spin flop in one-dimensional quantum antiferromagnets

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Spin flop in one-dimensional quantum antiferromagnets To ˆ ru Sakai Faculty of Science, Himeji Institute of Technology, Ako, Hyogo 678-1297, Japan ~Received 19 February 1999! The field induced transition corresponding to the spin flop in the quantum antiferromagnetic chains in the presence of the Ising-like single-ion anisotropy D( ,0) is studied by the finite-cluster analysis. It is found that for S > 3 2 a first-order metamagnetic transition occurs even in one dimension for small negative D except for the Haldane phase, while there exist two second-order transitions and an incommensurate phase appears between them for intermediate D, as already found for S 51. We also discuss some experiments of CsNiCl 3 related with the present work. @S0163-1829~99!00133-2# Introduction. Magnetization measurements of antiferro- magnets detect various macroscopic quantum effects. One of interesting examples appear in the spin flopping ~SF! process of low-dimensional materials. SF is one of field induced metamagnetic phase transitions of anisotropic Heisenberg antiferromagnets. 1 In terms of the classical spin systems, the transition brings about an abrupt change in the direction of the Ne ´ el order from parallel to perpendicular with respect to the easy axis under the applied field H along the axis and it is a first-order transition with a jump in the magnetization m as a function of H. In quantum systems the jump generally shrinks due to quantum fluctuation and such an effect be- comes larger in lower dimensions and for smaller S. For S 5 1 2 antiferromagnets with the Ising-like anisotropic ex- change coupling, the magnetization jump was revealed to survive down to two dimensions by the recent numerical analysis, 2 while the exact solution 3 by the Bethe ansatz sug- gested that it changes into a second-order transition descibed by the asymptotic behavior m ;~ H 2H c ! 1/d , ~1! where d 52 in one dimension ~1D!. Thus the large quantum fluctuation in 1D seems to change the order of the transition in the system. In view of experimental realization, one of the most im- portant anisotropy is the single-ion anisotropy ~SIA! de- scribed by D ( j ( S j z ) 2 which is relevant only for S >1. In the classical limit in any dimensions the negative D makes the Ne ´ el order oriented along z axis and the first-order SF tran- sition is induced by the applied field parallel to the axis. The recent numerical analysis 4 suggested that instead of SF two successive second-order transitions occur in 1D S 51 anti- ferromagnet for small negative D except for the Haldane phase. 5 This is because the magnetic phase of the system consists of the two gapless Luttinger liquid phases with dif- ferent elementary excitations created by ( S j 1 ) 2 and S j 1 , re- spectively. The first transition, described by the form ~1! with d 52, occurs from the Ne ´ el state to the first massless phase characterized by the dominant spin-correlation func- tion ^ S 0 z S r z & ;cos(2k F r)r 2h z , where 2 k F 52 p m , and the next transition with d 51 leads to the second massless phase where ^ S 0 1 S r 2 & ;( 21) r r 2h . The behavior of the spin- correlation function suggests that the first massless phase, which lies between the two transitions, has no correspon- dence in the classical systems. It implies that the appearance of the phase is an intrinsic quantum effect. Is such a strongly quantized behavior general in 1D, or peculiar to S 51? The question encourages us to investigate 1D larger-S systems. In this paper, using the finite-cluster analysis, we study the magnetization process of the 1D S 5 3 2 and 2 antiferromag- nets in the presence of the Ising-like SIA, in comparison with the case of S 51. Model and numerical calculations. The 1D Heisenberg antiferromagnet with SIA under magnetic field parallel to the easy axis of the Ne ´ el order is described by the Hamiltonian H5H 0 1H Z , H 0 5 ( j S j S j 11 1D ( j ~ S j z ! 2 , H Z 52H ( j S j z , ~2! where S j 2 5S ( S 11) and the periodic boundary condition are applied. We restrict ourselves on the Ising-like anisotropy D ,0 to investigate the transition corresponding to SF. For L-site systems, the lowest energy of H 0 in the subspace where the eigenvalue of ( j S j z is M ~the bulk magnetization is m 5 M / L ) and the momentum is k, is denoted as E k ( L , M ). In addition we define E ( L , M ) as the lowest one among E k ( L , M )’s. Using Lanczos algorithm, we calculated E k ( L , M ) for even-site systems up to L 520, 14, and 12 for S 51, 3 2 , and 2, respectively. Two gapless phases. The magnetic ground state ~GS! for S .1 is divided into two massless phases 6 in the phase dia- gram on the m / S -D ( D ,0) plane as well as S 51. The argument on the two phases for S 51 developed in the Ref. 4 is easily generalized for arbitrary S as follows: ~i! Large negative D phase; the gapless excitation is created by ( S j 1 ) 2 S and the soft mode has the momentum 2 k F 5(1 2m / S ) p . ~ii! Small negative D phase; the S j 1 excitation is gapless and the soft mode has 2 k F 8 52 mS p . The dominant spin-correlation functions are ~i! ^ S 0 z S r z & ;cos(2k F r)r 2h z and ~ii! ^ S 0 1 S r 2 & ; ( 21) r r 2h , respectively. Thus in quasi-1D the coherent in- terchain interaction is expected to result in the long-range ~i! incommensurate SDW along the z axis and ~ii! Ne ´ el order in the xy plane ~called the canted Ne ´ el order in some texts!, respectively. In this paper we call the two massless phases ~i! ‘‘z -SDW’’ and ~ii! ‘‘xy -Ne ´ el’’ phases to emphasize the long-range order expected to be observed in real quasi-1D systems, whereas pure 1D systems have some power-law decays in the associated spin correlation functions. PHYSICAL REVIEW B 1 SEPTEMBER 1999-I VOLUME 60, NUMBER 9 PRB 60 0163-1829/99/60~9!/6238~4!/$15.00 6238 ©1999 The American Physical Society

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PHYSICAL REVIEW B 1 SEPTEMBER 1999-IVOLUME 60, NUMBER 9

Spin flop in one-dimensional quantum antiferromagnets

Toru SakaiFaculty of Science, Himeji Institute of Technology, Ako, Hyogo 678-1297, Japan

~Received 19 February 1999!

The field induced transition corresponding to the spin flop in the quantum antiferromagnetic chains in thepresence of the Ising-like single-ion anisotropyD(,0) is studied by the finite-cluster analysis. It is found thatfor S> 3

2 a first-order metamagnetic transition occurs even in one dimension for small negativeD except for theHaldane phase, while there exist two second-order transitions and an incommensurate phase appears betweenthem for intermediateD, as already found forS51. We also discuss some experiments of CsNiCl3 related withthe present work.@S0163-1829~99!00133-2#

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Introduction. Magnetization measurements of antiferrmagnets detect various macroscopic quantum effects. Oninteresting examples appear in the spin flopping~SF! processof low-dimensional materials. SF is one of field inducmetamagnetic phase transitions of anisotropic Heisenbantiferromagnets.1 In terms of the classical spin systems, ttransition brings about an abrupt change in the directionthe Neel order from parallel to perpendicular with respectthe easy axis under the applied fieldH along the axis and itis a first-order transition with a jump in the magnetizationmas a function ofH. In quantum systems the jump generashrinks due to quantum fluctuation and such an effectcomes larger in lower dimensions and for smallerS. For S5 1

2 antiferromagnets with the Ising-like anisotropic echange coupling, the magnetization jump was revealedsurvive down to two dimensions by the recent numerianalysis,2 while the exact solution3 by the Bethe ansatz suggested that it changes into a second-order transition descby the asymptotic behavior

m;~H2Hc!1/d, ~1!

whered52 in one dimension~1D!. Thus the large quantumfluctuation in 1D seems to change the order of the transiin the system.

In view of experimental realization, one of the most important anisotropy is the single-ion anisotropy~SIA! de-scribed byD( j (Sj

z)2 which is relevant only forS>1. In theclassical limit in any dimensions the negativeD makes theNeel order oriented alongz axis and the first-order SF transition is induced by the applied field parallel to the axis. Trecent numerical analysis4 suggested that instead of SF twsuccessive second-order transitions occur in 1DS51 anti-ferromagnet for small negativeD except for the Haldanephase.5 This is because the magnetic phase of the sysconsists of the two gapless Luttinger liquid phases with dferent elementary excitations created by (Sj

1)2 and Sj1 , re-

spectively. The first transition, described by the form~1!with d52, occurs from the Ne´el state to the first masslesphase characterized by the dominant spin-correlation fution ^S0

zSrz&;cos(2kFr)r2hz, where 2kF52pm, and the next

transition with d51 leads to the second massless phwhere ^S0

1Sr2&;(21)r r 2h. The behavior of the spin

correlation function suggests that the first massless phwhich lies between the two transitions, has no correspdence in the classical systems. It implies that the appearof the phase is an intrinsic quantum effect. Is such a stron

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quantized behavior general in 1D, or peculiar toS51? Thequestion encourages us to investigate 1D larger-S systems. Inthis paper, using the finite-cluster analysis, we studymagnetization process of the 1DS5 3

2 and 2 antiferromag-nets in the presence of the Ising-like SIA, in comparison wthe case ofS51.

Model and numerical calculations.The 1D Heisenbergantiferromagnet with SIA under magnetic field parallel to teasy axis of the Ne´el order is described by the Hamiltonia

H5H01HZ , H05(j

Sj•Sj 111D(j

~Sjz!2,

HZ52H(j

Sjz , ~2!

whereSj25S(S11) and the periodic boundary condition a

applied. We restrict ourselves on the Ising-like anisotroD,0 to investigate the transition corresponding to SF. FL-site systems, the lowest energy ofH0 in the subspacewhere the eigenvalue of( jSj

z is M ~the bulk magnetization ism5M /L) and the momentum isk, is denoted asEk(L,M ).In addition we defineE(L,M ) as the lowest one amonEk(L,M )’s. Using Lanczos algorithm, we calculateEk(L,M ) for even-site systems up toL520, 14, and 12 for

S51, 32 , and 2, respectively.

Two gapless phases.The magnetic ground state~GS! forS.1 is divided into two massless phases6 in the phase dia-gram on them/S-D (D,0) plane as well asS51. Theargument on the two phases forS51 developed in the Ref. 4is easily generalized for arbitraryS as follows: ~i! LargenegativeD phase; the gapless excitation is created by (Sj

1)2S

and the soft mode has the momentum 2kF5(12m/S)p. ~ii !Small negativeD phase; theSj

1 excitation is gapless and thsoft mode has 2kF852mSp. The dominant spin-correlationfunctions are~i! ^S0

zSrz&;cos(2kFr)r2hz and ~ii ! ^S0

1Sr2&;

(21)r r 2h, respectively. Thus in quasi-1D the coherentterchain interaction is expected to result in the long-range~i!incommensurate SDW along thez axis and~ii ! Neel order inthe xy plane ~called the canted Ne´el order in some texts!,respectively. In this paper we call the two massless phase~i!‘‘ z-SDW’’ and ~ii ! ‘‘ xy-Neel’’ phases to emphasize thlong-range order expected to be observed in real quassystems, whereas pure 1D systems have some powerdecays in the associated spin correlation functions.

6238 ©1999 The American Physical Society

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Phase diagram.Consider the two energy gaps defined

D15E~L,M11!1E~L,M21!22E~L,M !,~3!D2kF

5E2kF~L,M !2E~L,M !.

D1 is the sum of the gaps corresponding todM51 and21 for the total HamiltonianH. SinceD1 (D2kF

) is open~gapless! in the z-SDW phase, while gapless~open! in thexy-Neel phase, the crossing point of the two gaps for fixedmis a good estimation of the phase boundary. Since only stwith M52Sn (n50,1,2, . . . ) contributes to the magnetization process in thez-SDW phase, our analysis is focusedsuch states. The crossing points of the finite clusters wvarious magnetizations are plotted on them/S-D plane asopen symbols in Figs. 1~a! S5 3

2 and ~b! S52. Little sizedependence of the curves justifies that the fitted solid liare the phase boundary of the bulk systems.Dc0 denotes theboundary between the Ne´el ordered and disordered phasesthe nonmagnetic GS. It corresponds to the boundary ofHaldane phase for integerS, while Dc050 for half odd in-tegerS. Using the phenomenological renormalization,7 Dc0of the S52 system is estimated asDc0520.00160.001.The phase boundary atm/S51 denotesDc2. It was deter-mined as the crossing point ofE(L,SL)2E(L,SL21) and

FIG. 1. m/S-D phase diagrams for~a! S532 and~b! S52. Open

symbols are the crossing point of the two gapsD1 and D2kFfor

finite systems. The fitted solid curves are the boundaries betwthe z-SDW andxy-Neel phases. Solid circles are the points wie9(m)50 and the curvaturee9(m) is negative in the regions surrounded them.~Dashed curves are guides for the eyes.!

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E(L,SL)2E(L,SL22S) which is almost independent ofL.The results areDc2521.31 and21.25 for S5 3

2 and 2, re-spectively. Them/S-D phase diagrams suggest that besidthe two ordinary critical fieldsHc1 and Hc2, which are thestarting and saturation points of the magnetization, resptively, the magnetization process has an intermediate critfield Hc3 (Hc1,Hc3,Hc2) corresponding to the boundarof the two massless phases forDc0.D.Dc2. According tothe argument forS51, the phase transitions at the three crical points are all second order and have the asymptotic fo

m;~H2Hc1!1/2, ~4!

m2mc3;uH2Hc3u, ~5!

S2m;~Hc22H !1/2, ~6!

respectively, wheremc3 is the magnetization correspondinto the boundary between the two massless phases. Inclassical limit the first-order SF transition occurs andmjumps from 0 to @2D/(D12)#1/2S @long-dashed lines inFigs. 1~a! and ~b!# for 0.D.21, while 0 to S for D,21. The phase diagrams including only the two masslphases might lead to the conclusion that even for largerS, aswell as S51, two second-order transitions occur insteadSF for Dc0.D.Dc2 and the phase boundary between ttwo massless phases will go to the classical line asS in-creases towards .

Spin flop.In contrast to the case ofS51, the followinganalysis will reveal that the first-order SF transition can ocur for S> 3

2 . The lowest eigenvalue of the HamiltonianH0for 0,m,S per site in the thermodynamic limit, denotede(m), can be estimated using the form8

~1/L ! E~L,M !;e~m!1C~m!1/L2, ~7!

where the second term is the size correction. The GS mnetization curve is derived from minimizing the total enere tot(m)[e(m)2Hm with respect tom. As far as the deriva-tive e8(m) is a monotonously increasing function ofm, thatis e9(m)>0, the curve is determined by the relationH5e8(m) and no discontinuity appears in the curve. On tother hand, if there exists a region wheree9(m) is negative,a first-order transition occurs and the magnetization jumover the region. Thus the presence of such an areae9(m),0 is a sufficient condition of the appearance of tfirst-order SF transition. Since onlyM52Sn is available inthe z-SDW phase, the form to estimate the derivativee8(m)is given by

E~L,M12S!22E~L,M !;e8~m!1Se9~m!1/L1O~1/L2!.~8!

Thene9(m) is estimated from the form

R~L,M ![L@E~L,M12S!1E~L,M22S!

22E~L,M !#/~2S!2

;e9~m!1O~1/L2!. ~9!

R(L,2S) is plotted vsD for S51, 32 , and 2 in Figs. 2~a!, ~b!,

and ~c!, respectively. They indicate the existence of thegion with R(L,2S),0 for S5 3

2 and 2, and it tends to extentowardsD50 with increasingL, in contrast with the case oS51 whereR(L,2S) is always positive. Neglecting the sizcorrection in Eq.~9! and takingR(L,M ) for the largestLavailable for eachm ase9(m), the points withe9(m)50 areplotted as solid circles in Figs. 1~a! and ~b!. In the area sur-

en

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6240 PRB 60BRIEF REPORTS

FIG. 2. R(L,2S) vs D of the finite systems for~a! S51, ~b! S5

32 , and~c! S52. R(L,2S) vs 1/L

with fixed D; ~d! S51 D520.8, ~e! S532 D5

20.6, and~f! S52 D520.6. These values ofDare around the minima~local minimum for S51) of the largest-L curves in~a!, ~b!, and ~c!.Dashed lines are based on the least-square fitt

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rounded by theme9(m) is negative. The fitted dashed lineare guides for the eyes drawn assuming that the negae9(m) region continues toDc0. Only for S51, e9(m)>0 issatisfied everywhere on them-D plane within the sameanalysis up toL520. For S5 3

2 and 2 the existence of tharea withe9(m),0 implies that the first-order SF transitiooccurs andm jumps over the region. Figures 1~a! and ~b!suggest that SF can appear only in the small negativDregion. This is because the large negativeD strongly quan-tizes the system and makes it equivalent to theS5 1

2 XXZmodel which is proved to have no first-order transition. Fures 1~a! and~b! also reveal that the negativee9(m) region islarger forS5 3

2 thanS52. It is reasonable because smalleS systems are expected to have larger quantum fluctuawhich suppresses the magnetization jump. The large ntive D border of the SF region, denoted asDc1, is roughlyestimated from the boundary of the negativee9(m) regionfor the finite systems as shown in Figs. 1~a! and ~b!. Theresults areDc1.20.9 and21.1 for S5 3

2 and 2, respec-tively. The size correction neglected on the estimation ofDc1is not supposed to be so large as to change the followqualitative argument, because Figs. 2~b! and~c! suggest thatthe left border of the region withR(L,M ),0 has no signifi-cantL dependence in comparison with the right border. FDc0.D.Dc1 the magnetization process is expected to ha jump fromm50 to thexy-Neel phase. Figures 1~a! and~b!might suggest the possibility of the jump from nonzeromnearDc1, but it is difficult to determine the point of the jumwithin the present small cluster analysis because the fiorder transition is ruled by the Maxwell construction, whicannot be applied until a more precise curve ofe(m) isgiven.

To convince of the existence of SF forS5 32 and 2 in the

thermodynamic limit, we investigate the behavior ofe(m)around the limitm˜01. Assuming that the size correctioof e(m) is at mostO(1/L2) even in the massive phase (m50), the size dependence ofR(L,2S) is expanded aroundm50 as

R~L,2S!;e9~0!1~2S!e-~0!1/L1O~1/L2!. ~10!

Around the minimum~local minimum for S51) of theR(L,2S)-D curve in Figs. 2~a!–~c! for the largestL (D5

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20.8 for S51 andD520.6 for S5 32 and 2!, the size de-

pendence ofR(L,2S) is investigated by the plot ofR(L,2S)vs 1/L in Figs. 2~d! S51, ~e! S5 3

2 , and ~f! S52, wheredashes lines are the results from the least-square fitting. Csidering the form~10!, the plots forS5 3

2 and 2 both suggese9(0),0 and e-(0).0 which is a sufficient condition ofthe appearance of the magnetization jump fromm50 atsome critical field in the thermodynamic limit. On the othhand, the plot forS51 implies e9(0)50 and e-(0).0which is consistent with the second-order transition atHc1described by the asymptotic form~4!. These results also support the existence of SF forS> 3

2 in the thermodynamiclimit. Since the quantum fluctuation becomes smallerlargerS, the SF region should become larger with increasS and in the limit S˜` the region would occupy all theregion at the left-hand side of the classical SF line inm/S-D phase diagrams in Figs. 1~a! and ~b!.

Possible magnetication processes.Based on them/S-Dphase diagrams in Figs. 1~a! and~b!, we summarize all pos-sible cases of the magnetization process of 1D Heisenbantiferromagnet with arbitraryS(. 1

2 ) in the presence of theIsing-like SIA. In the following discussion the critical fieldHc1 and Hc2 always stand for the starting and saturatipoints of the magnetization. The schematic magnetizacurves are shown in Figs. 3~a!–~d!, and their features arelisted as follows: ~a! D.Dc0 ~only for integer S); thesecond-order transition with the asymptotic form~4! fromthe Haldane toxy-Neel phases occurs atHc1. ~b! Dc0.D.Dc1; the first-order SF transition occurs atHc1 from theNeel to xy-Neel phase. The present analysis leads toabsence of the case only forS51. ~c! Dc1.D.Dc2; thesecond-order transition from the Ne´el to z-SDW phases atHc1 described by the form~4! is followed by another secondorder transition to thexy-Neel phase atHc3 with the form~5!. ~d! D,Dc2; the second-order transition from the Ne´el toz-SDW phases and the latter phase continues until the sration. The last case~d! in the large negativeD region isequivalent to the Ising-likeS5 1

2 XXZ chain. In any case thetransition at the saturation pointHc2 is a second-order onedescribed by the critical behavior~6!. As S increases towardsthe classical limit, the boundariesDc1 andDc2 will go to the

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PRB 60 6241BRIEF REPORTS

FIG. 3. Schematic magnetization curvesfour possible cases. Solid and dashed curvesnify the xy-Neel and z-SDW phases, respectively, except for massive regions.

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same limit 1 and the case~c! will disappear in the limit. ForD,Dc2 Hc1 and Hc2 will become close to each other ancoincide in the limitS˜`.

Discussion.Among the above magnetization processthe most interesting case is~c! in Fig. 3 where the twosecond-order transitions take place and thez-SDW phase ap-pears between them. The most realistic system where sumagnetization process can be realized is theS51 chain. Oneof good candidates to observe the process is expectedCsNiCl3, a quasi-1DS51 antiferromagnet with the Isinglike SIA, which has the Ne´el order at low temperatures.9 Thehigh-field magnetization measurements10,11on CsNiCl3 indi-cated a SF-like transition. However, the experimental mnetization curve also looks like a second-order transitionscribed by the critical behavior~4!, or two successivetransitions accompanied by an intermediate phase.12 In addi-tion the NMR measurement13 on CsNiCl3 indicated two spinstructures oriented in different directions around the trantion. It may imply that another transition from thez-SDW toxy-Neel phases occurs soon after the first one. It woulddifficult to detect the latter transition atHc3 in the measuredmagnetization curve, because the transition brings abousignificant anomalous behavior in the curve.4 As well asNMR, the ESR or neutron scattering would be a betmethod to detect the change in the direction of the dominspin correlation or the long-range order from parallel to p

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pendicular with respect toH on the transition atHc3. Itwould be interesting to observe thez-SDW phase which is anessential quantum effect.

Since the quantum effect is generally larger for smasystems, the first-order SF transition is more difficult to ocur in finite clusters, treated in the present study, than ininfinite system. Therefore the conclusion that SF due to Scan occur forS> 3

2 in 1D is expected to be valid even in ththermodynamic limit. In contrast, the absence of SF forS51 should be checked by some other methods likeMonte Carlo or density matrix renormalization group14 totreat larger chains. It would be also interesting to performmagnetization measurement on some 1D larger-S materialswith negativeD (uDu,J), where the first-order transition iexpected to occur, and to compare the curve with that ofS51 chain CsNiCl3.

Summary.The finite-cluster analysis on 1DS5 32 and 2

antiferromagnets suggested that the first-order SF transcan occur forS> 3

2 in the presence of the Ising-like SIA. Walso discussed on all possible magnetization processesD<0 including the case when two successive second-otransitions appear instead of SF.

It is a pleasure to thank K. Totsuka, H. Nojiri, and YAjiro for helpful discussions. The numerical computatiowas done using the facility of the Supercomputer CenInstitute for Solid State Physics, University of Tokyo.

d

1L. Neel, Ann. Phys.~Paris! 5, 232 ~1936!.2M. Kohno and M. Takahashi, Phys. Rev. B56, 3212~1997!.3C.N. Yang and C.P. Yang, Phys. Rev.151, 258 ~1966!.4T. Sakai, Phys. Rev. B58, 6268~1998!.5F.D.M. Haldane, Phys. Lett. A93, 464 ~1993!; Phys. Rev. Lett.

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M. Chiba ~unpublished!.14S.R. White, Phys. Rev. Lett.69, 2863~1992!.