an investigation of the quantum j 1 - j 2 - j 3 model on the honeycomb lattice

Eur. Phys. J. B 20, 241–254 (2001) THE EUROPEANPHYSICAL JOURNAL Bc©

EDP SciencesSocieta Italiana di FisicaSpringer-Verlag 2001

An investigation of the quantum J1-J2-J3 modelon the honeycomb lattice

J.B. Fouet1,2, P. Sindzingre2,a, and C. Lhuillier2

1 Laboratoire de Physique Theorique et Modelisation, 5 mail Gay-Lussac Neuville sur Oise, 95031 Cergy-Pontoise, France2 Laboratoire de Physique Theorique des Liquidesb, Universite P. et M. Curie, case 121, 4 Place Jussieu, 75252 Paris Cedex,

France

Received 17 November 2000 and Received in final form 21 January 2001

Abstract. We have investigated the quantum J1-J2-J3 model on the honeycomb lattice with exact diag-onalizations and linear spin-wave calculations for selected values of J2/J1, J3/J1 and antiferromagnetic(J1 > 0) or ferromagnetic (J1 < 0) nearest neighbor interactions. We found a variety of quantum effects:“order by disorder” selection of a Neel ordered ground-state, good candidates for non-classical ground-states with dimer long range order or spin-liquid like. The purely antiferromagnetic Heisenberg model isconfirmed to be Neel ordered. Comparing these results with those observed on the square and triangularlattices, we enumerate some conjectures on the nature of the quantum phases in the isotropic models.

PACS. 71.10.Fd Lattice fermion models (Hubbar model, etc.) – 75.10.Jm Quantized spin models – 75.40.-sCritical-point effects, specific heats, short-range order – 75.50.Ee Antiferromagnetics

1 Introduction

Frustrated quantum antiferromagnetic (AF) spin systemson low dimension (D) lattices have attracted a great dealof interest in recent years. Quantum fluctuations, largestfor small values of the spin S of the magnetic ions, lowD and small coordination of the lattice, are expectedto lead to novel magnetic behaviors. Their effects havebeen predominantly seen in 1D. They have been inves-tigated on a few 2D systems. The most studied modelsare the AF Heisenberg model on the triangular [1,2] orKagome lattice [3] which are geometrically frustrated sys-tems, the J1-J2 model on the square lattice [4–6] wherefrustration is introduced by 2nd neighbor interaction, theJ1-J2 model [7–9] and the multi-spin exchange model(MSE) [10], on the triangular lattice.

Less studied [11–14], spin models on the honeycomblattice deserve attention due to the special properties ofthe lattice and because there are experimental realiza-tions. A first feature of the lattice is that, like the squarelattice, it is not geometrically frustrated for AF nearestneighbor interactions but has lower coordination. Thusquantum fluctuations are expected to be larger than forthe square lattice. For this reason the spin-1/2 Heisen-berg antiferromagnet on the honeycomb lattice, has beenstudied theoretically by various methods [11–14] which allpredicted that Neel long range order (LRO) subsists butwith an order parameter smaller than for the square lat-tice case. This also motivated a Schwinger-boson study of

a e-mail: [email protected] UMR 7600 of CNRS

the effects of frustrating second neighbor interactions inthe J1-J2 model [15].

A major incentive to study frustrated magnets on thehoneycomb lattice is the availability of experimental datain the family of compounds BaM2(XO4)2 (M= Co, Ni; X=P, As) obtained, some years ago, by Regnault and Rossat-Mignod [16]. The magnetic ions M have small spins (itis supposed to be S = 1/2 for the Co oxide and S =1 for Ni), disposed in weakly coupled layers where theysit on a honeycomb lattice. The simplest model relevantto these quasi-2D compounds is a J1-J2-J3 model on ahoneycomb lattice with first, second and third neighborinteractions and either on site if (S = 1) or XXZ (ifS = 1/2) anisotropy.

So far the J1-J2-J3 model was only investigated withinfirst order linear spin-wave theory (LSW) [16,17], and toour knowledge the renormalization of the order parame-ter by quantum fluctuations has not been calculated evenin this simplest approach. The experimental results moti-vated us to do this calculation in the large S limit and thenattack the S = 1/2 problem with exact diagonalizations(ED).

In this paper, as a first step, we consider the caseof purely isotropic interactions. The Hamiltonian of themodel reads:

H = J1

∑〈i,j〉1

Si · Sj + J2

∑〈i,k〉2

Si · Sk + J3

∑〈i,k〉3

Si · Sk

(1)

where the first, second and third sums run on the first,second and third neighbor pairs of spins, respectively

242 The European Physical Journal B

Fig. 1. The honeycomb lattice. The full and empty circlesdifferentiate the two sublattices. t1 and t2 are the two vectorsof the triangular Bravais lattice. Top right: arrows show theinteractions between a site and its 1st, 2nd and 3rd neighbors.

(see Fig. 1). The coupling constants Ji can be either AF(Ji > 0) or ferromagnetic (Ji < 0). Depending of thevalues of the parameters Ji, this model displays variousclassical ground-states: a collinear AF ground-state, twodegenerate manifolds of planar helimagnetic ground-stateswith four or eight sublattices and a ferromagnetic ground-state. The classical phase diagrams of the isotropic andanisotropic XXZ models display only minor differences.In particular the classical ground-states are the same forthe parameters believed to be relevant to BaCo2(AsO4)2.In this paper we concentrate on the quantum effects in theisotropic model. The study of the quantumXXZ J1-J2-J3

model with parameters appropriate to BaCo2(AsO4)2 willbe presented in a separate paper [18].

We investigated quantum effects for selected values ofthe Ji chosen to give a broad picture of the quantum ef-fects encountered in the model. The restriction to isotropicinteractions limits the number of parameters and enablesus to separate the effects of anisotropy. In addition to theJ1-J2 models on the square and triangular lattices, thepresent model may be compared with the J1-J2-J3 modelon the square lattice which has a similar variety of clas-sical ground-states and to which a few studies have beendevoted [19–24].

This paper is divided into five parts. In Section 2, werecall the classical phase diagram of the model, obtainedby Rastelli et al. [17], and identify the degeneracies of theclassical ground-states not considered by these authors,we also discuss the stability of the first order spin-waveapproximation for these different phases. The ED resultsfor the case of antiferromagnetic and ferromagnetic near-est neighbor coupling are presented in Section 3 and 4respectively. In Section 5 we draw conclusions, and enu-merate some conjectures relative to the appearance of thevarious generic two-dimensional spin-liquids. We describe

-1 1

-1

1

0.5 0

Fig. 2. Classical phase diagram for antiferromagnetic nearestneighbor interactions. In the T = 0 classical approximationregions II and IV have in fact a degenerate manifold of non-planar ground-states. Thermal fluctuations or quantum fluc-tuations do select the collinear configurations shown in thisfigure.

in an appendix the various technical features specific toour present ED calculations on different samples.

2 Classical phase diagram and semi-classicaldeviations

2.1 Planar ground-state configurations

The classical model was studied by Rastelli et al. [17].They searched for planar or uniformly canted configura-tions minimizing the classical energy Ecl. The former werefound energetically favored over the latter. They representspiral configurations, characterized by a wave-vector Q.The classical spin (of length S) sitting at cell R of thetriangular Bravais lattice on sublattice α is given by:

SR,α = S(cos (Q ·R + φα) u + sin (Q ·R + φα) v) (2)

where u and v are two orthogonal unit vectors definingthe plane of the spins, φα can be chosen to be zero on onesublattice and will be denoted as φ on the other. The setof spiral wave-vectors Q minimizing the classical energywill be denoted as {Q}.

The phase diagram of planar solutions of this type isreproduced in Figure 2 (J1 > 0) and Figure 3 (J1 < 0).There is a mapping between the two phase diagrams: thetransformations J1 → −J1, J3 → −J3 and Si → −Si fori ∈ on the black triangular sublattice of Figure 1 leavethe Hamiltonian unchanged, and maps the ground-statefor J1 > 0 on that for J1 < 0.

J.B. Fouet et al.: An investigation of the quantum J1-J2-J3 model on the honeycomb lattice 243

-1 1

-1

1

-0.5 0

Fig. 3. Classical phase diagram for ferromagnetic nearestneighbor interactions. Same comments on regions II and IVas in Figure 2.

There are six regions in each phase diagram: fourcollinear phases (I, II, IV, VI) and two spiral ones (III,V). In the collinear regions I and VI, the wave-vector ofthe magnetic order is Q = 0, whereas in regions II andIV, Q = Ki, where Ki are the three inequivalent middlesof edges of the Brillouin zone (see Fig. 6). The phases are:In I, φ = π (0) if J1 > 0 (J1 < 0), in VI, φ = 0 (π) ifJ1 > 0 (J1 < 0), in II φ = π (0) if J1 > 0 (J1 < 0), in IVφ = 0 (π) if J1 > 0 (J1 < 0).

The separation lines I-III, I-V, II-III, IV-V representcontinuous transitions; the others are first order phasetransitions.

2.2 Non planar ground-states manifolds

Ansatz (Eq. (2)) is usually generic to find all allowed clas-sical ground-states [26,27]. It assumes that, up to the triv-ial degeneracy associated to a global spin rotation, theground-state is unique or exhibits at most a discrete de-generacy. It is valid if a linear combination of the differentQ modes of the same set {Q}, can be excluded as violat-ing the constraint |SR,α| = S on every site. Exceptionsoccur for special sets {Q}, in particular if Q is half or onefourth of a reciprocal lattice vector G [26]. This is the casein region II and IV where Q = G/2 = Ki (see Fig. 6).Here, the linear combination of three Ki solutions

SR,α =3∑i=1

S cos (Ki ·R + φα) ui (3)

with unnormalized ui, is submitted to the constraintsui ·uj = δi,j and

∑u2i = 1. The ground-state is a two di-

mensional manifold continuously connecting the three Ki

Fig. 4. Top: four-sublattice classical ground-state in regionIV in Figure 2 for antiferromagnetic first neighbor coupling(J1 > 0). Bottom: the collinear solutions with the three possi-ble arrangements (in this case, classical spins in sublattices Aand B are antiparallel).

solutions (there are nine degrees of freedom for choosingthe three ui, minus three for global rotations of the spinsand four constraints). This gives birth to the non planarground-states manifolds described below.

In regions IV for J1 > 0 or II for J1 < 0 ( φ = 0), theground-state manifold is the set of four-sublattice orderedsolutions such as SA + SB + SC + SD = 0. This could beseen directly from the expression of the classical energy ofthese configurations:

Ecl =2N

(J1 + 2J2) (SA + SB + SC + SD)2

− 2N

(J1 + 2J2 − 3J3)(S2A + S2

B + S2C + S2

D

). (4)

In this equation, N represents the total number of spinsof the sample. The generic four-sublattice configurationsare shown in Figure 4, as well as the three collinearconfigurations, which appear as special cases of it, withSA = SB = −SC = −SD and the two other com-binations of parallel spins. The situation is reminiscentof the J1-J2 model on the triangular lattice [7–9] with1/8 < J2/J1 < 1.

In regions IV for J1 < 0 or in II for J1 > 0 (φ = π), theground-state manifold is the set of eight-sublattice solu-tions shown in Figure 5 where sublattices labelled by thesame letter are paired (partner sublattice is over-headedby a bar) and

Sα = −Sα (5)

for α ∈ {A,B,C,D} and

SA + SB + SC + SD = 0. (6)


Fig. 5. Same as Figure 4 but for ferromagnetic first neighborcoupling (J1 < 0), region IV of Figure 3.

This minimizes the classical energy:

Ecl =8NJ2(SA+SB +SC +SD + SA + SB + SC + SD)2

+8N

(J1 − 2J2) (SA + SB + SC + SD)

× (SA + SB + SC + SD)

+4N

(3J3 − J1) [(SA + SA)2 + (SB + SB)2

+ (SC + SC)2 + (SD + SD)2]

− 4N

(−J1 + 2J2 + 3J3) (S2A + S2

B + S2C + S2

D

+ S2A + S2

B + S2C + S2

D). (7)

It is highly probable that thermal fluctuations will sta-bilize the collinear solutions as they do in similar situa-tions on the square and triangular lattice. We will showbelow that quantum fluctuations indeed do it.

Continuous degeneracy of the ground-state also occurswhen Q = G/4 in III and in V but since this happens onlyon lines (for instance if J2 = 0.5 in III) and not in full re-gions, we shall skip their description which is not essentialto our present goal. The transition line III-V between thetwo spiral regions is very special. It has an infinite degen-eracy of spiral ground-states corresponding to:

cos(Q ·t1)+cos(Q ·t2)+cos(Q ·(t1−t2)) = 1/8J22 − 3/2.

(8)

The lines of Q solutions of equation (8) are shown inFigure 6 for J2 = 0.2, 0.4, 0.5.

Fig. 6. Brillouin zone of the triangular Bravais lattice. Lightsolid, dashed and dotted lines are the solutions of equation (8),for J2 = 0.2, 0.4, 0.5 respectively.

Fig. 7. LSW values (triangles) of the order parameter m† forthe purely AF Heisenberg model (J1 = 1, J2 = 0,J3 = 0), asa function of N1/2. The asymptotic behavior ∼ N1/2 (dashedline fitted to the LSW values) is only reached for quite largesamples, much larger than the sizes studied in exact diagonal-izations.

2.3 Stability of the quasi-classical phase diagramin the large S limit: LSW results

The renormalization of the order parameter m† in the firstorder spin-wave approximation is already large in the puremodel (J1 = 1, J2 = 0, J3 = 0): m† ≈ 0.48 i.e. a valuereduced to ≈ 48% of its classical value (Fig. 7).

For J1 > 0, the interplay of quantum fluctuations andfrustration quickly destroys Neel LRO: m† goes to zerofor J2 ≈ 0.1 or J3 ≈ −0.1. The helimagnetic LRO of zoneV disappears: near the J3 = 0 axis, this is mainly dueto the large classical degeneracy of the ground-state andnear the J2 = 0 axis, the main cause is the vanishing of thespin-wave velocity at the point J1 = 1, J2 = 0, J3 = 0.25.The zone V being very small, we conclude that the Neelhelimagnetic ground-state does not survive in region V forantiferromagnetic J1.


For J1 < 0 the ground-state is ferromagnetic in zone Iof Figure 3. The ferromagnetic state is an exact eigenstateof the Hamiltonian, quantum fluctuations do not destroyit. However, the classical degeneracy on the boundary be-tween zones III and V implies a whole branch of soft modesand a disappearance of the helimagnetic LRO, on and nearthis line (as in the AF J1 case). The main difference withthe AF first neighbor case is the persistence of Neel orderin zone V near the J1 = −1, J2 = 0, J3 = 0.25 point andin the vicinity of the J2 = 0 axis.

The nature of the quantum spin-1/2 phase for small J2

and J3 appears an open problem that we will now attackwith the help of exact diagonalizations (ED).

Exact diagonalizations were performed on samples ofN = 18, 24, 26, 28, 30, 32 sites with appropriate bound-ary conditions (see Appendix). The technical problems en-countered in such approaches have already been studiedin detail in previous references [1,10] and will not be de-scribed in detail in this paper. We briefly discuss in theAppendix the different characteristics of the studied sam-ples with respect to the present model. We will now pro-ceed to the analysis of the ED results.

3 J1 > 0: AF nearest neighbor interactions

3.1 The purely Heisenberg model,and phase I of the quantum model

The AF Heisenberg point (J1 = 1, J2 = J3 = 0) was previ-ously investigated by exact diagonalization calculations onsmall samples [11], Monte Carlo simulations [12], series ex-pansions around the Ising limit [13], spin-wave theory upto second order [14], and Schwinger-boson mean field the-ory [15]. All concluded that the quantum system exhibitsNeel LRO but with a large reduction of the order parame-ter due to quantum fluctuations (the Monte Carlo result,rather close to the spin-wave value, is m† = 0.44± 0.06).Our ED results for sizes up to N = 32 are consistent withthis conclusion. The approach is however different. In an-tiferromagnets with linear Goldstone modes, the scalinglaw for m† is an expansion in 1/N1/2 [29,30]. For thesizes encountered in ED, the asymptotic 1/N1/2 law isnever reached [1] and the extrapolation of the order pa-rameter to the thermodynamic limit remains uncertain.A qualitative idea of the finite size scaling of m† can beobtained from the LSW results: they show that the asymp-totic 1/N1/2 regime cannot be expected for sizes smallerthan N ∼ 400 (see Fig. 7).

Nevertheless confirmation of Neel LRO can be ob-tained thanks to characteristic features of the spectrumitself which have more favorable scaling behaviors:

– For a given sample size, the lowest eigenlevels in eachsector of the total spin S evolve as E0(S,N) ∝ S(S+1)up to S ∼

√N , as shown in Figure 8. They are the

eigenlevels associated with the collective dynamics ofthe order parameter, the so-called Quasi DegenerateJoint States (QDJS) [1], which can be described by

Fig. 8. AF Heisenberg model, scaling of the QDJS with S andN for N = 18, 24, 26, 28, 32.

the effective Hamiltonian:

Hcoll.dyn. =12∆(N) S 2 (9)

where ∆(N) is the finite-size difference in total energybetween the absolute ground-state and the first tripletexcitation.

– The number of QDJS and their symmetries are thoseexpected for the projections of the classical Neel orderon the irreducible representations (IR) of SU(2) ⊗ G(G: lattice symmetry group): There is just one statefor each S value since there is just one way to coupletwo spins of magnitude N/4 in a total spin S. Thesestates are invariant under lattice translations (theirwave-vector is k = 0), under 2π/3 rotations around thecenter of an hexagon, they are even (odd) under inver-sion with respect to the center of an hexagon for evenS and N = 4p (respectively odd S and N = 4p+ 2).

– In the thermodynamic limit, the QDJS collapse on thesinglet ground-state as 1/N (Fig. 10b). The QDJS re-main distinct from the softest magnon excitation whichcollapse to the ground-state as 1/N1/2. The QDJS andthe softest magnon are shown for the N = 32 samplein Figure 9.

– The asymptotic ∼ 1/N behavior of ∆(N) is not yetreached for our largest sizes as seen in Figure 10b. Thenext order term in the 1/N1/2 expansion reads [29,30]:

∆(N) =1

4χN(1− β c

ρ√N

) +O(1N2

) (10)

where χ is the spin susceptibility, c is the spin-wavevelocity, ρ the spin stiffness and β is a number of orderone. A fit of the spin-gaps to this scaling law is shownin Figure 10b. The importance of the term in 1/N3/2 is


Fig. 9. Low energy part of the AF Heisenberg spectrum forN = 32: eigenenergies are plotted versus eigenvalues of S2. Fulltriangles represent QDJS; empty squares describe the softestmagnon. The dashed-line and the dotted-line are guides to theeye for the QDJS and the softest magnon respectively.

Fig. 10. AF Heisenberg model, (a) energy per site e0: the lineis a fit to the leading term of equation (11). (b) spin-gap: thefull line is a fit to equation (10), the dashed line if a linear fitin 1/N .

not unexpected since this term is ∝ c/ρ and quantumfluctuations which reduce ρ with respect to its classicalvalue are strong (as already shown by the reduction ofthe order parameter).

Fig. 11. Low energy spectrum for J1 = 1, J2 = 0, J3 = −2and N = 32. Full triangles represent states of the family {2E},empty triangles represent states belonging to {4E} and notto {2E}, and empty squares represent the softest magnon. Alltheses states have the symmetries predicted in Tables 2 and 3.

– The energy per site e0(N) = E0(0, N)/N of theground-state scales as:

e0(N) = e∞ −α′

N3/2(1− β′ c

ρ√N

) +O(1

N5/2) (11)

for a Neel order. Figure 10a shows that the leadingterm of order O(1/N3/2) is enough to describe the sizeeffects in this range.

A rapid analysis of the quantum phase diagram in re-gion I does not reveal new phases, but both LSW cal-culations and ED confirm that a weak antiferromagneticsecond or ferromagnetic third neighbor coupling are suf-ficient to kill LRO: the boundary between phase I andphases III and V is shifted upwards by quantum effects.

3.2 Region IV

In region IV the classical model presents a degeneratemanifold of four-sublattice ordered ground-states. Thefinite-size spectra clearly show that this degeneracy islifted by quantum fluctuations which favor a collinear two-sublattice order (see Fig. 11): the low lying levels of thesespectra, below the magnon excitations, exhibit a largefamily {4E} of QDJS associated to four-sublattice solu-tions. At the bottom of this family there appears a lineof eigenlevels with definite symmetries: these levels con-stitute the family {2E} of QDJS states associated to acollinear symmetry breaking.

The situation, seen here, is very similar to the one pre-viously studied for the J1-J2 antiferromagnet on the trian-gular lattice [9]. The expected number of states 4NS , 2NS


Table 1. Character table of the permutation group S4. Firstline indicates classes of permutations. Second line gives an el-ement of the space symmetry class corresponding to the classof permutation. These space symmetries are: t the one step

translation (A → B), R2π/3 (R′2π/3) the three-fold rotationaround a site of the D (B)-sublattice, and σ the axial symme-try keeping invariant C and D. Nel is the number of elementsin each class.

S4 I (A,B)(C,D) (A,B,C) (A,B) (A,B,C,D)

G I t R2π/3 σ R′2π/3σNel 1 3 8 6 6

Γ1 1 1 1 1 1

Γ2 1 1 1 −1 −1

Γ3 2 2 −1 0 0

Γ4 3 −1 0 1 −1

Γ5 3 −1 0 −1 1

Table 2. Number of occurrences nΓi(S) of each irreduciblerepresentation Γi in {4E} as a function of the total spin S.

N = 32

S 0 1 2 3 4 5 6 7 8

nΓ1(S) 2 0 3 1 4 2 4 2 4

nΓ2(S) 1 0 2 1 2 1 2 1 1

nΓ3(S) 3 0 5 2 6 3 6 3 5

nΓ4(S) 0 4 4 7 6 8 7 8 6

nΓ5(S) 0 4 3 6 5 7 5 6 4

in {4E} and {2E} and their space symmetries are easilydetermined for each value of the total spin S. The eigen-states of {4E} can be labelled by the five irreducible rep-resentations Γi (i = 1, 5) of S4 (permutation group of fourelements). The mapping between the space group opera-tions on the four-sublattice solutions and permutations ofS4 is described in Table 1, together with the character ta-ble of S4. The four-sublattice order is invariant in two-foldrotations (Rπ): thus the eigenstates of {4E} belong to thetrivial representation of C2. Since it is also invariant undera two-step translation of the Bravais lattice they have ei-ther a wave-vector k = 0 or a wave-vector Ki. Γ1, Γ2 andΓ3 belong to the k = 0 subspace, whereas Γ4 and Γ5 be-long to the subspace {Ki}. Γ1 and Γ2 are invariant underthe three-fold rotations R2π/3 of C3, whereas Γ3 is associ-ated with the two-dimensional representation of C3v. Thenumber of replicas of Γi that should appear for each Svalue can be computed as in reference [9]. The results areshown in Table 2 for the N = 32 sample. Analysis of thetwo-sublattice order can be done similarly: the collinearsolution has a three-fold degeneracy, the set of eigenstates{2E} maps on Z3. It is characterized by the IR Γ1, Γ3 andΓ4. The number of replicates are shown in Table 3 for theN = 32 sample.

The “order out of disorder” phenomenon [28] is clearlyseen for J1 = 1, J2 = 0, J3 = −2. In Figure 11 we show

Table 3. Number of occurrences nΓi(S) of each irreduciblerepresentation Γi in {2E} as a function of the total spin S.

N = 32

S 0 1 2 3 4 5 6 7 8

nΓ1(S) 1 0 1 0 1 0 1 0 1

nΓ3(S) 1 0 1 0 1 0 1 0 1

nΓ4(S) 0 1 0 1 0 1 0 1 0

Fig. 12. J1 = 1, J2 = 0, J3 = −2, energy gaps measuredfrom the absolute ground-state versus 1/N for N = 24, 32.Full squares connected by the dashed line: gap to the lowestenergy state in the triplet sector (it belongs to {2E}). Fulltriangles: gap to the 2 nd singlet state of symmetry Γ3 (itbelongs to {2E}). Open triangles: gap to the 3 rd singlet stateof symmetry Γ3 (this state belongs to {4E} and not to {2E}).

the lower part of the N = 32 spectrum at this point.The lowest eigenstates in each S sector are the states of{2E}, describing collinear order. Further support for thisassumption is given by the finite size effects of the energygaps. As shown in Figure 12, a plot of the spin-gap ofthe N = 24, 32 samples versus 1/N is consistent with avanishing value for N → ∞. On the other hand the gapbetween the two states Γ1 and Γ3 of {2E} of the S = 0sector tends to close when the size goes to infinity, whereasthe gaps between the levels of {2E} and the other levelsof {4E} increase with N .

We have investigated the scaling behavior of thespectra at some other points of region IV not too closeto the classical boundaries and found essentially the samebehavior and a selection of collinear LRO by quantumfluctuations.

Closer to the boundary between region IV and V,the separation between the {4E} states and the magnonstates decreases. This is an indication of a softening of the


Fig. 13. Spin-spin correlations as a function of distance forthe pure Heisenberg model on N = 24 (triangles) and N=32samples (squares) and for J1 = 1, J2 = 0.3 on N = 24 (three-legged star) and N = 32 samples (four-legged star).

magnons and the neighborhood of a 2nd order phase tran-sition towards another phase. The behavior of the spin-gaps at J1 = 1, J2 = 0.5, J3 = −0.5 and J1 = 1, J2 =0, J3 = −1, similar to Figure 12, nevertheless indicatesthat these points of the quantum phase diagram are stillin the collinear phase IV.

Various studies of the spectra of the N = 18, 26, 28samples under suitable boundary conditions confirm theseresults for the quantum phase IV, and indicate that thequantum boundary between phase IV and V is probablyslightly shifted down relatively to the classical boundaryshown in Figure 2.

Results of ED calculations (not shown) in region II forJ1 < 0 (which has the same classical manifold of degen-erate ground-states as in IV for J1 > 0) suggest a similarselection of the collinear solution there too. In conclusionup to a slight motion of the boundaries , the semi classicalbehavior in regions I, II, IV and VI, is not qualitativelyaffected by the strong quantum fluctuations of the spins1/2.

3.3 Quantum phases between I and IV

The intermediate phases between the two collinear Neelphases cover region V and part of region III. In this partof the quantum phase diagram, SU(2) symmetry is unbro-ken, and there is a gap to triplet excitations: these phasesonly support short range order in the spin-spin correla-tions (see Fig. 13). Our LSW and ED calculations indi-cate that this quantum region likely extends in regions Iand IV1. In this work we study region V, region III close

1 LSW calculations predict non vanishing order parametersfor the classical spiral solutions inside III away from the bound-aries but a vanishing order parameter in the whole region V.

Fig. 14. Low energy spectrum for J1 = 1, J2 = 0.4, J3 = 0 andN = 32. Full triangle: ground state; empty triangle: first singletexcited state (these two states have a wave vector k = 0);empty square: second singlet excited state; full square: firsttriplet state.

to I, and the transition line III-V with ED calculationsusing TBC on N = 18, 24 samples (see Appendix) andPBC on the N = 24 and N = 32 samples. A thoroughsearch of the ED spectra, sweeping the twist angles at thesample boundaries, did not yield evidence of incommen-surate helical LRO, neither with the wave-vectors of theclassical solutions nor at other wave-vectors. In all casesno tower of QDJS was found. The ED results corrobo-rate the conclusion of LSW calculations that the classicalspiral solutions are destabilized by quantum fluctuations.This seems a rather general statement in systems wherethe quantum fluctuations are strong enough [20,21].

Is this quantum phase a quantum disordered one? Toanswer this question we performed extended ED calcula-tions on N = 24, 32 samples on different points of thetransition line III-V where the classical model has an infi-nite set of spiral ground-states, and the LSW calculationdiverges. Along this line, we found evidence of two differ-ent phases both with a gap.

Let us begin by the phase around J2 = 0.4: this pointis very close to the point where the energy versus J2 isthe largest and may be considered as a point of maximumfrustration. The spectrum of the N = 32 sample is shownin Figure 14. This spectrum differs from the spectra of thecollinear ordered system in IV:

– the lowest states are not IR of {2E}and features associated with Neel LRO are missing:

– The lowest eigen-energies for each S value do not in-crease as S(S + 1) with S

– The lowest states in each S sector are not separatedfrom the others as the QDJS are separated from the


Fig. 15. J1 = 1, J2 = 0.4, energy gaps measured from theabsolute ground-state versus 1/N for N = 20, 24, 28, 32. Fullsquares: spin gap, i.e. gap to the first triplet excitation; opentriangles: gap to the first singlet excitation (k = 0, IR Γ3);open squares: gap to the second singlet excitation. The dashedline is the estimated gap at the thermodynamic limit.

magnons, instead there is a dense continuum of statesin each S sector except the S = 0 one.

– Furthermore a plot of the spin-gap of the N = 20 to32 samples versus 1/N , displayed in Figure 15, showsthat the scaling law characteristic of a Neel orderedsystem is not obeyed and indicates a large spin-gap∼ 0.3 for N →∞.

Most likely however the system is not fully disorderedbut exhibits dimer LRO (see Fig. 16). The dimer oper-ator on a pair of sites (i, j) is di,j = (1− Pi,j) /2 wherePi,j = 2(Si · Sj + 1/4) is the spin permutation opera-tor. This projector is greater (less) than 0.25 when thespin-spin correlation is negative (positive), equal to 1 ona singlet and to 0 on a triplet. For J1 = 1, J2 = 0.4,on the N = 32 sample, the first neighbor correlation is〈dk,l〉 = 0.4899. On the symmetry breaking Spin-Peierlsstate (pure product of ordered dimers), the average valueof the dimer operator is 1 on the bonds where there is adimer, and 1/4 on the other bonds. As the exact eigenstatedoes not break C3 symmetry the number 0.4899 shouldbe compared with the result obtained on the symmet-ric superposition of the three Spin-Peierls states alignedalong the three main directions of the lattice: in this sym-metrized Spin-Peierls state this correlation is dΨsym

k,l = 0.5.The average value of the dimer operator in the exactground-state is thus very close to the Spin-Peierls value.

The dimer-dimer correlation between a reference bond(i, j) and the bond (k, l) is D(i,j),(k,l) = 〈di,jdk,l〉 −〈di,j〉〈dk,l〉. As in reference [10], we normalized D(i,j),(k,l)

by its maximum value which is achieved when the twobonds are completely correlated. We thus measured dimer

12

3456

78

91011

1213

16

1718

19202122

2324

25

272829

3132

78

1920

26

34

78

1415

2324

30

3456

Fig. 16. J1 = 1, J2 = 0.4, singlet-singlet correlations p(1,2),(k,l)

between the reference bond (1, 2) and bonds (k, l) in theground-state of the N = 32 sample. The numbers above bonds(k, l) are the values of p(1,2),(k,l) truncated to the two first sig-nificant digits. Full (dashed) lines indicate positive (negative)values of p(1,2),(k,l), The width of the lines is proportional to themagnitude of |p(1,2),(k,l)|. The number above the bond (1, 2) is〈d1,2〉 (see text).

correlations by

p(i,j),(k,l) =D(i,j),(k,l)

〈dk,l〉 − 〈di,j〉〈dk,l〉(12)

=〈di,jdk,l〉 − 〈di,j〉〈dk,l〉〈di,j〉 (1− 〈dk,l〉)

·

If p(i,j),(k,l) = 1 the presence of a dimer on bond (i, j) im-plies the existence of a dimer on bond (k, l); if p(i,j),(k,l) =0 there is an absence of correlations between singlets onbonds (i, j)) and (k, l). If p(i,j),(k,l) is negative, a singleton bond (i, j) induces a tendency towards ferromagneticcorrelation on bond (k, l).

The correlation pattern for dimers on first neighborbonds is displayed in Figure 16. The calculation of dimer-dimer correlations on the state ΨS.P

sym gives p(i,j),(k,l) =+0.5 if (i, j) and (k, l) are parallel and p(i,j),(k,l) = −0.25if (i, j) and (k, l) are non parallel bonds. The exact dimer-dimer correlations are not too far from these values anddecay very slowly with distance. This is in favor of acolumnar LRO of dimers with a C3 symmetry breaking,previously proposed by Einarsson et al. [32].

Moreover the degeneracy of the ground-state for N →∞ points to the same conclusion: Figure 15 indicates thatthe gap between the (Γ1 and Γ3, S = 0) lowest states,which are both k = 0 states, closes for N →∞, while the


12 34

56 8

910 1112

1314

1516

171819 20

212223

1314

171819 20

34

7

1112

17 20

24

910 1112

1516

910

1819

Fig. 17. Singlet-singlet correlations for J1 = 1, J2 = 0.3 andN = 24 (same legend as in Fig. 16).

gap between these states and the upper levels increaseswith the size. Γ1 is non degenerate and Γ3 twice degener-ate: this allows the building of the three columnar dimerpatterns with a C3 symmetry breaking and no transla-tion breaking. In this picture the finite size ground-stateis the symmetric combination of these three states. Thisdegeneracy corresponds to a true symmetry breaking witha local non zero order parameter (dimer LRO): this is aValence-Bond Crystal (VBC).

The honeycomb lattice could a priori accommodatea different kind of VBC with alternation of hexagonswith three dimers and hexagons without dimers: this pat-tern breaks both C3 and translational symmetry. In theirlarge N approach, Read and Sachdev [31] found thatthis structure might be the ground-state. In the rangeJ2 ≈ 0.3 − 0.35, we find a short range structure roughlyreminiscent of this arrangement. In fact the short rangedimer-dimer correlations are even more symmetric thanin this VBC crystal and would be more compatible witha crystal of hexagon-plaquettes in a symmetric S = 0state [33] (Fig. 17). For example the correlation (1-2)(7-6)should be negative and equal to −0.25 in the Read andSachdev VBC state whereas the exact value on the N = 24sample is +0.05, much closer to the expected value of 0.01in the pure hexagon-plaquette VBC. In fact in our SU(2)model ll correlations decrease noticeably with distance(Figs. 13, 17) and the pattern does not seem to propagateat large distances. The ground state in this range of pa-rameter is probably a RVB spin liquid. This conclusion isqualitatively substantiated by the study of the energy gapsto the ground-state: plausibly none of them goes to zeroat the thermodynamic limit, which would be consistentwith the RVB hypothesis. Unfortunately the finite-sizeeffects are rather chaotic: the ground-state energy of the

N = 18 and N = 30 samples (samples on which the Readand Sachdev dimer pattern is not frustrated) are largerthan the ground-state energy of the N = 26 and N = 32samples, which frustrate it. This is probably related to thefact that the N = 18 and N = 30 samples do not allowthe system to take full advantage of the second neigh-bor antiferromagnetic coupling, whereas the N = 26 andN = 32 samples do. But the building of singlets on secondneighbor bonds tends to destroy VBC patterns and favora RVB ground-state. All these arguments point in favor ofan RVB phase at this coupling parameter: unfortunatelythe sizes that can be studied do not allow a quantitativedetermination of the gaps.

The quantum AF J1-J2 model on the square lat-tice close to the point of maximum frustration exhibitsdimer [5] or plaquette [4] LRO; the same kind of conclusionhas been drawn for the J1-J3 model [24], and for the MSEmodel on the square lattice [25]. These phases share qual-itative properties with the phase identified for J2 = 0.4:in each cases a collinear LRO is destabilized by frustra-tion giving birth through a 2nd-order phase transitionto a massive phase with dimer LRO. Such VBC phasesappear in many models on bipartite lattices: Rokhsarand Kivelson [34] in the Quantum Dimer approach (QD),Dombre and Kotliar [35] for the Hubbard model, Readand Sachdev [21] in the SU(N) approach of the Heisen-berg model found VBC phases. These phases, as the firstone described here (for J2 = 0.4), have a gap for all exci-tations, a discrete degeneracy of the ground-state, expo-nential decrease of the two points spin-spin correlationsbut LRO in higher correlation functions; they have onlyconfined spinons.

Up to now we only know few spin-1/2 models exhibit-ing true RVB phases with a clear-cut gap: the MSE modelon the triangular lattice [10] and the Quantum Dimermodel on the triangular lattice [38]. More work is neededto know if the excitations of these different RVB states aresimilar and in particular if they sustain deconfined spinonsexcitations.

4 J1 < 0: ferromagnetic nearest neighborinteractions

As already underlined above in Section 3.2 the classicalcollinear phases (F or AF) observed for large J2 and J3

are likely to survive to quantum fluctuations. We thusfocus our study on the region of maximum frustration,0 < |J2|, |J3| < 0.5, corresponding to region V and part ofregion III of Figure 3. In this situation LSW calculationspredict a non vanishing order parameter of the spiral solu-tions for values of J2 and J3 not too close to the transitionlines. However extensive ED calculations performed on theN = 18, 24 samples with twisted boundary conditions donot yield any evidence of spiral LRO.

We thus studied samples up to N = 32 spins to in-vestigate the nature of the ground-state for a few sets ofparameters. The most extensive calculations were done atthe point J2 = 0.25, J3 = 0 on the transition line III-V


Fig. 18. Low energy spectrum for J1 = −1, J2 = 0.25, J3 = 0and N = 32. Full triangle: ground state; empty triangle: firstsinglet excited state (these two states have a wave vector k =0); empty square: second singlet excited state; full square: firsttriplet state.

which may be considered as highly frustrated (for this J2

value the ground-state energy is close to its maximum,and in the LSW approach quantum fluctuations destroyLRO). Strong indications that the model has a RVB spin-liquid ground-sate, were found at this point:

– The spectra do not exhibit a tower of QDJS as shownin Figure 18 for N = 32, and E0(S) clearly does notevolve as S(S + 1) with S.

– A plot of the spin-gap versus 1/N , shown in Figure 19,indicates that the spin-gap is small but finite whenN →∞2.

But contrary to the case with positive J1 and J2 = 0.4:

– The correlations display a strong short range order butplausibly no LRO. The short range pattern is original:the first neighbor spin-spin correlation is ferromagnetic

2 In view of Figure 19, one may object to our extrapola-tion to N → ∞ on two numerical samples. In fact our con-clusion is supported by examination of both the gap and theenergy per bond of the samples with sizes 18, 24, 28, 32with various boundary conditions (available on request at:[email protected]). This study shows that the varia-tions of these quantities with the size is very small and mainlydue to the frustration of the short range antiferromagnetic or-der between third neighbors due the boundary conditions (seebelow) and not to the cut-off in the low-energy long wave-length quantum fluctuations. Notice that the energy per bondon the two non frustrating sizes 24 and 32 does not increasewith the system size but decreases by a very small amount(∼ 10−3). We thus conclude that we are in the cross-overregime for both sizes 24 and 32 and the extrapolation of thespin gap in Figure 19 is reasonable.

Fig. 19. J1 = −1, J2 = 0.25, energy gaps measured from theabsolute ground-state vs. 1/N for N = 24, 32. Black squaresshow the spin-gap. Open triangles (squares) the gap to the first(second) excitation in the singlet sector.

12

3456

78

91011

1213

16

1718

19202122

2324

25

272829

3132

78

1920

26

34

78

1415

2324

30

3456

Fig. 20. J1 = −1, J2 = 0.25, first neighbor triplet-triplet cor-relations on the N = 32 sample. The spin-spin correlation onthe reference bond (1, 2) is ferromagnetic: the number abovethis bond measures the projection of the two-spin state of theexact ground-state on the pure triplet state.

(〈Si · Sj〉 = 0.10), the second (third) neighbor spin-spin correlations are antiferromagnetic (〈Si · Sj〉n.n.=−0.13, 〈Si · Sj〉n.n.n.= −0.25), but no long range pat-tern does emerge from this picture. The dimer-dimercorrelations equally show a strong short range pat-tern and apparently no LRO. Figure 20 representsfirst-neighbor dimer-dimer correlations: they are muchweaker than in the AF case (remark that triplet-triplet


1

2

3

4

5

6

7

89

10

11

12

13

14

15

1617

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

12

13

21

22

29

7

8

12

13

14

15

19

20

23

26

27

30

3

4

5

7

14

15

1617

23

24

30

31

32

4

5

6

1725

31

32

22

29

Fig. 21. J1 = −1, J2 = 0.25, second neighbor singlet-singletcorrelations on the N = 32 sample. The reference bond is(1, 10): the spin-spin correlation on this bond is antiferromag-netic, the number above the bond measures the projection ofthe two spin-state of the exact ground-state on a pure sin-glet i.e.: 〈d1,10〉; the value of this observable in a symmetrizedwave function of products of second-neighbor singlets is 0.375.Non-parallel singlet-singlet correlations have been omitted forclarity, all of them are very small and negative.

correlations are equal to singlet-singlet ones and com-pare with Fig. 16). Figure 21 displays second-neighbordimer-dimer correlations which also decrease with dis-tance. The third-neighbor dimer-dimer correlations de-crease even quicker. The strength of the short rangecorrelations explains the finite size results on smallfrustrating sizes (see footnote 2).

– The ground-state is probably unique in the thermody-namic limit. The first two singlet excitations are showntogether with the first triplet excitation in Figure 19.The spin-gap and the ground-state energy per spin dis-play a very small sensitivity to the size for N = 24, 32.The gap to the third excitation seems more sensitive tothe size but this might due to the fact that the differentsizes do not accommodate the same wave vectors. Inview of the results it seems probable that the singletexcitations will not collapse to the absolute ground-state in the thermodynamic limit.

We thus conjecture that the quantum ground-state ofthis system does not break any symmetries of the Hamilto-nian or of the lattice: it is a quantum spin-liquid where allexcitations are gapped. Results obtained for the N = 24and 32 samples for J2 = 0.5 also indicate a finite spin-gapwhen N → ∞. This suggests that there is a spin-liquidphase in a finite range of parameters.

Such a quantum massive phase, without LRO, is highlyreminiscent of the spin-liquid phase found in the MSEmodel on the triangular lattice [10]. Curiously enough itappears in the two cases in the vicinity of a ferromagneticphase destabilized by antiferromagnetic couplings. Thereis a difference in the degeneracy of the ground-state in thetwo cases: whereas the ground-state on the honeycomb lat-tice is unique, it has a 4-fold degeneracy on the triangularlattice. This is easily understood as the honeycomb lat-tice is not a Bravais lattice and has two spin-1/2 per unitcell. Thus the uniqueness of the ground-state in this lattercase does not contradict the Lieb-Schultz-Matthis-Affleckconjecture [36], or the topological approach of Read andChakraborty [37].

5 Conclusions and conjectures

This study of the spin-1/2 J1 − J2 − J3 model on thehoneycomb lattice has brought the following new results:

– For small frustrations J2/J1 or J3/J1 less than ∼ 0.15or larger than 1, the system remains essentially clas-sical: when various kinds of LRO are possible, quan-tum fluctuations, as well as thermal fluctuations in theclassical case, select the LRO with the most symmetricorder parameter amongst the various possibilities.

– The classical symmetry between the phase diagram forferromagnetic J1 and the phase diagram for antiferro-magnetic J1 discussed in Section 2.1 is destroyed byquantum fluctuations.

– For the largest frustrations these models exhibitgapped phases.

– For an antiferromagnetic first neighbor coupling, a Va-lence Bond Crystal phase has been clearly evidencedaround J2/J1 = 0.4.

– For an intermediate frustration J2/J1 = 0.3, an RVBspin-liquid appears between the Neel ordered phaseand the VBC phase.

– For a ferromagnetic first neighbor coupling, the presentresults favor the hypothesis of a RVB spin-liquid phasefor a large range of parameters. No VBC has beenfound in that case.

This study of the spin-1/2 J1 − J2 − J3 model on thehoneycomb lattice, when compared to similar approachesof SU(2) Hamiltonians leads us to formulate some conjec-tures on the generic behavior of such models on differentlattices.

– In 2D the pure S=1/2 Heisenberg model is Neel or-dered on any bipartite lattices with coordination num-ber ≥ 3. It is disordered on the triangular-basedKagome lattice which has a coordination number equalto 4.

– Non-coplanar classical ground-states are not robustagainst quantum fluctuations in the isotropic models.

– Neel order or ferromagnetism disappears around thepoints of maximum classical frustration giving birthto phases with spin-gap and short range spin-spin cor-relations.


Fig. 22. The N = 18, 24, 26, 28, 30, 32 samples.

– Disappearance of a ferromagnetic phase due to anti-ferromagnetic frustrations leads generically to a spin-liquid phase, with short range correlations in all ob-servables and a gap to all excitations.

– Disappearance of a collinear antiferromagnetic phasemight lead to a VBC phase either directly (J1-J2 modelon the square lattice), or through an intermediate RVBphase (this study). The spin-liquid phase observed bySantoro et al in the spin-orbital model [39] might berather similar to the RVB phase described here.For completion we might add:

– Disappearance of a non-collinear phase (3-sublatticeNeel phase) takes place through a phase with a spingap but a continuum in the singlet sector [40,41].

Computations were performed at The Centre de Calcul pourla Recherche de l’Universite Pierre et Marie Curie and at theInstitut de Developpement des Recherches en Informatique Sci-entifique of CNRS under contract 990076.

Appendix: Special properties of the studiedsamples, and boundary conditions

Most ED calculations were performed as in references[1,10] on systems ofN = 18, 24, 26, 28, 30, 32 sites shownin Figure 22 (additional calculations were performed onN = 16, 20 samples). The N = 18, 24, 32 samples havethe full point group symmetry of the lattice, whereas theN = 26 sample misses axial but still has rotational C3

symmetry, the N = 28, 30 have neither.With periodic boundary conditions (PBC), all the

samples are of course compatible with the Q = 0 orderin region I. In region IV, however only the N = 24 and 32

samples have the full symmetry of the classical order. TheN = 28 sample is compatible with one collinear solutionbut frustrates the two others as well as the non coplanarsolutions. The other samples are frustrating but can allowa collinear order if twisted boundary conditions (TBC) areused. This is the case for the N = 18 sample if a twist ofπ is applied along t1 or t2.

To search for spiral order, we used TBC and sweep thewhole range [0, 2π] of twist angles φ1,2 in the t1 and t2

directions. These specific boundary conditions are definedas:

S(Ri + tj) = eiφjSz(Ri)S(Ri)e−iφjSz(Ri). (A.1)

This allows one to look for boundary conditions whichwould not frustrate helical ground-states. This approachwas found effective for the Heisenberg model on the tri-angular lattice to deal with samples frustrating the three-sublattice Neel order [1]. For such samples the ground-state energy was found to reach its minimum for the twistswhich release the frustration: at that point the spectra re-cover the characteristic features of Neel order.

A VBC with the pattern of Read and Sachdev [31](considered in Sect. 3.3) fits in the N = 30 sample butnot on a N = 32 sample.

References

1. B. Bernu, C. Lhuillier, L. Pierre, Phys. Rev. Lett. 69, 2590(1992); B. Bernu, P. Lecheminant, C. Lhuillier, L. Pierre,Phys. Rev. B 50, 10048 (1994).

2. L. Capriotti, A.E. Trumper, S. Sorella, Phys. Rev. Lett.82, 3899 (1999).

3. P. Sindzingre et al., Phys. Rev. Lett. 84, 2953 (2000) andreferences therein.

4. L. Capriotti, S. Sorella, Phys. Rev. Lett. 84, 3173 (2000)and references therein.

5. R.R.P. Singh et al., Phys. Rev. B 60, 7278 (1999).6. V.N. Kotov, J. Oitmaa, O. Sushkov, Z. Weihong, Phys.

Rev. B 60, 14613 (1999) and cond-mat/9912228.

7. A. Chubukov, T. Jolicoeur, Phys. Rev. B 46, 11137 (1992).8. S.E. Korshunov, Phys. Rev. B 47, 6165 (1993).9. P. Lecheminant, B. Bernu, C. Lhuillier, L. Pierre, Phys.

Rev. B 52, 9162 (1995).10. G. Misguich, B. Bernu, C. Lhuillier, C. Waldtmann, Phys.

Rev. Lett. 81, 1098 (1998); G. Misguich, C. Lhuillier, B.Bernu, C. Waldtmann, Phys. Rev. B 60, 1064 (1999).

11. J. Oitmaa, D.D. Betts, Can. J. Phys. 56, 897 (1978).12. J.D. Reger, J.A. Riera, A.P. Young, J. Phys. Cond. Matt.

1, 1855 (1989).13. J. Oitmaa, C.J. Hamer, Z. Weihong, Phys. Rev. B 45, 9834

(1992).14. Z. Weihong, J. Oitmaa, C.J. Hamer, Phys. Rev. B 44,

11689 (1991).15. A. Mattsson, P. Frojdh, T. Einarson, Phys. Rev. B 49,

3397 (1994).16. L.P. Regnault, J. Rossat-Mignod in Phase transitions in

quasi two-dimensional planar magnets, edited by L.J. DeJongh (Kluwer Academic Publishers, 1990), pp. 271–320.


17. E. Rastelli, A. Tassi, L. Reatto, Physica B 97, 1 (1979).18. J.-B. Fouet, P. Sindzingre, C. Lhuillier (in preparation).19. F. Figueirido, A. Karlhede, S. Kivelson, S. Sondhi, M.

Rocek, D.S. Rokhsar, Phys. Rev. B 41, 4619 (1990).20. A. Moreo, E. Dagotto, T. Jolicoeur, J. Riera, Phys. Rev.

B 42, 6283 (1990).21. N. Read, S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991);

Phys. Rev. B 42, 6283 (1990).22. H.A. Ceccatto, C.J. Gazza, A.E. Trumper, Phys. Rev. B

47, 12329 (1993).23. F. Ferrer, Phys. Rev. B 47, 8769 (1993).24. P.W. Leung, N. Lam, Phys. Rev. B 53, 2213 (1996).25. A. Chubukov, E. Gagliano, C. Balseiro, Phys. Rev. B 45,

7889 (1992).26. J. Villain, J. Phys. France 38, 385 (1977).27. E.F. Bertaut, in Spin Arrangements and Crystal Structure,

Domains and Micromagnets, edited by T. Rado, H. Suhl,Magnetism Vol. III (Academic Press, New York, 1963),p. 149.

28. J. Villain, R. Bidaux, J.P. Carton, R. Conte, J. Phys.France 41, 1263 (1980).

29. H. Neuberger, T. Ziman, Phys. Rev. B 39, 2608 (1989);D.S. Fisher, Phys. Rev. B 39, 11783 (1989).

30. P. Hasenfrantz, F. Niedermayer, Z. Phys. 92, 91 (1993).31. N. Read, S. Sachdev, Phys. Rev. B 42, 4568 (1990).32. T. Einarsson, H. Johannesson, Phys. Rev. B 43, 5867

(1991).33. We thank R. Moessner for attracting our attention to this

possibility and giving us information on work in progress.34. D.S. Rokhsar, S.A. Kivelson, Phys. Rev. Lett. 61, 2376

(1988).35. T. Dombre, G. Kotliar, Phys. Rev. B 39, 855 (1989).36. I. Affleck, Phys. Rev. B 37, 5186 (1988).37. N. Read, B. Chakraborty, Phys. Rev. B 40, 7133 (1989).38. R. Moessner, S.L. Sondhi, Phys. Rev. Lett. 86, 1665

(2001), cond-mat/0007378.39. G. Santoro et al., Phys. Rev. Lett. 83, 3065 (1999).40. P. Lecheminant, B. Bernu, C. Lhuillier, L. Pierre, P.

Sindzingre, Phys. Rev. B 56, 2521 (1997).41. W. LiMing, G. Misguich, P. Sindzingre, C. Lhuillier, Phys.

Rev. B 62, 6372 (2000).

an investigation of the quantum j 1 - j 2 - j 3 model on the honeycomb lattice

Documents