PHYSICAL REVIEW B VOLUME 36, NUMBER 4 1 AUGUST 1987

Quantum spin chains with alternation

H. W. J. BloteLaboratorium voor Technische Natuurkunde, Technische Universiteit,

P O. .Box 5046, 2600 GA Delft, The Netherlands

Jill C. BonnerDepartment of Physics, University of Rhode Island,

Kingston, Rhode Island 0288I -08I 7(Received 13 March 1987)

ANeck has made predictions for the critical exponents of a class of integrable spin chains ofgeneral spin s. The expressions are s dependent. Our interest here is a finite-size scaling analysisof spin-1 systems to determine the validity of the prediction for the alternation critical exponent.We discuss these results in the light of our numerical results for the conformal anomaly c, the ex-ponent g, and the mass-gap exponent associated with biquadratic exchange. The numerical re-sults are not generally in good agreement with theory, and hence the possible presence of logarith-mic corrections is discussed.

INTRODUCTION

A new way to derive the critical properties of quantumspin chains of arbitrary spin s has been described byAfBeck, ' building on ideas of Haldane. AfBeck has pro-posed a number of mappings between various types ofspin-chain and two-dimensional (2D) continuum fieldtheories, namely, nonlinear o. models and related Wess-Zumino models. He extends the classification of criticaltheories in 2D by constraints of conformal invariance onsystems with continuous symmetries. Conformal invari-ance in these Wess-Zumino models allows computation ofthe anomalous dimension of operators from which thecritical exponents of the quantum spin chains are de-duced.

A prediction of particular interest relates to the locali-zation length problem of the 2D quantum Hall system.AfBeck' adds a staggering interaction

a g ( —1)"S„S,+,

to an integrable, i.e., gapless, quantum chain of spin s. Asuitable set of integrable Hamiltonians is provided byBethe-ansatz integrable models of SU(2) symmetrywhose Hamiltonians take the form of an expansion of de-gree 2s in powers of (S; S, + t ). For convenience, wehereafter refer to these as Takhtajan-Babujian models.AfBeck concludes that a mass gap m is generated accord-ing to the expression

4(1+s)/(5+ 8s) (2)

4 —(E —E, ) (3)

The critical exponent thus monotonically decreases from avalue of 3 for s =

2 to a value of —,' for s =~. AfBeck'

conjectures that the exponent v in the expression for thelocalization length for the Hall eA'ect states E (Ref. 7),

equals the s =~ limit of the exponent in (2), namely, —,' .

A study of the alternating version of the Takhtajan-Babujian models is thus of interest not only in the contextof the quantum Hall eA'ect, but also to investigate the va-lidity of the rather complicated series of mappings andlimiting processes on which the AfBeck predictions rely.

A number of other studies are available which addressthe problem of the validity of AfBeck's predictions,and the results are not altogether consistent either amongthemselves or with AfBeck. One problem already ad-dressed concerns the spin-1 integrable SU(2) model,perturbed by biquadratic exchange. The Hamiltonian is

NH=J g [(S; S;it) —P(S;.S;+t)'i . (4)

A finite-size scaling study based on a gap from singletground state to triplet excited state is in qualitative agree-ment with AfBeck, but the numerical evidence indicatesthat the gap opens more slowly, i.e., the relation is foundto be'

A concurrent numerical study is consistent with a slowopening of the gap for P & 1. For P ) 1, however, indica-tions are that an excited singlet excitation extrapolatesbelow the excited triplet to become degenerate with thesinglet ground state. The situation is both complicatedand unusual, and is currently under investigation. "Hence, convincing agreement with the prediction ofAfBeck for biquadratic perturbations is not at present es-tablished.

The problem of the spin- —,' alternating antiferromag-

netic chain has already received considerable attention on

AfBeck' conjectures that a mass gap opens on each side ofthe integrable point P = I such that

(5)

36 2337 1987 The American Physical Society

2338 BRIEF REPORTS 36

account of its relevance to spin-Peierls theory, ' and as amodel for solitonic excitations, in analogy with fractional-ly charged solitons in polyacetylene. ' An early theory ofCross and Fisher ' gave an exponent of —,

' for the gap, in

agreement with Affleck [see Eq. (2) with s = —,' ]. Subse-

quent renormalization-group ' and finite-size scaling '

calculations on alternating chains up to N =12 spins pro-duced a larger value of —0.70-0.75 for the gap exponent.This problem stimulated a considerable amount of subse-quent numerical work, including studies on finite chainsup to N =21 spins. ' Values for the gap exponent, howev-er, varied considerably, but consistently yielded valuesgreater than 0.75. As discussed by Bonner and Blote, ' amapping involving a staggered Ashkin- Teller model,equivalent to the Potts model with the critical number ofcomponents (q, =4), indicated the presence of a marginaloperator with concomitant logarithmic corrections. '

Logarithmic corrections are well known to slow conver-gence of numerical methods, presumably accounting forthe considerable variation in gap exponent results. '

G(a, n) =1 'G(l 'a, n/I)

where y, is the exponent associated with the scaling fielda. Differentiating G(a, n) twice with respect to a, andputting I =n and a =0, gives

d'G(a, n) 2y. —)=anJQ 0

(8)

FINITE-SIZE SCALING CALCULATIONS

The present study features calculations on a spin-1 al-ternating chain comprised of Hamiltonian (4) with P= i

and the perturbing term of Eq. (I) to induce alternation.Finite-size data for the ground-state energies E(n) andthe gaps G(n) between the singlet ground state and thetriplet first excited state in the close vicinity of the uni-form point were obtained numerically for periodic systemsup to n =12 spins. Derivatives of these quantities withrespect to the alternation parameter a were calculated bymeans of numerical differentiation and are shown in TableI. To find the exponent, a finite-size scaling analysis wasthen conducted along the same lines as in Ref. 10. Weconsider the gap and the ground-state energy in the pres-ence of the relevant scaling field a which is equal to zeroin the "critical" case where the gap vanishes for n

Neglecting irrelevant scaling fields, we expect the follow-ing behavior of the gap under a renormalization transfor-mation with a linear scale factor I:

TABLE I. Values of d E(n)/da and d G(n)/da' computedfor values n =2 through 12.

d E(n)/da G (n )/da

24681012

0—22.773 550—58.778 675—109.216 997—174.121 502—253.497 293

017.773 55041.853 39074.261 417

115.254 036164.938 282

where a is a constant. Similarly, we expect the derivativesof the ground-state energy E =E/n to behave asymptoti-cally as

r

d E( an) 2y, —2=e+bnJQ ~~0

(9)

where e and b are constants. Equations (8) and (9)have been used as the basis for our finite-size analysis,since the exponent y, predicts the behavior of the gap andthe ground-state energy in the vicinity of the critical pointfor n =~. Choosing l =a in Eq. (7) yields

G(a, )-a'"' . (ioa)

Similarly, one obtains for the ground-state energy

E(a, ) —a~a'"' . (iob)

d E(am)( )+b( ) 2y (n 1) —2

Ja(i2)

for m =n, n+ 2, and n+ 4. The "iterated" entriesy, (n, 2) for n =2, 4 (three-point fit), or n =2,4, 6 (two-point fit) were obtained by fitting the relation

y, (m, i) y, (n, 2)+P„m "

Hence, we have analyzed the second derivatives of G andE in order to obtain estimates of the exponent y as fol-lows. So-called two-point fits yield estimates a(n) and

y, (n, l ), respectively, in Eq. (8) by requiring that

d G(am) r i 2y (n 1) —)

JQ

for m =n and n+2. Similarly, three-point fits yield esti-mates E(n), b(n), and y, (n, l) of the parameters in Eq.(9) from the equation

TABLE II. Numerical results for the alternation exponent as obtained by two-point fits and iteratedfits to d 2E(n)/da2 and d2G (n )/da2 data as explained in text.

From d E(n)/day. (n, l ) y. (n, 2)

From d G(n)/day. (n, l) y.(n, 2)

246810

1.671.581.551.53

1.581.511.50

1.5561.4971.4851.483

1.4941.4781.482

36 BRIEF REPORTS 2339

TABLE III. Numerical results for the alternation exponentas obtained by three-point fits and iterated fits to d E(n)/daand d G(n)/da2 data as explained in text.

y. (n, 1 )

1.191.411.451.47

From d E(n)/da2y. (n, 2)

1.491.51

(where the P„are constants and x„ is a correction-to-scaling exponent) to the values y, (m, 1) for m =n, n+2,n+4, or m =n, n+2, respectively. In Table II we presentresults from a two-point fit to d E (n )/da andd G(n)/da . In the first main column of Table II, we ob-serve a smoothly behaved sequence for y, (n, 1 ), tending toa value of —1.50 for large n. The iterated fits [y, (n, 2)]show rapid convergence indicating that y =1.50 may bean upper bound. In the second main column of Table II,we again obtain a smoothly behaved sequence for y, (n, 1 ),indicating a limiting value y =1.49. The correspondingy, (n, 2) show a slight minimum before resuming an up-ward trend, also consistent with a limiting value ofy, = 1.49.

In the first column of Table III, which shows three-point fits to d E(n)/da, values for y, (n, 1 ) show a mono-tonic increasing trend with n, and the correspondingiterated fits y, (n, 2) suggest a limiting value of y, =1.505.

DISCUSSION

The finite-size scaling results in the various situationsdiscussed above are mutually consistent and converge rap-idly to an exponent value very close to y =1.50. Hencethe ground-state energy of an infinitely long chain behavesas a =a' . Since this exponent (——', ) is smallerthan 2 (the exponent associated with the lattice elastic en-ergy) we can immediately conclude that a spin-Peierlstransition will occur in spin systems described by the s =1Takhtajan-Babujian Hamiltonian at sufficiently low tem-peratures. The Aflleck formula [Eq. (2)], however, pre-dicts the value y = —", =1.625, corresponding to an alter-nation exponent y

' =0.615. One might therefore drawadditional conclusions from this study to the efrect that(a) the spin-1 exponent value y, = 1.5 + 0.02 issignificantly diff'erent from the ANeck prediction of 1.625;and (b) the spin-1 exponent found here numerically is ac-tually close to the predicted value, " for spin —,', ap-parently contradicting AfBeck's prediction of s depen-dence in the critical exponents associated with the class ofSU(2) Takhtajan-Babujian models.

These surprising results impel closer scrutiny and fur-ther consideration. Let us first discuss the discrepancywith the spin-1 AfBeck prediction. The data in Tables IIand III are consistent and show no sign of a trend to1.625. However, a situation like this has been observedbefore. We have performed a comparable s'.udy on thespin- 2 alternating chain' and observed apparent conver- g =3/(2+2s) (14)

gence to the value 1.33, significantly lower than the valueof 1.5 predicted by Cross and Fisher' and AfBeck. ' Thisdiscrepancy has been attributed' to an essential singular-ity in the case of the spin- 2 uniform Heisenberg chain,present in the mass gap which opens up under theinfluence of Ising-like XLZ spin anisotropy. In therenormalization-group approach, this corresponds, in ad-dition to anisotropy, to the presence of another, marginal,operator. This operator results in the presence of logarith-mic corrections (which are known to influence the ap-parent value of exponents obtained numerically) in thelimit of vanishingly small alternation. As noted in the In-troduction, the presence of logarithmic connections wasdeduced from a mapping to the staggered Ashkin-Tellermodel. ' The question is, therefore, whether a marginaloperator in the spin-1 alternating chain and associatedlogarithmic corrections are afrecting the numerical resultfor the exponent. Using the Yang-Baxter generalizedstar-triangle relations, Sogo ' has obtained a spin-sBethe-ansatz integrable analogue of the spin- —,

' AAZmodel. The Sogo model has both a uniaxially anisotropicand planar anisotropic regime, and the analytic solutionfor spectral and thermodynamic properties difTers fromone family member to another only by a factor s. TheSogo model Hamiltonian in the case s =

& reduces to thewell-known s =

2 XAZ Hamiltonian, but is much morecomplicated even in the next simplest case ' of s =1, ex-cept when no anisotropy is present and the spin-1Takhtajan-Babujian model is recovered. An essentialsingularity does appear in the Sogo solution ' at the iso-tropic point: In fact, the essential singularity explicitlyappears in earlier work by Babujian in an expression forthe zero-point susceptibility. Hence, we must considerseriously the possibility of marginal operators also in thecase of spin 1, although their presence is not indicated bythe AfBeck' analysis, nor is there an analogous mappingto a four-state Potts model.

Let us now, therefore, consider the question of s depen-dence which appears in the exponent predictions ofAflleck. Our prima facie result that the alternation ex-ponents for s =

2 and 1 are the same is, in consequence,surprising, and warrants further investigation. A study ofthe case of s =

2 would be interesting, but unfortunatelyour present method would limit us to @=10, i.e., to aninsufhcient number of n values for a reasonably accuratefinite-size scaling analysis. We do note that our presentspin-1 exponent value of 1.50 is significantly diferentfrom our spin- 2 value of 1.33. If both results are on anequivalent footing, i.e. , if marginal operators are presentin both cases, this would indicate s dependence. Further,note that the difference in th two values is easilysuflicient to cover the discrepancy between 1.50 and theAfBeck prediction of 1.625.

Hence, we examine numerical s =2 and s =1 values

for other critical parameters to investigate further theAfBeck prediction of s dependence. In addition to the s-dependent expression for the alternation exponent [Eq.(2)], Aflleck also predicts the following relation for thecorrelation exponent g:

2340 BRIEF REPORTS 36

c =3s/(I+s) . (IS)

Hence c =1 for s = 2, c =1.5 for s =1, and c =3 fors =~. These predictions have been verified numericallyto within 3% accuracy. Their validity may also be es-tablished directly, using exact analytic results.

Hence, in the light of these other critical parameterstudies, our surprising result that alternation exponents

Numerical estimates for g, though not highly accurate,tend to support AfHeck. For the s =

2 Heisenberg chain,the numerical predictions yield ri —0.85 (in place ofthe exact value g =1); and for the s =1 model, q —0.65[also low compared with the prediction of 0.75 from Eq.(14)]. Nevertheless, an s dependence is apparent in thenumerical results, and there are suggestive similarities tothe case of the alternation exponent.

Note that universality is not violated by these exponentrelations, since the Takhtajan-Babujian models form aclass of integrable models characterized by values of theconformal anomaly c ~ 1 such that

for s = 1 and s =2 are the same may be misleading. If

marginal operators are present in all members of theTakhtajan-Babujian family (as in the case of spin —, ), theproblem would be resolved, i.e., our numerical studieswould not be inconsistent with theoretical expectations.There is evidence this could be the case.

ACKNOWLEDGMENTS

We are grateful to M. P. Nightingale, H. W. Capel,and V. J. Emery for valuable discussions and, particularly,to I. Aleck for comments in connection with marginaloperators. Financial support for H. W.J.B. from theStichting voor Fundamental Ondersoek der MaterieResearch Program of the "Nederlandse Organisatie voorZuiver-Wetenschappelijk Onderzoek (ZOEK), " and forJ.C.B. by a National Science Foundation Grant No.DMR86-03036 is gratefully acknowledged, as also is sup-port by NATO, Grant No. 198/84.

'I. AIIIeck, Nucl. Phys. B265, 409 (1986).2I. AIIIeck, Phys. Rev. Lett. 56, 746 (1986).SF. D. M. Haldane, Phys. Lett. 93A, 464 (1983); Phys. Rev.

Lett. 50, 1153 (1983).4L. A.Takhtajan, Phys. Lett. 87A, 479 (1982).sH. M. Babujian, Phys. Lett. 90A, 479 (1982).The "mass gap" in a quantum spin chain refers to the energy

gap between ground state and first excited state of a finite sys-tem.

7AfHeck argues that all states are localized except at a single en-

ergy E, near the center of each Landau band.D. Kung (unpublished).J. Oitmaa, J. B. Parkinson, and J. C. Bonner, J. Phys. C 19,

L595 (1986).'oH. W. J. Blote and H. Capel, Physica A 139, 387 (1986).' 'J. C. Bonner, J. B. Parkinson, J. Oitmaa, and H. W. J. Blote,

J. Appl Phys. 61, 4432 (1987).' J. W. Bray, L. V. Interrante, I. S. Jacobs, and J. C. Bonner, in

Extended Linear Chain Compounds, edited by J. S. Miller(Plenum, New York, 1982), Vol. 3, p. 353.

' T. Nakano and H. Fukuyama, J. Phys. Soc. Jpn. 49, 1679(1980); 50, 2489 (1981).

'4M. C. Cross and D. S. Fisher, Phys. Rev. B 19, 402 (1979).'5J. N. Fields, H. W. J. Blote, and J. C. Bonner, J. Appl. Phys.

50, 1808 (1979);J. N. Fields, Phys. Rev. B 19, 2637 (1979).' H. W. J. Blote, in International Conference on Physics in One

Dimension, Fribourg, Switzerland, 1980 (unpublished). Dis-cussed in J. C. Bonner and H. W. J. Blote, Phys. Rev. B 25,6959 (1982).

'7Z. G. Soos, S. Kawajima, and J. E. Mihalick, Phys. Rev. B 32,3124 (1985), and references therein.See Bonner and Blote, Ref. 16.

~9J. L. Black and V. J. Emery, Phys. Rev. B 23, 429 (1981);M. P. M. den Nijs, ibid 23, 6111. (1981).

2oEquations (8) and (9) are, in general, valid only in the limitn ~ because we have ignored the irrelevant scaling fieldswhich produce terms with smaller powers of n.

VK. Sogo, Phys. Lett. 104A, 51 (1984); and (private communi-cation).

22However, I. AIIIeck (private communication) has recentlyclaimed to find such marginal operators for the Takhtajan-Babujian models.The two-dimensional four-state Potts model is in the sameuniversality class as the s = —,

' antiferromagnetic chain (c =1)whereas the s =1 chain has c =1.5. This provides a basis forour belief that a mapping to a q, =4 Potts model does not ex-ist for the spin-1 system.

Z4J. C. Bonner (unpublished). See also Ref. 9.2SH. Schulz and T. A. L. Ziman, Phys. Rev. B 33, 6545 (1986).26S. Takada and K. Kubo, J. Phys. Soc. Jpn. 55, 1671 (1986).27L. V. Avdeev and B.-D. Dorfel, J. Phys. A 19, L13 (1986);

and (unpublished).