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VOLUME 75, NUMBER 5 PH YS ICAL REVIEW LETTERS 31 JULY 1995 Spin Dependence of Correlations in Two-Dimensional Square-Lattice Quantum Heisenberg Antiferromagnets N. Elstner, ' A. Sokol, R. R. P. Singh, M. Greven, and R. J. Birgeneau ~Service de Physique Theorique, Commissariat a l'Energie Atomique Sac-lay, 91191 Gif sur 1'vet-te Cedex, France Department of Physics, University of Illinois at Urbana Cha-mpaign, Urbana, Illinois 61801-3080 and L.D. Landau Institute, Moscow, Russia Department of Physics, University of California, Davis, California 95616 4Department of Physics, Massachusetts In stitute .of Technology, Cambridge, Massachusetts 02139 (Received 28 February 1995) We present a series expansion study of spin-S square-lattice Heisenberg antiferromagnets. The numerical data are in excellent agreement with recent neutron scattering measurements. Our key result is that the correlation length g for S ) 1/2 strongly deviates from the exact T ~ 0 (renormalized classical) scaling prediction for all experimentally and numerically accessible temperatures. We note basic trends with S of the experimental and series expansion correlation length data and propose a sequence of crossovers from renormalized classical to classical to Curie-Weiss regimes to explain them. PACS numbers: 75. 10. Jm, 75.25.+z, 75.30. Ds, 75. 40. s In recent years much attention has focused on two- dimensional (2D) square-lattice quantum Heisenberg an- tiferromagnets, described by the Hamiltonian (2) H = JQS;. SJ, (1) (1j) where (ij) denotes summation over all pairs of near- est neighbors. In their seminal work [1], Chakravarty, Halperin, and Nelson (CHN), utilizing a mapping of the low-energy spectrum of Eq. (1) onto the quantum nonlinear sigma model (QNLo. M), have shown that the low-temperature properties of these systems obey renor- malized classical (RC) scaling, where the correlation length $ = 0. 31(c/2~p, ) exp(2vrp, /T) for T && p, (p, is the T = 0 spin stiffness and c the spin-wave velocity). Subsequently, Hasenfratz and Niedermayer calculated for the 2D QNLo. M the exact value of the prefactor and the leading O(T/2mp, ) correction [2]: e c 2~p, l exp 8 2~p, T T T x I + 4' p, 2' p, ) The neutron scattering measurements of s (T) in S = 1/2 layered Heisenberg antiferromagnets such as LazCu04 [3] and SrzCuOzC12 [4] reveal a remarkable agreement with Eq. (2). At first sight, one would expect the RC description to improve as the value of the spin is increased. If S is formally regarded as a continuous variable, then the Neel order is expected to vanish for some S ( 1/2. At the critical point, where p, vanishes, Eq. (2) no longer applies and the correlation length is, in fact, inversely proportional to the temperature; this is the quantum critical regime (QC), discussed in Refs. [1, 5]. [It was argued earlier [5, 6] that the S = 1/2 Heisenberg model (1) exhibits certain signatures of QC behavior for T ) 0.51; we do not discuss RC to QC crossover effects in this Letter. ] Naively, increasing the value of spin would move the system away from this limit so that the RC behavior would be more pronounced. However, such an expectation does not hold. Recently, Greven and co-workers [4,7] have reported a significant discrepancy between the neutron scattering measurements of the correlation length in the S = 1 systems K2NiF4 and La&Ni04 and the RC prediction. Preliminary experiments on the S = 5/2 system RbzMnF4 reveal an even larger dis- crepancy [8]. As is evident from Fig. 1, series expansion results for S = 1 [11] are in excellent agreement with the experimental data in the region of overlap, and also deviate substantially from the RC prediction (here, $ is in units of the lattice constant a). In order to investigate the origin of this deviation, we have calculated high-temperature expansions for the Fourier transform of the spin-spin correlation function (S' S') for all spin values in the range from S = 1/2 to S = 5/2. We present the data for the antiferromagnetic structure factor So = S(Q), where Q = (7r/a, ~/a), and for the second moment correlation length s, defined by = 8 S(Q + q)/Bq ~~ o/2Sti =The s. eries are ana- lyzed either by performing a direct Pade approximation or by taking the logarithm of the series first and then calcu- lating the Pade approximants. The latter is likely to show better convergence if the correlation length increases ex- ponentially fast at low temperatures. We restrict ourselves to temperature ranges where different methods of extrapo- lation agree within a few percent. The ratio between the calculated g and the Hasenfratz and Niedermayer formula s HN given by Eq. (2) is plotted in Fig. 2 for different spins as a function of T/p, Contrary to the naive expectation that the RC behavior becomes more pronounced as S increases, we find that for the range of temperatures probed here $ monotonically 938 0031-9007/95/75(5)/938(4)$06. 00 1995 The American Physical Society

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Page 1: Spin Dependence of Correlations in Two-Dimensional Square-Lattice Quantum Heisenberg Antiferromagnets

VOLUME 75, NUMBER 5 PH YS ICAL REVIEW LETTERS 31 JULY 1995

Spin Dependence of Correlations in Two-Dimensional Square-LatticeQuantum Heisenberg Antiferromagnets

N. Elstner, ' A. Sokol, R. R. P. Singh, M. Greven, and R. J. Birgeneau~Service de Physique Theorique, Commissariat a l'Energie Atomique Sac-lay, 91191 Gif sur 1'vet-te Cedex, France

Department of Physics, University of Illinois at Urbana Cha-mpaign, Urbana, Illinois 61801-3080and L.D. Landau Institute, Moscow, Russia

Department of Physics, University of California, Davis, California 956164Department of Physics, Massachusetts In stitute .of Technology, Cambridge, Massachusetts 02139

(Received 28 February 1995)

We present a series expansion study of spin-S square-lattice Heisenberg antiferromagnets. Thenumerical data are in excellent agreement with recent neutron scattering measurements. Our key resultis that the correlation length g for S ) 1/2 strongly deviates from the exact T ~ 0 (renormalizedclassical) scaling prediction for all experimentally and numerically accessible temperatures. We notebasic trends with S of the experimental and series expansion correlation length data and propose asequence of crossovers from renormalized classical to classical to Curie-Weiss regimes to explain them.

PACS numbers: 75.10.Jm, 75.25.+z, 75.30.Ds, 75.40.—s

In recent years much attention has focused on two-dimensional (2D) square-lattice quantum Heisenberg an-

tiferromagnets, described by the Hamiltonian

(2)

H = JQS;. SJ, (1)(1j)

where (ij) denotes summation over all pairs of near-est neighbors. In their seminal work [1], Chakravarty,Halperin, and Nelson (CHN), utilizing a mapping ofthe low-energy spectrum of Eq. (1) onto the quantumnonlinear sigma model (QNLo. M), have shown that thelow-temperature properties of these systems obey renor-malized classical (RC) scaling, where the correlationlength $ = 0.31(c/2~p, ) exp(2vrp, /T) for T && p, (p,is the T = 0 spin stiffness and c the spin-wave velocity).Subsequently, Hasenfratz and Niedermayer calculated forthe 2D QNLo. M the exact value of the prefactor and theleading O(T/2mp, ) correction [2]:

e c 2~p, lexp

8 2~p, T

T Tx I — +4' p, 2' p, )The neutron scattering measurements of s (T) in

S = 1/2 layered Heisenberg antiferromagnets such asLazCu04 [3] and SrzCuOzC12 [4] reveal a remarkableagreement with Eq. (2). At first sight, one would expectthe RC description to improve as the value of the spinis increased. If S is formally regarded as a continuousvariable, then the Neel order is expected to vanish forsome S ( 1/2. At the critical point, where p, vanishes,Eq. (2) no longer applies and the correlation length is, infact, inversely proportional to the temperature; this is thequantum critical regime (QC), discussed in Refs. [1,5].[It was argued earlier [5,6] that the S = 1/2 Heisenbergmodel (1) exhibits certain signatures of QC behavior for

T ) 0.51; we do not discuss RC to QC crossover effectsin this Letter. ] Naively, increasing the value of spinwould move the system away from this limit so that theRC behavior would be more pronounced.

However, such an expectation does not hold. Recently,Greven and co-workers [4,7] have reported a significantdiscrepancy between the neutron scattering measurementsof the correlation length in the S = 1 systems K2NiF4 andLa&Ni04 and the RC prediction. Preliminary experimentson the S = 5/2 system RbzMnF4 reveal an even larger dis-crepancy [8]. As is evident from Fig. 1, series expansionresults for S = 1 [11]are in excellent agreement with theexperimental data in the region of overlap, and also deviatesubstantially from the RC prediction (here, $ is in units ofthe lattice constant a).

In order to investigate the origin of this deviation,we have calculated high-temperature expansions for theFourier transform of the spin-spin correlation function(S' S') for all spin values in the range from S = 1/2 toS = 5/2. We present the data for the antiferromagneticstructure factor So = S(Q), where Q = (7r/a, ~/a), andfor the second moment correlation length s, defined by

= —8 S(Q + q)/Bq ~~ o/2Sti =The s.eries are ana-lyzed either by performing a direct Pade approximation orby taking the logarithm of the series first and then calcu-lating the Pade approximants. The latter is likely to showbetter convergence if the correlation length increases ex-ponentially fast at low temperatures. We restrict ourselvesto temperature ranges where different methods of extrapo-lation agree within a few percent.

The ratio between the calculated g and the Hasenfratzand Niedermayer formula s HN given by Eq. (2) is plottedin Fig. 2 for different spins as a function of T/p,Contrary to the naive expectation that the RC behaviorbecomes more pronounced as S increases, we find that forthe range of temperatures probed here $ monotonically

938 0031-9007/95/75(5)/938(4)$06. 00 1995 The American Physical Society

Page 2: Spin Dependence of Correlations in Two-Dimensional Square-Lattice Quantum Heisenberg Antiferromagnets

VOLUME 75, NUMBER 5 PH YS ICAL REVIEW LETTERS 31 JUr v 1995

~ K2NIF4

La2Ni04

Series

-——Renarmalized ClassicalI

1.3I

1.5T/J

I

1.7I

1.9 2.1

deviates from

KAHN

as 5 increases. As noted above, thissame systematic trend is seen experimentally [3,4,7,8].

In an attempt to understand these data, we have con-sidered various possible scaling scenarios. To high-

FIG. 1. Semilogarithmic plot of the series expansion result forthe correlation length in the 5 = 1 Heisenberg antiferromagnetversus T/J (solid line) together with the experimental data forK2NiF4 (solid circles, [4]) and La2Ni04 (open circles, [7]).Numerical and experimental results agree with each other, andall deviate from the exact RC prediction [2] evaluated usingknown values for c and p, [9,10] (dashed line).

light the expected spin dependence, we note that for asquare-lattice nearest-neighbor Heisenberg antiferromag-net p, = Z~(5)J52 and c = 23/ aZ, (5)JS. The quantumrenormalization factors Z~(5) and Z, (5) are known fromspin-wave theory [9] (for all values of S), T = 0 se-ries expansion [10] (for 5 = 1/2 and S = 1), and MonteCarlo studies [12] (for 5 = 1/2). For S = 1/2 and 5 =1 these different methods yield good agreement.

Equation (2) may then be written as

5$ eZ, /2~Zp la 2 / vrZ &T/JS jexp 1—

(3)which suggests plotting 5$/a vs T/JS to elucidate thedependence on S. The results so obtained are shown inFig. 3. For S = 1/2 we also show Monte Carlo data[13],which extend to somewhat lower temperatures thanthe series expansion result. Surprisingly, over the rangeof 5$/a from 1 to at least 50, the data to a goodapproximation fall on the same curve. For the sake ofclarity, we have omitted experimental data for La2Cu04(5 = 1/2), LazNi04 (5 = 1), and Rb2MnF4 (5 = 5/2),al) of which fa11 on the approximate "scaling" curve ofFig. 3 to within experimental error.

It was demonstrated in Ref. [4] that Eq. (3) describesthe correlation length data in absolute units in the S =1/2 system Sr2Cu02C12 extremely well. Interpretednaively, Fig. 3 would then suggest that RC behavior holdsfor all 5, but with quantum renormalization factors Z~(5)and Z, (5) that are nearly 5 independent for 1/2 ~ 5 ~

0.4- S =3/

S = 1/2

C$

V)

20

5

I I

Sr2Cu02CI2 (S=1/2) .

~ K2N]F, (S=1)SE (S=1/2)

——SE (S=3/2)~ ~ ~ ~

&MR

~ MC (S=1/2)

S = 5/2

I

1.0I

1.5T/JS'

I

2.0 2.5

FIG. 2. Series expansion results for the correlation lengthplotted as g/KAHN vs T/p, Here KAHN is the exact . RCprediction [2], Eq. (2), with c and p, from Refs. [9,10].The ratio g/KAHN, which is fairly close to unity for S =1/2, gradually decreases as S increases, suggesting increasingdisagreement with RC theory for larger S.

FIG. 3. Semilogarithmic plot of Sg/a vs T/JS ofSr2Cu02C12 (S = 1/2, open circles, [4]) and K2NiF4 (S = 1,solid circles, [4]) together with the series expansion results(1/2 ~ S ~ 5/2, represented by the various lines) as well asMonte Carlo data (S = 1/2, solid squares, [13]). Howeverimpressive, this data collapse is inconsistent with Eq. (2) withc and p, from Ref. [9].

939

Page 3: Spin Dependence of Correlations in Two-Dimensional Square-Lattice Quantum Heisenberg Antiferromagnets

VOLUME 75, NUMBER 5 PHYSICAL REVIEW LETTERS 31 JUr v 1995

5/2 and close to their values at 5 = 1/2, Z~(1/2) =0.72 and Z, (1/2) = 1.18. However, this is very unlikelySpin-wave theory [9] predicts a substantial S dependence:Z~(5/2) = 0.95 and Z, (5/2) = 1.03, which is alreadyclose to the classical limit (5 ~ ~) in which ZP, Z, ~ 1.Serious errors in these values seem very unlikely, giventheir good agreement for 5 = 1/2 and 5 = 1 with thoseobtained by other methods [10,12].

In an attempt to understand these data, it is helpful torecall that at any fixed T/p, RC theory will inevitably failfor sufficiently large values of spin. Indeed, a straight-forward application of Eq. (2) to the 5 ~ ~ limit taken atT/JS —1 would predict se = 0. However, this limit cor-responds to the classical Heisenberg magnet, where g/a isknown to be nonzero and of order unity for T —JS .

One may understand where Eq. (2) might fail by fol-lowing CHN in their derivation of the leading asymptoticbehavior in the RC regime, but taking into account thatS may be large. CHN have shown that for T « p, themagnetic correlations can be calculated using classical dy-namics, except that all wave vector integrations shouldbe limited to ~q~

~ q, = T/c rather than taken over thewhole Brillouin zone. Here "classical dynamics" simplymeans that for ~q~

~ q„all Bose factors for spin wavescan be approximated assuming cq « T. A key resultof CHN is to show that such a calculation yields correct$(T) and other observables for the quantum Heisenbergmodel. In other words, $(T) for T (& p, can be obtainedfrom the expression for the correlation length scL(T) ofthe classical Heisenberg model by substituting the latticespacing of the classical model, acL, by const X (c/T)We now evaluate the cutoff wave vector as

T p, T 5 Tq C

C C PS a PS

This dependence of q, on 5 arises because the spinstiffness is proportional to the square of 5, but the spin-wave velocity is only linear in 5.

For 5 » 1 and T —p, —JS, the cutoff wave vectorq, —5/a » m/a is outside the Brillouin zone. Hence,the requirement that cq ~ T, or equivalently q ~ q„places no further restrictions on the q integrations whichare already limited by the Brillouin zone. In this caseall of the integrals are the same as those of the classicalHeisenberg magnet, and the classical 5 ~ cc limit isrecovered.

The crossover temperature T„between the RC andclassical regimes depends on S, and its order of magnitudecan be estimated as the temperature where q, —aThis yields

T„———JS, while ps —JS .C ~ 2

a

By substituting T„ into Eq. (2), one concludes that thecrossover from RC behavior at low temperatures toclassical behavior at higher temperatures should occur fora s „=g(T„) that is larger for larger S.

940

10

7 8

5 e

4

CJ

3

0.6 0.8 1.0T/(JS(S+1))

1.2

FIG. 4. Semilogarithmic plot of the correlation length ob-tained from series expansion plotted versus T/JS(S + 1) forS = 1/2, 1, and 5/2. For larger spins g/a is close to theclassical (S ~ ~) limit, which provides evidence that clas-sical (S ~ ~) magnetic behavior holds for JS && T && JS~,in agreement with our proposed scaling crossover scenario.Note, that in most of the temperature range shown, the 5model is not in the scaling limit and its correlation lengthdeviates from the expected T ~ 0 behavior g/a = const X(T/JS2) exp(2m. JS2/T).

In order to test this scenario, we plot the correlationlength as a function of T/JS(5 + 1) in Fig. 4, whereJS(5 + 1) is the classical (not T = 0) spin stiffness. Inreplacing S2 by 5(5 + 1), we follow a purely empiricalobservation that the correlation length at T » J5 for5 ) 3/2 depends on 5 primarily through the combination5(5 + 1) (for T (& JS, g depends on 5 only through p,and c). We find from our series expansion results that forS & 1 at intermediate temperatures T ~ JS

$(5, T) = scL(TcL), where Tcr = T/JS(S + 1),(6)

as shown in Fig. 4. This result supports the hypothesisthat the deviations from asymptotic RC behavior evidentin Figs. 1 and 2 are primarily driven by RC to classicalcrossover effects.

Before we conclude, we would like to point out somepuzzling features regarding the Lorentzian amplitudeof the spin-spin correlator, Sn/g, where So is thecorrelator magnitude at Q = (7r/a, vr/a), defined suchthat So = S(5 + 1)/3 for T ~ ~. This ratio has onlya power-law temperature dependence, and is thereforeless sensitive to the model parameters than g or Soseparately. The scaling prediction for this quantity is[1]So/$ = Zs2vrNO(T/2' p,), where Z3 is a universalnumber and Wo is the T = 0 sublattice magnetization

Page 4: Spin Dependence of Correlations in Two-Dimensional Square-Lattice Quantum Heisenberg Antiferromagnets

VOLUME 75, NUMBER 5 PH YS ICAL REVIEW LETTERS 31 JULY 1995

defined such that in the classical limit Np = S. The valueof Z3 can be estimated by substituting the numerical datafor the two limiting cases S = 1/2 [11,13] and S ~ ~[14,15] into the above formula. Here we estimate Z3 atthe lowest temperature accessible to us, T;„. We getZ3 = 3.2 for S = 1/2 at T;„=0.35J, and Z3 = 6.6for S ~ ~ at Tm;„= 0.8JS . This strong disagreementfor a parameter, which was shown in Refs. [1,5] to beuniversal, implies that at least one of the models, andpossibly both of them, are not in the scaling limit fortemperatures of order their respective T;„[16].

Here we speculate on two possible causes of thisdiscrepancy. First, if the classical model is in the classicalscaling limit and the S = 1/2 model is not in the RClimit, this discrepancy may be due to the vicinity ofthe classical to RC crossover. In this case, So/$ Tfor the S = 1/2 model should increase at temperatureslower than those studied numerically, in order to reach itspresumed larger scaling limit for T ~ 0. This scenariomay be consistent with neutron scattering measurementswhich give So/s —const [3,4,7].

Alternatively, the finite size of the Brillouin zone (i.e.,lattice corrections) may be important in determining theratio So/g for all values of spin at any numerically orexperimentally accessible temperature, including the tem-perature range where the correlation length for S = 1/2is itself in agreement with the universal RC prediction.

In summary, we present and analyze high-temperatureseries expansion data for the spin-S square-lattice Heisen-berg model for all S in the range S = 1/2 to S = 5/2.In agreement with neutron scattering measurements, wefind that the correlation length deviates from the low-temperature RC prediction of Eq. (2) for S & 1/2, andthat the deviation becomes larger for larger S. We suggestthat this deviation results from a sequence of crossovers,observed for large spin, from renormalized classical [T «JS, Eq. (2)], to classical (JS « T « JS2), to Curie-Weiss (T » JS ) regime. We find that in the proposedclassical regime, the correlation length is given by Eq. (6),i.e., it is nearly equal to the correlation length of the clas-sical S ~ ~ model with the same lattice spacing, whenplotted versus T/ JS(S + 1).

We would like to thank S. Chakravarty, A. V.Chubukov, B.I. Halperin, M. A. Kastner, M. S. Makivic,D. R. Nelson, S. Sachdev, and U.-J. Wiese for helpfuldiscussions. We thank J. Tobochnik and D. N. Lambethfor providing their numerical data for comparisons. A. S.is supported by an A. P. Sloan Research Fellowship.R. R. P. S. is supported by the NSF under Grant No.

[21

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[10]

[11]

[12][13]

[14]

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S. Chakravarty, B.I. Halperin, and D. R. Nelson, Phys.Rev. B 39, 2344 (1989).P. Hasenfratz and F. Niedermayer, Phys. Lett. B 26S, 231(1991);Z. Phys. B 92, 91 (1993).B. Keimer et al , Phy. s. Rev. B 46, 14034 (1992); R. J.Birgeneau et al. (unpublished).M. Greven et al. , Phys. Rev. Lett. 72, 1096 (1994);Z. Phys. B 96, 465 (1995).A. V. Chubukov, S. Sachdev, and J. Ye, Phys. Rev. B49, 11 919 (1994); A. V. Chubukov and S. Sachdev, Phys.Rev. Lett. 71, 169 (1993).A. Sokol, R. L. Glenister, and R. R. P. Singh, Phys. Rev.Lett. 72, 1549 (1994).K. Nakajima et al. , Z. Phys. B 96, 479 (1995).Y.S. Lee, M. Greven, B.O. Wells, and R. J. Birgeneau(unpublished).T. Oguchi, Phys. Rev. 117, 117 (1960); J. Igarashi, Phys.Rev. B 46, 10763 (1992); C. J. Hamer, Z. Weihong, andJ. Oitmaa, Phys. Rev. B 50, 6877 (1994).R. R. P. Singh, Phys. Rev. B 39, 9760 (1989); R. R. P.Singh and D. H. Huse, Phys. Rev. B 40, 7247 (1989);C. J. Hamer, Z. Weihong, and J. Oitmaa, Phys. Rev. B 50,6877 (1994).N. Elstner, R. L. Glenister, R. R. P. Singh, and A. Sokol,Phys. Rev. B 51, 8984 (1995).U.-J. Wiese and H. -P. Ying, Z. Phys. B 93, 147 (1994).We used the data by M. Greven, U.—J. Wiese, andR. J. Birgeneau (unpublished); similar results were earlierobtained by M. S. Makivic and H. -Q. Ding, Phys. Rev. B43, 3562 (1991).M. Liischer and P. Weisz, Nucl. Phys. B300, 325 (1988);D. N. Lambeth and H. E. Stanley, Phys. Rev. B 12, 5302(1975); S. H. Shenker and J. Tobochnik, Phys. Rev. B 22,4462 (1980).S. Tyc, B.I. Halperin, and S. Chakravarty, Phys. Rev. Lett.62, 835 (1989).Our estimate of Z3 from the S ~ oo model agreeswith earlier Monte Carlo estimates; see P. Kopietz andS. Chakravarty, Phys. Rev. B 40, 4858 (1989), fordiscussion and references. Z3 can also be calculateddirectly using I/N expansion. As pointed out in a recentpreprint by A. Sandvik, A. V. Chubukov, and S. Sachdev,

1INZ3 = 2.15, which is 40% smaller than our estimatebased on the S = 1/2 model. With increasing S theagreement becomes worse: there is about a factor of 3

]/Ndifference between Z3 and our S ~ oo estimate.

DMR 93-18537. The work at MIT was supported by theNSF under Grant No. DMR 93-15715 and by the Centerfor Materials Science and Engineering under NSF GrantNo. DMR 94-00334.

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