g. misguich et al- spin liquid in the multiple-spin exchange model on the triangular lattice: ^3-he...

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VOLUME 81, NUMBER 5 P H Y S I C A L R E V I E W L E T T E R S 3 AUGUST 1998

Spin Liquid in the Multiple-Spin Exchange Model on the Triangular Lattice: 3He on Graphite

G. Misguich,1 B. Bernu,1 C. Lhuillier,1 and C. Waldtmann21Laboratoire de Physique Thorique des Liquides, UMR 7600 of CNRS, Universit P. et M. Curie,

case 121, 4 Place Jussieu, 75252 Paris Cedex, France2 Institut fr Theoretische Physik, Universitt Hannover, D-30167 Hannover, Germany

(Received 6 March 1998)

Using exact diagonalizations, we investigate the T 0 phase diagram of the multiple-spin exchange(MSE) model on the triangular lattice, we find a transition separating a ferromagnetic phase froma nonmagnetic gapped spin liquid phase. Systems far enough from the ferromagnetic transitionhave a metamagnetic behavior with magnetization plateaus at mmsat 0 and 12. The MSE hasbeen proposed to describe solid 3He films adsorbed onto graphite, thus we compute the MSE heatcapacity for parameters in the low density range of the 2nd layer and find a double-peak structure.[S0031-9007(98)06770-2]

PACS numbers: 75.10.Jm, 75.50.Ee, 75.40.s, 75.70.Ak

An increasing number of experiments on 3He filmshave enforced the triangular lattice multiple-spin ex-change (MSE) picture of the solid second layer ([1,2], andreferences therein). Following Thouless [3,4], the mag-

netic properties of a 2-dimensional quantum crystal aredescribed in spin space by the effective Hamiltonian:

H 2X

P

21sgnPJPP , (1)

where P is any permutation operator of the spins ofthe lattice, and JP equals one half of the (positive) tun-neling frequency associated to the exchange process P.Whereas exchange of an even number of spins favors an-tiferromagnetism, odd processes are ferromagnetic. Onthe triangular lattice with spin 12, 2- and 3-body ex-change reduce to a Heisenberg Hamiltonian with an ef-fective Jeff2 J2 2 2J3. In 2-dimensional

3He, becauseof stoichiometric hindrance, the 3-body process is much

more efficient than the 2-body one, and the effective cou-pling constant is ferromagnetic. However, it was sus-pected since a long time that n-particle terms with n . 3could not be ignored [1,5,6]. Very recently, a thoroughanalysis of susceptibility and heat capacity measurementsenabled Roger and the Grenoble group [7] to establish thedensity dependence of the Js and the importance of the4-spin exchange J4 in a large density range of the 2ndlayer: from the 2nd layer solidification [antiferromagnetic(AF) solid] to the third layer promotion [ferromagnetic(F) solid]. This is in accordance with recent path inte-gral Monte Carlo calculations which estimate Jeff2 J4 22 6 1, J

5J

4 1 6 0.5 and J

6J

4 1 6 0.5 [8].

As first suggested by Roger in 1990 [9], 4-spin exchangestrongly frustrates the system. Momoi, Kubo, and Nikihave recently studied the Jeff2 -J4 model in the classical andsemiclassical (spin-wave) limits and found numerous or-dered phases: a ferromagnetic, a 4-sublattice ferrimagneticphase, 3- and 4-sublattice AF phases, and a chiral orderedphase [10,11]. Our SU(2) Schwinger-Boson analysis ofthe Jeff2 -J4-J5 Hamiltonian also pointed to a rich phasediagram with Nel as well as helicoidal phases [12].

In the present work, exact diagonalizations results showthat none of these T 0 long range ordered (LRO) AFphases survive spin-12 quantum fluctuations. Instead, inthe AF region, the system is a quantum spin liquid (QSL)

with short range spin-spin correlations, as suggested byIshida et al. [13] and Kubo et al. [11].

We truncate Eq. (1) to the following simplest cyclicexchange patterns:

H Jeff2

XP2 1 J4

XP4 1 P

214

2 J5X

P5 1 P215

1 J6

XP6 1 P

216

. (2)

The spin-12 permutation operators can be rewritten withusual spin operators (Pauli matrices): Pij 2Si .Sj 112. P1234 1 H.c. and P12345 1 H.c. (P123456 1 H.c.)are polynoms of degree two (three) in Si .Sj [12,14].The quantum phase diagram of this model is studiedthrough the analysis of the finite size scaling of the lowenergy spectra of samples subjected to various boundaryconditions (twisted or periodic with different shapes).

Ferromagnetic-antiferromagnetic transition.We firstlook at the T 0 ferromagnetic-antiferromagnetic tran-sition line of Eq. (2). This line (see Fig. 1) is determinedfrom about 50 spectra (N 19) in the Jn space, with mostpoints near the transition. For all cases, the ground stateis either an S 0 or an S N2 state. This excludesthe possibility of a uuud phase found in the classical cal-culations of Kubo et al. [11], which is ferrimagnetic andhas a total spin S N4. The line where Jx , the 1T

2

coefficient of the susceptibility, vanishes stands roughlyparallel to the T 0 F-AF line, inside the AF region.The density dependence of the Jns proposed by Rogeret al. [7] for the second layer (see crosses in Fig. 1) leadsus to conclude that a T 0 transition to ferromagnetismoccurs at r2 6.8 6 0.3 nm

22 whereas Jx is zero atr2 6.5 nm

22 [7]. No Long Range Order.The nature of the nonmag-

netic or antiferromagnetic (AF) phase is a more challeng-ing question. We first look for signatures of Nel long

1098 0031-9007 98 81(5)1098(4)$15.00 1998 The American Physical Society


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FIG. 1. Phase diagram for Jeff2 , 0. The solid line is the T

0 transition line between ferromagnetic and antiferromagneticphases. The crosses are Roger et al.s results (7) for four

2nd layer densities (nm22): ra2 6.5 (J6J4 0.7), rb2 7.0,r

c2 7.65, and r

d2 7.8. For b, c, and d J6J4 0.4 (the

size of the crosses may underestimate the uncertainties in theparameters). Two of the black triangles indicate the sets ofcoupling parameters corresponding to Figs. 2, 3, and 4. Theupper one is the point close to the frontier mentioned inthe text.

range order (NLRO). NLRO is characterized on finitesystems by very stringent spectral properties [15]: (i)The symmetry breakings associated to the order parame-ter are embodied in a family of Na low lying lev-els collapsing to the ground state in the thermodynamic

limit (a is the number of magnetic sublattices). Theselevels with definite space and SU(2) symmetries and dy-namical properties should appear directly below the firstmagnon excitations. (ii) The finite-size scalings of theselevels are known. In particular, the ground-state energyper site ES 0,NN has corrections scaling as N232,the DS 1 spin gap ES 1,N 2 ES 0,N goesto zero as N21. None of these prescriptions is obeyedby the MSE spectra in the S 0 ground state regiondisplayed Fig. 1, whatever the twisted or shape boundaryconditions may be. Thus, we exclude any commensurateor noncommensurate NLRO.

Jeff

22 J

4model.We have analyzed in detail the

Jeff2 22, J4 1 model for nearly all possible systemsfrom N 6 to 30 and for N 36. We noticed twodifferent scaling behaviors: samples with N multiple of4 or 6 have a low ground-state energy increasing with Nwhereas others samples have a high ground-state energydecreasing with N. The energies of both families mergefor N0 40. Our interpretation is the following: N0 isa crossover size above which the system is not anymoresensitive to boundary conditions and this is the signature

of a finite length scale j p

N0 in the ground-statewave function. This is supported by two facts: a fastdecay of the spin-spin correlations with distance and anonvanishing spin gap D in the thermodynamic limit.DS 1 gaps are plotted in Fig. 2: The two familiesof samples (squares for N multiple of 4 or 6 and starsfor others) have gaps of the order of 1 for N . 24.An estimation of the N ` gap is possible using the

strong correlation between EN and D, the result isD 1.1 6 0.5 (details will be given elsewhere).

These data point to a quantum spin liquid state witha gap of the order of 1 for J

eff2 22 and J4 1.

This gap and the sensitivity to the geometrical shapesand boundary conditions of the small samples suggest avalence-bond picture of the ground state and of the firsttriplet excitation.

Because of the strong frustration between the effectivefirst neighbor ferromagnetic Heisenberg term (J

eff2 ) and

the 4-spin antiferromagnetic exchange, it is the triangular6-spin plaquette which is the first system with a paramag-netic (S 0) ground state and a significant gap. [For

N 24 (N 12) one can compare the spectrum of thesample built with four (two) 6-site triangles with the spec-tra computed from other shapes. The triangle-compatibleshape gives the lowest energy and the largest gap (oth-ers have energies and gaps comparable with the frustratedfamily ones).] However, the ground-state wave functionis not a naive tensorial product of S 0 independent tri-angle wave functions: such an approximation gives a veryhigh ground-state energy (22.8 to be compared to 24)and largely underestimates the gap (0.4 to be compared to1). This quasiindependent triangle picture is thus deeplyrenormalized by resonances.

Analysis of the low lying levels in the S 0 subspaceleads to the same conclusion: The number of singletsbelow the first triplet level is very small (#10 alldegeneracy taken into account). As a comparison, this isvery different from the Kagom case where a continuumof singlets state is found in the magnetic gap [16,17].Thus, this system seems a very example of a short rangeresonating valence bond state with a clear cut gap in atranslationally invariant 2-dimensional spin-12 model.Consequently, both the low temperature specific heat

FIG. 2. Spin gap plotted as a function of 1N for Jeff2 22,J4 1. The vertical bar is the N ` extrapolation made outof the EN $ D correlation analysis.

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and spin susceptibility are thermally activated [xT e2DT, CVT e

2DT]. However, the gap decreasesrapidly when approaching the AF-F transition and we arenot yet able to decide if the gap vanishes at the transitionto ferromagnetism or before. Indeed, experimental resultsof Ishida and collaborators seem to indicate either a verysmall gap or a quantum critical behavior [13].

2nd layer heat capacity. We compute the MSE heat

capacity for Jeff2 J4

22, J5J4

0.2, J6J4

0.08,

with N 16, 20, and 24 (Fig. 3) and find a clear lowtemperature peak. The entropy at TJCy 0.5 is0.4Nln2 and the low temperature peak is thus likelyto remain in the thermodynamic limit. It also subsistsfor a relatively large range of competing Jn. Such alow temperature peak is characteristic of different highlyfrustrated systems [9,18]. The high temperature peak islocated at a temperature of the order of JCy (JCy is theleading coefficient of the 1T expansion of the specificheat: Cy 94JCyT

2 1 O 1T3, its expressionas a function of the Js is given in [19]). The lowtemperature peak height and location do not only depend

on JCy but also on the relative values of the couplingparameters. A better agreement between present resultsand experimental results [13] is expected by a fine tuningof the coupling parameters. Indeed, moving towards theboundary line between AF and F phases both decreasesthe gap and shifts the low temperature peak towardslower temperature.

Low energy degrees of freedom.To understand theexcitations responsible for this low energy peak we lookedfor possible common properties of low lying levels:It appears that most of these levels have a significantprojection on the subspace engendered by spin-1 diamondtilings. This subspace

Eis defined by the nonorthogonal

family of wave functions:

jC O

d1,...,N4

jSd 1,Szd [ 21,0,1 , (3)

FIG. 3. Heat capacity versus temperature for three sizes.Coupling parameters are J

eff2 J4 22, J5J4 0.2, J6J4

0.08. For these values JCy 0.93J4 and the N ` spin gap isof the order of J42. Because of finite-size effects, on N 20and 24 the high temperature peak (T JCV) only shows up asa shoulder.

where jSd 1,Szd is the S 1,Sz Szd state, symmet-ric with respect to the small diagonal of the dth diamond.For N 16 the projections of the exact low lying levelson E range from 10% to 50% for nearly all the states in-volved in the peak (and drop under 1% for higher energystates). These numbers are very large compared with theexpectation values of the projection of a random S 0 orS 4 wave function, which are, for the same lattice size,

of the order of1%. Since each tiling of the lattice gives3N4 independent states, the entropy associated with E isat least ln3N4 0.4Nln2, in agreement with the lowtemperature peak entropy found in our samples.

These results lead to the following picture: At hightemperature down to T JCy the degrees of freedomare essentially random spin 12. For T less than JCy ,the thermal wavelength increases and near neighborspins behave coherently as weakly ferromagnetic entities(pseudo spin 1). This explains the low temperature peak.At T D (spin gap), the 4-spin exchange couplingcreates larger clusters and the system organizes itself asa QSL.

Magnetization.Now we apply a magnetic field andlook for the spin S of ground state (T 0 magnetiza-tion). The large gap between the sectors of total spinsS Smax2 and S Smax2 1 1 indicates that excitingone diamond to its S 2 state costs a large energy. Thisfeature gives rise to a low temperature plateau at magneti-zation m 12 (Fig. 3) which has also been found in theclassical variational picture by Kubo et al. [11]. (Relatedphenomena have already been encountered in the Heisen-berg model with Ising anisotropy [20], in the MSE modelon the square lattice [21], on Heisenberg ladders or chains[22,23]. An m 0 plateau has been observed in thespin-ladder compound Cu2C5H12N22Cl4 [24,25].) For

the same coupling parameters as Fig. 3, Fig. 4 shows twometamagnetic transitions at HC1 and HC2 and the magneti-zation is completely quantized: m 0, 12, or 1. Thanks

FIG. 4. Magnetization versus magnetic field B. Exchangeparameters are those of Fig. 3. Since in 3He, J4 2 mK andgm 1.5 mK T21, HC1 is in the telsa range.

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g. misguich et al- spin liquid in the multiple-spin exchange model on the triangular lattice: ^3-he...

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