f. devreux and j. p. boucher- solitons in ising-like quantum spin chains in a magnetic field : a...

8/3/2019 F. Devreux and J. P. Boucher- Solitons in Ising-like quantum spin chains in a magnetic field : a second quantization

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Solitons in Ising-like quantum spin chains in a magnetic field : a secondquantization approach

F. Devreux and J. P. Boucher

DRF-G/Service de Physique/DSPE (ER CNRS 216), Centre dEtudes Nuclaires de Grenoble, 85 X-38041 Grenoble Cedex, France

(Reu le 14 mai 1987, accept le 24 jccin 1987)

Rsum. 2014 Nous proposons un formalisme de seconde quantification pour les modes de solitons dans leschanes antiferromagntiques dIsing de spin s =1/2. Nous donnons les expressions exactes 2014 dans le cadredune approximation un soliton 2014 des facteurs de structure dynamique en prsence dun champ magntique,parallle ou perpendiculaire la chane. Nous montrons que le champ magntique donne lieu unddoublement des modes de solitons. Nous calculons aussi la fonction de corrlation de spin relative au mode

antiferromagntique dans le cas de chaines finies ou infinies.

Abstract. 2014 We propose a second quantization formalism for the soliton modes in the s =112 Ising-likeantiferromagnetic chain. We give the exact expressions, within the one-soliton approximation, for the

dynamical structure factors in presence of a magnetic field, either parallel or perpendicular to the chain. Weshow that the field gives rise to a splitting of the soliton modes. We also derive the spin correlation functionrelative to the antiferromagnetic mode for finite and infinite chains.

J. Physique 48 (1987) 1663-1670 OCTOBRE 1987,

Classification

PhysicsAbstracts75.25 - 75.30 - 76.50 .

1. Introduction.

Antiferromagnetic Ising-like chains are known tooffer good examples of soliton excitations.As shownfirst by Villain, an exact derivation for the fluctua-tions induced by the solitons can be obtained in thecase of quantum spins s = 1/2 [1]. For classical spinsdifferent derivations have been proposed, which allrefer to the sine-Gordon model. Recently, Haldane

has established that, in this case, additional internalmodes are expected to occur in the solitons [2].Moreover, the nature of these internal modes should

depend on whether the spin value is integer or halfinteger. In the same line,Affleck has related these

predictions to the dyon concept which is used in

particle physics. It was also suggested that the

application of an external magnetic field in the spindirection might offer a useful way of disentanglingthe dyon levels [3].

In the present work, we consider explicitely thecase of quantum spin chains in an external magneticfield which is applied either parallel or perpendicularto the spins. The second quantization approach thatwe propose provides a clear understanding of theeffects of such a field on the solitons. Rather

different behaviours are actually observed dependingon the field direction. In the following, the dynamicalstructure factors sa ( q, w) (a = x, y, z ) associatedwith the solitons are calculated within the one-

soliton approximation in the full Brillouin zone

(- 7T q 7r). Concerning the antiferromagneticmode near q = ir, resulting from the soliton-indu-ced spin flippings in the magnetic sublattices, newstatistical arguments are given, which reinforce the

phenomenological description proposed by Maki [4].

2. Second quantization.

The starting point is the following Hamiltonian :

with s, Hi IJ and H_LIJ 1. The first term where J isthe exchange coupling describes the Ising energy. Itis responsible for the antiferromagnetic ground statewith the spins aligned along the z direction (whichhere coincides with the chain axis). Following Vil-lain [1], we consider chains with an odd number of

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198700480100166300
http://www.edpsciences.org/http://dx.doi.org/10.1051/jphys:0198700480100166300http://dx.doi.org/10.1051/jphys:0198700480100166300http://www.edpsciences.org/


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spins and cyclic conditions. This situation forces the

presence of an odd number of alternance defects or

topological solitons in the chains. The diagonaliza-tion of the Hamiltonian in equation (1) has beenperformed by Shiba andAdachi [5] in order tocalculate the

dynamical susceptibilityat q = 0. Our

second quantization approach allows an extension ofthis work to any value of q.

LetIp) (p half integer) be the state with onesoliton between sites p - 1/2 and p + 1/2. In the

subspace with one soliton, the matrix elements ofthe spin operators are :

The phase convention chosen in equation (2) is suchthat the soliton is positive (extra spin up) for

p - 1/2 odd and negative (extra spin down) forp - 1/2 even (see Fig. 1). We now introduce cp and

Fig. 1. -Antiferromagnetic ground state (a), positivesoliton (b) and negative soliton (c).

cp the creation and annihilation operators for asoliton at the position p and their Fourier transforms

given by :

The spin operators are expressed as :

and using equation (2), evaluated to be :

Similarly, the Hamiltonian can be rewritten as :

The corresponding eigenvalues are obtained bydiagonalization in the k) ,k - 7T ) subspace [5] :

With no magnetic field, the k) andI 7T - k) statesare degenerate and the Brillouin zone can be reduced

to - r k -- - ,T. The effect of a magnetic field is2 2

g

seen to lift this degeneracy.As a result, one obtainsa splitting of the soliton energy in two branches asshown in figure 2.

Fig. 2. - Band-structure of the one-soliton excitationswithout magnetic field (a), and with a magnetic fieldparallel (b) and perpendicular (c) to the chain.

3. Parallel field.

For a field parallel to the chain axis (Hl = 0), theHamiltonian (Eq. (4)) becomes :

where

(a = - I ) states correspond to a combination of


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positive (negative) solitons as shown by the followingdecomposition :

Thus, one obtains for H parallel, a simple Zeeman

splitting favoring the negative solitons in the ( k - )states with respect to the positive solitons in the

k + ) states. Moreover, the energy splitting isindependent of the wave vector k (Fig. 2b).Using equations (3) and (8), the spins operators

are now given by :

and the spin correlation functions can be expressedas :

where nku defines the thermal population of the

I ku) state. It can be expressed as nku = ns P kU/ ZI ,

where ns ~ exp(- (3J) is the soliton density and :

In the last equation, Io (x ) is the zero-order modifiedBessel function. The dynamical structure factors

SI(q, ( ) are obtained by Fourier transformation of

equation (10).After an integration over k, one gets :

are thermal weighting factors which are given by :

two energy branches are uniformly occupied and thethermal factors are simplified inA( (q) =A( (q) =ns(1 - ns) and B (q) = 0.At lower temperature,ns can be neglected compared to 1 and for Hjj =

0, one recovers the results previously given byVillain for the z-component [1] and by Nagler et al.for the transverse spin component [6]. Finally, at

very low temperature (kT .c Hi, EJ), only the lowerstates of the lower band are occupied : nku =

ns 5 k, ,,, /2 8 u, - and equations (11) are replaced by&-functions which can be easily derived from

equations (10).All the dynamical structure factors in

equation (11) exhibit a square root divergence. Thiscorresponds to the nesting condition characteristic ofone-dimensional bands. It is also worthnoting that

Sf (q, w ) is unaffected by the external magnetic


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field. This is expected as this function corresponds totransitions inside each energy band of figure 2b. Onthe opposite, for the transverse spin fluctuationswhich describe transitions from one band to the

other, the position of the divergence is shifted by theinterband

splitting : f2 q 6=

(12 q + HI) with = 1. This results in a doubling of the soliton modes,as shown in figure 3 where we have drawn thefunction :

Fig. 3. - Dynamical structure factor Sy(q, w ) in theabsence of magnetic field (a) and with a magnetic field

parallel (b) and perpendicular (c) to the chain. The

following parameters have been used : J = 100 K, T =

25 K, EJ = 10 K. The wavevectors are labelled in reciproc-al lattice unit : q = q/7r.

notices that, at q = 0 and q = w (i.e. for f2q = 0),the spreaded distribution of frequencies merges into&-functions. In particular the ESR mode Sr(q =

0, w ) (a = x or y), which corresponds to verticaltransitions between the two subbands in figure 2b is

peaked at the conventional Larmor frequency:a) = HII -

4. Perpendicular field.

In perpendicular field (HII = 0), the Hamiltonian(Eq. (4) can be diagonalized in the initialk) basis :

with - 7T -- k 7r.Again, two distinct energy branc-hes are obtained in the reduced Brillouin zone as

shown in figure 2c. The states in the upper branch

correspond to kI 1 7r in equation (13) ; they canbe viewed as triplet states (positive combination ofadjacent positive and negative solitons), while the

lower branch k :> 7r in equation (13)) can beconsidered as singlet states. Using equation (3), thecorrelation functions are expressed as :

Exact calculations of the dynamical structure factorscan also be performed (seeAppendix). However,their expressions are too complicated to allow a

simple formulation.As an example, the function

Sy L (q, has been calculated; it is shown in

figure 3c for H 1. = &/.A doubling of the solitonmode is also observed, which, however differs

drastically from the case of parallel field. The

position of the square root singularities can be

expressed (see Appendix) as a function of thescattering wave vector q as :

where

For h -- 1, four singularities are obtained whichcorrespond to 7y = 1 in equation (15) ; however,for hq > 1, there are only two singularities obtainedfor Ty = + 1. This remarkable result is displayed in

figure 4 where the dispersion relation of the sing-ularities of Sy L (q, l) and S L (q, w ) are shown fordifferent values of H_Ll eJ. The twice degeneratedsingularity line for H 1. = 0 is splitted into twodistinct lines for H1. > 0. While a gap with frequencyw = 2 H 1. is open at q = 0 for one line, the second

line is now restricted to the interval comprisedbetween q = 2 sin-1 (H1./8 EJ) and q = ir. With

H1. increasing, this second line is reduced to a


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Fig. 4. - Dispersion relation of the singularities of

Sl (q, w ) and Sz L (q, - ) for different values of the perpen-dicular field : H 1.. = 0

(fullline), H 1.. = 2 e7 (dotted

line),H 1.. = 4 EJ (dotted-broken line) and H 1.. = 8 sJ (brokenline). The singularity lines for S(q, w ) are obtained bychanging q in 7r - q.

smaller and smaller zone near q = ir, where it

condensates for H.1. = 8 EJ. For larger field, only thefirst singularity line remains present.A similarbehaviour is observed for the function Sl (q, m ) bychanging q in zr - q. For this function, the gapoccurs at q = 7r and the condensation point at

q = 0. Finally, one notices that, contrary to the

parallelfield

case,the ESR modes

Sr (q=

0, w )and

Sz L (q = 0, w ) consist of a continuum of resonance

frequencies and present square-root divergences atthe somewhat unusual values : co = 2 H.1. which

actually correspond to the nesting condition forvertical transitions in figure 2c.

5.Antiferromagnetic mode.

The one-soliton approximation becomes insufficientto describe the fluctuations in the z direction near

the antiferromagnetic point q = 7T, as evidenced by

the divergent 1/cos2 1 q factor in equations (10) and2(14). To go beyond this approximation, we calculatethe fluctuations resulting from the flipping induced

by the solitons, considered as independent randomevents. The z-component of the spin at site n is fixed

by two conditions : i) the parity of n and ii) the totalnumber of solitons on the left hand side of the site n.

The corresponding correlation function can be writ-ten as :

In this equation, we have used (- )N = ei7TN and wehave definedAN (n, t) = N (n, t ) - N (0, 0), whereN (n, t ) is the operator which describes the numberof solitons present at time t between the left end ofthe chain and the site n. This number is the result of

a random process, which accounts for the presenceor the absence of solitons in a given k) state.Atany time,AN can be considered as the sum of two

termsAN=AN + +AN - , corresponding to solitons

which give positive (+ 1 ) and negative (-1 ) con-tributions. In equation (17), one can actually replacethis sum by the differenceAN = N+ - N_, whichremains a positive quantity. The correlation function

appears therefore to be the characteristic function

(or the Fourier transform) 0 (n, t ; u ) of the proba-bility distribution p (AN ) for u = 7r :

In order to evaluate the moments of this dis-

tributions,AN is

expressedas a function of the

creation-annihilation operators :

In an infinite system, the second term in the

parenthesis occuring in equation (19) can be droppedout since it gives vanishing contribution when inte-

grated over K if N -+ oo.As the contributions of the

different k) states are statistically independent, thefirst two moments of p (,N) are given by :


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Using equation (19), one obtains :

where vx = dEx/dk. In the last step of equation(21), we have made the approximations 1-

nk , K == 1 and Ek + x - Ek = Vk K. Otherwise theintegration over K is exact [7]. It appears thereforethat the two first moments of the distribution

p (AN) are equal. This result agrees formally withthe assumption of a Poisson distribution and justifiesin the quantum case, the description first proposedby Krumhansl and Schrieffer [8] for solitons in theclassical 04 model. Since the characteristic functionassociated with the Poisson distribution is 0 (u)exp[M1(eiu -1)], we obtain :

This expression was first given by Maki [4], but not

explicitely demonstrated. It can be viewed as anextension to this quantum s = 1/2 case of the soliton

gas model preiously used for classical s = oo sine-Gordon solitons [9, 10]. In this derivation, the effectof the magnetic field has not been considered.However, it is easy to realize that the only effects arerelated to changes in the thermal population nk andsoliton velocity vk in perpendicular field.

Finally, we consider the case of finite chains. Thesecond term in parenthesis of equation (19) can nomore be dropped out. It follows that the time-

dependent contribution to the first moment van-

the fact that the k) states are no more progressive,but stationnary waves. To simplify, we consider onlythe autocorrelation function (n = 0), which is relev-ant for neutron spin echo and NMR techniques. Therelated second moment is given by :

where we have again made the approximations :1- nk + x =1 and Ek+K - Ek = vk K. The minimumfunction Min (x, y ) results from an exact integrationover K [11]. To account for a zero first moment anda finite second moment, we can tentatively assume aGaussian distribution for

AN(0,t ). The correspond-

ing characteristic function is :

At short time, the time dependence of the corre-lation function is the same for finite and infinite

chains. However, at long times, it goes to a constant

in finite chains : Sz 0, t ) =1 exp (- 7T2 ns l ), whichdepends on the average number of solitons ns llocated between the end of the chain and the

considered site. Of course, in real systems, it wouldbe

necessaryto take into account the random

distribution of the segment lengths [12].

6. Conclusion.

The present description in terms of creation-annihi-lation soliton operators provides a comprehensivepicture of the effects of an external magnetic field onthe soliton excitations in Ising-like quantum spinchains. New specific features have been establishedfor both field parallel and perpendicular to the spins.For parallel field, an additional Zeeman splitting isfound. For perpendicular field, one is led to a

simultaneous changeof

boththe

energy andthe

velocity of the solitons. The corresponding dynami-cal structure factors have been calculated in the one-

soliton model. It is known that this approximation is

quite insufficient to describe realistic spin systems [1,10, 13]. Interactions between solitons, betweensolitons and magnons and with impurities are ex-

pected to round off the square-root singularities [14].However, the main feature - the doubling of thesoliton modes - established by our formulationwould be maintained. The lineshapes displayed in

figure 3 have been evaluated for the physical par-ameters J = 100 K, T = 25 K and EJ = H =10 K,which are realistic values for the compoundsCsCoCl3 and CsCoBr3. For these compounds thedoubling of the soliton modes should be accessible

experimentally. In principle, neutron scatteringmeasurements would permit a direct observation ofthe S"(q, w ) in the full Brillouin zone [6, 14-16].

Alternatively, the magnetic resonance techniquescan provide a check of the Zeeman splitting and ofthe gaps of the singularity lines expected at q = 0,for HII and H1- respectively [17, 18].The present results have been obtained in the case

of quantum spin chains. We may wonder if they can

be extrapolated to classical spin chains. The Zeemansplitting obtained for Hp might be the analog of the

dyon levels described byAffleck. It would remain to


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find the classical analog of the soliton doubling for

H 1.. Similar calculations in Ising-like classical spinchains are highly desirable. Finally, the present workwhich concerns a spin 1/2 system can be a firstcontribution to the crucial question concerning thedifferent behaviours expected for antiferromagneticchains with integer and half-integer spins.

AppendixA : dynamical structure factors in perpen-dicular field.

To get the Fourier transforms of equations (14), oneuses the relation:

with ki being the solutions of f (k ) = 0. This gives :

where ki are the solutions of the energy conservation

equations :

4 EJ sin q sin 2 k+ 2 H sin1q sin ki +2

4 EJ sin q sin 2ki +2Hcos 1 q cos ki +2

and

By changing ki into Ir k,. for Sx and takingu = cos ki, one gets an unique equation :

where n = ( /2 n q and hq are defined by

equations (16). It can be solved by standartmethods, but only the solutionsI u I -- 1 should beretained. As the solutions of this fourth degreeequation are quite complicated, we will not give the

explicit expressions for the resulting dynamical struc-

ture factors ; it is, however, easy to have themcalculated with a computer by replacing ki in

equation (A.1) with the solutions obtained from

equation (A.3).On the other hand, the singularities are obtained

as the poles of equation (A.1) which are given by :

with v = sin ki (again ki has been changed into1 7T - k for S").Among the two existing solutions :

with 71 = 1, both are valid for h -- 1, but onlythat with = + 1 is correct for hq > 1 because ofthe conditionI v ,1. The general expression for

the singularity frequencies (Eq. (15)) immediatelyfollows using equation (A.2). Moreover, it can bechecked by making a development near d2 that thesingularities are of square root type as in parallelfield.

References

[1] VILLAIN, J., Physica B 79 (1975) 1.[2] HALDANE, F. D. M., Phys. Rev. Lett. 50 (1983) 1153.[3]AFFLECK, I., Phys. Rev. Lett. 57 (1986) 1048.

[4] MAKI, K., Phys. Rev. B 24 (1981) 335.[5] SHIBA, H. andADACHI, K., J. Phys. Soc. Jpn 50

(1981) 3278.

[6] NAGLER, S. E., BUYER, W. J. L., ARMSTRONG,A. L. and BRIAT, B., Phys. Rev. B 28 (1983)3873.

[7] GRADSHTEYN, I. S. and RYZHIK, I. M., Table ofIntegrals, series and products (Academic Press,New York) 1965, p. 371.


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[8] KRUMHANSL, J.A. and SCHRIEFFER, J. R., Phys.Rev. B 11 (1975) 3535.

[9] MIKESKA, H. J., J. Phys. C 13 (1980) 2913.

[10] MAKI, K., J. Low Temp. Phys. 41 (1980) 327.

[11] Reference [7], p. 451.

[12] NAGLER,S.

E., BUYERS, W. J. L., ARMSTRONG,R. L. and RITCHIE, R.A., J. Phys. C 17 (1984)4819.

[13] SAZAKI, K. and MAKI, K., Phys. Rev. B 35 (1987)257.

[14] BOUCHER, J. P., REGNAULT, L. P., ROSSAT-MIG-NOD, J., HENRY, Y., BOUILLOT, J. and STIR-

LING, W. G., Phys. Rev. B 31 (1985) 3015.

[15] YOSHIZAWA, H., HIRAKAWA, K., SATIJA, S. K. andSHIRANE, G., Phys. Rev. B 23 (1981) 2298.

[16] BUYERS, W. J. L., HOGAN, M. J., ARMSTRONG,

R. L. and BRIAT, B., Phys. Rev. B 33 (1986)1727.

[17]ADACHI, K.,J. Phys. Soc. Jpn 50 (1981) 3904.

[18] BOUCHER, J. P., RIUS, G. and HENRY,Y., to appearin Europhys. Lett.

f. devreux and j. p. boucher- solitons in ising-like quantum spin chains in a magnetic field : a...

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