PHYSICAL REVIEW B VOLUME 45, NUMBER 1 1 JANUARY 1992-I

Spectrum of magnetic excitations in the spin-Peierls state

A. M. TsvelikDepartment ofPhysics, Uniuersity ofFlorida, Gainesuille, Fiorida 32611

(Received 28 June 1991)

The spin-Peierls state is usually described by the spin-2 Heisenberg model with alternating exchange

integrals. In the case of weak dimerization, this model is equivalent to the sine-Gordon model [T.Nakano and H. Fukuyama, J. Phys. Soc. Jpn. 50, 2489 (1981)] with the definite coupling constant(2m )' . We call attention to the fact that the one-particle excitation spectrum of this model consists offour branches of excitations with gaps M, M, M, 1.73M. The expression for the imaginary part of thespin-spin correlation function is provided. It is directly measurable by inelastic neutron scattering.

has a definite scaling dimension 6 and therefore may berepresented via exponents of the bosonic field y:

e(x, t) =A cospy(x, t), p /8m=6, .(3)

The spin-Peierls transition is a kind of magnetoelastictransition. It occurs in a system of one-dimensionalspin- —,

' chains coupled to three-dimensional phonons be-

cause the energy-energy correlation function of a singlechain has a singularity at the wave vector ~q~ =sr/a (a isa lattice distance, q is parallel to the chain). At low tem-peratures it leads to dimerization of the lattice, formationof a gap in the spectrum of magnons and exponentialtemperature drop of the magnetic susceptibility. Allthese efFects have been observed for various one-dimensional compounds (see Ref. 1 and referencestherein).

In this paper we discuss excitation spectrum and spin-spin correlation functions in the spin-Peierls state inmore details.

The usual rn.odel for the spin-Peierls state is the spin- —,'

Heisenberg chain with exchange integrals of alternatingmagnitude:

NH=J g [(S„S„+,)+5( —I)"(S„S„+t)].

n=1

The quantity 5 is usually considered as a fixed parameter.It implies that fluctuations in the phonon subsystem areweak which is possible if phonons are three dimensional.We consider the case ~5~ &&1 (weak dimerization) whenthe gap is small in comparison with the magnon's bandwidth and discuss only low-energy (large-distance) prop-erties. All equivalences between difFerent models will beunderstood as equivalences of low-energy properties.

The requirement of weak dirnerization is satisfied insuch materials as CuClz (5=0.25, J =22 K), piperazini-um CuC13 (5=0.25, J=20 K), Cu-octadecyltrichlorosilane (5=0.05, J=40 K).

At 5=0 the model (1) is exactly solvable and wasshown to be equivalent to the model of the free bosonicfield. ' The operator

e(x) =( —1)"(S„.S„+,)

(A is nonuniversal constant).According to Ref. 4 the scaling dimension of the

operator (2) is equal to —,'. The same result may be ex-tracted from Ref. 5. The authors calculated the 5 termin the ground-state energy as a function of length L of thesystem. They got

[E(5)—E(0)]/L — 5L . — (4)

H= —fdx +v +m cost/8m. ppU 2 2

p= —,', m -5,[vr(x), y(y)] =i 5(x —y),

(7)

where u is the magnon velocity.Because only the scaling properties of the spin operator

coupled with phonons are essential the present descrip-tion is valid not only for insulating spin chains but alsofor metallic systems described by the Hubbard model. Itcan be shown that the scaling dimension of the corre-sponding spin field does not depend on the electron con-centration. Therefore the model (7) describes the magnonspectrum of conducting spin-Peierls systems as well.

The sine-Gordon model is exactly solvable, and at the

From another side the quantity (4) can be expressedthrough the correlation function:

[E(5)—E(0)]/L ——5'f dt f dx(e( xt)e( 0, 0)) .00 0

(5)

The correlation function of the bosonic exponent onthe strip of width L is equal to '

( e(x, t )e(0,0) )— (m /L)[sinh (srtv/L) —sin (srx/L)]

(6)

Substituting (6) into (5) we get that the L dependence atL «g (correlation length) should be L' '. The com-parison with (4) gets b, = —,'.

Thus the model (1) in its low-energy sector isequivalent to the sine-Gordon model

45 486 1992 The American Physical Society

45 BRIEF REPORTS 487

point P= —,' its spectrum consists of kink and antikink

modes and two bound states (breathers). The dispersionlaws at small momenta are given by

e„(q)=Qv q +M„(n =1,2, s,s),

Examples). In order to separate matrix elements for Byand cos( V 8n Py ) one has to use the fact that they havedifferent properties under the operation of charge conju-gation:

M, =M =M, (8)

7T 7rM =2sin —M=M M =2sin —M=1.72M .j. 6 & 2

The particles s, s represent kinks and antikinks, and par-ticles n =1 and n =2—their bound states. The kink'smass gap M is related to the critical magnetic field

gLp&H, =2M at which the system acquires a nonzerosusceptibility.

Because the quantity 5 is a coefftcient in front of theoperator with the scaling dimension 6=—,

' it has a scalingdimension 1 —6=—,

' (5-L ~ } and therefore the mass

gap which has the dimension —,' should be of order of

M-5 . The formula (4) is valid up to L -(=M ' andtherefore using (4) we get for an infinite system the well-known result (see, for example, Ref. 4):

[E(5) E( 0)]/L———5

If the approach based on the model (1) is correct, theexcitations (8) must be observable by inelastic neutronscattering which measures the imaginary part of thespin-spin correlation function.

According to Refs. 2 and 3 the spin operators of theisotropic Heisenberg chain are expressed in terms of thebosonic field as follows:

CZ,+(8)C '=Z,+(8),

CZ+(8}C '=( —1) Z+(8),

(13)

Imli(co, q)=yoq 5(cv —&q v +M )Qq2v~+M~

++co qv —4M—f (cv, q}

co(3M . (14)

where Z;+(8) is an operator creating a one-particle statewith rapidity 8 (rapidity is related to momenta asfollows: M; sinh8=qv). These operators satisfy theZamolodchikov-Faddeev algebra but normalizeddifferently [see two equations without numbers just beforeEq. (11) in Ref. 9].

These results allow us to calculate the imaginary partsof the two-spin correlation functions at T=O. Near

q =0 one has

S'(x)= ' v'"B„y(x)+(—1)"'~cosPy(x),~2~S'(x)=( —1)"'~e*'t'~'"'

and therefore we get

G (cv, n q)= ,'G+ (c—v, n q—)+II(co,q)—,2V

11(co,q) = (y(cv, q)y( —co, —q) )2'

(loa)

(lob)

This is the correlation function (B„yB„y) which hasnonvanishing matrix elements with the state Z&+(8)~0)containing a single first breather and the state with kinkand antikink Z,+(8&)Z,+(82}~0) (single kinks never ap-

pear). The second breather does not appear because thestate Zz+(8}~0) does not change sign under the chargeconjugation (13}.

(the factor —,' is present because the system is isotropic).

It is reasonable to suggest that the relations (10) and(11) are still valid for the dimerized model (1) if the di-merization is weak. Then one can calculate the spin-spincorrelation functions using the exact results derived forthe sine-Gordon model by Smirnov. '

Smirnov calculated the operator g+(x, t;a) which isthe generating functional for matrix elements of physicalfields of the sine-Gordon model. In particular, the fol-lowing is valid [see Eq. (Al) from Ref. 9]:

r rrr/ /

/// /

//

Continuum

2M1.73M

g (x, t;a~ ao )

X+=e (0~ B„y+rn f (cos&8m. Py —1}do.

+O(e } (12)

where x+ =vt+x and the state (0~ is the exact groundstate. The matrix elements of g+(x, t;a) for the stateswith lowest energy are presented in Ref. 8 (see the section

FIG. 1. The qualitative picture of the singularities of thespin-spin correlation function on the (co, q) plane. The solidlines correspond to singularities given by Eqs. (14) and (15); thedashed lines are extrapolations for large wave vectors.

488 BRIEF REPORTS 45

Near q =m. one has

ImGt(cv, ~—q) = &1 ti( —V q v +3M )

Qq 2v 2 +3M 2

e, (q)=+q v +M + ,'gipzH .—(16)

+ g( q' (3M.Pro —

q v —4M

(1&)

The functions f (rv, q), g (ro, q) in Eqs. (14) and (15) aresome known smooth nonzero functions and yo is a sus-

ceptibility of the ordinary Heisenberg chain.We give the qualitative picture of the singularities for

the G (co,q) on the (rv, q) plane in Fig. l.Additional singularities related to multiparticle emit-

tance arise at larger energies. We do not discuss them be-cause they are weak and hardly can be observed.

The spectrum changes if a magnetic field is applied.We call attention to the fact that all bound states do notreact on the magnetic field; in the same time the field liftsthe degeneracy between kink and antikink states:

~, (q) =+q'v'+M' ,'gL priH— —,

As we have mentioned above when the magnetic fieldexceeds the critical value gz p&H, =2M the system ac-quires a finite magnetization. Therefore the mass gap canbe independently determined as the critical magnetic fieldfrom measurements of magnetic susceptibility.

Thus according to our analysis the spin-spin correla-tion function of the dimerized Heisenberg chain (1) hasthe pole below the continuum. This is the maindi6'erence between this case and the ordinary isotropicHeisenberg chain which has only the continuum (there-

fore it has only square-root singularities in spin-spincorrelation functions' ' "). A strong magnetic field

gL p&H )2M returns the dimerized system to this type ofbehavior.

I would like to express my gratitude to Roser Valenti,who attracted my attention to this problem. I am alsothankful to Nicos Nicoupolos for help in preparing themanuscript and discussion. This work was supported byDARPA under Contract No. MDA 972-85-1006 and

by the University of Florida Division of SponsoredResearch.

'Current address: Department of Physics, Princeton Universi-

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