Spin nematic order in multiple-spin exchange models on the triangular lattice
aa r X i v : . [ c ond - m a t . s t r- e l ] N ov Spin nematic order in multiple-spin exchange models on the triangular lattice
Tsutomu Momoi, Philippe Sindzingre, and Kenn Kubo Condensed Matter Theory Laboratory, RIKEN, Wako, Saitama 351-0198, Japan Laboratoire de Physique Th´eorique de la Mati`ere Condens´ee, UMR 7600 of CNRS,Universit´e P. et M. Curie, case 121, 4 Place Jussieu, 75252 Paris Cedex, France Department of Physics and Mathematics, Aoyama Gakuin University,5-10-1 Fuchinobe, Sagamihara, Kanagawa 229-8558, Japan (Dated: November 7, 2018)We figure out that the ground state of a multiple-spin exchange model applicable to thin filmsof solid He possesses an octahedral spin nematic order. In the presence of magnetic field, it isdeformed into an antiferro-quadrupolar order in the perpendicular spin plane, in which lattice Z rotational symmetry is also broken. Furthermore, this system shows a narrow magnetization plateauat half, m/m sat = 1 /
2, which resembles recent magnetization measurement [H. Nema et al. , Phys.Rev. Lett. , 075301 (2009)].
PACS numbers: 75.10.Jm, 75.40.Cx
Frustrated quantum antiferromagnets have long beena subject of active research [1], since Anderson[2] sug-gested that a spin-1/2 Heisenberg antiferromagnet onthe triangular lattice would have a gapless spin liq-uid ground state named as resonating-valence-bond(RVB) state. Recent experimental studies of quasi-two-dimensional compounds, such as solid He films ab-sorbed on graphite [3–5], the organic Mott insulator [6] κ -(BEDT-TTF) Cu (CN) and the transition metal chlo-ride Cs CuCl [7], have further prompted theoretical re-search of quantum spin liquid and competing exotic or-ders in triangular lattice antiferromagnets [8–14].Among these, solid He films offer a perfect realizationof a spin-1/2 triangular lattice. They contain a uniquecharacter in spin exchange interactions. The nearest-neighbor interaction is ferromagnetic (FM) and com-petes with antiferromagnetic (AF) multiple-spin cyclicexchange [15, 16]. As such, He on graphite belongs toa new class of quasi-two dimensional systems to exhibit“frustrated ferromagnetism” [17, 18]. It is also uniquein the possibility of tuning the ratio of the competinginteractions continuously by varying the density of Heatoms. An additional strong motivation for understand-ing this system comes from the fact that, for a range ofdensities bordering on ferromagnetism, spins in solid Hefilms exhibit anomalous double-peak structure in specificheat [4] with gapless excitations [5].The effective Hamiltonian for nuclear magnetism of He thin films is given by the S = 1 / H eff = J X h i,j i P + J X ✔ ✔ r rr r ( P + P − ) − J X ✔ ❚ r rr r r ( P + P − ) + J X ✔❚ ❚✔ rr rrr q r ( P + P − ) − h X i S zi , (1)where P n denotes the cyclic permutation operator of n spins. The summations in front of the permutation op-erators run over all minimal n spin clusters. The firstand the second summations, for example, run over allpairs of nearest neighbors and all four-spin diamond clus-ters, respectively. In solid He on graphite, the effectivetwo-spin coupling J is negative (FM) and the other cou-plings J n ( n = 4 , ,
6) are positive. Experimental esti-mates suggest that two-spin and four-spin exchange in-teractions are dominant, but five-spin and six-spin inter-actions are also not small [16]. We set J = − J /J = 2 throughout this letter.Magnetism induced by multiple-spin exchange on thetriangular lattice has been studied mostly in the J – J model [9–14], which contains two-spin and four-spin ex-changes. When FM J strongly competes with AF J ,it was found that, in a finite range bordering on ferro-magnetism, three magnon bound states become the moststable magnetic particles, giving rise to an octupolar or-dered phase in applied magnetic field [14]. However thenature of the ground state in the absence of magneticfield was not identified in this regime. When four-spinexchange J is dominant, exact diagonalization analysisconcluded a quantum disordered state with a large spingap [12, 13]. The strong J also supports a wide mag-netization plateau at half of saturation ( m/m sat = 1 / uuud structure [9, 11, 12].In this letter, we study S = 1 / (cid:11)(cid:68)(cid:12) h (cid:11)(cid:69)(cid:12) h =0 FIG. 1: Director configurations in spin nematic order in S =1 / J J (Γ, R π/3 =1) Γ K W (Γ , R =j, j ) (W) (Γ, R π/3 =−1) (K) d+id −wave FIG. 2: Magnon instabilities to the fully polarized stateat the saturation field. Parameters are set as J = − J = 2 J . The wave vectors in the Brillouin zone (Γ, K, W,defined in the inset) and some space symmetries are shown.FM denotes the stable ferromagnetic phase. triangular lattice containing up to six-spin exchange cou-plings [Eq. (1)], which is directly applicable to solid Hefilms. Inclusion of these multiple spin exchanges helpsus to identify that the ground state is an octahedralspin nematic state [Fig. 1(b)], which has both liquid-like character and antiferro-quadrupolar orders on bonds.In applied field, this spin state is continuously deformedinto a spin nematic state with antiferro-quadrupolar or-der [Fig. 1(a)], where lattice Z rotational symmetry isalso broken. This state is understood as condensation of d + id -wave two magnon bound states.The occurrence of this spin state may indeed be mosteasily understood from the channel of magnon instabil-ity at saturation field. As shown in Fig. 2, magnonpairs form stable bound states in a wide range near theFM phase boundary. As previously seen [19–21], con- densation of bound magnon pairs leads to a spin ne-matic state [22], which has quadrupolar order in the per-pendicular component to the applied field and breaksthe U(1) symmetry of the Hamiltonian. In the tri-angular lattice J – J model (i.e. J = J = 0), in-stabilities given by two magnon bound states [20] andthree magnon bound states [14] are competing. Inclu-sion of five and six-spin exchange interaction removesthis competition, making the d x − y + id xy -wave twomagnon instability most dominant in a wide parame-ter range near FM phase boundary, as shown in Fig. 2.The d + id -wave magnon pairing operators are given by Q + = P i ( S − i S − i + e + jS − i S − i + e + j S − i S − i + e − e ) andits complex conjugate Q − , where j = exp( i π/
3) and e = (1 , e = (1 / , √ / He also belongs to this instability region.The d + id -wave bound magnon pairs Q ± | FP i , where | FP i denotes the fully polarized state, are degeneratewith chiral degrees of freedom as the eigenvalues to thespace rotation by 2 π/ R π/ = j, j . In such a case,strong repulsion between different species of bosons caninduce density imbalance between the condensates of twospecies, breaking additional chiral Z symmetry.To identify the nature of symmetry breaking, we haveperformed numerical exact diagonalization of Eq. (1) forclusters up to N = 36 spins. For parameter sets in thetwo-magnon instability regime, the special stability oflowest energy states in even S sector at large magneti-zation (high S states), as seen in Fig. 4(b), signals theformation of bound magnon pairs with repulsive interac-tions, which leads to jumps by | ∆ S | = 2 in the magneti-zation process as shown Fig. 3 and points to spin nematicordering in the perpendicular spins [21].From the analysis of irreducible representations (ir-reps) and quasi-degeneracy of these low-lying states, weconclude to non-chiral spin nematic ordering. A quasithree-fold degeneracy is observed in low-lying states inthe even spin S sector (for S < m s − Z symmetry is broken.The order parameter of this non-chiral nematic stateis identified as O U (1) = Q + − Q − = i √ X i ( S − i S − i + e − S − i S − i + e ) , (2)by expansion of the coherent state exp[ − λ O U (1) ] | FP i intoirreps, which gives the numerically observed irreps. Therequired low-lying states for this order are listed in Ta-ble I(a). Note that space rotational Z symmetry is bro-ken in this order parameter, corresponding to the choiceof two bonds out of three. A schematic figure of nematic-directors for this state is depicted in Fig. 1(a). The =0.8 h J =J /2=0.15 h J=−2, J =0.75J =J /2=0.15 m/m s (a) (b) uuud state FIG. 3: Magnetization process m/m s of the MSE model for N = 36 spin cluster. The exchange parameters are J = − J = J = 0 .
15 with (a) J = 0 .
75 and (b) J = 0 .
8. Jumpsof | ∆ m | = 2, corresponding to spin nematic order, are clearlyvisible. In (b), a narrow plateau structure appears at m/m s =1 / h > U (1) symmetric case m m s − m s − n m s − n + 1)Irreps Γ Γ , Γ Γ , Γ (b) h = 0, SU (2) symmetric case S Γ Γ Γ , Γ Γ Γ , Γ , Γ TABLE I: Irreps of low lying states in each total spin S (mag-netization m ) sector, needed for antiferro-quadrupolar order-ing, (a) in applied field and (b) at zero field. The symbols aredefined as Γ ≡ ( R π/ = 1 , R π = 1 , σ = 1), Γ ≡ ( R π/ =1 , R π = 1 , σ = − ≡ ( R π/ = j, j , R π = 1), and allof them have the wave vector k = (0 , ground state manifold has spin U (1) /Z symmetry andspace Z symmetry.In the absence of magnetic field, SU (2) symmetry isrestored and the signature of SU (2) symmetry breakingis the existence of an “Anderson tower” of quasidegener-ate joint states (QDJS) belonging to different spin sectorswhich form the N = ∞ ground state with energies scaledas E QDJS ∼ S ( S +1) N and are well separated (at finite N )from the lowest magnon excitations (scaled as ∼ / √ N ).The energy spectrum around S = 0 is shown in Fig. 4(a)for the parameter set J = − J = 0 . J = J = 0. Asimilar spectrum structure is also found for finite J , forexample in J = − J = 0 . J = J = 0 .
15. The spin S = 1 sector has high energy, which excludes conven-tional spin ordering. Comparing irreps of the low energystates with those expected for various possible spin struc-tures, we found that these states perfectly match with theAnderson tower of a spin nematic state with three sets oforthogonal directors, whose order parameter is given by O SU (2) = X i { Q xxi ( e ) + Q yyi ( e ) + Q zzi ( e − e ) } , (3)where Q ααi ( r ) = S αi S αi + r − h S i · S i + r i . SS(S+1) E ne r g y Γ Γ Γ others(a) (b) FIG. 4: (Color online) Energy spectrum of the MSE modelfor N = 36 spin cluster with (a) J = − J = 0 . J = J = 0 near the singlet ( S = 0) and (b) J = − J = 1, J = J = 0 . S = 18). The symbolsΓ , Γ , and Γ are defined in Table I. This order parameter has octahedral spin symmetry O , if one combines spin transformations with the spacegroup C v of the triangular lattice. To obtain irreps spe-cific to this spin state, one may decompose the irreps D S of SU (2) into irreps of O in each spin S sector [23] andremove irreps odd under π rotation ( C ), i.e. the threedimensional irreps F and F , which should be absent asthe nematic order parameter is invariant under C (sincedirectors are headless). This allows projection only on A , A , and E . For N = 36 spins, A , A , and E maponto Γ , Γ , and Γ of C v , respectively (Γ µ are definedin Table I). The irreps for N = 36 in each spin S sectorare shown in Table I(b), which agrees well with symme-tries of QDJSs found in Fig. 4(a). The mapping of A and A to Γ and Γ may interchange depending on thenumber of spins N . (Note that, only the relative sym-metries between QDJSs are important for constructingsymmetry broken states). With this assignment, irrepsaround S = 0 smoothly connect with those of low-lyingstates determined by the two-magnon instability at sat-uration field. While the dynamics of the order param-eter of a non-collinear N´eel is that of a symmetric top,which has 2 S + 1 states in each total spin S and mag-netization m sector, here the ground state manifold hasspin SU (2) /D symmetry and so the Anderson tower has S/ S − /
2] states for even (odd) S sector, thesame as a symmetric top which are even under π rota-tions around the principal axes.In applied field, two orthogonal nematic directors be-come perpendicular to the field and the other doesn’thave any order in the perpendicular plane, leading to thespace Z symmetry breaking. The QDJSs around S = 0smoothly merge with the series of low-lying states at high uuud non−plateau J J plateauuuud plateaunon−plateauFM FIG. 5: Phase diagram of the MSE model in m/m sat = 1 / J = − J = 2 J .Spin nematic phase spreads over most of the“non-plateau”region. magnetization. Thus the octahedral spin nematic stateat zero field is continuously deformed into the spin ne-matic state under the field.The order parameter O SU (2) has the same spin symme-try as the antiferro-quadrupolar (AFQ) state discussed inthe triangular lattice S = 1 bilinear-biquadratic model[24, 25]. The irreps of QDJSs for the S = 1 AFQstate, obtained from the decomposition of the wave func-tion [26], are equivalent to the Anderson towers of the S = 1 / S = 1 / S = 1AFQ state. In S = 1 / S = 1 degrees offreedom are formed on bonds and moving around frombond to bond. The ground state is translationally in-variant at any field, i.e., it doesn’t show any spin densitywave, which is also different from the S = 1 system [25].Thus, the S = 1 / S = 1 AFQ stateis formed by the product of the quadrupolar momentslocalized on sites, having three sublattice structure withthe wave vector k = (4 π/ , S = 0 was well detected in a wide parameterrange J . . d + id -wave two-magnoninstability region at saturation field slightly extendingto the smaller J regime. In most of the region, the spinnematic state exists from zero magnetization to the satu-ration. Near J = J = 0 regime, three magnon instabil-ity induces an octupolar order under magnetic field [14],but, at very low magnetization, the octahedral spin ne-matic order overcomes it, as shown in Fig. 4(a). For the J > . m/m s = 1 / uuud state inthe strong J regime [9, 11, 12]. Recent magnetiza-tion measurement in 2D solid He observed a narrowplateau structure at m/m s = 1 / m/m s = 1 / uuud phase is easily de-tected by the four-fold quasi degenerate low-energy statesand a large energy gap above them. As shown in Fig. 5,the plateau phase spreads close to the FM phase bound-ary. Near its edge, the magnetization process has a verynarrow plateau at m/m s = 1 / He.It is our pleasure to acknowledge stimulating discus-sions with Hiroshi Fukuyama, H. Ishimoto, M. Morishita,H. Nema, K. Penc, N. Shannon, and R. Shindou. Numer-ical calculations were conducted on RICC in RIKEN andat IDRIS. This work was supported by KAKENHI No.17071011, No. 22014016, and No. 23540397. [1] G. Misguich and C. Lhuillier, in
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