat- an SU(N) generalization of the usual (N = 2) Hubbard o form to realize exotic phases. However, so far, reach- model, c ing the required low temperatures remains a challenge. [ Previous work predicted a topologically ordered chiral H =−t (cid:88) eiφijc†α,icα,j + U2 (cid:88)(ni−1)2 (1) 1 spinliquid(CSL)groundstateinfermionicalkalineearth (cid:104)i,j(cid:105),α i v atoms (AEA) in a deep square optical lattice [5, 6]. In 6 where c is the fermionic annihilation operator for nu- this Letter we show, within a slave-rotor approximation, α,i 8 clear spin state α at lattice site i, (cid:80) indicates a that by applying a synthetic gauge field to this system (cid:104)i,j(cid:105) 0 sum over nearest neighbors i and j; φ = φ is 4 it is possible to enhance the parameter space where the ij − ji the (externally imposed) lattice gauge field. We define 0 CSLexists,toincreasethecorrespondingspingap,andin 1. turn to increase the temperatures at which CSL physics ni = (cid:80)αc†α,icα,i, and t and U are the hopping energy 0 manifests. In addition, without a synthetic gauge field, and on-site interaction energy, whose ratio can be tuned 5 away from the strongly insulating limit we find a gapless bymodifyingtheopticallatticedepth. InthisLetter,we 1 quantum spin liquid with a spinon Fermi surface. take the average fermion number per site to be one. : Thegaugefieldφ dependsbothontheartificialelec- v ij i Recently, experiments have trapped and cooled AEA tromagnetic field as well as the gauge choice. We are X to quantum degeneracy and loaded them in an optical interestedinthephysicsofatwo-dimensionalsquarelat- r lattice [7–17]. Moreover, experiments [18–20] have con- tice with a spatially uniform, time-independent artificial a firmed the predicted SU(N) spin symmetry in the colli- magnetic field, and use the Landau gauge where sional properties of fermionic AEA [21–23]. This SU(N) (cid:40) symmetry generalizes the usual SU(2) symmetry, and N Φx δ if i,j bond is vertical φ = j yj−1,yi { } (2) canbecontrollablyvariedbyinitialstatepreparationup ij 0 otherwise, to 2I +1, with I the nuclear spin (as large as N = 10 for 87Sr with I =9/2). The low temperatures reached in x is the x coordinate of site j measured in lattice units, j recentexperiments[24–26],atwhichshortrangespincor- andΦisthefluxpenetratingasinglesquareplaquetteof relations should begin to develop, makes it particularly thelattice[55]. WefocusonthecaseΦ=2π/N,because timely to study quantum magnetism in these systems. thischoiceofΦisfavorablefortheexistenceofthechiral 2 no gauge field c a 12 N=4 N=3 (cid:83) (cid:83) valence bond 10 VBS CSL solid (VBS) 8 t chiral spin Fermi liquid 6 (cid:31) U SFS liquid (CSL) (FL) 3.0 4 k 2.9 2.8 2 FL spinon Fermi integer 2.7 2.6N(cid:31)3 0 surface (SFS) quantum 3 4wit5h ga6uge7 fiel8d 9 10 k Hall (IQH) 12 b 10 VBS N d 8 CSL ( ) t 6 (cid:31) U 4 magnetic unit cell 2 IQH 0 3 4 5 6 7 8 9 10 N FIG. 1. Phase diagram, calculated with a slave-rotor mean-field approximation, as a function of spin degrees of freedom N andinteractionstrengthU/tinthe(a)absenceand(b)presenceofanartificialuniformmagneticfieldwithfluxperplaquette Φ=2π/N, illustrated in panel (d) for N =3. Thin black lines are second order phase transitions, while thick black lines are firstorderphasetransitions. Thestatesfoundarethevalencebondsolids(VBS),chiralspinliquid(CSL),spinonFermisurface (SFS),Fermiliquid(FL),andintegerquantumHall(IQH)states. Thesearedescribedinthetextandillustratedinpanel(c). spin liquid. We note that the magnetic unit cell associ- these new degrees of freedom, giving atedwiththetranslationalinvarianceoftheHamiltonian (cid:88) U (cid:88) is enlarged from the one imposed by the optical lattice H = t eiφijei(θi−θj)f† f + L2. (5) − α,i α,j 2 i potential. Figure 1(d) shows the system with this flux (cid:104)i,j(cid:105),α i and gauge choice, and the enlarged magnetic unit cell, Although the rewritten Hamiltonian Eq. (5) together for N =3. with the constraint Eq. (4) is exactly equivalent to We calculate the phase diagram and properties of this Eq. (1), to make further progress we make a mean- system within a slave rotor mean-field approximation field approximation to decouple the rotor and spinon de- [56, 57], which we describe briefly. This technique is de- greesoffreedom. Wethenobtainthecoupledmean-field signed to match on to the previous large-N solution in Hamiltonians for the rotors and the spinons, thelargeU/tlimit, andiswell-suitedfordescribingnon- magnetic ground states in proximity to the Mott tran- H = (cid:88)J eiθi−iθj +(cid:88)UL2+h (L +1), (6) sition. First we expand the Hilbert space to include a r − ij 2 i i i (cid:104)i,j(cid:105) i U(1) bosonic rotor degree of freedom on each site, θ , j (cid:88) (cid:88) and new fermionic spinon degrees of freedom associated Hf =− t˜ijeiφijfα†,ifα,j − hifα†,ifα,i, (7) with operators f , which are defined by (cid:104)i,j(cid:105),α i,α α,j where h is a Lagrange multiplier that enforces on aver- c =e−iθjf . (3) i α,j α,j age the constraint Eq. (4), t˜ij t eiθi−iθj r, and Jij InordertoreproducetheoriginalHilbertspace,wemust teiφij(cid:80)α(cid:104)fα†,ifα,j(cid:105)f. Here the≡su(cid:104)b-index(cid:105)r (f) refer≡s impose the constraint to taking the expectation value in the rotor (spinon) mean-field ground state ψ (ψ ). The Hamiltoni- Lj =(cid:88)fα†,jfα,j −1 (4) ans Hr and Hf are invaria|n(cid:105)trun|de(cid:105)rfa U(1) gauge trans- α formation, fα†,i → fα†,ie−iχi,θi → θi + χi, and t˜ij → that the rotor angular momentum Lj is uniquely de- t˜ijeiχi−iχj,Jij Jije−iχi+iχj. WesolveHr andHf self- → termined by the particle number. Here, L satisfies consistently for several variational ansatz [58] and find j [θ ,L ] = i. We rewrite the Hamiltonian in terms of the ground state by optimizing the total energy ψ H ψ j j (cid:104) | | (cid:105) 3 where H is given by Eq. (5) and ψ ψ r ψ f is the Phases (cid:104)eiθ(cid:105) rotor flux spinon gap spinon flux | (cid:105) ≡ | (cid:105) | (cid:105) mean-field state. FL (cid:54)=0 0 0 0 Results.— Figure 1(a, b) shows the slave-rotor mean- SFS 0 0 0 0 field phase diagram as a function of U/t and N; the top panelshowsthephasediagramintheabsenceofagauge CSL 0 −2π/N (cid:54)=0 2π/N fieldandthebottomshowsthephasediagramforagauge SU(3)-VBS 0 −π (cid:54)=0 π field with flux Φ = 2π/N. We find five phases: Fermi SU(4)-VBS 0 0 (cid:54)=0 0 liquid (FL), integer quantum Hall (IQH), valence bond solids (VBS), a gapless spin liquid with a spinon Fermi IQH (cid:54)=0 0 (cid:54)=0 2π/N surface(SFS)[59],andachiralspinliquid(CSL)[60,61]. CSL 0 0 (cid:54)=0 2π/N Thin black lines indicate second order transitions and SU(3)-VBS 0 π/3 (cid:54)=0 π thick black lines indicate first order phase transitions. Generically, the role of the Hubbard U interaction is to SU(4)-VBS 0 π/2 (cid:54)=0 0 localize the atom on lattice sites. Such Mott localization is signalled in the rotor sector; when the bosonic rotor is TABLE I. Parameters that characterize the obtained phases. gappedanduncondensedwith eiθ =0, thesystemisin The upper five (lower four) rows describe phases in the ab- (cid:104) (cid:105) sence (presence) of the synthetic gauge field. The rotor a Mott insulating state. The mean-field parameters and (spinon)fluxreferstothefluxthatisexperiencedbytherotor some key properties of the different phases are listed in (spinon)inthemean-fieldHamiltonianH (H ). FortheFL, TableI.Asweshowinthetable,therotorandthespinon r f SFS,IQH,andCSLstates,thefluxisdefinedfortheelemen- may experience different, even opposite, gauge fluxes in tarysquareplaquette. ForSU(3)-VBS[SU(4)-VBS]state,the their mean-field Hamiltonians for different phases. Since flux is defined through the 6-site [4-site] cluster [58]. the rotor and the spinon must form a whole atom, the total gauge flux experienced by the rotor and the spinon should be equal to the synthetic gauge flux that is exter- strongly interacting limit, the Hubbard model reduces nally imposed on the atom. to an SU(N) Heisenberg model, and the phase diagram TheFLphaseisverysimilartotheusualSU(2)Fermi coincides with previous slave-fermion mean-field calcu- liquid, and its structure and instabilities are essentially lations of the Heisenberg model [5]: for N = 3,4 the thosedescribedintheabsenceofalattice[22]. TheVBS ground state is a VBS, while for N 5 the ground state ≥ aretranslation-symmetrybreakingphaseswithrepeating is a CSL. This is true both with and without a synthetic units of SU(N) singlets spread across multiple sites. In gauge field, as in the the U/t limit the physics → ∞ particular,asweplotinFigure1(c),thesystemisdecou- is governed by two-site nearest neighbor superexchange, pledinto6-siterectangular(4-sitesquare)clustersinthe which is insensitive to the gauge flux. SU(3)-VBS[SU(4)-VBS]state. TheSFSspinliquidstate In the intermediate U/t regime, the gauge field causes is characterizedbya gaplessspinon Fermi surfacewith a more significant differences. Without a gauge field, we gappedbosonicrotorinthemean-fieldtheory. Goingbe- find that an SFS phase intervenes between the non- yond the mean-field description, we need to include the interacting FL and Heisenberg-limit CSL or VBS for all U(1) phase fluctuation of the spinon hopping t˜ . This is N except N = 4, in which case there is a direct transi- ij theinternalgaugefluctuation[57]; itisdynamicallygen- tion between the FL and VBS ground states. The FL- erated and is unrelated to the synthetic gauge field that SFS transition is second order and is expected to remain is imposed externally. At low energies, the SFS spin liq- continuousbeyondmean-fieldtheory[67],whiletheSFS- uidisdescribedbythespinonFermisurfacecoupledbya CSL and FL-VBS are first order phase transitions. In fluctuating internal U(1) gauge field [57, 62–66]. Due to contrast, in the presence of the Φ = 2π/N gauge flux, the spinon-gauge coupling, the overdamped U(1) gauge a direct second order transition occurs between the non- fluctuationscattersthespinonsontheFermisurfaceand interacting IQH phase and the CSL phase within our destroysthecoherenceofthespinonquasi-particles. The mean-field theory, and the CSL exists at intermediate resultingstateisanon-Fermiliquidoffermionicspinons. U/t even for N =3 and 4. The CSL is distinct from the SFS in that the spinons Thegaugefieldincreasestheparameterspaceforwhich form an integer quantum Hall state in the CSL. Upon the CSL occurs: in addition to persisting down to N = couplingtoU(1)gaugefluctuations,thisleadstoachiral 3,4, the CSL occurs for a broader range of U/t values. topologically ordered phase with anyon excitations, and Inparticular,theminimumU/tforwhichtheCSLexists gapless chiral edge states that carry spin but no charge decreases from about U/t 5.5 to U/t 3.5 (the exact ≈ ≈ [60]. values depend on N). To understand the global structure of the phase dia- IntheCSL,boththespinonsectorandtherotorsector gram, it is useful to consider the two limits U/t=0 and are gapped. Figure 2 illustrates the excitation gap ∆’s U/t . The FL and IQH states are simply the non- dependence on U/t, N, and the gauge flux in the CSL → ∞ interacting ground states occurring at U/t = 0. In the where ∆ is the smaller of the spin gap and the rotor 4 has a 100s natural lifetime, and is therefore stable on ∼ the timescale of the system. Because a single laser can directly drive tunneling of a g atom to an e atom at an adjacent lattice site while imprinting a phase φ , one ij avoidsthecomplexityofdrivingRamanprocesses. How- ever, when this proposal is implemented in the context of interacting quantum phases additional considerations arise that were not accounted for in the prior analysis. First, two e-state atoms on the same site can inelasti- cally collide and be lost from the trap. We have found that this problem can be largely mitigated when using a checkerboard g-e pattern [69]. Second, the interactions areinhomogeneous, beingdifferentforthesitesoccupied FIG.2. TheexcitationgapoftheCSLphase,∆,asafunction by g atoms and e atoms. This issue can modify the dis- of interaction strength, U, both in units of the tunnelling t . cussed phase diagram. Third, the flux generated in the ThecurvesillustratetheN-andmagneticfluxΦ-dependence. simplestimplementationofthisproposalisstaggeredand Frombottomtotop,weshow(N =10,Φ=0);(N =10,Φ= thusrequiresrectificationtechniquestomakeithomoge- 2π/10); (N = 5,Φ = 2π/5); and (N = 5,Φ = 2π/5). The neous. turning points at U/t ≈ 4 are the locations below which the Preparationanddetection.—Reachingthetemperature rotor gap becomes smaller than the spin gap. regimes to observe the phase diagram Figure 1 is chal- lenging. However, the expected advantage of the SU(N) gap. In the slave-rotor mean-field approximation, the symmetry for cooling [24, 33, 39, 70] together with the spin gap is simply the band gap of the spinon spectrum, less stringent temperature requirements to observe CSL and the rotor gap is set by the Hubbard U interaction phases in the presence of the synthetic gauge field might andthusstaysmuchlargerthanthespingapintheMott help achieve the required conditions. Other potentially insulating regime except near the Mott transition. For favorable aspects of the gauge field are the absence of a given U/t, the spin gap slightly increases when the an intermediate SFS phase and that all transitions are gauge field is turned on. An even more favorable effect second order in the mean-field analysis. Consequently, of the gauge field for the spin gap occurs because the adiabaticallygoingfromweaktostronginteractionsmay CSL persists to lower U/t. Since ∆ increases as U/t be easier than in the absence of the gauge field. On the decreases, the gauge field increases the maximum ∆ by other hand, the gauge field itself introduces further con- about a factor of 1.5. Because ∆ sets the temperature straintssuchastherequirementtouseadeeplatticepo- to which the CSL’s characteristics remain, we therefore tential and a complex band structure even in the weakly expect the gauge field to increase the temperature range interacting regime. Consequently, determining optimal over which the CSL behavior is accessible. preparation is beyond the scope of this work. Gauge field implementation.—Many proposals to im- To conclude, we briefly outline methods to detect the plement artificial gauge fields exist. Here we suggest one CSL and SFS. Although it is premature to analyze pro- scheme,whichusesRaman-inducedtunnelingindeeplat- tocolsindetail, asthesewilldependsubstantiallyonthe tices subject to a uniform potential gradient [49, 50]. A specific experimental implementation, it is useful to de- Raman process is on resonant with the energy splitting scribe the basic ingredients that would be required. To between adjacent lattice sites, and the atoms acquire a detect the CSL Ref. [6] suggests methods to probe two phase kick each time they hop, imprinting the phase φ characteristic properties of topological phases: looking ij inEq.(1). Thisschemeisnaturalforourcurrentconsid- for topologically protected, chiral edge currents and in- erations,sinceitutilizestheopticallatticeandgenerates troducing a weak attractive optical potential that is lo- theHamiltonianEq.(1)withstronggaugefluxes. Gauge calized to a few lattice sites, which should bind the any- fields have been recently demonstrated in bosonic alkali onic quasiparticles. Braiding or interfering these quasi- atoms using this technique [49, 50], although we note particles can manifest their anyonic nature. To detect that these experiments have observed unexplained heat- the SFS state, one can perform spin-dependent Bragg ing,whichcouldbeproblematicforrealizinglowtemper- spectroscopytodetectthe2-spinoncontinuuminthedy- ature phases. namic spin structure factor; the most basic signature of We also mention the alternative scheme proposed in the exotic nature of this phase is the lack of order and Ref. 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Phys. 12, 033007 where we have replaced the rotor variable eiθi by a uni- (2010). modular operator Φ such that Φ 1. Since the oper- i i [55] This is a non-dimensionalized flux: it is divided by a ator L = (cid:80) f† f 1, h is |the|n≡thought as a chem- unitfluxquantumsoΦisidenticaltothephaseacquired i α α,i α,i− (cid:80) ical potential. Because L = 0, h must vanish for around a single plaquette. (cid:104) i i(cid:105) the mean-field solutions. With a 2π/N flux per square [56] S. Florens and A. Georges, Phys. Rev. B 70, 035114 (2004). plaquette and one fermion per site, the Hamiltonian Hf [57] S.-S. Lee and P. A. Lee, Phys. Rev. Lett. 95, 036403 gives a spinon band structure with N bands. Only the (2005). lowestbandisfilled(foreachspecies)andseparatedfrom [58] See the supplementary material. the others by a gap. Moreover, the lowest spinon band [59] P. W. ANDERSON, Science 235, 1196 (1987). has Chern number C = 1 for each fermion flavor α. As [60] X. G. Wen, F. Wilczek, and A. Zee, Phys. Rev. B 39, we shown in Table I, whether the system is in the IQH 11413 (1989). or the CSL is determined by the behavior of the rotor [61] V.KalmeyerandR.B.Laughlin,Phys.Rev.B39,11879 (1989). sector. [62] P.A.LeeandN.Nagaosa,Phys.Rev.B46,5621(1992). TosolvetherotorHamiltonianHr,weimplementaco- [63] S.-S. Lee, Phys. Rev. B 78, 085129 (2008). herent state path integral formalism in imaginary time. [64] S.-S. Lee, Phys. Rev. B 80, 165102 (2009). We integrate out the conjugate variable L and obtain i [65] D.F.Mross,J.McGreevy,H.Liu, andT.Senthil,Phys. the partition function that is written as a functional in- Rev. B 82, 045121 (2010). tegration over the Φ variable, [66] D. Dalidovich and S.-S. Lee, Phys. Rev. B 88, 245106 (2013). (cid:90) [67] T. Senthil, Phys. Rev. B 78, 045109 (2008). Φ† Φ λe−S−(cid:82)τλi(|Φi|2−1). (10) [68] A. J. Daley, M. M. Boyd, J. Ye, and P. Zoller, Phys. Z (cid:39) D D D Rev. Lett. 101, 170504 (2008). Hereλ istheLagrangemultiplierintroducedtotheuni- [69] K. R. A. Hazzard, Unpublished. i [70] K. R. A. Hazzard, V. Gurarie, M. Hermele, and A. M. modular condition for the rotor variable at every lattice Rey, Physical Review A 85, 041604(R) (2012). site. The effective action is given by [71] X.-G. Wen, Phys. Rev. B 65, 165113 (2002). [72] E. Tang, M. P. A. Fisher, and P. A. Lee, Phys. Rev. B (cid:90) 1 (cid:88) (cid:88) S = ∂ Φ 2 2J (cosk +cosk )Φ 2, 87, 045119 (2013). 2U | τ k| − x y | k| dτ k∈BZ k∈BZ (11) where“BZ”referstotheBrioullinzoneofthesquareop- ticallatticeandwehavesetthelatticeconstanttounity. SUPPLEMENTARY MATERIAL In a standard spherical approximation for a mean-field (orsaddlepoint)analysis,weassumeauniformLagrange multiplier λ λ. We integrate out the variable Φ and i ≡ Slave rotor mean-field theory: no translational obtain the saddle point equation for λ, symmetry breaking 1 (cid:88) U =1, (12) N ω Syntheticgaugefieldcase.—Herewegiveadetailedde- s k k∈BZ scriptionoftheslaverotormean-fieldtheoryinthepres- ence of the synthetic gauge flux. To study the energetics where ω =[2U(λ 2J(cosk +cosk ))]1/2 is the band k x y − as well as the phase transition from the IQH to the CSL dispersion of the rotor and N is the number of lattice s in Figure 1(b), we first choose the variational ansatz for sites. We solve the saddle point equation Eq. (12) self- the IQH and the CSL such that t˜ = t˜,J = J. More- consistentlywiththespinonmean-fieldHamiltonianH . ij ij f over, we assume an uniform Lagrange multiplier such When the rotor band touches zero energy, the rotor is that h h. This is equivalent to replacing the local condensed and the internal U(1) gauge field picks up a i ≡ 7 mass due to the Anderson-Higgs mechanism. The ro- surface. Again, whether the system is in the FL or the tor and the spinon are then bound together and form a SFS is determined by the rotor sector. Since the FL and fermionic atom. The resulting phase is the IQH. When the SFS only occur for the model without the synthetic the rotor band is gapped and the rotor is not condensed, gauge flux, the rotor sector Hamiltonian is identical to theinternalU(1)gaugefieldisgappedoutbytheChern- Eq. (8). When the rotor is condensed, the system falls Simons term and the resulting phase is the CSL. into the FL. When the rotor is gapped, the system is in the SFS whose low energy property is described by In the IQH, the system has N chiral edge modes that thespinonFermisurfacecoupledwithafluctuatingU(1) transport spin quantum numbers as well as atoms. The gauge field. CSL, however, is a Mott insulating state. The atoms are localized by the interaction in the CSL. The chiral edge states in the CSL only carry spin quantum number Variational ansatz: VBS states and cannot transport charge. The effect of the phase TOPOLOGICALLIQUIDSANDVALENCECLUSTER... PHYSICALREVIEWB84,174441(2011) transitionfromtheIQHtotheCSLontheedgestatesis As we described in the main text, VBS states become to gap out the mode that transports atoms. favorable in the strongly interacting limit for the SU(3) (a) (b) andSU(4)models. Asexpected,asimilarconclusionwas foundinTthOePpOrLeOviGoIuCsAsLlavLeIQ-fUerImDSioAnNstDudVyA,Li.EeN.CthEeCgLrUouSnTdER... PHYSICALREVIEWB84,174441(2011) stateoftheSU(3)[SU(4)]Heisenbergmodel–theU/t → π π ∞[6] lfiamvoitreodftthh(eeaV)HBuSbbsatradtemsho(odbwel)n–inonFitghueresq(3ub[a)Friegularteti4c]e. (a) In our slave rotor mean-field calculation, we have chosen H suchthatthespinonshavethesamehoppingandfeel f the same mean-field gauge fluxes as those oneπs shown in π (b) Figure3andFigure4,whiletherotorsectorhasthesame (a) (c) 0 0 cluster structure as the spinon sector but experiences a different flux (see Table. I). π π FIG. 5. (Coloronline)IllustrationoftwoN clusterground =∞ statesonthesquarelatticefornc(c)2and0k 4,whichsa0turatethe = = 0 0 lowerbound(94).Squareclustersofonecoloraremarkedwithsolid lines (red online), while those of the other color are marked with π π FIG. 3. The VBS state for the SU(3) model. Tdahsehesdpilninoens(blueonline).Anyconfigurationwhereclustersofthe FIG. 5. (Coloronline)IllustrationoftwoN clusterground boχotrhru′eFnrIhdsGa.(s.I9n44c).othonCensltkutahsnettesrmhdfiH3qsoaeuaatslpraagmtdktarpneetigiibseltnlat,uao(goudnttnthngecidieciesa=.eoflnflnTfzu1oueHxht)rxrheo(wtfeafhπ.ei)rodtlokhTtnflaurue=bhgktnxyehhe2btrtefie,oghhag(nirlebceuidoh)gsrfsukehessgratai=mhixntsu-bdtisr3aoohaii,dtnntseeaiaidnnbczppgsdeoltsart(lepaohqdcdineu)nordekfloenrotc=ntowiettssmhae4iiesnrn.nRgotlneehf-.eizsntuee6wmnpctr.ehooo=erneacgoonym1cnl.-oocefiamratesshnlpesde-,uetpthianergadteepgleyernttuielrrebatachytievalemato1tin/cNgeitshceaosNreres=tc0attie∞osnisgsretooxupntehdcetsetgdarttoeo0u;bansedi-lnisftttahetdee llsiotnaweteessr(boreondutnhoden(ls9iqn4ue)a).,rSewqluahtaitlrieecetchlfuoossrteenrocsf=otfh2oenaoentdchoeklro=croa4lr=o,erwm∞aharierckhmedsaawrtkuierthadteswotilhtihde dashed lines (blue online). Any configuration where clusters of the = π,whileitiszeroforeachfour-siteplaquetteinthek 4state.For bitesoayectthxhhepevegatcylrtuopeeudenootdoff-kbcs,tleauintlseitfHFteteherLnaesdNemsNaurohgpinoyl=ostdwyno(nR∞ntncheitoishafeml.ainem3Sptg4iuFiric)ttso,.iSunge,mngvadweuporsegeydtreatitcufitihefilrie.bonedTaglodthsiobviecsfeyatlt1hashpr/eege=Nu.se—qtsdctpueoiWaingrnrreegoientlcnaehφttritomianijcncoyees→agna-ufi0giWk.esel!deiFflmkauo5pnnrxooossttwanshhtitezebohlfeensfoqoFhdfiurcoaeIasllGprrudukep.scgitblhn4eaao.grutvntgisadiboχcetsla.eotTurhfltr,uzeeT′heFuneassreIhxrhdnsGooa.tedV(fhsf.I9e0onaBwl44kctnt).otS.flehbconCuIteaycnssnhxltntokutateathntshnhst=ehtjtleeaeieirssgtrmcfo3qufehsstouuarruttsiraagagmttrutatrhnebteeetiaihseootttltt,uenhnai(hhodnttiadetheScnecitsecU,bsb=seopafloia(ontf4inlut1uondu)idxt)rnsohrs(wmadnteviaqhtssi)eiorufdtookihordananynurerroe=gkeltnn.hhe-2ibezsreT,ogeam(tnirhbcehdohee)sesaksossnmaaipn=xn-tiuefi-dnatsr3eoahinli,ttndseea-innzpgdelart(ohqceu)oeklnot=twethe4iesr. ntuewnpceoo=rncgoy1cl.oocamrsspes,uetpthianergadteepgleyernttuielrrebatachytievalemato1tin/cNgeitshceaosNreres=tcattie∞osnisgsretooxupntehdcetsetgdarttoeou;bansedi-lnisftttahetdee such that t˜ij t˜,Jij J and hi h. ThcehaslpleinngoinnsgtorHigaomroilutsolnyπiad,newtHehrifmle.iiTntehisetzhfieergoluarfroegreise-Naacdhgafrpootuuernd-sdiftrseotmpatlaeRq,ueef.tte6.inthek=4state.For ≡ ≡ ≡ eachvalueofk,intheN limit,everytilingofthesquarelattice istrivialtoseethpaatrEtiM′aFllTy=filEl btohuendbiafnadnds aonnldygifivceonrdisietiotno(t1h)e spianopnroFbelermmiwe do not cubryretnhtelytypkenoowfclhuostwerstoshso=owlvn∞eis.aIngsrtoeuandd,state.Thislargedegeneracy We know of no cluster states that can saturate the bound for holds. we employ a systematicisneuxmpeecritecdaltosebaerlcifhtefdourpgornocuonmdpsuttaintegsp,erturbative1/Ncorrections k ! 5onthesquarelattice,andweconjecturethatsaturation As before, saturation of the bipartite bound is impossible which,whilenotfoolprootof,thaellgorwosunuds-stotatdeeetenremrgiyne(Rtehfe.3g4ro).und is impossible for such values of k. In this situation, it is very forlargeenoughk.Again,weconsideralatticewhereeither statewithsomeconfidence. challengingtorigorouslydeterminethelarge-N groundstate, Jbornr′d=sinJmthaexloarttiJcerrw′ =ith0n,oannzderloeteNxcbhabnegteh.eFotortkal>nu4mNbbe/rNosf, minHimeriez,atiwoen (fiSrCstMd)espcrihrosiocbtleerddivsu.oirauelr,townshueimechethriawctaeEl M′dseFevTlfe=-lcoopEnesbdoisutnaednnditfandonlyifcondition(1) awepreomblpelmoywaesydsotenmotatcicurnreunmtleyrikcanlowseahrocwh ftoorsgorlovuen.dInssttaetaeds,, the bound (87) is stricter than Eq. (94), so saturation is employed in Ref. 35 for the case n 1. A very similar As before,cs=aturation of the bipartite bound is impossible which,whilenotfoolproof,allowsustodeterminetheground impossibleforsuchvaluesofk. procedure was later used by Foss-Feig and Rey to study for largeenoughk.Again, we consider alatticewhere either statewithsomeconfidence. Sinceaflatenergyspectrumofthemean-fieldHamiltonian the Kondo lattice model, in collaboration with one of us is necessary to saturate the bipartite bound, we expect that it (M.H.),101 and subsequeJntrlry′ =wiJthmabxotohroJfrru′s=.1020,DaunedtloetthNeb be the total number of Here, we first describe our numerical self-consistent will only be saturated by VCS states. VCS states saturating local constraint, the SCMbonpdrsocinedtuhreelaisttincoetwsiitmhpnloynzaetrroiveixalchange.Fork >4Nb/Ns, minimization (SCM) procedure, which we developed and the bound (87) is stricter than Eq. (94), so saturation is employed in Ref. 35 for the case n 1. A very similar theboundonthesquarelatticefornc 1areshowninFig.4 iteration of a self-consistent equation, and to our knowledge c = = impossibleforsuchvaluesofk. procedure was later used by Foss-Feig and Rey to study and were also reported in Ref. 35. For k 2, the bound is ithasnotbeenusedpreviouslybyothers;therefore,weshall = Sinceaflatenergyspectrumofthemean-fieldHamiltonian the Kondo lattice model, in collaboration with one of us saturated by any dimer state, and for k 4 it is saturated by describe the SCM procedure here in some detail. Following = is necessary to saturate the bipartite bound, we expect that it (M.H.),101 and subsequently with both of us.102 Due to the four-clusterstatesofthetypeshown.Fork 3,theboundis this discussion, we shall describe the results of SCM on the = will only be saturated by VCS states. VCS states saturating local constraint, the SCM procedure is not simply a trivial actuallysaturatedbyaclassofsix-clusterstates. squarelatticeforn 1,2. Whenever a given lattice admits a n 1 cluster state c = theboundonthesquarelatticefornc 1areshowninFig.4 iteration of a self-consistent equation, and to our knowledge c = saturating the bound, it is easy to see that=the same lattice and were also reported in Ref. 35. For k 2, the bound is ithasnotbeenusedpreviouslybyothers;therefore,weshall = (i.e.,samesetofexchangecouplings )alsoadmitsn >1 saturated by any dimer state, and for k 4 it is saturated by describe the SCM procedure here in some detail. Following rr c = cluster states saturating the bound. TJhes′e n >1 states have A. Self-consistefonutrm-cinluimstiezratsitoantepsroocfetdhuerteypeshown.Fork 3,theboundis this discussion, we shall describe the results of SCM on the c = diagonalχab asinEq.(83),andeachχa ischosentogivea We first describe theaScCtuMallaylgsaotruitrhamtedinbythaecsliamssploefrscixa-scelusterstates. squarelatticefornc 1,2. clusterdecorrm′ positionofthetypethatsartru′ratestheboundfor of n 1; modifications inWthheenenver a2gciavseenalraettidceescardibmeidts a nc 1 cluster state = c c = n 1. Examples of such states (for k 4 and n 2) are below.=Thebasicideaisssaimtupralytintogitteh=reatbeotuhneds,elift-cisonesaisstyentcoysee that the same lattice c c illu=stratedforthesquarelatticeinFig.5.= = condition(13).However,(iif.eth.,issaismaellsoenteodfoeexsc,hthaenngtehceofuerpmliinogns rr)alsoadmitsnc >1 density will be nonunifcolrumstearndstaEteqs. s(a1t4u)rawtiinlgl btheevbiooulantedd..TJhes′e nc >1 states have A. Self-consistentminimizationprocedure Instead, the idea is to itdeiraagteonEaql.χ(ra1rb3)aswinithEinq.a(8c3o)n,satnradineeadchχrar ischosentogivea We first describe the SCM algorithm in the simpler case VI. LARNGUEM-NERRIECSAULLTGSROONUNSQDU-SATRAETELASETATIRCCEHAND saectcoomfpχlrisrh′ atnhdis,µtrh,esaolgtcnohlcrauittshtEem1qr..dpE(er1cox4oca)emm′eipdpsosleasslaiwtsiooaffnyossloulofscawhtthisess:fitta(ey1tdep).seAT(thfonoartskatu′r4ateasndthnecboun2d) aforer obfelonwc.=Th1e; bmaosidcifiidceaatioisnssiminpltyhetonitce=rat2etchaeseselafr-ecodnessisctreibnecyd = = = In this section, we focus on the square lattice and, in initialχ ischosenrandilolumsltyra.tIendofuorrcthaelcsuqlautaiorensla,twtiececihnosFeig.5. condition(13).However,ifthisisallonedoes,thenthefermion rr ′ particular,onthecasek ! 5.ThediscussionofSec.VBabove χrr χrr eiφrr′,where χ waschosenintheinterval[0.03, density will be nonuniform and Eq. (14) will be violated. ′ =| ′| | | establishesthat,fork 2,3,4,thelarge-Ngroundstatesonthe 0.18] and φ in the interval [0,2π], both with a uniform Instead, the idea is to iterate Eq. (13) within a constrained = VI. LARGE-N RESULTSONSQUARELATTICEAND squarelatticeareVCSstatesofthetypeshowninFigs.4and5. distribution.(2)Givenχ ,thepotentialµ ischosensothat set of χ and µ , so that Eq. (14) is always satisfied. To rr′ NUMErRICALGROUND-STATESEARCH rr′ r accomplish this, the algorithm proceeds as follows: (1) An 174441-13 In this section, we focus on the square lattice and, in initial χrr ischosen randomly. In our calculations, we chose ′ particular,onthecasek ! 5.ThediscussionofSec.VBabove χrr χrr eiφrr′,where χ waschosenintheinterval[0.03, ′ =| ′| | | establishesthat,fork 2,3,4,thelarge-Ngroundstatesonthe 0.18] and φ in the interval [0,2π], both with a uniform = squarelatticeareVCSstatesofthetypeshowninFigs.4and5. distribution.(2)Givenχ ,thepotentialµ ischosensothat rr r ′ 174441-13