dynamical effects of phonons on soliton binding in quasi-one-dimensional spin–peierls systems

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Physica B 259261 (1999) 10151016 Dynamical effects of phonons on soliton binding in quasi-one-dimensional spinPeierls systems D. Augier!,*, D. Poilblanc!, E. S+rensen!, I. Affleck" !Laboratoire de Physique Quantique, Universite & P. Sabatier, 31062 Toulouse, France "Department of Physics and Astronomy, and CIAR, University of British Columbia, Vancouver, BC, Canada V6T 1Z1 Abstract The role of dynamical magneto-elastic coupling in spinPeierls chains is investigated by various numerical techniques. We show that, generically, a Heisenberg spin chain coupled to dynamical optical phonons exhibits a transition towards a spontaneously dimerized state. The low energy excitations are characterized as topological solitons. No binding between solitons occurs in the isolated spinphonon chain. However, elastic interchain coupling can lead to the formation of bound states. ( 1999 Elsevier Science B.V. All rights reserved. Keywords: SpinPeierls transition; Solitons; Spinphonon coupling; Quantum spin Quasi-one-dimensional (quasi-1D) magnetic systems have recently received renewed experimental and theor- etical attention with the observation of the spin-Peierls transition in the inorganic CuGeO 3 compound. The spinPeierls materials are widely described in the litera- ture as static 1D antiferromagnetic frustrated dimerized Heisenberg chains. At the MajumdarGhosh (MG) point, the ground state (GS) is doubly degenerate corre- sponding to two simple dimer patterns (called A and B). The elementary soliton s (antisoliton s 6 ) excitations can be depicted as unpaired spins separating two dimer patterns A (B) and B (A). The static dimerized model has however some draw- backs. The dimerization is introduced de facto and conse- quently the model ignores phonon dynamics which are expected to be important when the phonon frequency and the exchange energy become comparable. We consider here a model where the exchange integral is dynamically modulated by the relative atomic displace- ments along the chains. Using independent phonon * Corresponding author. Fax: 05 61 55 60 65; e-mail: augier@irsamc2.ups-tlse.fr. creation (destruction) operators bs i (b i ), this model reads affleck [1], H"J+ i [(1#g(b i #bs i ))(S i ) S i‘1 !1 4 )#aS i ) S i‘2 ] #X+ i bs i b i #H M , where g is the magneto-elastic coupling constant and H M the interchain coupling. We assume dispersionless optical phonons of frequency X. An approximate treat- ment of the phonon dynamics is possible by retaining a few phonon modes only [2]. Here we use instead the variational treatment introduced by Fehrenbacher [3] based on phononic coherent states, which enables us to keep all phonon modes. The numerical results are based on Lanczos exact diagonalization (ED) of closed rings with up to L"14 sites supplemented by finite size scaling analysis and a comparison with Bethe ansatz exact re- sults of the Heisenberg chain and density matrix renor- malization group (DMRG) calculations. The existence of a spontaneous symmetry breaking in the isolated chain (H M "0) is analytically and numer- ically established in a large region of parameter space [4]. The dimerized phase is signalled, in the thermodynamic 0921-4526/99/$ see front matter ( 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 0 6 3 7 - 1

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Physica B 259—261 (1999) 1015—1016

Dynamical effects of phonons on soliton bindingin quasi-one-dimensional spin—Peierls systems

D. Augier!,*, D. Poilblanc!, E. S+rensen!, I. Affleck"

!Laboratoire de Physique Quantique, Universite& P. Sabatier, 31062 Toulouse, France"Department of Physics and Astronomy, and CIAR, University of British Columbia, Vancouver, BC, Canada V6T 1Z1

Abstract

The role of dynamical magneto-elastic coupling in spin—Peierls chains is investigated by various numerical techniques.We show that, generically, a Heisenberg spin chain coupled to dynamical optical phonons exhibits a transition towardsa spontaneously dimerized state. The low energy excitations are characterized as topological solitons. No bindingbetween solitons occurs in the isolated spin—phonon chain. However, elastic interchain coupling can lead to theformation of bound states. ( 1999 Elsevier Science B.V. All rights reserved.

Keywords: Spin—Peierls transition; Solitons; Spin—phonon coupling; Quantum spin

Quasi-one-dimensional (quasi-1D) magnetic systemshave recently received renewed experimental and theor-etical attention with the observation of the spin-Peierlstransition in the inorganic CuGeO

3compound. The

spin—Peierls materials are widely described in the litera-ture as static 1D antiferromagnetic frustrated dimerizedHeisenberg chains. At the Majumdar—Ghosh (MG)point, the ground state (GS) is doubly degenerate corre-sponding to two simple dimer patterns (called A and B).The elementary soliton s (antisoliton s6 ) excitations can bedepicted as unpaired spins separating two dimer patternsA (B) and B (A).

The static dimerized model has however some draw-backs. The dimerization is introduced de facto and conse-quently the model ignores phonon dynamics which areexpected to be important when the phonon frequencyand the exchange energy become comparable. Weconsider here a model where the exchange integral isdynamically modulated by the relative atomic displace-ments along the chains. Using independent phonon

*Corresponding author. Fax: 05 61 55 60 65; e-mail:[email protected].

creation (destruction) operators bsi(b

i), this model reads

affleck [1],

H"J+i

[(1#g(bi#bs

i))(S

i) S

i`1!1

4)#aS

i) S

i`2]

#X+i

bsibi#H

M,

where g is the magneto-elastic coupling constant andH

Mthe interchain coupling. We assume dispersionless

optical phonons of frequency X. An approximate treat-ment of the phonon dynamics is possible by retaininga few phonon modes only [2]. Here we use instead thevariational treatment introduced by Fehrenbacher [3]based on phononic coherent states, which enables us tokeep all phonon modes. The numerical results are basedon Lanczos exact diagonalization (ED) of closed ringswith up to L"14 sites supplemented by finite size scalinganalysis and a comparison with Bethe ansatz exact re-sults of the Heisenberg chain and density matrix renor-malization group (DMRG) calculations.

The existence of a spontaneous symmetry breaking inthe isolated chain (H

M"0) is analytically and numer-

ically established in a large region of parameter space [4].The dimerized phase is signalled, in the thermodynamic

0921-4526/99/$ — see front matter ( 1999 Elsevier Science B.V. All rights reserved.PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 0 6 3 7 - 1

Fig. 1. (a) Minimum energy of the *Sas a function of the inverse

square of the chain length 1/L2 for a"0, g"0 (L: BetheAnsatz), a"0.5, g"0 (h: DMRG, j: ED) and a"0, g"0.45,X"0.3J (e: ED). (b) Binding energy for a"0.5, g"0 (h:DMRG, j: ED)), a"0, g"0.45, X"0.3J (e: ED) and a"0.5,g"0.4, X"0.3J (n: ED) as a function of 1/L.

limit, by (i) the two fold degeneracy of the GS, (ii) theopening of a spin gap and (iii) a lattice dimerization.

The minimum energy *Sof the elementary excitations

in the dimerized GS is shown in Fig. 1a. *Sappears to be

finite and the spin-12excitation spectrum is massive indic-

ating the existence of solitons in contrast to the case ofthe Heisenberg chain where the spin-1

2spinon excitations

are gapless, as explicitly shown in Fig. 1a using the Betheansatz solution. Our results for the spin—phonon modelare also compared to the case of a purely magnetic

frustrated Heisenberg chain using DMRG data (we esti-mate *

SK0.1170(2)J at the MG point).

The soliton—antisoliton binding energy is shown inFig. 1b. The results for the spin—phonon model are verysimilar to the ones of the MG chain, strongly suggestinga vanishing binding energy. We conclude that solitonsand antisolitons are not bound in the 1D spin—phononmodel.

In order to show that the interchain coupling is crucialto bind solitons and antisolitons, we have considered theelastic interchain coupling H

M[1] at the mean field level,

still retaining the full dynamics of the phonons. In thatcase, the low energy excitations become well defined,suggesting the formation of magnons which can be seenas ss6 bound states [4].

We thank IDRIS (Orsay) for allocation of CPU timeon the C94 and C98 CRAY supercomputers. The re-search of I.A. is supported in part by NSERC of Canada.

References

[1] I. Affleck, in: Proc. NATO ASI: Dynamical Properties ofUnconventional Magnetic Systems, 1997, to be published.

[2] D. Augier, D. Poilblanc, Eur. Phys. J. B 1 (1998) 19.[3] R. Fehrenbacher, Phys. Rev. Lett. 77 (1996) 2288.[4] D. Augier, D. Poilblanc, E. S+rensen, I. Affleck, Phys. Rev.

B 58 (1998) 9110.

1016 D. Augier et al. / Physica B 259—261 (1999) 1015—1016