We study the magnetic properties of a frustrated Heisenberg spin chain with a
dynamic spin-phonon interaction. By Lanczos diagonalization, preserving the
full lattice dynamics, we explore the non-adiabatic regime with phonon
frequencies comparable to the exchange coupling energy which is e.g. the
relevant limit for the spin-Peierls compound $CuGeO_3$. When compared to the
static limit of an alternating spin chain the magnetic properties are strongly
renormalized due to the coupled dynamics of spin and lattice degrees of
freedom. The magnitude of the spin triplet excitation gap changes from a strong
to a weak dimerization dependence with increasing phonon frequencies implying
the necessity to include dynamic effects in an attempt for a quantitative
description of the spin-Peierls state.Comment: 4 pages, 5 figure

A. J. Heeger

A. M. Tsvelik

A. P. Kampf

B. Bäuml

B. Büchner

D. Augier

D. Khomskii

F. D. M. Haldane

F. H. L. Essler

G. Bouzerar

G. Castilla

G. Wellein

H. Fehske

J. P. Boucher

J. Riera

J. W. Bray

K. Fabricius

K. Okamoto

L. Bulaevskii

L. P. Regnault

M. Braden

M. C. Cross

M. Hase

R. Chitra

R. E. Peierls

S. R. White

T. Barnes

English

arXiv

Crossref

Physical Review Letters

Peierls Dimerization with Nonadiabatic Spin-Phonon Coupling

Wellein, G.

Fehske, H.

Kampf, Arno P. (Prof.)

OPUS Augsburg

VOLUME 81, NUMBER 18 P H Y S I C A L R E V I E W L E T T E R S 2 NOVEMBER 19983956Peierls Dimerization with Nonadiabatic Spin-Phonon CouplingG. Wellein,1 H. Fehske,1 and A. P. Kampf21Physikalisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany2Institut für Physik, Theoretische Physik III, Elektronische Korrelationen und Magnetismus,Universität Augsburg, 86135 Augsburg, Germany(Received 9 April 1998)We study the magnetic properties of a frustrated Heisenberg spin chain with a dynamic spin-phononinteraction. Using Lanczos diagonalization that preserves the full lattice dynamics, we explore thenonadiabatic regime with phonon frequencies comparable to the exchange coupling energy, which is,for example, the relevant limit for the spin-Peierls compound CuGeO3. Compared to the static limit ofan alternating spin chain, the magnetic properties are strongly renormalized due to the coupled dynamicsof spin and lattice degrees of freedom. The dependence of the magnitude of the spin-triplet excitationgap changes from a strong to a weak dimerization with increasing phonon frequencies implying thenecessity to include dynamic effects in a quantitative description of the spin-Peierls state in CuGeO3.[S0031-9007(98)07570-X]PACS numbers: 75.10.Jm, 71.30.+h, 75.40.Mg, 75.50.EeLow dimensional quantum spin systems have attractedconsiderable attention of theorists over the past decades.Most of the remarkable features observed in these systemsare pure quantum effects uniquely due to their low dimen-sionality. Among the current experimental and theoreticalefforts for understanding the magnetic properties of linearchain and ladder materials, the recent discovery of a spin-Peierls (SP) transition at TSP ­ 14.3 K in the inorganiccompound CuGeO3 (CGO) [1] has received particular at-tention. In analogy to the Peierls instability towards dimer-ization in quasi-one-dimensional metals [2], the energy ofspin chains is lowered by dimerizing into an alternatingpattern of weak and strong bonds. The SP transition istherefore driven by the magnetic energy gain which over-compensates the lattice deformation energy [3]. In the SPphase of CGO, the copper moments form singlet dimersalong the chains and there is an energy gap to spin tripletexcitations. Experimentally, the SP nature of the transi-tion and the spin gap have been firmly established by in-elastic neutron scattering (INS), susceptibility, x-ray, andelectron-diffraction experiments [4].The Peierls transition also naturally involves the lat-tice degrees of freedom due to the alternating distor-tion of atomic positions. A rather common situationhas been the clear separation of the electronic or mag-netic energy scales from the frequencies of those phononswhich couple most effectively to the SP lattice distor-tion. This adiabatic limit is, e.g., realized in transpoly-acethylene [5] or organic SP materials [6]. In contrast,in CGO two weakly dispersive optical phonon modesinvolved in the SP transition have been identified withfrequencies h̄v ø J and h̄v ø 2J, where J is the su-perexchange coupling between neighboring Cu spins [7].These hard phonons require a very strong spin-phononcoupling for a SP transition to occur [8]. A strong mag-netoelastic coupling has indeed been observed in ther-mal expansion and magnetostriction experiments [9], and0031-9007y98y81(18)y3956(4)$15.00inferred from large uniaxial pressure (pi) derivatives≠Jy≠pi [10]. The SP physics in CGO is therefore in anonadiabatic regime.Despite this unusual situation, previous theoretical stud-ies have commonly adopted an alternating and frustratedantiferromagnetic (AF) Heisenberg spin-chain model [11],H ­ JXihf1 1 ds21digSi ? Si11 1 aSi ? Si12j , (1)with a static dimerization parameter d, thus representingthe extreme adiabatic limit of a SP chain. i denotes thesites of a chain with length N , and Si are S ­ 1y2 spinoperators. a determines the strength of a frustrating AFnext-nearest-neighbor coupling. The model (1) containstwo independent mechanisms for spin-gap formation. Atd ­ 0 and for a , ac, the ground state is a spin liq-uid and the elementary excitations are massless spinons[12]. ac ­ 0.241 was accurately determined by numeri-cal studies [11,13]. For a . ac the ground state is spon-taneously dimerized, the spectrum acquires a gap, and theelementary excitations are massive spinons [14,15]. Onthe other hand for any finite d, the singlet ground stateof (1) is also dimerized, but the elementary excitation is amassive magnon [12,16].From fits to magnetic susceptibility data in the uni-form high temperature phase [10,17] and from therequirement to reproduce the experimental triplet excita-tion gap [17,18], the parameter set J ­ 160 K, a ­ 0.36,and d ­ 0.014 was estimated for CGO within the staticmodel (1). However, the amplitude of the dimerization issubstantially underestimated when compared to estimatesfrom structural data in the SP phase [7,19]. In this Letterwe show that, due to the nonadiabatic, phonon dynamics,a much larger dimerization is needed to achieve the samemagnitude of the triplet gap and thus offer a possibleroute towards a quantitatively consistent description ofboth the structural distortion and the INS data.© 1998 The American Physical SocietyVOLUME 81, NUMBER 18 P H Y S I C A L R E V I E W L E T T E R S 2 NOVEMBER 1998The dimerization parameter d in Eq. (1) may be viewedto result from a mean-field treatment of an interchaincoupling which is generated by a purely elastic mecha-nism [20]. In fact, this coupling to neighboring dimerizedchains provides an external potential, lifting the degen-eracy between the two possible dimerized ground states[21]. Thereby, soliton and antisoliton excitations formspin triplet or spin-singlet bound states [22]; the formeris the elementary massive magnon of the spin Hamilton-ian (1) for d . 0.In this Letter we choose a different starting pointand use a Heisenberg chain model in which the spinsare explicitly coupled to the lattice degrees of freedommaintaining their full quantum dynamics. Specifically, westudy the magnetic properties of the Hamiltonian [21],HSP ­ JXihf1 1 gsb1i 1 bidgSi ? Si11 1 aSi ? Si12j1 h̄vXib1i bi , (2)with the local phonon creation and annihilation operatorsb1i and bi , respectively. For the optical phonon withfrequency v we thus choose a dispersionless Einsteinmode—an appropriate choice for modeling those phononsin CGO which are most relevant to the Peierls distortion[7]. In the nonadiabatic (h̄v , J) and intermediate cou-pling (g , J) regimes the spin and the phonon dynamicsare intimately coupled, and integrating out the phononsin order to obtain an effective renormalized spin-onlyHamiltonian is, in general, not possible [23].This situation thus requires a numerical approach. Wehave performed exact diagonalizations on finite chains upto N ­ 16 sites with periodic boundary conditions. Sincethe Hilbert space associated with the phonons is infiniteeven for a finite system, we apply a controlled truncationprocedure retaining only basis states with, at most Mphonons [24]. M is fixed by requiring a relative error inthe ground-state energy less than 1027. In our calculationsall possible phonon modes are taken into account.Figure 1 demonstrates, for a 12-site chain, how manyphonons have to be included to achieve convergencefor the ground-state energies of the singlet S ­ 0 andthe triplet S ­ 1 sectors. Although the specific valueof a is not crucial for our arguments, we keep thefrustration parameter a ­ 0.36 in all calculations fixedto its most likely value in CGO. Figure 1 contrasts theresults obtained if phonons only with wave vector q ­ por q ­ p and q ­ 0 are used to the case where allphonon modes are included. While it has been arguedin Ref. [25] that keeping the q ­ p mode only shouldalready capture the dominant dimerization effect of thespin-phonon interaction, we find in Fig. 1, that e.g.,the singlet-triplet excitation gap is strongly renormalizedwhen phonons of all q are taken into account, implyingthat a spin-triplet excitation is accompanied by a localdistortion in the lattice.0 5 10 15 20M-7.5-6.5-5.5-4.5E 0S , E 0T  [J]π-mode0&π-modesall modesN=12α=0.36g=0.5ω=0.3FIG. 1. Energies of the singlet ground state (solid symbols)and the lowest triplet excitation (open symbols) vs the numberof phonons used in the numerical diagonalization. The differentsymbols indicate the results obtained by including phonons(i) only with q ­ p (circles), (ii) with q ­ 0 and q ­ p(diamonds), and (iii) at all wave vectors (squares). The singlet-triplet gap is given by DST ­ ET0 2 ES0 . As in all subsequentfigures, energies are measured in units of J.The renormalization of the singlet-triplet gap DST isshown in the scaling plot in Fig. 2. It is contrasted to theresult with a restriction to the q ­ p phonons only. Inspite of the limited system sizes of up to 16 sites the dataare well fitted by the scaling function [18,26]DST sNd ­ DST 1ANe2NyN0 . (3)It appears that the restriction to the p modes substantiallyoverestimates the size of the gap for N ! `.The triplet dispersion shown in Fig. 3 supports the ex-pectation—as inferred from the neutron scattering stud-ies of the phonons [7]—that the frequencies of the0.00 0.05 0.10 0.151/N0.00.20.4∆ST(N) [J]N=8,10,12,14,16  (π-mode)N=8,10,12,14,16 (all modes)g=0.11ω=0.3α=0.36FIG. 2. Finite-size scaling behavior of the triplet excitationgap DST for a ­ 0.36. The lines result from the fit to thescaling function (3). The (almost adiabatic) phonon parametersg ­ 0.11 and v ­ 0.3 were used in order to make contact withprevious numerical studies [25].3957VOLUME 81, NUMBER 18 P H Y S I C A L R E V I E W L E T T E R S 2 NOVEMBER 19980 π/4 π/2 3π/4 πQ0.00.51.01.52.0ET0(π-Q)-ES 0(π) [J]g=0.11 ω=0.3g=0.50 ω=1.0g=1.00 ω=2.0Experiment0.05.010.015.020.025.0[meV]N=16M=N/2α=0.36FIG. 3. Dispersion of the elementary triplet excitation ( leftscale) for a chain with N ­ 16 sites for different phononfrequencies and spin-phonon coupling constants. Experimentaldata (p, right scale) are also included from Ref. [29].relevant optical phonon modes in CGO are comparableto, or larger than, the exchange coupling J. Becauseh̄v , J, the phononic character of the excitation domi-nates, leading to a flattening of the dispersion [27,28].This, however, is incompatible with the magnetic INSdata [29] which do not show any signature for a flatdispersion of the triplet excitation branch. We have in-cluded in Fig. 3 experimental data on CGO from Ref. [29]which, however, are not intended as a fit to the data—ourresults in Fig. 3 are for a fixed chain length with N ­ 16site—but rather show the shape of the triplet dispersion.We stress that the upturn at Q ­ 0 is a finite-size effectwhich also occurs for the pure spin model and rapidly de-creases with increasing N [30]; for N ! `, ET0 spd andET0 s0d become degenerate.The Peierls ordering structure of the ground state has tobe demonstrated in different ways for the static (1) and thedynamic model (2). For the Heisenberg spin-chain model(1) the magnetic order parameter,D ­*1NXis21disSi21 ? Si 2 Si ? Si11d+. (4)is the proper choice to describe the formation of localsinglets. For the translationally invariant quantum spin-phonon model (2), D vanishes [31] and the Peierlsdistortion of the lattice is reflected in the spatial structureof the displacement correlation function,C1,i ­ ksb1 1 b11 d sbi 1 b1i dl 2 d1,i . (5)The alternating structure of the correlation function C1,ishown in Fig. 4 implies the Peierls formation of shortand long bonds and thus alternating strong and weak AFexchange interactions, i.e., a dimerized ground state.Naturally the dimerization is enhanced (weakened) by in-creasing the spin-phonon coupling (phonon frequency).We note that for any finite phonon frequency there exists39580.00.3C 1,ig=0.11 ω=0.30.00.3C 1,ig=0.50 ω=1.00.00.3C 1,ig=0.50 ω=2.07 8 9 10 11 12 1 2 3 4 5 6 7i0.00.3C 1,ig=1.00 ω=2.0α=0.36 N=12 M=NFIG. 4. Displacement correlation function C1,i as defined inthe text for different phonon frequencies and spin latticecoupling constants for a chain with N ­ 12 sites.a critical coupling constant gc [32]; in the infinite systemquantum lattice fluctuations will destroy the Peierls dimer-ization for couplings g , gc. For nonadiabatic phononfrequencies h̄v , J also the critical coupling gc is com-parable to J —a situation which matches the strong mag-netoelastic coupling and the SP physics in CGO [19].In order to quantify these results we obtain the mag-nitude of the Peierls dimerization order parameter d fromthe static ( lattice) structure factor at wave number q ­ p ,d2 ­g2N2Xi,jkuiujleipsRi2Rjd , (6)where ui ­ bi 1 b1i . As in the ordinary Peierls phe-nomenon, a finite dimerization d . 0 necessarily leadsto a gap DST in the magnetic excitation spectrum. Forevaluating the relation between the dimerization and theresulting magnitude of the spin-triplet excitation gap,we keep the phonon frequency fixed, vary the couplingstrength g, and calculate for each parameter set DST (seeFig. 5) and the dimerization d from Eq. (6).The results for the static (spin-only) and the dynamicspin-phonon model are compared in Fig. 5 for three dif-ferent chain lengths. For vanishing lattice dimerization,i.e. in the absence of any spin-phonon coupling sg ­ 0d,the results for the static and dynamic model naturallyagree (note that for d ­ 0 there remains a spin excita-tion gap even for N ! ` due to the frustration drivensinglet dimer ordering D fi 0). The important messageof Fig. 5 is that a substantial enhancement of the dimer-ization is needed to achieve a given triplet excitation gap,if the dynamic phonons have frequencies comparable toJ. This partially resolves the quantitative problem en-countered within the alternating and frustrated Heisenbergspin-chain model (1), because fixing parameters J and a,by fitting the high temperature susceptibility, and d, bymatching DST to the INS data in CGO, the dimerizationparameter d is underestimated by roughly a factor of 5VOLUME 81, NUMBER 18 P H Y S I C A L R E V I E W L E T T E R S 2 NOVEMBER 19980.00 0.25 0.50δ0.00.20.40.60.81.0∆ST [J]N=8 (static model)N=10N=12N=8 (dynamic model)N=10N=120.00 0.02 0.04 0.06δ0.410.460.51∆ST[J]ω=1.0ω=0FIG. 5. Triplet excitation gap DST vs dimerization d forthe static spin-only (v ­ 0; ¶, h, ±) and the dynamic spin-phonon model (solid symbols) with phonon frequency v ­ 1for different chain lengths at a ­ 0.36. The inset shows theresults for the dynamic 8-site model [filled symbols: v ­ 0.1(0.3) triangles up (down); v ­ 1.0, s≤d] in comparison to thestatic case (±).[33]. From the uniaxial pressure derivatives of the ex-change coupling J [10] and the structural distortion inthe dimerized phase [7], it is possible to show that thedimerization of J is in fact significantly stronger thand ­ 1.4%. The result of a straightforward data analy-sis leads to an estimate d ø 4% 5% [19]. The inset inFig. 5 shows the evolution of the DST vs d results withincreasing phonon frequency v at a fixed chain length.With decreasing v the data smoothly merge with the re-sult for the static spin-only model.In conclusion, we have numerically analyzed the Peierlsdimerized ground state and the spin–triplet excitationsin a Heisenberg chain model with dynamic spin-phononcoupling. The magnetic excitations inherently includea local lattice distortion requiring a multiphonon modetreatment of the lattice degrees of freedom. A spin-phonon coupling-induced flattening of the triplet disper-sion is avoided for a strong spin-phonon coupling in thenonadiabatic regime. The nonadiabatic phonon dynamicsstrongly renormalizes the magnetic excitation spectrumand the dimerization dependence of the triplet excitationgap. These features are in accordance with experimentaldata on CGO, suggesting the necessity for a nonadiabaticspin-phonon approach to the SP physics in this material.We acknowledge useful discussions with G. Bouzerar,M. Braden, D. Khomskii, W. Weber, and A. Weiße. Weare particularly indebted to B. Büchner for informationabout unpublished estimates for the dimerization of J inthe SP phase of CGO. The numerical calculations wereperformed on the vector-parallel supercomputer FujitsuVPP700 at the LRZ München.[1] M. Hase et al., Phys. Rev. Lett. 70, 3651 (1993).[2] R. E. Peierls, Quantum Theory of Solids (Clarendon,Oxford, 1955).[3] M. C. Cross and D. S. Fisher, Phys. Rev. B 19, 402 (1979).[4] For a review, see J. P. Boucher and L. P. Regnault, J. Phys.I (France) 6, 1 (1996).[5] A. J. Heeger et al., Rev. Mod. Phys. 60, 781 (1988).[6] For a review, see J. W. Bray et al., in Extended LinearChain Compounds, edited by J. S. Miller (Plenum, NewYork, 1983), Vol. 3, pp. 353-415.[7] M. Braden et al., Phys. Rev. B 54, 1105 (1996); Phys.Rev. Lett. 80, 3634 (1998).[8] L. Bulaevskii et al., Solid State Commun. 13, 125 (1973).[9] B. Büchner et al., Phys. Rev. Lett. 77, 1624 (1996).[10] K. Fabricius et al., Phys. Rev. B 57, 1102 (1998).[11] G. Castilla et al., Phys. Rev. Lett. 75, 1823 (1995).[12] F. D. M. Haldane, Phys. Rev. B 25, 4925 (1982).[13] K. Okamoto and K. Nomura, Phys. Lett. A 169, 433(1993).[14] R. Chitra et al., Phys. Rev. B 52, 6581 (1995).[15] S. R. White and I. Affleck, Phys. Rev. B 54, 9862 (1996).[16] A. M. Tsvelik, Phys. Rev. B 45, 486 (1992).[17] J. Riera and A. Dobry, Phys. Rev. B 51, 16 098 (1995).[18] G. Bouzerar et al., cond-mat /9701176.[19] B. Büchner et al., cond-mat /9806022.[20] F. H. L. Essler et al., Phys. Rev. B 56, 11 001 (1997).[21] D. Khomskii et al., Czech. J. Phys. 46, Suppl. S6, 32(1996).[22] G. Bouzerar et al., Phys. Rev. B 58, 3117 (1998).[23] An attempt in this direction has recently been formulatedby G. Uhrig (cond-mat /9801185).[24] B. Bäuml et al., Phys. Rev. B 58, 3663 (1998).[25] D. Augier and D. Poilblanc, Eur. Phys. J. B 1, 19 (1998).[26] T. Barnes and J. Riera, Phys. Rev. B 50, 6817 (1994).[27] We emphasize that the flattening of the dispersion, inprinciple, cannot be obtained if only phonon modes withq ­ p are included as in Ref. [25].[28] G. Wellein and H. Fehske, Phys. Rev. B 56, 4513 (1997).[29] L. P. Regnault et al., Phys. Rev. B 53, 5579 (1996).[30] G. Wellein, Ph.D. thesis, Universität Bayreuth, 1998.[31] The local singlet formation nevertheless shows up inmodel (2) by a strong decrease of the long-ranged spin-spin correlations kSz1Szj l with increasing g.[32] D. Augier et al., cond-mat /9802053.[33] M. Braden (private communication).3959

Peierls dimerization with nonadiabatic spin-phonon coupling

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Peierls Dimerization with Non-Adiabatic Spin-Phonon Coupling

Peierls Dimerization with Non-Adiabatic Spin-Phonon Coupling

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