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VOLUME 86, NUMBER 11 P H Y S I C A L R E V I E W L E T T E R S 12 MARCH 2001

Quantum Frustration in the “Spin Liquid” Phase of Two-Dimensional 3He

E. Collin,1 S. Triqueneaux,1 R. Harakaly,1,2 M. Roger,3 C. Bäuerle,1 Yu. M. Bunkov,1 and H. Godfrin1,*1Centre de Recherches sur les Très Basses Températures, Centre National de la Recherche Scientifique,

BP 166, 38042 Grenoble Cedex 9, France2Department of Experimental Physics, P.J. Safárik University, Park Angelinum 9, 04154 Kosice, Slovakia

3Service de Physique de l’Etat Condensé, Commissariat à l’Energie Atomique, Centre d’Etudes de Saclay,91191 Gif sur Yvette, France(Received 6 September 2000)

We have measured the ultralow temperature and low field magnetic susceptibility of the 4

7phase of

two-dimensional 3He adsorbed on graphite preplated by one layer of 4He. The experiments are performedby progressively adding 4He to the system, thus suppressing in a controlled way the 3He atoms trappedin substrate heterogeneities. This procedure enables us to determine the intrinsic properties of this spin 1

2model magnet in the zero field limit. The results show quantitatively that the system is strongly frustratedby multiple spin exchange interactions. A characteristic gapped spin liquid behavior is observed atultralow temperature.

DOI: 10.1103/PhysRevLett.86.2447 PACS numbers: 75.70.Ak, 67.80.Jd, 75.10.Jm

Strongly frustrated magnets are of fundamental interestdue to the possible existence of disordered ground states.Recent studies have concentrated on spin chains and lad-ders, Mott insulators with orbital degeneracy, disorderedsystems, and also geometrically frustrated systems wherethe incompatibility of the local antiferromagnetic inter-actions with the global symmetry of the lattice leads tounconventional magnetic properties at low temperatures[1,2]. The main features of frustrated magnets are the ex-istence of a large density of low energy states evidencedby heat capacity experiments and the extremely weak de-parture from the Curie-Weiss susceptibility even at verylow temperatures [3]. The expectation that highly frus-trated systems may have a disordered spin liquid groundstate, characterized by a finite correlation length at zerotemperature and a singlet-triplet spin gap in the excitationspectrum has motivated an active search of suitable experi-mental model magnets.

A unique example of a strongly frustrated two-dimensional S �

12 nuclear magnet is provided by a solid

layer of 3He adsorbed on a graphite substrate [4–9]. Inthis quasilocalized quantum system, the large zero-pointmotion favors cyclic permutations of the 3He atoms.Such exchange processes lead to an effective magneticinteraction between the nuclear spins formally describedby the Hamiltonian H �

Pn2�21�nJnPn where Pn is the

operator for cyclic permutation of n particles. High orderspin exchanges are particularly important in a hard-coresolid like 3He; indeed, a quantitative description of themagnetic properties involves several cyclic processes upto six particle exchange [9].

The cyclic exchange of an even number of particlescontributes to an antiferromagnetic Heisenberg coupling,while that of an odd number of particles has the oppo-site (ferromagnetic) sign [10]. In addition, multiple spinexchange processes involving four or more particles intro-

0031-9007�01�86(11)�2447(4)$15.00

duce new physics due to higher order effective couplingterms in the Hamiltonian [5]. The impossibility to satisfysimultaneously the spin configurations favored by differentmultispin exchange processes gives rise to a sophisticatedquantum frustration in addition to the geometrical frustra-tion associated with the triangular lattice structure.

We will concentrate in this Letter on the properties of thesecond layer of adsorbed 3He which forms a low densitysolid commensurate with respect to the first layer triangularsolid in a 4

7 density ratio [11]. The second layer formsstrictly speaking a triangular Bravais lattice with a basis.However, quantum Monte Carlo calculations show that theexchange constants are essentially site independent, and itcan be considered as a simple triangular lattice [12].

Ishida et al. [8] have shown that the second layer heatcapacity displays a double peak structure and an unusualtemperature dependence at the lowest temperatures (T �90 mK). This feature led the authors to the suggestion ofa disordered ground state for the 4

7 phase.Recently, we have been able to determine experimen-

tally the multispin exchange constants up to six particleexchange [9] as a function of the second layer areal density,thus completely determining the Hamiltonian of this sys-tem. High order ring exchanges, in particular four spin ex-change, are found to be of the order of several millikelvin.According to exact diagonalizations of finite clusters forthe multiple spin exchange (MSE) model, a gapped spinliquid ground state is expected for the experimentally de-termined exchange parameters [13].

The first measurement of the nuclear susceptibility ofthe 4

7 phase at temperatures well below the exchange ener-gies showed a substantial deviation from the Curie-Weisslaw at temperatures on the order of 100 mK [14]. It wasnot clear, however, whether the very peculiar temperaturedependence observed in this experiment was an intrinsicproperty of this strongly frustrated system, or an artifact

© 2001 The American Physical Society 2447


due to the presence of substrate heterogeneities or to thelarge field used in this experiment, where kBT was on theorder of mB at the lowest temperatures.

In this Letter we report the determination of the intrinsicnuclear magnetic susceptibility of the 4

7 phase of 3He downto temperatures of 100 mK in low magnetic fields. Themeasurements are done by preplating a Papyex exfoliatedgraphite substrate by one layer of 4He and introducing aquantity of 3He slightly lower than that needed to form thecommensurate solid 4

7 phase. Then, and this is an essentialfeature of this work, 4He is added to the system whilekeeping the 3He quantity fixed at 2.97 ccSTP (cm3 of gasin standard temperature and pressure conditions).

This procedure allows us to finely tune the formation ofthe 4

7 phase taking advantage of the preferential adsorptionof 4He to suppress progressively the magnetic signal of the3He atoms trapped at heterogeneities [15].

A run at a submonolayer coverage (9.28 atoms�nm2)where the magnetic susceptibility follows a Curie law en-ables us to calibrate the NMR spectrometer and to verifythat the cell is in good contact with the thermometers in theentire temperature range (0.1–400 mK). The temperatureis measured using a pulsed Pt-NMR thermometer and acalibrated carbon resistance thermometer. A magnetic fieldof 30.5 mT was used in order to ensure that mB ø kBTfor all temperatures. Further experimental details are givenelsewhere [14].

The evolution of the nuclear susceptibility in the vicin-ity of the 4

7 phase as we add 4He is shown in Fig. 1. Athigh temperatures all curves display the Curie behaviorcorresponding to the constant 3He amount. At interme-diate temperatures one observes a decrease in magneti-zation as we increase the 4He quantity. This is due tothe promotion of 3He atoms into the third layer liquidwhich has a smaller magnetization than the solid. Thelow temperature magnetization is emphasized in the inset.In this plot we subtracted the contribution of the liquidand normalized the magnetization to the quantity of solid3He remaining in the second layer. We observe initially(for amounts of 4He equal to 6.56, 6.71, and 6.92 ccSTP)a rapid decrease of the magnetization. We then reach aregime where the magnetization of the second layer solidremains remarkably unaffected for a large range of 4Hecoverages (open symbols: 6.92, 7.13, 7.34, 7.64 ccSTP;see also Fig. 2). Finally, at a 4He quantity of 8.04 ccSTP(filled circles) the system changes its nature, the sud-den increase of the susceptibility indicating the onset offerromagnetism.

This evolution can be understood by considering the re-arrangement of the second layer as 4He atoms are added.The second layer contains initially atoms in the 4

7 phase co-existing with some 3He atoms located at substrate hetero-geneities. These strongly bound atoms are preferentiallyreplaced by 4He atoms, giving rise to the initial decreaseof the low temperature susceptibility. When these defectatoms have been removed, the effect of the addition of 4Heis to promote atoms from the 4

7 phase to the third layer liq-

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0.1

0.1 1

10- 3

10- 2

10- 1

0.1 1 10 100χ

T (mK)

4He (ccSTP) : 6.566.716.927.137.347.648.04

(a.u

.)

FIG. 1. Nuclear susceptibility of 2D 3He in the vicinity of the4

7phase as 4He is added to the system. Inset: the liquid contri-

bution has been subtracted and the magnetization normalized tothe quantity of 3He solid in the second layer.

uid. In this regime, the magnetization normalized to theamount of solid 3He does not change; only the amount of47 phase varies.

This scenario is supported by a quantitative analysis.By fitting the magnetization at temperatures T $ 2 mKwe are able to determine the quantity of 3He atoms pushedinto the liquid third layer (Fig. 2a). We can then extractthe quantity of 3He atoms trapped in the substrate hetero-geneities (Fig. 2b) by fitting the low temperature data asa coexistence between the 4

7 phase and defects having atemperature dependence close to Curie law [16].

At point B in Fig. 2, all the trapped 3He atoms have beenreplaced. It is interesting to note that both graphs show adiscontinuity at this point.

The total amount of atoms trapped in substrate hetero-geneities when the second layer has completely solidifiedis of the order of 16% of the second layer. This is consis-tent with heat capacity measurements of Ishida et al. [8].

The system then enters a region (from points B to E)where all magnetization curves of the second layer solidare identical (see open symbols in the inset of Fig. 1).All defects have been replaced by 4He atoms, and wethus measure the intrinsic magnetic properties of the 4

7phase. Here one 4He atom replaces one 3He atom. In thisplateau region, which is approximately 3 times wider indensity than for the pure 3He system [17,18], the amount


0

0.1

0.2

0.3

0.4

0.5

6.5 7 7.5 8 8.5 9 9.5

Def

ects

3H

e (c

cST

P)

Added 4He (ccSTP)

b)

0

0.5

1

1.5

2

2.5

Liqu

id 3H

e (c

cST

P)

a)

B

E

FIG. 2. Evolution of different properties as we add 4He.(a) Amount of 3He atoms promoted to the third (fluid) layer.A slope of one (solid line) corresponds to a situation whereone 4He atom replaces one 3He atom in the second layer.(b) Quantity of second layer paramagnetic defects. The pointsB (beginning) and E (end) define the region where adding 4Hedoes not change the magnetic behavior of the solid layer. Theerror bars represent 2% of the 3He quantity in the second layer.

of 4He in the second layer varies from 20% to 40%. Thisdemonstrates that the 4

7 phase is very stable.Another important conclusion can be drawn from the

fact that in the plateau region the amount of 3He present inthe third layer grows substantially as we add 4He, whereasthe magnetism of the second layer remains unchanged.This provides a direct proof that the third layer liquid doesnot affect the second layer exchange. The same conclu-sion was reached for films of much higher density in ourprevious work [9]. The evolution of the magnetism of 3Hefilms as a function of density can therefore be describedconsistently by the in-plane MSE model.

At a proportion of roughly 40% of 4He atoms in thesecond layer (point E) the system becomes ferromagnetic.Here the number of 3He atoms promoted into the thirdlayer is smaller than the number of 4He atoms added to thesecond layer, which is a signature of the compression of thesecond layer solid. The system then passes a ferromagneticanomaly as already seen in the pure 3He system [19] andother preplated systems [20,21]. In Fig. 3 we show themagnetization of the 4

7 phase obtained after subtraction ofthe liquid and defect contributions and normalization to thesame quantity of 3He solid. All our data collapse onto asingle curve, showing that the measured magnetization isan intrinsic property of this strongly frustrated system.

For comparison we also display previous measurementsin higher magnetic fields [14], to which we apply the sameanalysis. We observe no significant difference in the tem-perature dependence for the two different magnetic fields

which indicates that the measurements are performed inthe zero field limit of the MSE Hamiltonian and confirmsour previous findings. In recent measurements on a dif-ferent system, the 4

7 phase of 3He on a Grafoil substratepreplated by HD, a field independent susceptibility wasalso observed [22].

The susceptibility follows a Curie-Weiss law downto very low temperatures, in agreement with previousresults. Indeed, the system almost acts like a collection offree spins in spite of the strongly interacting environmentdue to multiple spin exchange. We are able to analyzequantitatively our data using a Padé approximant of thehigh temperature series expansions (HTSE) of the MSEHamiltonian [23] down to temperatures of the order of2 mK. With this approach, we determine the different ex-change constants with an accuracy of 0.1 mK. We assumethe hierarchy 2Jeff

2 . J4 . J6 . J5 predicted by theoryand verified for pure 3He films. We also limit the parame-ter space to a physically sound range, consistent with pure3He films data [9]. Within these hypotheses we find reason-able exchange constants: Jeff

2 � 22.8 mK, J4 � 1.4 mK,J5 � 0.45 mK, J6 � 1.25 mK. The leading term Jx of theMSE-HTSE is found to be very small, and even slightlyferromagnetic: Jx � 0.07 6 0.1 mK. Such a smalleffective exchange constant is a signature of a stronglyfrustrated system. The corresponding Curie-Weisstemperature which can be deduced (QCW � 3Jx �10.2 mK) is quite different from the value obtained by asimple empirical Curie-Weiss fit and has the opposite sign(QCW � 20.9 mK). This clearly demonstrates the strong

0.2

0.4

0.6

0.81

0 2 4 6 8

χ

1/T (mK-1)

T (

a.u.

)

10-3

10-2

10-1

0.1 1 10 100

χ

T (mK)

This work 30.5 mT

Bäuerle et al. 113 mT

(a.

u.)

FIG. 3. Magnetic susceptibility of the pure 4

7phase. The solid

lines correspond to the [2,3] Padé approximant of the HTSE ofthe MSE Hamiltonian [23] (upper curve), and to a Curie-Weissfit (lower curve). The inset emphasizes the low temperaturebehavior of the susceptibility multiplied by temperature. Thesolid line shows an exponential decrease.

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cancellation of the Heisenberg term due to multispinexchange.

The leading term of the heat capacity series expan-sion deduced from the present susceptibility data is JC �1.6 mK. The exchange parameters are all slightly smallerthan those found in pure 3He films [9] as would be ex-pected since the latter have a slightly lower density.

We do not observe any feature in the susceptibility downto temperatures of 100 mK which could be interpreted asa finite temperature phase transition [5,8,13]. Our suscep-tibility results suggest that the system displays a very puretwo-dimensional behavior [24]. Note, however, that herewe use a small magnetic field, and that susceptibility andheat capacity are not equally sensitive to a possible phasetransition.

The magnetization is large at very low temperatures, asignature of the large density of low energy magnetic ex-cited states typically found in strongly frustrated systems[2,3]. However, we observe that xT decreases exponen-tially at the lowest temperatures as emphasized in the insetof Fig. 3. Such a behavior is expected for a spin liquidground state having a magnetic gap.

Exact diagonalizations predict two different groundstates for the MSE model [13]. In the region wherepredominantly ferromagnetic (FM) interactions are ofimportance the ground state is simply a FM S � N�2ordered state. However, in a parameter range where anti-ferromagnetic (AF) interactions are dominant (S � 0), nolong range correlations are observed even at T � 0 and aquantum spin liquid ground state is proposed.

For reasonable exchange parameters an excitation gapof the order of J4 is predicted. Approaching the AF/FMtransition line the gap is expected [13] to become muchsmaller than J4.

From our measurements, we can deduce a magnetic gapof approximately 100 mK. Such a small value is consistentwith the theoretical predictions, since our experimentallydetermined exchange parameters fall close to the AF/FMtransition line; clearly, more detailed calculations wouldbe desirable.

It is important to note that our experiment rules outthe possibility that the observed gap is due to trivial sizeeffects. As already mentioned, while the size of the solid3He domains is reduced by approximately 30% in theplateau region, the normalized magnetization remainsunaffected.

Valuable information on the nature of the spin gapshould be obtained by susceptibility measurements at lowertemperatures while complementary heat capacity mea-surements would provide an independent determinationof the exchange constants of this very pure model system.

In conclusion, we have measured the intrinsic magneticsusceptibility of the 4

7 phase of two-dimensional 3He byprogressively suppressing the magnetization of the 3Heatoms trapped in substrate heterogeneities. We have clear

2450

evidence that in this system, the third layer liquid is notparticipating in the magnetic exchange and hence in-planemultispin exchange determines the magnetic properties ofthis 2D system. The unusual magnetization curve obtainedhere corresponds to a unique strongly frustrated two-dimensional S � 1

2 quantum magnet whose properties arequantitatively described by the MSE model. Our resultsare consistent with a gapped spin liquid ground state. Thegap is found to be 1 order of magnitude smaller than theexchange constants. The nature of the AF�FM transitionas well as the value of the excitation gap at the transitionis still an open and challenging question.

We acknowledge valuable discussions with H.Fukuyama, H. Ishimoto, C. Lhuillier, G. Misguich,M. Morishita, D.D. Osheroff, and P. Schiffer.

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