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Journal of Magnetism and Magnetic Materials 310 (2007) 1487–1493

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Spin–chirality decoupling in Heisenberg spin glasses and related systems

Hikaru Kawamura�

Faculty of Science, Osaka University, Toyonaka 560-0043, Japan

Available online 30 November 2006

Abstract

Recent studies on the spin and the chirality orderings of the three-dimensional Heisenberg spin glass (SG) and related systems are

reviewed with particular emphasis on the possible spin–chirality decoupling phenomena. Chirality scenario of real SG transition and its

experimental consequence on the ordering of Heisenberg-like SGs are discussed.

r 2006 Published by Elsevier B.V.

PACS: 75.10.Nr; 05.10.Ln; 05.70.Fh; 64.60.Fr

Keywords: Spin glass; Chiral glass; Chirality; Spin–chirality decoupling; Anomalous Hall effect

1. Introduction

Ordering of spin-glass (SG) have been studied quiteextensively as a typical example of ‘‘complex’’ systems.Experimentally, convincing evidence has now been ob-tained for the existence of an equilibrium phase transitionat a finite temperature in typical SG magnets, e.g.,canonical SG. The true nature of the SG transition andof the SG ordered state, however, still remains to be atissue [1].

In theoretical or numerical studies of SG, a simplifiedmodel called the Edwards–Anderson (EA) model haswidely been used [1]. For the case of the Ising EA modelin three dimensions (3D) corresponding to an infinitelystrong magnetic anisotropy, it is now well established thatthe model exhibits an equilibrium SG transition at a finitetemperature [1]. One should bear in mind, however, thatthe magnetic interactions in many real SG materials arenearly isotropic, being well described by an isotropicHeisenberg model. Although earlier numerical studies onthe 3D Heisenberg EA model suggested that the Heisen-berg SG exhibited a SG transition only at T ¼ 0 [2–6],recent numerical studies tend to suggest in common thatthe Heisenberg SG in 3D exhibits a finite-temperaturetransition [7–22]. Yet, the nature of the transition still

- see front matter r 2006 Published by Elsevier B.V.

/j.jmmm.2006.10.1128

6 6850 5543; fax: +66850 5494.

ddress: kawamura@ess.sci.osaka-u.ac.jp.

remains controversial. Obviously, in order to understandthe true nature of the experimental SG ordering, it iscrucially important to elucidate the nature of the orderingof the 3D Heisenberg SG.Some time ago, the present author proposed a scenario,

a chirality scenario, for the ordering of real Heisenberg-likeSG [7,8]. Chirality is a multispin variable representing thehandedness of the noncollinear or noncoplanar structuresinduced by spin frustration. A key notion in this scenario isthe ‘‘spin–chirality decoupling’’, which might possiblyoccur in certain frustrated magnets including the Heisen-berg SG. In this article, I wish to review the present statusof research on the spin and the chirality orderings of the3D Heisenberg SG and related systems.

2. Chirality

Two types of chirality have been discussed in theliterature, a vector chirality and a scalar chirality. Thetwo-component XY spin system ordered in a noncollinearmanner possesses a twofold Z2 chiral degeneracy, as thenoncollinear spin structure is either right- or left-handed, inaddition to the SO(2) spin-rotation degeneracy. The vectorchirality k is defined as a vector product of the twoneighboring spins by k ¼

PSi � Sj. The sign of its z-

component tells which chiral state the system takes.The three-component Heisenberg spin system ordered in

a noncoplanar manner also possesses a twofold Z2 chiral

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Fig. 1. The temperature dependence of the correlation length x and the

correlation time t of the 3D Heisenberg SG both for the spin and for the

chirality, expected from the chirality scenario.

H. Kawamura / Journal of Magnetism and Magnetic Materials 310 (2007) 1487–14931488

degeneracy, in addition to the SO(3) spin-rotation degen-eracy. The scalar chirality w is defined by the product ofthree neighboring spins by w ¼ Si � Sj � Sk.

As is evident from the definition of the local chirality, thechirality is a composite operator of the spins locally, notindependent of the spin. The spin–chirality decoupling, ifany, means that, on sufficient long length and time scales,say, beyond a certain crossover length and time scale,chiral correlations might outgrow spin correlations, i.e., thechirality correlation length gets much longer than the spincorrelation length, x (chirality) b x(spin).

In terms of the phase transition, the spin–chiralitydecoupling might lead to either of the following twosituations: in one, the spin and the chirality might order atthe same temperature, say, at zero temperature, wherethere appear two distinct diverging length scales, eachassociated with the spin and with the chirality. Moreprecisely, the chirality correlation-length exponent isgreater than the spin correlation-length exponent, n(chirality) 4 n (spin). In the other, the spin and thechirality might order at two distinct temperatures. Bydecreasing the temperature, the chirality orders first at ahigher temperature followed by the spin order at a lowertemperature, Tc (chirality) 4 T c (spin).

3. The spin–chirality decoupling in regularly frustrated XYantiferromagnets

We review briefly the spin and the chirality orderings ofregularly frustrated XY antiferromagnets. First example isthe classical XY (plane rotator) model on the one-dimensional (1D) triangular-ladder lattice. The model isexactly solvable [23]. While both the spin and the chiralityorder only at T ¼ 0, the associated correlation-lengthexponents are mutually different. Indeed, the spin correla-tion-length exponent is equal to unity ns ¼ 1, while thechiral correlation-length exponent is equal to nk ¼ 1,meaning that the chiral correlation length divergesexponentially toward T ¼ 0 [23]. Hence, in this particular1D model, the spin–chirality decoupling is rigorouslyshown to occur.

Another example might be the classical XY antiferro-magnet on the 2D triangular lattice. Although there hadbeen some controversy concerning how the chiral Z2 andthe spin-rotation SO(2) order in this system, consensus nowappears that separate spin and chirality transitions occursuccessively [24–29]. With decreasing temperature, thechirality orders first at a higher temperature into thelong-range ordered state, while the spin orders at a lowertemperature into the quasi-long-range ordered state.

4. Spin and chirality orderings of the three-dimensional

Heisenberg spin glass

According to the chirality scenario [7,8], the 3DHeisenberg SG exhibits the spin–chirality decoupling. Bydecreasing the temperature, chiral correlations outgrow

spin correlations at some crossover temperature T ¼ T�,and at a lower temperature T ¼ TCG the chirality exhibitsa glass transition into the chiral-glass ordered state withoutaccompanying the standard SG order. The SG transitiontemperature is lower than the chiral-glass transitiontemperature, TSGoTCG, TSG being either zero or nonzero.The basic picture is summarized in Fig. 1 in terms of thetemperature dependence of the spin and the chiralitycorrelation lengths (correlation times).As mentioned, the issue of whether the spin–chirality

decoupling really occurs in the 3D Heisenberg SG remainscontroversial. While several numerical results in favor ofthe occurrence of the spin–chirality decoupling werereported in Refs. [9–14], a simultaneous spin and chiralitytransition without the spin–chirality decoupling wasclaimed in other works [15–22].Here, we wish to report on our recent Monte Carlo

results on the spin and the chirality orderings of the 3DHeisenberg SG with the nearest-neighbor �J coupling(done in collaboration with Dr. K. Hukushima). Details ofthe simulation, including the precise definitions of variousphysical quantities, have been given in Ref. [14].In Fig. 2, we show the temperature dependence of the

spin and the chirality autocorrelation times on a semi-logplot. At higher temperatures spin correlations dominateover chiral correlations, where the system is in thespin–chirality coupling regime. By decreasing the tempera-ture beyond a crossover temperature T�, chiral correlationsexceed spin correlations, and the system gets into thespin–chirality decoupling regime at TtT�. Thus, thespin–chirality decoupling appears to be realized in thissystem.In order to clarify the transition behavior of the model,

we investigate the Binder ratio of the spin and of thechirality. As shown in Fig. 3(a), the chirality Binder ratio

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102

103

104

105

106

107

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

corr

elat

ion

time

τ

T/J

SG, L=16L=20

CG, L=16L=20

SG, L=16(Ref.11)CG, L=16(Ref.11)

Fig. 2. The temperature dependence of the chirality and the spin

autocorrelation times of the 3D �J Heisenberg SG on a semi-log plot

(taken from Ref. [14]).

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

g CG

ChiralityL=8

L=12L=16L=20

0

0.2

0.4

0.6

0.8

1

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

g SG

T/J

SpinL=8

L=12L=16L=20

a

b

Fig. 3. The temperature dependence of the chirality Binder ratio (a), and

of the spin Binder ratio (b), of the 3D �J Heisenberg SG (taken from Ref.

[14]).

H. Kawamura / Journal of Magnetism and Magnetic Materials 310 (2007) 1487–1493 1489

exhibits a negative dip which deepens with increasingsystem size L. The data of different L cross on the negative

side of gCG. These features strongly suggest the existence ofa finite-temperature transition in the chiral sector. Byextrapolating the dip temperature to L ¼ 1, we get anestimate of the chiral-glass transition temperatureTCG ’ 0:19. We note that the observed shape of thechirality Binder ratio resembles the one observed in systemsexhibiting a one-step replica-symmetry breaking (RSB).The corresponding spin Binder ratio, by contrast, exhibitsno signature of a phase transition, no crossing nor merging.Even at T ¼ TCG, gSG stays completely off-critical. Hence,the Binder ratio suggests the occurrence of a chiral-glasstransition at a finite temperature, T ¼ TCG ’ 0:19 withoutaccompanying the standard SG order.The overlap distribution function in the ordered state is

shown in Fig. 4 for both cases of the chirality (Fig. 4(a))and of the spin (Fig. 4(b)) at a temperature T ¼ 0:15. Thechirality overlap distribution PðqwÞ exhibits symmetric sidepeaks at qw ¼ �qEA

w corresponding to the long-range chiral-glass order, which grow with increasing L. On top of it,PðqwÞ also exhibits a centra peak at qw ¼ 0, which alsogrows with increasing L. The existence of such apronounced central peak is a characteristic feature of thesystem exhibiting a one-step-like RSB, never seen in theIsing SG. The data strongly suggest that the chiral-glassordered state exhibits a one-step-like RSB. By contrast, thespin overlap distribution PðqdiagÞ, calculated for thediagonal component of the spin-overlap tensor qdiag ¼P

mqmm (m ¼ x; y; z), shows an entirely different behavior:although PðqdiagÞ exhibits symmetric peaks at finite valuesof qdiag for smaller lattices, suggesting the appearance ofthe SG long-range order, these peaks gradually go away forlarger lattices, and PðqdiagÞ tends to a single-peak functionaround qdiag ¼ 0, which is a characteristic of the disorderedphase.In Fig. 5, we show the dimensionless correlation lengths

both for the chirality and for the spin, xCG=L and xSG=L,the data for smaller lattices (L ¼ 8 and 12) in upper paneland those for larger lattices (L ¼ 16 and 20) in lower panel.For smaller lattices, both the spin and the chiralitycorrelation lengths cross at a more or less commontemperature, which seems consistent with the observationof Ref. [19]. By contrast, for larger lattices, while the chiralcorrelation length still exhibits a crossing at the expectedchiral-glass transition point T ’ 0:19, the spin correlationlength does not quite cross any longer, only a merging-likebehavior being observed below TCG. Thus, our observationfor the spin correlation length is that the crossing tendencyis more and more weakened if one goes to larger lattices. Itis not clear at the present stage what is a true asymptoticbehavior of the spin correlation length for large enoughlattices. Since the spin–chirality decoupling, if any, shouldmanifest itself beyond a certain crossover length L�, thebehavior of xSG=L observed here seems consistent with thespin–chirality decoupling with L� ’ 20. We also note that,although it is sometimes argued that the normalized

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0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

-0.06 -0.04 -0.02 0 0.02 0.04 0.06

P( q

χ)

qχ

Chirality

8121620

0

0.005

0.01

0.015

0.02

-1 -0.5 0 0.5 1

P(q

diag

)

qdiag

Spin

8121620

a

b

Fig. 4. The overlap distribution function for the chirality (a), and for the

diagonal component of the spin overlap tensor (b), of the 3D� J

Heisenberg SG (taken from Ref. [14]). The temperature is T ¼ 0:15 well

below the chiral-glass transition point T ’ 0:19.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.16 0.18 0.2 0.22 0.24

ξ CG

/L, ξ

SG

/Lξ C

G/L

, ξSG

/L

T/J

chiral-glass L=2016

spin glass L=2016

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.16 0.18 0.2 0.22 0.24

T/J

chiral glass 128

spin glass 128

a

b

Fig. 5. The temperature dependence of the dimensionless spin and

chirality correlation lengths of the 3D� J Heisenberg SG, for smaller

lattices L ¼ 8 and 12 (above), and for larger lattices L ¼ 16 and 20 (below)

(taken from Ref. [14]).

H. Kawamura / Journal of Magnetism and Magnetic Materials 310 (2007) 1487–14931490

correlation length would be the best quantity in probingthe ordering behavior [19], there exists an occasion wherexSG=L overestimates the ordering tendency [30]. It is thusvery important to examine the ordering behavior of thesystem by comparing the dimensionless correlation lengthwith various other independent quantities, e.g., the orderparameter, the Binder ratio or the overlap distribution.

Very recently, Campos et al. studied the 3D HeisenbergSG with the Gaussian coupling for larger lattices up toL ¼ 32, though the temperature range was limited to justbelow the transition temperature [22]. Their data of xCG=L

exhibits a weak crossing at around T ’ 0:15, whereas thoseof xSG=L for larger lattices LX16 do not cross in theinvestigated temperature range. Hence, at least the rawdata of the dimensionless correlation lengths are consistentwith the spin–chirality decoupling picture. Nevertheless,the authors of Ref. [22] interpreted the data as suggesting asimultaneous spin and chirality transition of Koster-litz–Thouless (KT) type, by invoking a large correction-to-scaling term. This interpretation, however, seems notpersuading. We note that, in the type of systems whoseordered state exhibits a one-step-like RSB, the correspond-ing dimensionless correlation length xðLÞ=L might tend to afinite value even below Tc at L!1, not diverging to

infinity, disguising a KT transition, in sharp contrast to thestandard system where xðLÞ=L!1 in the ordered state.This is because, in a one-step RS broken state, the orderedstate is quite exotic consisting of many pure states whichare mutually dissimilar with vanishing overlaps.We also estimate the chiral-glass exponents via the

standard finite-size scaling analysis. The exponents

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obtained are nCG ’ 1:2 and ZCG ’ 0:8, etc., which differsignificantly from the standard 3D Ising SG values, n ’2�3 and Z ’ �0:35�� 0:4 [1]. The results indicate that thechiral-glass transition belongs to a universality classdifferent from the one of the 3D Ising SG. Possible long-range and/or many-body nature of the chirality–chiralityinteraction might be the cause of this difference.

Fig. 6. The schematic phase diagram of the weakly anisotropic Heisen-

berg spin glass in the anisotropy (D) versus temperature (T) plane. CG and

SG stand for the chiral glass and the spin glass, respectively.

5. Chirality scenario for the weakly anisotropic Heisenberg

SG

On assuming that the spin–chirality decoupling occurs inthe 3D isotropic Heisenberg SG, we now ask: What doesthis mean for real Heisenberg-like SG where the weakrandom magnetic anisotropy inevitably exists ? Thechirality scenario claims that the weak random anisotropyinherent to real SG magnets ‘‘recouples’’ the spin to thechirality, and the chiral-glass transition of the isotropicsystem is revealed as the standard SG transition in realweakly anisotropic Heisenberg SG [7,8].

Such a ‘‘spin–chirality recoupling’’ can be understoodbased on a simple symmetry consideration. The isotropicHeisenberg SG possesses both the chiral Z2 symmetry andthe spin-rotation SO(3) symmetry, i.e., Z2� SO(3). Due tothe spin–chirality decoupling, only the chiral Z2 isspontaneously broken in the isotropic system at thechiral-glass transition with keeping the SO(3) symmetryunbroken, which leaves the spin to be paramagnetic evenbelow TCG. Suppose that the weak random anisotropy isadded to the isotropic system. It energetically breaks theSO(3) symmetry with keeping the chiral Z2 symmetry.(Note that the invariance under the spin inversionS!�S, which flips the chirality, is kept in the presenceof the random magnetic anisotropy.) Since the chiral Z2

has already been decoupled from the SO(3) in the isotropicsystem, it would be natural to expect that the Z2 chiral-glass transition of the anisotropic system occurs essentiallyin the same manner as that of the isotropic system. As soonas the Z2 chiral-glass transition takes place, however, thereis no longer any symmetry left in the anisotropic system,which forces the spin to order below TCG. This is aspin–chirality recoupling due to the magnetic anisotropy.

The situation might be summarized in the schematicphase diagram in the anisotropy (D) versus temperature (T)plane of Fig. 6. In the isotropic limit D ¼ 0, due to thespin–chirality decoupling there, the chiral-glass transitionoccurs at a temperature higher than the SG transitiontemperature, TCG4TSG. A crucial observation is that theSG (simultaneously chiral-glass) transition of the aniso-tropic system with D40 is a continuation of the chiral-glass fixed point of the isotropic D ¼ 0 system, not acontinuation of the SG fixed point of the isotropic system.The SG transition of real Heisenberg-like SG with weakrandom anisotropy is governed by the same chiral-glassfixed point all the way along the transition line, includingboth D ¼ 0 and D40. In this way, the D! 0 limit is not

singular, and there is no Heisenberg-to-Ising crossover inthe SG critical properties even in the D! 0 limit.Such a picture leads to the following interesting

predictions on the properties of experimental Heisenberg-like SG. (i) The SG transition temperature T ¼ Tg dependson the anisotropy D in a regular manner, asTgðDÞ�TCGð0Þ þ cDþ � � � (c is a numerical constant). (ii)The SG critical exponents are given by the chiral-glasscritical exponents of the isotropic system, which differsignificantly from the 3D Ising SG exponents. They areb�1, g�2, d�3 and Z positive. Furthermore, even theweakly anisotropic SG does not show Heisenberg-to-Isingcrossover in its critical behavior. (iii) The SG ordered stateof Heisenberg-like SG exhibits a one-step-like RSB. As acorollary of this, experimental Heisenberg-like SG isexpected to exhibit an equilibrium SG transition evenunder magnetic fields as an RSB transition. This providesan interesting, and somewhat unexpected possibility thatan in-field ordering behavior of the weakly anisotropicHeisenberg-like SG might entirely differ from that of thestrongly anisotropic Ising SG. For the latter, recenttheoretical studies suggest that there is no equilibrium in-field SG transition [31,32]. (iv) The magnetic phasediagram of experimental Heisenberg-like SG resemblesthe one of the corresponding mean-field model: in the highfield regime, the SG transition line behaves as the mean-field Gabay–Thouless (GT) line with an exponent 1=2, i.e.,Hg�jTgðHÞ � Tgð0Þj

1=2 [11], while, in the low field regime,it behaves as the mean-field de Almeida-Thouless (AT)line, i.e., Hg�jTgðHÞ � Tgð0Þj

ðbCGþgCGÞ=2, where bCG andgCG are the corresponding chiral-glass exponents of theisotropic system. Since one has bCG�1 and gCG�2, one getsan exponent close to 3

2, which happens to be close to the

corresponding AT-line exponent 32.

Basically, these predictions from the chirality scenarioare compared favorably with the existing experimental datafor real Heisenberg-like SG including canonical SG.Namely, (i) experimentally observed anisotropy depen-dence of the SG transition temperature is close to oneexpected from the chirality scenario [33]. (ii) The SG

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critical exponents observed by various researchers forcanonical SG, b ’ 1, g ’ 2 and Z ’ 0:5, are in goodagreement with each other, but deviate significantly fromthe Ising SG values: See, e.g., Ref. [34] and references citedtherein. By contrast, these experimental values agree verywell with the chiral-glass values. Furthermore, the absenceof the expected Heisenberg-to-Ising crossover in the criticalbehavior of the weakly anisotropic Heisenberg-like SG isthe property which has puzzled SG researchers for years[1]. (iii) Experimentally, the in-field properties are oftensignificantly different between in the Ising-like SG and inthe Heisenberg-like SG [35]. (iv) While the experimentallydetermined magnetic phase diagram of the Heisenberg-likeSG is often well described by the mean-field phase diagramincluding the GT and AT lines, the true origin of thiscoincidence has long been a mystery. Remember that,generally mean-field theory does not give exponent valuesof real systems correctly. In contrast, the chirality scenariogives a natural alternative explanation for the apparentlymean-field-like phase diagram widely observed in experi-mental Heisenberg-like SG.

The most stringent experimental test of the chiralityscenario would be to directly measure the chirality,particularly, the chiral susceptibility X w and the nonlinearchiral susceptibility X nl

w . This has long remained to be anextremely difficult task, since the chirality is a higher-orderquantity in spins, cubic in spins. Recently, however, it hasbeen recognized that the chirality might be measurable byusing the anomalous Hall effect as a probe. In fact, G.Tatara and the present author analyzed the chiralitycontribution to the anomalous Hall effect of metallic SGbased on the perturbation analysis [36,37]. The anomalousHall coefficient Rs is then given by

Rs ¼ r=M

¼ � ðArþ Br2Þ � CDðX w þ X nlw ðDMÞ2 þ � � �Þ. ð1Þ

It consists of two kinds of terms. The first part is thestandard contribution to the anomalous Hall effect, whichis proportional to the resistivity r or its squared r2. Sincethe resistivity does no show any anomaly at Tg, this firstpart can be regarded as a regular background. The secondpart is the chirality contribution, which is proportional tothe chiral susceptibility X w. It even contains the informa-tion of the nonlinear chiral susceptibility X nl

w as a higher-order contribution.

Inspired by this theoretical suggestion, several experi-mental groups tried to measure the chirality contributionto the anomalous Hall effect in metallic SG. Thesemeasurements observed a sharp cusp-like anomaly at T ¼

Tg in the temperature dependence of Rs [38–41], followedby the deviation between the field-cooled and the zero-field-cooled data below Tg [39,40]. Furthermore, Taniguchiet al. very recently observed a singular behavior of thenonlinear chiral susceptibility at T ¼ Tg characterized bythe exponent dCG ’ 3, which is rather close to thecorresponding chiral-glass exponent [41]. All these

observations indicate that the chirality in metallic SGindeed exhibits a strong anomaly at the SG transition,providing strong experimental support to the chiralityscenario of SG transition. We stress that, if the orderparameter of the SG transition were not the chirality butwere the spin itself as in the case of the mean-field(Sherrington–Kirkpatrick) Heisenberg SG, the chiral sus-ceptibilities would not exhibit such a strong singularity: forexample, the nonlinear chiral susceptibility of the Heisen-berg SK model does not diverge at Tg [42]. This is simplydue to the fact that the chirality is a composite operator,being of higher order in the spin. Hence, in the absence ofthe spin–chirality decoupling, a power-counting argumentshould apply as a first-order approximation, which leads tonCG ¼ nSG, bCG ’ 3bSG and gCG ’ gSG � 4bSG, etc. If onesubstitutes here the experimental SG exponents forcanonical SG, bSG ’ 1 and gSG ’ 2, one gets the chiral-glass susceptibility exponent gCG ’ �2o0, meaning thatthe nonlinear chiral susceptibility should not diverge !

6. Summary

Recent studies on the spin and the chirality orderings ofthe 3D Heisenberg SG and related systems were reviewed,with particular emphasis on the possible spin–chiralitydecoupling phenomena. Our Monte Carlo results supportthe view that the 3D isotropic Heisenberg SG exhibits aspin–chirality decoupling, i.e., a finite-temperature chiral-glass transition not accompanying the standard SG order.Chirality scenario of real SG transition and its experi-mental consequence on the ordering of real Heisenberg-likeSG were discussed. The scenario appears to explain someof the long-standing experimental puzzles concerning theHeisenberg-like SG, thereby getting some support fromexperiments, particularly from the recent Hall measure-ments. Then, the chirality might be a ‘‘missing link’’,playing a crucial role in the ordering of SG.

Acknowledgments

The author is thankful to Dr. K. Hukushima, Dr. G.Tatara, Dr. D. Imagawa and Mr. A. Matsuda for theircollaboration, and to Dr. I. Campbell, Dr. T. Taniguchi,Dr. E. Vincent, Dr. M. Ocio, Dr. H. Yoshino, Dr. M.Picco, Dr. M. Sato and Dr. H. Takayama for usefuldiscussion.

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