An international workshop on Elementary Excitations and Fluctuations in Magnetic Systems was held in Turin for five days beginning 25 May, 1987. The workshop followed much the same format as the one with the same title held in San Miniato in 1984 (proceedings: Springer Series in-Solid-State Sciences, Vol. 54), that most participants contributed talks and provided papers for the proceedings. While many of the participants had attended the first workshop, 15 of the 40 invited review papers were presented by scientists who had not. The majority of the talks reported theoretical work concerned with the introduction of new techniques. However, experimental work was also well represented, not least because many of the reported theoretical studies were motivated by experimental findings, and a highlight of the workshop was an extremely stimulating session devoted to recent neutron scattering measurements, on various systems, that exploited polarization analysis. The fine venue of the workshop, Villa Gualino, with its excellent facilities and spacious accommodation, helped to produce a delightful relaxed and friendly atmosphere. For the use of Villa Gualino and significant financial support we are indebted to our host organization, the Institute for Scientific Interchange (lSI). Additional financial support came from the Consiglio Nazionale delle Ricerche (CNR), Centro Interuniversitario di Struttura della Material del Ministero della Pubblica Istruzione (CISM-MPI) and Gruppo Nazionale di Struttura della Materia (GNSM-CNR)

Bonner, Jill C.

Müller, Gerhard

Parkinson, John B.

English

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University of Rhode IslandDigitalCommons@URIPhysics Faculty Publications Physics1988Excitations and Critical Behavior in GeneralizedHeisenberg Spin ChainsJill C. BonnerGerhard MüllerUniversity of Rhode Island, gmuller@uri.eduSee next page for additional authorsFollow this and additional works at: https://digitalcommons.uri.edu/phys_facpubsTerms of UseAll rights reserved under copyright.This Book Chapter is brought to you for free and open access by the Physics at DigitalCommons@URI. It has been accepted for inclusion in PhysicsFaculty Publications by an authorized administrator of DigitalCommons@URI. For more information, please contact digitalcommons@etal.uri.edu.Citation/Publisher AttributionJill C. Bonner, Gerhard Müller and J.B. Parkinson. Excitations and critical behavior in generalized Heisenberg spin chains in MagneticExcitations and Fluctuations II, U. Balucani, S.W. Lovesey, M.G. Rasetti, and V. Tognetti (Eds.), Springer-Verlag New York 1988, pp95-100.Available at: http://dx.doi.org/10.1007/978-3-642-73107-5_19AuthorsJill C. Bonner, Gerhard Müller, and John B. ParkinsonThis book chapter is available at DigitalCommons@URI: https://digitalcommons.uri.edu/phys_facpubs/146Excitations and Critical Behavior in GeneralizedHeisenberg Spin ChainsJill C. Bonner, 1 Gerhard Müller, 1 and John B. Parkinson 21 Department of Physics, University of Rhode Island, Kingston RI 02881, USA2 Department of Applied Mathematics, UMIST, Manchester M601QD, United Kingdom1. IntroductionHeisenberg spin chains which represent the simplest realistic models for magnetic insulatorswere thought to be well understood and generically similar for any spin-value s. This is expressedin the spin-wave approach to Heisenberg spin chains [1]. Consequently surprise and some degreeof controversy resulted from recent work of Haldane [2,3], who proposed a dramatically differentpicture. Consider the spin-s XXZ Hamiltonian with anisotropy parameter ∆:H = JN∑l=1{Sxl Sxl+1 + Syl Syl+1 + ∆Szl Szl+1}. (1)For half-integer s, the region 0 ≤ ∆ < 1 is a gapless phase with power-law decay of the two-spincorrelation functions, terminating in an essential singularity at ∆ = 1. For ∆ > 1, the ground stateconsists of two degenerate singlet states associated with long-range order and a gap to an excitationcontinuum. For integer s, on the other hand, the gapless phase associated with planar anisotropyextends only over a range 0 ≤ ∆ ≤ ∆1 (∆1 < 1) and the phase with gap and ordered ground stateextends over ∆ ≥ ∆2 (∆2 > 1). A new phase (called hereafter the Haldane phase) appears in theregion ∆1 < ∆ < ∆2 encompassing the Heisenberg point at ∆ = 1. The Haldane phase groundstate is a non-ordered singlet with exponentially decaying spin correlation functions, and there isa gap to an excitation continuum, which has its maximum value at ∆ = 1. The spin-dependentgap at ∆ = 1 is given by ∆E/J ∼ s2 exp(−pis).Generalized Heisenberg spin chains are not exactly solvable (Bethe Ansatz integrable) for s >1/2, except in very special situations. Hence Haldane’s conjecture has, perforce, been investigatedwith a variety of numerical techniques, including scaled-gap and finite-size scaling calculations,finite-chain extrapolations, variational approaches and various correlation function calculations[4]. Obtaining reliable numerical results turns out to be quite difficult and great care must betaken. For example, while the first numerical, finite-size scaling calculation on spin-1 XXZ chains[5) revealed the predicted Haldane phenomena, it was subsequently demonstrated that a "pseudo-Haldane picture" is obtained also for the spin-1/2 XXZ model [6] in contradiction to exact analyticresults. However, it is now the consensus of a large body of numerical work that the Haldane pictureis, nevertheless, correct, and experimental support for this conclusion is starting to appear [7].On the basis of a comprehensive numerical study and survey we have concluded that thevarious classes of excitations predicted by Haldane for the spin-1 model are all present and behaveas conjectured [8]. However, we observe additional interesting features, in particular, classes ofexcitations which have the potential for modifying somewhat the basic Haldane picture, as we willdiscuss.The addition to the basic Heisenberg Hamiltonian of exchange anisotropy, single-ion anisotropy,biquadratic exchange, or a magnetic field, generates a rich and complicated phase diagram forchains with s > 1/2 [4]. Unusual spectral features in generalized spin-1 chains will be discussed inthe context of nonintegrability effects and quantum chaos [9,10].1Excitations and Critical Behavior in Generalized Heisenberg Spin Chains2. Spin-1 Heisenberg Antiferromagnetic ChainSince the predicted singlet-triplet Haldane gap is a maximum at the isotropic Heisenberg point,∆ = 1.0, numerical attention has focussed on this limit. While several numerical techniques havebeen devised to study this problem [8], the most direct approach is to examine the behavior of theHaldane gap for a sequence of finite systems of increasing size and examine the trend as N →∞.Fig. 1 shows finite-N gaps for up to N = 20 spins with s = 1/2 as a function of 1/N . The gapsextrapolate convincingly to a value very close to zero, in agreement with the exact result that thissystem should be gapless in the limit N → ∞. Exact results up to N = 14 for the s = 1 case,on the other hand, show concave upwards curvature, consistent with a nonzero gap in the limitN → ∞. However, since earlier studies [6] have pointed to the importance of obtaining data forvery long chains to be sure of observing a reliable large N trend, a quantum Monte Carlo approachwas developed to obtain data out to N = 32 spins [11]. The Monte Carlo data continue the concaveupward trend of the exact finite-N data and predict a limiting singlet-triplet gap of magnitude∆E/J ∼ 0.41. (2)This gap occurs at the Brillouin zone boundary, since the excited triplet is at k = pi and theground state is always a k = 0 state. The Haldane conjecture implies that a gap of equal magnitudeshould occur at the zone center. In Fig. 1, the finite-N triplet gaps for the k = 2pi/N mode areshown, for N ≤ 14. The convergence is quite regular, and the extrapolated limit is in reasonableagreement with Eq. (2).Figure 1. Various excited modes for the s = 1 antiferromagnetic Heisenberg chain as discussedin the text.There does, however, appear to be one remarkable feature associated with the spectral exci-tations in the Heisenberg limit and vicinity. Numerical studies have revealed a crossover in thecharacter of the spectral excitations as a function of field [12]. At high fields, the low-lying dis-persion spectra are qualitatively similar to those for s = 1/2, i.e. have quantum character. Atlow fields, on the other hand, the dispersion spectra display notable classical character, with onecomplication. An additional set of modes occurs which appears unrelated to the classical spectra.When extrapolated as a function of 1/N , these anomalous modes project below all other excita-tions with the same value of SzT . For this reason these states have been termed”supersoft” modes2Excitations and Critical Behavior in Generalized Heisenberg Spin Chains[12]. In particular the SzT = 1 mode lies at k = pi/N , and its excitation energy is included inFig. 1. Extrapolating below all other excited states it appears to extrapolate below the triplet atk = pi which has been used to determine the Haldane gap! The curvature is consistently concavedownwards and increases with increasing N , making it unlikely that the curve could develop aninflection point for larger N and tend to the value (2).Detailed information on the T = 0 phase behavior of quantum spin chains may be inferred fromthe integrated intensityI(k) =+∞∑R=−∞eikR〈~Sl · ~Sl+R〉. (3)At zero temperature, I(k) is a property of the ground state solely, and yet also contains informationon the excited states, at wave number k. It can be shown [8] that in the presence of a gap ∆Ebetween the ground state and the lower edge of the excitation continuum, the integrated intensitymust satisfy the following inequality:I(k) ≤ 2|EG|(1− cos k)/3∆E, (4)where EG is the ground-state energy per spin. For the Heisenberg spin-1 chain calculations of Blöteand Nightingale [11] give EG/J ∼ −1.4015. Together with (3) this yields the following lower boundfor the inverse integrated intensity: I−1(pi) ≥ 0.219. A plot of 1/I(pi) versus 1/N is consistent withthis bound in the limit N → ∞, but a plot of the inverse intensity 1/I(pi − 2pi/N), which isdominated by the SzT = 1 supersoft mode, fails appreciably to satisfy this criterion and is notconsistent with an energy gap (2).Hence present numerical evidence out to N−16 spins is consistent with the presence of a specialclass of excitations not predicted by Haldane. These isolated excitations probably have insufficientthermodynamic weight to affect the thermal properties, but since there exist N/2 such modes(corresponding to SzT = 1, 2, . . . , N/2), they should be experimentally observable in the T = 0magnetization isotherm at low magnetic fields. In particular, the T = 0 magnetization isothermshould become zero at a much lower field than that corresponding to the Haldane gap (2).3. Spin-1 XXZ ModelOf all the excitation phenomena predicted by Haldane to occur for the s = 1 XXZ model,the one which is most difficult to confirm numerically has been the behavior in the vicinity ofthe critical point ∆2 [13]. In fact, doubt has even been expressed concerning the validity of thisparticular aspect of the conjecture. Haldane predicts that the transition at ∆ = ∆2 should be in theuniversality class of the spin-1/2 transverse Ising model. It has already been established [4,8] thatat ∆2 ∼ 1.18− 1.20, the Haldane gap disappears and an excited SzT = 0 state becomes degeneratewith the SzT = 0 ground state for ∆ > ∆2. The mapping to the transverse Ising model impliesthe existence in the limit N →∞ of an infinite continuum of scaling states quasi-degenerate withthe ground state (states) at, and only at, ∆ = ∆2. The development of this scaling continuum isnot very apparent in the spectra for small finite systems. A search for the scaling states impliedby the Haldane conjecture revealed a class of high-lying SzT = 0 excitations at k = 0. Plottedas a function of ∆, these excitations are shown in Fig. 2. A minimum develops in the vicinityof ∆ ∼ 1.1 − 1.2 which intensifies with increasing N . Fitting a polynomial through data in thevicinity of the minimum and plotting the excitation gap at the polynomial minimum versus 1/Nyields an extrapolated value well below the lower edge of the triplet (SzT = 1) continuum, andeven consistent with a value zero. We conclude these are the Haldane scaling states at ∆2. Thisconclusion is reinforced by a detailed study of corresponding excitations for the spin-1/2 transverseIsing model near the critical field. The transverse Ising model picture shows a striking resemblanceto Fig. 2.A surprising feature of the Haldane prediction [3] is that the zz-correlation function decays asa power law, |〈Szl Szl+R〉| ∼ R−1/4, whereas the correlation function 〈Sxl Sxl+R〉 decays exponentiallyfor R→∞. This unusual prediction implies that the fluctuations are critical in the longitudinal (z)3Excitations and Critical Behavior in Generalized Heisenberg Spin ChainsFigure 2. SzT = 0 “scaling” states for N = 4, 6, 8, 10 and 12 (denoted by ◦). The excitationdenoted by  is the extrapolated limit of the upper component of the ∆ ≥ ∆2 ground state.direction but not in the transverse (x) direction. Such a situation has not been observed previously.Hence we examined also the SzT = 1 excitations at k = 0 as a function of ∆, and again observedthe development of minima with increasing N in the vicinity of ∆2. These minima also displaya potential for extrapolating to zero, analogous to the SzT = 0 minima, and in contrast to k = 0excitations in the same class corresponding to SzT = 2, etc. If gapless SzT = 1 excitations occurat ∆ = ∆2, the transverse xx-correlations will also display power-law decay, in disagreement withHaldane’s specific predictions for the behavior of the two-spin correlation functions [2].4. Nonintegrability Aspects of the SpectraThe above numerical studies of the spin-1 XXZ model, including the Heisenberg point, show thatthe Haldane prediction represents a remarkably successful mapping. All classes of states predictedby Haldane appear to be present and behave generally as predicted. It appears, however, thatwe have discovered additional classes of states whose presence have the potential of playing animportant role in the extended T = 0 phase diagram. The |SzT | = 1 “scaling” states at ∆ ∼ ∆2are perhaps less surprising since anaIogous states occur for the transverse Ising model near thecritical field. The "supersoft" modes at ∆ = 1, on the other hand, are a remarkable featurenot encountered previously in any integrable model. Here we discuss in more detail the possibleoccurrence in generalized s ≥ 1 Heisenberg spin chains of spectral features which are characteristicof nonintegrable models.Consider the spin-1 Heisenberg antiferromagnetic chain generalized by addition of biquadraticexchange:H/J =N∑l=1~Sl · ~Sl+1 − βN∑l=1(~Sl · ~Sl+1)2. (5)4Excitations and Critical Behavior in Generalized Heisenberg Spin ChainsSpecial limits of this bilinear-biquadratic exchange Hamiltonian are:β = 0 Heisenberg model (nonintegrable)β = 1 “Russian” model [14] (integrable)β =∞ pure biquadratic (nonintegrable)Figure 3. A plot of the values of β−1 at which the lowest excited state in the range 0 ≤ β−1 ≤ 1changes from a singlet to triplet versus N−1. The rough extrapolated limit of the crossing pointsis consistent with the value unity, implying that the singlet is ultimately the dominant excitedstate in this range.Hamiltonian (8) at the special point β = 1 is an example of an s > 1/2 model which is BetheAnsatz integrable and gapless [14,15]. Hence it is of interest to investigate the parameter range0 ≤ β ≤ ∞ to determine the extent of the gapless region. A prediction of Aﬄeck [16] is that only thepoint β = 1 is gapless and that a gap opens on either side of the Russian point as ∆E ∼ |1− β|l.Numerical studies including scaled-gap [17] and other finite-size scaling calculations [18,19] areconsistent with Aﬄeck in predicting the opening of a singlet-triplet gap at β = 1, but differ inpredicting that the gap opens up more slowly than linearly. Furthermore, numerical calculationsrevealed a very curious phenomenon in the Russian-biquadratic regime (0 ≤ 1/β ≤ 1) [18]. Thisphenomenon is illustrated in Fig. 3. Close to the biquadratic limit, the first excited state is nolonger the triplet at k = pi for sufficiently large N , but instead becomes a singlet at pi. A crossovereffect occurs in this regime, illustrated in Fig. 3, which implies that as N →∞, the lowest excitedstate is a singlet over the whole regime. Traditional finite-size scaling approaches which assume,as an act of faith, that the dominant excited states for small finite N remain the dominant statesas as N →∞, can be seriously misled by a situation such as the above. Just as in the case of thesupersoft modes, a new class of important modes appears at sufficiently large N . This triplet-to-singlet crossover phenomenon does not occur for integrable systems (e.g. the spin-1/2 XXZ model),and is presumably a spectral nonintegrability effect.Acknowledgment: We acknowledge contributions by J. Oitmaa. Support has been providedby the US NSF grant DMR 86-03036 and the Research Corporation. We have used a modifiedcmpj.sty style file.5Excitations and Critical Behavior in Generalized Heisenberg Spin ChainsReferences1. See, e.g., R. Kubo, Phys. Rev. 87, 586 (1952).2. F. D. M. Haldane, Phys. Lett. 93A, 464 (1983); F. D. M. Haldane, Phys. Rev. Lett. 50, 1153 (1983).3. F. D. M. Haldane, ILL. Report No. SP81/95 (unpublished).4. J. C. Bonner, J. Appl. Phys. 61, 3941 (1987) (invited review).5. R. Botet and R. Jullien, Phys. Rev. B 27, 613 (1983).6. J. C. Bonner and G. Müller, Phys. Rev. B 29, 5216 (1984).7. J. P. Renard et al., Europhys. Lett. (in press). See also W. J. L. Buyers et aI., Phys. Rev. Lett. 56,371 (1986).8. J. C. Bonner, J. B. Parkinson and G. Müller, unpublished work.9. G. Müller, J. Bonner and J. Parkinson, J. Appl. Phys. 61, 3950 (1987).10. G. Müller, Phys. Rev. A 34, 3345 (1986).11. M. P. Nightingale and H. W. J. Blöte, Phys. Rev. B 33, 659 (1986).12. J. B. Parkinson and J. C. Bonner, Phys. Rev. B 32, 4703 (1985) and J. Phys. C (in press).13. See, e.g., K. Kubo and S. Takada, J. Phys. Soc. Jpn. 55, 438 (1986).14. L. A. Takhtajan, Phys. Lett. 87A, 479 (1982); H. M. Babujian, Phys. Lett. 90A, 479 (1982).15. J. B. Parkinson and J. C. Bonner, J. Phys. C 19, 6063 (1986).16. I. Aﬄeck, Nucl. Phys. B265, 409 (1986).17. J. Oitmaa, J. B. Parkinson and J. C. Bonner, J. Phys. C 19, L595 (1986).18. J. C. Bonner, J. B. Parkinson, J. Oitmaa and H. W. J. Blöte, J. Appl. Phys. 61, 4432 (1987).19. H. W. J. Blöte and H. W. Capel, Physica 139A, 387 (1986).6

Excitations and Critical Behavior in Generalized Heisenberg Spin Chains

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Excitations and Critical Behavior in Generalized Heisenberg Spin Chains

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