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Octupolar order in the multiple spin exchange model on a triangular lattice Tsutomu Momoi,1 Philippe Sindzingre,2 and Nic Shannon3 1Condensed Matter Theory Laboratory, RIKEN, Wako, Saitama 351-0198, Japan 2Laboratoire de Physique Théorique de la Matière Condensée, UMR 7600 of CNRS, Université P. et M. Curie, case 121, 4 Place Jussieu, 75252 Paris Cedex, France 3H. H. Wills Physics Laboratory, University of Bristol, Tyndall Ave, BS8-1TL, UK 7 (Dated: 24 October2006) 0 Weshowhowagaplessspinliquidwithhiddenoctupolarorderarisesinappliedmagneticfield,in 0 amodelapplicabletothinfilmsof3Hewithcompetingferromagneticandantiferromagnetic(cyclic) 2 exchange interactions. Evidence is also presented for nematic — i.e. quadrupolar — correlations n bordering on ferromagnetism in theabsence of magnetic field. a J PACSnumbers: 75.10.Jm,75.40.Cx,67.80.Jd 5 2 The triangular lattice occupies a special place in the ] history of quantum magnetism. Although it is the sim- h/|J| l e plest lattice to exhibit geometrical frustration, this still Fully polarized CAF r- has highly nontrivial consequences — notably, that the 0.5 t z s Ising antiferromagnet (AF) on a triangular lattice re- t. mains disordered even in the limit of zero temperature. a ThisfactledAndersontosuggestthatthegroundstateof m TRIATIC thespin-1/2HeisenbergAFonatriangularlatticewould - be a gapless spin-liquid state with “resonating valence d FM 0 n bond” (RVB) character [1]. While this conjecture has 0.23 0.24 0.25 0.26 K/|J| o since been disproved [2], the idea of the RVB state con- c tinuestoexertastrongholdonthepopularimagination, [ andhasmotivatedasearchforspin-liquidstatesinawide FIG. 1: (Color online). (Left) Schematic phase diagram for thespin-1/2J-K multiplespinexchangemodelonatriangu- 2 variety of two-dimensional spin models [3]. v lar lattice in applied magnetic field, in the parameter range 5 Lately, attention has been refocused on the trian- where J <0 is FM and K ≥0 is AF. A ”triatic” phase with 7 gular lattice by the discovery of a number of exper- magnetic octupolar order appears in the shaded region be- 5 tween fully polarized and canted AF (CAF) phases. (Right) imental realizations of a quasi-2D spin-1/2 magnets 1 Schematicspinstructureofthistriaticorder,whichisinvari- with triangular coordination. These include the organic 1 ant underspin rotations of 2π/3 about thez axis. 6 (BEDTTTF)2Cu2(CN)3 [4], the transition metal chlo- 0 ride Cs CuCl [5], and solid films of 3He absorbed on 2 4 / graphite [6]. Of these, 3He films offer the most perfect relevant to 3He on graphite [7] t a realization of a spin-1/2 triangular lattice, but with an m interesting twist — the nearest–neighbor interaction is H=JXP2+K X(P4+P4−1)−hXSiz, (1) d- ferromagnetic (FM) and competes with an AF 4-spin hiji hijkli i n cyclic exchange [7]. As such, 3He on graphite is one of a where P and P (cyclically) permute spins on nearest o number of new quasi-twodimensionalsystems to exhibit 2 4 neighbor bonds ij and diamond plaquettes ijkl , re- c “frustrated ferromagnetism” [8, 9]. However it is unique h i h i : spectively. We include a coupling to magnetic field h, v inbothits purity,andinthe possibility oftuning the ra- and consider exclusively FM J <0 and AF K >0. i tio of the competing interactions continuously simply by X Our main finding is that a dynamical process unique varying the density of 3He atoms. An additional strong r to 4-spin exchange on the triangular lattice leads to the a motivationforunderstandingthissystemcomesfromthe formation of three magnon bound states which — un- fact that, for a rangeofdensities bordering onferromag- der applied magnetic field — condense, giving rise to netism, the magnetic ground state of 3He films is a fully a new, partially-polarized quantum phase in which ro- gapless spin-liquid [10, 11]. tational symmetry is spontaneously broken, but with- Recently, we proposed that the competition between out any long-range spin order perpendicular to the field FMandAFinteractionsinfrustratedspinsystemscould (Fig. 1). We identify the order parameter for this novel leadtoanewkindofgaplessspin-liquidstatewithhidden phase as a rank three tensor with octupolar character, multipolar order bordering on the FM phase [12, 13, 14] and confirm its existence through the exact diagonaliza- (for related, earlier work see [15]). In this Letter we ex- tion of finite-size clusters. tend these ideas to a triangular lattice spin-1/2 model To better understand the origins of this new quan- 2 h c1 0.6 Fully polarized state K K h/|J| 0.4 h c3 K K h 0.2 c2 0 0.24 K/|J| 0.26 FIG. 2: (Color online). The four-spin cyclic exchange operator can be used to propagate three flipped spins (•) in FIG. 3: (Color online). Critical fields hc1 (dotted line), hc2 a polarized FM background by tunneling between compact (dashedline),andhc3(solidline),whereone-,two-,andthree- triangular and straight line configurations. magnon instabilities destabilize the fully polarized state in theJ-K model with FM J <0. The lines of hc1 and hc2 are given from exact solutions. The variationally estimated line hc3 gives an exact lower bound for three-magnon instability line. tum phase, let us first reconsider the classical S → ∞ limit of Eq. (1). For K/J < 1/4, the ground state | | is a saturated FM. For 1/4 < K/J < 1 it is highly | | degenerate [16]. Freely propagating domain walls de- estimateofthethree–magnonboundstateislowerinen- stroy long range spin order, but nematic (i.e. quadrupo- ergy than any single magnon or two–magnon state for a lar) order persists [13]. This pathology is also evi- range of parameters 0.2347<K/J <0.2652. | | dent in the spin excitations for the quantum spin-1/2 To put these calculations in a more physical context, case. Within the FM phase, the one–magnon dispersion letusconsiderthecaseinwhichamagneticfieldhisused ω(q) = (J +4K)(cid:2)3 cos(qx) 2cos(qx)cos(√3qy/2)(cid:3) to fully polarize the system. This field is then reduced − − − is well defined. However exactly at the classical critical until a critical value h is reached, at which point the c point J = 4K it vanishes identically. magnetizationstartstochange. Wecancalculatecritical − Nondispersing — i.e. localized — magnons are also fields h and h at which one- and two-magnons start c1 c2 found at the border of a FM phase in the equivalent toBose-condense. Wecanalsoestimate—usingthetrial multiple spin exchange model on the square lattice. In wave function discussed above — the field h at which c3 this case, pairs offlipped spins can propagatecoherently thesystembecomesunstableagainstthecondensationof under the action of AF spin exchange, even where a three-magnon bound states. In the absence of any other single magnon is localized. Magnons therefore form ki- phase transition, the greatest of the three fields h , h c1 c2 netic energy driven bound states, which condense and andh willdetermine the firstinstability ofthe system, c3 give rise to a quadrupolar “bond–nematic” order [14]. and the nature of magnetic order near to saturation. The energy of two flipped spins in a FM background Performingthisanalysis,wefindthatthefirstinstabil- can also be calculated exactly for the multiple spin ex- ity for K/J & 0.27 occurs in the one–magnon channel, | | change model on a triangular lattice. We find that two- andisagainstathree-sublatticecantedAF,aspreviously magnon bounds states with total spin S = N/2 2 and foundinmeanfieldanalysis[16]. Howeverinthe param- − momenta q = (0,2π/√3),(π,π/√3),( π,π/√3) have eter range 0.2347 < K/J . 0.27, h > h h — { − } | | c3 c2 ≫ c1 a lower energy per spin than independent magnons for see Fig. 3. Since the estimate of h is variational(while c3 K/J >0.2349. the values of h and h are exact), it provides a strict | | c2 c1 However,thefrustratedgeometryofthetriangularlat- lower bound on h . c ticeintroducesanew,two-stepprocessbywhichagroup ExactdiagonalizationoffiniteclustersofuptoN =36 of three flipped spins can move together — as shown in spins confirms that the transition out of the saturated Fig.2. Thismotivates ustotry asimpletrialwavefunc- state is controlled by three–magnon bound states for tionforathreespinboundstate,withintherestrictedba- 0.24 . K/J . 0.28. Results for a 36–spin cluster with | | sisofstateswherethethreespinsformacompactstraight K/J = 0.25 are shown in Fig. 4. Jumps of ∆m = 3 | | | | line or triangle. Solving the Hamiltonian Eq. (1) in this are clearly visible in the magnetization process (inset) restricted Hilbert space, we obtain five eigenmodes for for fields close to saturation (H h ). The strong s c3 ≡ the three-magnon bound states. The lowest eigenvalue, AF coupling K ensure that interactions between three– E = 13J 50K √J2+4JK+28K2,correspondstoa magnon bound states are repulsive, and excitations con- − − − statewithmomentumq=(0,0),belongingtothe trivial taining more than three magnons therefore play no part irrep of the space group, i.e. even under the space ro- inthetransition. AtfiniteH ,thetransitionisofsecond s tations R , R and the reflection σ. This variational order,andcanbeunderstoodasaBose–Einsteinconden- 2π/3 π 3 whereRσ denotesarotationinspin–spacethroughangle θ θ about z axis. The three–magnon condensate breaks the remaining U(1) symmetry down to the symmetries of a triangle in the plane perpendicular to the applied field (Fig. 1). We therefore dub the resulting octupolar order “triatic”. We note that triatic order of this type does not require any breaking of translational symmetry — the condensed three-magnon bound states can move, as long as they remain phase coherent. Triatic order of varioustypeshaspreviouslybeendiscussedinthecontext of the Heisenberg AF on a kagomé lattice [17]. Period–3 structure in the energy spectrum persists down to low spin in all of the clusters studied. How- ever, new features appear in the structure of the spec- W trum for S .6 (see Fig. 4). These features are sensitive to cluster geometry, which suggests that different types K of order compete with octupolar order in the absence of magnetic field. While this makes interpretation of the spectra more difficult, it is nonetheless possible to place FIG. 4: Energy spectrum of the J-K model for N =36 spin strong constraints on possible ground states. cluster with J = −4 and K = 1, showing the oscillation in First,itispossibletoconcludethatanewgroundstate energiesofthelowestlyingstatesinspinsectorsintheperiod exists between the FM phase and the short-rangedRVB threefor large S. Theinset shows themagnetization process spin liquid phase with finite spin-gap ∆ J , which m/ms. Jumpsof|∆m|=3,correspondingtooctupolarorder, ∼ | | are clearly visible at large H.(Color online). is known to occur for K/J 0.5 [18]. For 0.235 . | | ≈ K/J .0.28 the ground state and low–lying excitations | | havedifferentsymmetriesfromtheRVBphase[20]. Fur- thermore, in this parameter range, the rapid decay of sation of three–magnon bound states. For H 0, this s → excitation energies with system size suggests a gapless gives way to a first order transition. ground state for h = 0, and thus a breaking of SU(2) Thesejumpscanbetracedtoasetofstateswhichlieat symmetry. the bottomofspinsectorsS =3n,belongingto the triv- Dipolar (Néel), quadrupolar (nematic) and octupolar ialirrepofthespacegroup. Eachjumpof∆m= 3cor- − (e.g. triatic) magnetic order all break SU(2) symmetry, responds to adding one more three–magnonbound state andeachoftheseformsoforderisassociatedwithaspe- tothesystem. Theconvexlocusofthesestatesasafunc- cific “Anderson tower” of low–lying states [14]. The An- tion of S implies that the transition at h is of second c3 dersontowercontainsstatesinallspinsectorsS forNéel order. Numericalandanalyticresultscloseto saturation order; only states with even S (period–2) for quadrupo- are therefore in perfect agreement. The condensation of lar order, and only states in sectors with S a multiple of three–magnon bound states will lead to a new form of three for pure octupolar order (period–3). quantum order. But what kind of order is it ? In view of the spectra, Néel order seems improbable. We consider first the situation in applied magnetic The number of low–lying states below the continuum in field, where full SU(2) spin symmetry is brokendown to each spin sector S is not compatible with what would U(1)rotationsaboutthez-axis. Inthiscasetheorderpa- be expected for dipolar order. Furthermore, a period–2 rameter is determined by the operator O† =S−S−S−, ijk i j k structure where the lowest lying states with odd S are whichcreatesathree–magnonboundstateinapolarized raised in energy relative to states with even S, is seen background. Both e O† and m O† are linear combi- R I for S . 6 in those N = 12 and N = 36 spin clusters nations of fully symmetrized rank-3 tensor operators whichpossessthefullsymmetryofthelattice,andinthe Oiαjβkγ = X SiaSjbSkc, (2) N =24clusterwiththelowestground-stateenergy. This period–2structure suggestsquadrupolarorderis present a,b,c∈P(α,β,γ) in the ground state. The period–3 structure, present at andthe resultingorderparametercorrespondstoamag- largerS,isstillclearlyvisibleforsmallS insomeN =24 netic octupole on a triangular plaquette ijk . clusters, but is hard to distinguish in the lowest–energy { } As expected, both the realand the imaginary parts of clusters. Moreover,the groundstates for the N =12,36 the order parameter transform with period 2π/3 clusters are not in the trivial irrep, which indicate that eO† = 4(Oxxx+Rσ [Oxxx]+Rσ [Oxxx]), the symmetry of the ground state differs from the pure R ijk 3 ijk 2π/3 ijk 4π/3 ijk octupolar order observed in magnetic field. mO† = 4(Oyyy +Rσ [Oyyy]+Rσ [Oyyy]), In fact, the symmetry and momenta of the low lying I ijk −3 ijk 2π/3 ijk 4π/3 ijk 4 .01 .01 exchange model on the triangular lattice, and that this state is therefore a strongcontenderto explainthe spin– .02 -.01 .03 .03 -.01 liquid seen in experiment. It is our pleasure to acknowledge stimulating discus- .01 .02 .02 .01 sions with P. Fazekas, T. Koretsune, K. Kubo, K. Penc andH.Tsunetsugu. ThisworkwassupportedbyMEXT Grants-in-Aid for Scientific Research (No. 17071011and .02 .02 No. 16GS0219), the Next Generation Super Computing Project, Nanoscience Program, MEXT, Japan, and EP- .01 .02 .02 .01 SRC grant (EP/C539974/1). 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Itseemsreasonable wavefunctioncontainsstateswithcrossedsinglet/triplet that a nematic phase (or octupolar phase) should inter- structurewhichisalsoseenintwo–spinboundstatesnear polatebetweenFMandRVBphasesinthe multiple spin to saturation [13].