classical one-dimensional ferromagnets in a magnetic field: static and dynamical properties

PHYSICAL REVIE% B VOLUME 26, NUMBER 9 1 NOVEMBER 19S2

Classical one-dimensional ferromagnets in a magnetic field: Static and dynamical properties

Umberto Balucani and Maria Gloria PiniIstituto di Elettronica Quantistica, I 501-27Firenze, Italy

Valerio TognettiIstituto di Elettronica Quantistica, I 50I2-7Firenze, Italy

and Istituto di Fisica Superiore dell'Uniuersita di Firenze, Firenze, Italy

Angelo RettoriIstituto di Fisica dell'Universita e Gruppo 1Vazionale di Struttura della Materia,

I-50125 Firenze, Italy(Received 16 June 1982)

An interacting-spin-wave theory is presented for describing the behavior of a classicalHeisenberg ferromagnetic chain in an external magnetic field. In this framework all thestatic and dynamical longitudinal properties can be analytically calculated by means of anonperturbative Green's-function approach. In particular, the features of recent computerexperiments showing, at intermediate wave vectors, spectra with a structure tentatively re-

ferred to as a "second-magnon" effect receive a very simple physical interpretation. It is

shown that a thermally induced spin-energy coupling causes transitions between different

regimes: from an ordered condition due to the field to a quasi-isotropic situation (i.e., to a"first-magnon" behavior) determined by the competitive role of the temperature. Explicitexpressions for static quantities, namely longitudinal susceptibility, correlation length, and

magnetization, are also given. The thermally induced crossover toward a nearly isotropicsituation is found to be again present. Exact numerical calculations by the transfer-matrixmethod have been also performed, and an excellent agreement with the theoretical evalua-

tion is found.

I. INTRODUCTION

Much interest has been aroused in the past threeyears about a series of computer experiments inclassical ferromagnetic chains subject to an externalmagnetic field H. ' The main feature observed inthose simulations was the striking influence of thefield on the dynamic correlations associated to spinfluctuations along H. In particular, instead of theusual spin-wave-like peak seen for H=O, the spec-tra showed a rather extended band comprising twobroad peaks. This structure was particularly evi-

dent at intermediate wave vectors. A first interpre-tation of the effect was already attempted byLoveluck and Balcar, ' who referred to the inter-mediate frequency peak as a "second-magnon"mode, bearing some analogy with, e.g., second-sound hydrodynamic modes. As remarked in Refs.1 and 2 the analogy cannot be taken too literally,because the simulations probe wave vectors whichare clearly outside the hydrodynamic regime.

The first theoretical attempt of explanation is due

to Lovesey and Loveluck' who started from a mul-

tivariable Mori approach developed previously andargued that the "second-magnon-like" mode is a re-sult of a static field-induced coupling between thelongitudinal spin fluctuation and the energy. Thispicture of "hybridization" as the relevant feature ofthe phenomenon was subsequently developed byLovesey in an evaluation of the collisional self-

energy which is expected to be valid at very lowtemperatures. As a result, spectra with two 5 peakssuperposed on a background were obtained. It isworthwhile to note that in these theories tempera-ture does not play any essential role, as the variablesleading to the mixed mode are treated on the samefooting.

On the other hand, on physical grounds we ex-

pect that in the presence of a field and at very lowtemperatures a simple noninteracting-boson ap-proach accounts for the dynamic properties of thesystem. As far as static quantities are concerned,the validity of such an approach in this range hasbeen shown previously ' by a comparison with ex-

26 4974 1982 The American Physical Society

CLASSICAL ONE-DIMENSIONAL FERROMAGNETS IN A. . .

act transfer-matrix results. The simulation spectrareported in Refs. 1 and 2 refer to intermediate-temperatures where the computer simulations areeasier to perform. In order to clarify which ap-

proach is correct, Balcar and Loveluck performednew experiments in which relatively low tempera-tures are also considered. Whereas no evidence wasfound for the two sharp features predicted in Ref.5, these low-temperature data supported the predic-tions of noninteracting-boson theory. Apart fromthe presence of two-peaked structures, the new dataindicate that thermal effects are quite important in

changing the spectral features, the intermediate-

frequency peak becoming more and more enhanced

as the temperature is increased.Meanwhile, the "second-magnon" interpretation

was criticized by Reiter eI; al. who showed that theoscillations seen in the energy correlation function

(previously thought to be an indication of a mixed

spin-energy mode) can be viewed at low tempera-tures as an effect due to two-magnon density ofstates. Similar results were also obtained byDe Raedt and De Raedt. ' However, the problemof a theoretical interpretation of the spin-

fluctuation spectra with a field remained unsolved.

In particular, the explanation of the pronouncedtemperature effects in the spectral shapes was un-

clear. Moreover, similar problems arise for staticquantities if finite temperature transfer-matrix dataare considered.

The purpose of the present paper is to give an in-

terpretation of such relevant features by means ofan interacting-boson approach in which the mainthermal effects are explicitly considered. Short ac-counts of the main results have been given before, "but the full detailed derivation along with new re-sults for the static quantities is presented here. Asshown in the following, the dynamical problem re-

quires a nonperturbative solution, a not uncommonfeature in other one-dimensional (1D) problems. ' 'This is explicitly done in Sec. II, where thenoninteracting-boson result is also discussed. Thedynamics of the system is found to be strongly af-fected by temperature through a thermally induced

coupling between spin and exchange energy fluctua-tions. The striking consequences on the spectralshapes at different temperatures, fields, and wavevectors are discussed in Sec. III where the differentphysical origin of the two peaks observed in thesimulation is clarified. The whole interpretationsupports the intuitive idea about the competing roleof temperature and magnetic field. In Sec. IV thetheoretical results for the static quantities are dis-

cussed and compared with new data obtained by thetransfer-matrix method.

II. THEORY

SJ+ =&2Saj,

SJ ——&2Saj.(1—aj aj/2S),

S=S—aazJ J J '

With the use of the Fourier transformation

(2a)

(2b)

(2c)

a =N '~2+aqexp(iqj)q

and omitting constant terms, the Hamiltonian (1)can be written as

A =y (nk+g priH )aka/,

quq p

)(Qq Qp Qq~ap~,

where yi, =cosk and Qi, =2JS(1—yq) is the mag-non frequency in the absence of a magnetic field.

The fluctuation of the quantity Sk (for k+0) isfound to be

Sk=—Sk —(Sk~= X k+e eq

Let us introduce the exchange energy of a spin withits neighbors:

EJ —,(SJ i SJ+SJ. SJ.+i) . .

The fluctuation of its space Fourier transform Ek(k+0) is expressed in terms of Bose operators by

We take the following ferromagnetic Hamiltoni-an:

N N

Jg—S, .SJ+, gps—iH g Sf,j=1 j=l

where J &0 is the nearest-neighbor exchange in-

teraction, and X is the number of spins in the chain.The presence of a magnetic field H which ordersthe spins along the z axis enables us to use thestandard Dyson-Maleev representation of the spinoperators in terms of boson operators. In the fol-lowing, limiting criteria for temperature and mag-netic field will be discussed. We have

4976 BALUCANI, PINI, TOGNETTI, AND RETTORI 26

Ek =E—k—(Ek )= S—N g( 1+yk —

yq—yk+q )ak+qaq

—1/2

q

+2N Q &k+p, k'+p'+k(yk k'+—yp —p' —yk' yk—+k')aka yak'ap'—3/2

A, ,A, ,p,p

In order to evaluate the spectrum (S' kS k )„ofthe longitudinal spin fluctuation we consider the associatedZubarev Green's function

gk(E):—((Sk,S'k ))E N——g((ak+qaq&a k+q aq )&E

The equation of motion of ((ak+qaq, a k+q aq ))E reads

(E+Qk~q Q, )((ak+qaq a k+q'aq—'))E

ak+qaq'a k+q'aq'—l & (J/N) Q &q +q, q +q„(yq q-

g ),$2, i(3,$4

X (([ak+qaq, aq aq aq aq ];a k+qa» ))E . (9)

Performing the commutators on the right-hand side (rhs) of Eq. (9) we obtain

(E+Qk+q Qq)« ak+qaq a k+q a—q »E''=(2 ) '(( ) —( ))&

q+q, ,q, +q. yq q, yq, -« -k+q q, q, q. ' k+q'q »-Ei2 V3 9'4

—J» g &q+q, ,q, +q, (yq, q, yq, )-«ak—+q q, q, q, k+q q' »-E&& &3 &4

+(J/N) y '5»/+»2, k+q+qg(yq/ —k —q yk+q)«aq]a»2a»4aq~a —k+q'aq'&&E0~ ~92~04

+(J/N) rf ~q/+»2, »3+k+»(yq, —q3 y»3)&& q] ta»3 qi —k+q' q'&&E (10)

It is worth noticing that all the terms are normal ordered, thus Eq. (10) is valid both in the quantum and clas-, sical case. Only the latter will be explicitly considered in the following. Decoupling the six-operator Green s

functions in the usual way, we obtain the following set of coupled equations which contain all the leading tem-perature terms in the classical limit:

(E+Qk+q —Qq)((ak+qaq;a k+qa» ))E ——(I/2m )(nk+q nq)5» k+q (2—J/N)(nk—+q n»)—Xg(yk yk+q ——

yp +yq —p )((ak+papla —k+q'aq' )&E i

where nq = (a»aq ) =(kET)/(h+ Qq) and Qq =2JSa(1—yq) is the Hartree-Fock renormalized magnon fre-quency. The self-consistent Hartree-Fock renormalization factor a—:a(T*,h) turns out to be

a=1—(NS) 'g(1 yq)nq—1 —(T"/2—a)(1 hT/ItT), —

with

h T ——h /a =gp~H/JSo. ,

ICr =br(h r+4),T*=I,Toss'.

(13)

CLASSICAL ONE-DIMENSIONAL FERROMAGNETS IN A 4977

Typical values of a versus temperature and field are shown in Table I. Clearly the decoupling scheme adoptedin Eq. (11)neglects any effect due to boson damping, a higher-order effect in temperature.

The "free-boson" result, which is a reasonable approximation at very low temperatures, is obtained lettinga =1 and neglecting the last term in Eq. (11). However, at finite temperatures this term is of the same orderas the Hartree-Fock correction and consequently cannot be neglected. For this reason, a nonperturbative solu-

tion of the integral equation (11) is required. Introducing the following quantities,

hk(E)=N gyq((ak+qaq'a k+q' —q' t)&]

tk(E)=N gsinq((ak+qaq, a k+q aq ))E ~

qq

(14)

from Eq. (11), we arrive at the following inhomogeneous linear system for the unknown quantities gk(E),h/, (E), t/, (E):

{1+ayk[gk (E) hk (E—)]+asinktk '(e) jgk(E) —a[gk '(E)—hk '(E)lhk(E)+atk '(E)tk(E) =gk '(E),

{ayk[hk '(E)—ck '(E)]+asinkrk '(E) jgk(E)+ {1 a[hk—'(E) ck '(E—)]}hk(E)+ark '(E)tk(E) =hk '(E),

{ay/, [t/, (E) r/, (E)]+—asjnk[g/, (E)—ck (E)]jg/, (E) a[t/', —(E) rk '(E—)]h/, (E)

+{1+a[gk '(E) ck '(E—)]}tk(E) =tk '(E),

(16)

where a =4m.J and

gk '(E) =(2n.N) 'QDk (E),q

hk (E):(2~N) gyqDk (E)q

tk" (E)=(2qrN) 'gsinqDkq(E),

ck (E) (2rrN) gyqDg (E)q

rk '(E) =(2nN) 'gyqsinqDk (E),

(17b)

(17c)

(17d)

(17e)

Dk (E)=(nk+q —nq)(E+Qk+q —Qq)

(19)

The complete expression for gk(E) obtained from the system (16) is written in Appendix A. Here we only re-

port the result which includes all the main temperature effects,

gk (E)+4/rJ{[gk (E)1 [hk (E)1 [tk

1+4m J{(1+yk)[gk '(E)—hk '(E)]+sinktk '(E) j

a(T*,h)

T~ =0.1

T*=0.2T*=0.3T*=0.4T*=0.5

h =0.2

0.9600.9160.8670.8130.750

h =0.50.9660.9290.8910.8490.803

h =1.00.9720.9420.9120.8790.846

TABLE I. Hartree-Fock renormalization factor vs

temperature T~ and magnetic field h.The quantity gk '(E)=((Sk,S'* k))g' with a=1represents the free-boson approximation for gk(E)It is quite important to note that the denominatorof Eq. (19) can be written as

1+4m.J{ j =1+(4nJ/S)((Ek, S' k))'k'. ,

(20)where only leading-order terms in the definition (7)of Ek are considered. This shows the role played on


In the classical 10 case al) the quantities in Eq.(19) can be analytically evaluated. LettingE =co+i 0+ and x =co/JSa, the unperturbedGreen's function gk '(x) for x &16sin (k/2) isgiven by

2m Regk '(x) =(T'/Ja )bk(hz+2)

x bk(Kz. +bk—)KrD(k, x)

2nImgI', '(x. )=(T*/Ja )bkx

x bk (Kr +—8 bk )—X

g (k,x)D (k,x)

(23a)

(23b)

where

bk ——4sin (k/2), g(k, x)=(4bk x)', (24—)

D (k,x)=x +bI, [2K& bk(Kr +2)]—x

+bk (Kr'+ bI', ) (25)

In a similar way the other quantities (17b)—(17e)can be obtained and their final expressions are givenin Appendix B. For a= 1, Eqs. (23) have alreadybeen derived by Lovesey. The presence of the fac-tor [g(k,x)] ' in Eq. (23b) causes a divergence atthe edge of the longitudinal spectrum (x =+2bk)This feature is due to a singularity in the two-magnon density of states for the 1D Heisenberg fer-romagnet and it has already been discussed else-where. ' The nonperturbative treatment which

the behavior of the system at finite temperatures bythe thermally activated dynamic coupling between

spin and energy fluctuations.With the use of the classical version of the spec-

tral theorem the physically interesting quantitiescan be calculated in terms of the Green's functiongk(E) = ((Sk,S' k))E. In particular, for the longi-tudinal spectrum we have

AT(S' „S'k)„=—2 Im((Sk;S' k))E „~,o+ ~

N

(21)and for the longitudinal static correlation function,

(S'kS k) = 2+kg T—Re((S k', S' k))

(22)

leads to Eq. (19) removes this unphysical divergencepresent in the free-boson result through the dynam-ic spin-energy coupling CI', '(E)=(J/S)X((Ek,'S' k)}E' in the denominator. This caneasily be seen from the explicit expressions for Reand Im Ck '(x),

2n. ReCk '(x) = — 2 sin k hrJa

x (br+3)+bk(Kr~+bk)X

KrD(k, x)(26a)

2~ Im Ck '(x) = — 2 sin k xJcz

x (br+1)+ bk(hr bk)—X

g (k,x)D (k,x)(26b)

As shown in Appendix 8, the second term in thenumerator of Eq. (19) does not present any diver-

gence. Therefore, the singularity of gk '(x) is elim-inated by the square-root divergence of Ck '(x). Itis important to note that both parts of Ck '(x) areproportional to T' and therefore the removal of thedivergence and the spectral modifications are due tothis thermally activated coupling. Furthermore, weobserve that in other 10 problems this removal hasbeen associated with bound-state effects. ' '

III. DYNAMICAL PROPERTIES

The free-boson limit for the spectrum of themagnetization fluctuations is obtained substitutingEq. (23b) into Eq. (21). The main feature of thesespectra is the presence of the aforementionedsquare-root divergence. Minor additional structuresat intermediate frequencies can occur owing to thepossible minimum of D (k,x) which, however, nevervanishes for finite fields. Since in absence of thefield, D(k, x) vanishes at the frequency of the spin-wave of the isotropic system, we can attribute thesepossible structures to the surviving effect of the ex-change when an ordering magnetic field is present.

When the temperature increases, the spin-waveinteractions must be considered and the nonpertur-bative solution, as given by Eq. (19), must be insert-ed into Eq. (21). After some cumbersome algebra,the spectrum of (S' kS k )„can be obtained:

g(k, x)

2' *'2J(S S )

Sbk T'KA CX

bk(Kg+8 bk) x —~—P(—k,x)+2 2 2 2

a SC&

T'hpg (k,x) D(k, x)— p"(k,x)

a Ez

'2

Ssin kpter

e Lz.2

'2

hz.P"(k,x)

(27)

CLASSICAL ONE-DIMENSIONAL FERROMAGNETS IN A. . . 4979

where

P(k,x)=8sin k+(hT+2)g (k,x),P'(k, x)=[(h,+3)x'+ bk(KT'+ bt„. )]

&(8sin k,P"(k,x)=(8sin k) (x +hTbk) .

(29)

(30) = 1 —T'/KS (32)

preliminary considerations can be made here. Thecorrection terms at the numerator of Eq. (27) have afully physical meaning only forT'«a KT(T'«2v h for low fields). This limitcorresponds to a situation where the magnetization,as expressed by the free-boson theory

Firstly, we note that now the spectrum vanishesat x '=4bI, showing the removal of the aforemen-tioned singularity. However, at low temperatures orhigh fields, a narrow peak is left very nearx =+2bI, as a remnant of the singularity, whereasthe remaining part of the spectrum is virtually un-

changed. When the temperature increases, the Tterm in the denominator tends to destroy this peak,and a new structure at intermediate frequency blows

up, corresponding to the new minimum at thedenominator. As far as the spectral shape is con-cerned, temperature is competing with the fieldthrough the thermally induced spin-energy cou-pling. Therefore, the disorder introduced by thetemperature opposes the ordering effect on the field,yielding a nearly isotropic situation: the observedsecond peak is simply due to the onset of this nearlyisotropic dynamics which ultimately leads to theusual first magnon. The same behavior can be ob-

tained decreasing the magnetic field at fixed tem-

perature: in this case, the tendency of the system toreach an isotropic situation is clearly evident.Moreover, we observe that at low temperatures forh ~0 the denominator of Eq. (27), near the frequen-

cy determined by the vanishing of D(k, x), gives aLorentzian shape with

uip„g ——Qk(1 —T'/2) (31a)

and a half width determined by the T term, name-

ly

0.4 h=0.5752=~~3 0.15

is substantially different from zero.Furthermore, whatever the intensity of the mag-

netic field, for high temperatures, expression (27) isnot more adequate because in it only the leadingtemperature effects have been considered. In prin-ciple, some of the higher terms could be consideredif the more general expression for gk(E} as given in(A5} and (A6) were used. However, this would notbe consistent, as our theory is at first order in theequation of motion and neglects all higher-orderterms leading to boson damping.

Our theoretical spectra at several values of T*,magnetic field h, and wave vector k, shown in thefollowing figures, clearly expound the competitionbetween field and temperature. In Figs. 1 —5 all

amplitudes have been normalized to the respectivevalues of the static correlation function(S' kS k(t =0)), whose expression will be given in

the following section. Figure 1 shows the tempera-ture dependence of the spectra for an intermediatefield h =0.575 and wave vector k =m /3. The un-

6=T*JSsink, (31b) Q2. 0i.e., the values found by Reiter and Sjolander for theisotropic case.

For a heuristic purpose, in the previous discus-sion we have considered a very large range of tem-

peratures and fields. However, we recall that ourapproach is valid for more restricted ranges ofparameters. First of all, the boson approach re-

quires the presence of an ordering direction. Whenthe magnetic field decreases, the temperature rangewhere the Bose approach is valid for nonrotational-invariant properties becomes smaller. Althoughthis range can be more brought out by discussingthe static properties in the following section, some

0.1.

0 I I

tu/ZS

FIG. 1. Temperature dependence of the normalizedspectrum (S' kS k )„/(S' kS k(t =0}) for wave vectork =m/3 and reduced field h =O.S75. The dashed line isthe unperturbed free-boson result for T*=0 and the ar-row indicates the peak position in the isotropic system.

498O

Qg.

+AQQ CANI PINI, TOO~ET II, AND RET O

b=020—

00FIG. 2. Same

2 m/JS

FI . . arne as Fig. 1 but for k =m=m/2 and h =0.25.

perturbed result exhibit p ed singularits the ex ectow a y o - e Ben

whichmes t e exchange e f

h 1'P gy

g as a remnant of it=0.25 the 1atter ss

occurs. Similar

gp

lower-frequuency peak w

and Balcar. A'

tt irst tf db

d' "h'1i e the other one w

parent that the lower fre-

O.g .

T=0.3 k=n/40—

0 0.5~/ss

1.5

a Temperature dependence of th Pp

~ 0

pen e spectra

quency peak is due to thee onset of a n

, i.e., irst-ma nonearly isotro ic

g ear-h

vi lower fr uen

feature and its fre ud ed an arrow Any trace'

g anty has now disa pp

igher terncription of th

earn ih peratures th d p g

effecte necessa

e 0-e main

e square-root sin 1gsg -frequency tail ob-

0.8

0.6 .

1,!I's!I Q 0 06 T*=0.2

h=0.3h=0.7

04.2m/3

0.40.2

0.2 .

00 1

FIG.. 3. Sameas F'2

s ig. 1 but foror =3m andh =0.2.

0! I

i, . I

Wave-vector de e

ra/JS

W r ependence of the se spectra for

~ 7( 1dl ~

)

26 CLASSICAL ONE-DIMENSIONAL FERROMAGNETS IN A. . . 4981

tained in the computer simulation.

Figure 2 shows the temperature dependence ofthe spectra for h =0.25 and k =n /2. The overallbehavior is rather similar to that shown in Fig. j.

but in this case, owing to the smallness of the fieldand the higher wave vector, the unperturbed resultexhibits, in addition to the expected singularity, abroad inaximum connected with the exchange ener-

Figure 3 clearly shows the transition from one

type of regime (T*=0.1), characterized by the verynarrow peak, remnant of the square-root singulari-

ty, to the other (T*=0.2 and 0.35) dominated bythe isotropiclike peak. Moreover, we observe thatin spite of the low value of the field (h =0.2) andthe higher value of wave vector (k = —,m), no addi-

tional structure is observed in the free-boson resultbecause it would occur at frequency very near to thesingularity, thus being hidden by the rapid growthof the factor 1/g (k,x) in Eq. (23b).

The competition between temperature and mag-netic field is shown in Figs. 4(a) and 4(b) for thesame wave vector k =~/4. It is possible to notethat the effect of increasing the temperature at fixedfield is qualitatively similar to decreasing the fieldat fixed temperature. In both cases "disorder" is in-

troduced in the dynamics of the system leading to anearly isotropic regime.

Finally the wave-vector dependence of the spectrais reported in Fig. 5. The rising of the low-

frequency peak together with the remnant of thesquare-root divergence occurs in the first half of theBrillouin zone. Within the boson framework theoverall behavior of the spectra, as described in thesefigures, is a clear evidence of the transition from asystem with a preferred direction to an isotropicone. Very recent computer-simulation data con-firm the validity of this interpretation, showing a

I

good qualitative agreement with the theoreticalspectra.

IV. STATIC PROPERTIES

The theory worked out in Sec. II permits us toobtain analytical expressions also for the staticlongitudinal properties, thus gaining a better physi-cal insight with respect to the transfer-matrixmethod which provides exact results, in any rangeof temperatures, only in a numerical way. We em-

ployed this technique to check the range of validityof our theory (see Appendix C for further detailsabout the transfer-matrix method).

The free-boson limit for the longitudinal staticcorrelation function (S' kS k(0)) can be obtainedsubstituting Eq. (23a) into Eq. (22):

&s' „s',(0) )"'=(ST*)'- +K(K +bl, )

(33)

2

(Sz S z (0) )(HF) h~+2KT(KT+bk )

(34)

It is instead necessary to insert in Eq. (22) the com-plete expression for Regk(E) as given in Eq. (19).After some algebra we find

This simple free-magnon approach was found togive good results in the very-low-temperature re-g&me.

For a correct description of (S' kS (k0)) in awider range of temperatures it is necessary to takeinto account the magnon interactions. However, aspreviously noted in Sec. II, the temperature correc-tions cannot be limited to the Hartree-Fock (HF) re-normalization. This would give

(s',s'„(0)) =2

ST

KT[(Kz +bk)

hT+2 —2a ET

hT(4 bk))—a ET

(35)

It is worth noticing that the parameter T*/a KT isagain recovered in the perturbation terms. A com-parison between Eq. (35) and the correspondingtransfer-matrix data is shown in Fig, 6. A verygood agreement is found also for values of parame-ters (T'=0.4,h =0.4) which are near the limits ofour first-order theory and where the free-boson ap-proach is clearly inadequate (a =0.84). Moreover,

I

the failure of the HF approximation is apparent,proving again that the effect of the spin-energy cou-pling must be taken into account for the correctdescription of (S' ksk(0) ).

The fundamental role played by these couplingterms is shown in a striking way in the case of thecorrelation length. The longitudinal and transversecorrelation lengths have been calculated exactly by


0.16

0.12

T =0.4following definition:

2 llmk-+0

we obtain

d2d ((S. S,(0)))

1

(S' kSk(0) ~

(36)

0.081+(T /a K7.)h

K& 1 —4(T /a Kr)(hr/Kr)(37)

0.040 0.5

FIG. 6. Wave-vector dependence of the static correla-tion function (S*kSk(0)) for T*=0.4 and h =0.4.The dashed and the dashed-dotted lines represent the pre-dictions of the free-boson theory and of the Hartree-Fockapproximation, respectively. The fu11 line is the result ofthe present theory. The circles are the exact transfer-matrix results.

the transfer-matrix method (Appendix C), and their

temperature dependence at several values of themagnetic field is reported in Fig. 7 where their verydifferent behavior is clearly shown. This was previ-ously discussed by Mcourn and Scalapino' in theirnumerical solution of the continuum classical fer-romagnetic Heisenberg chain in a magnetic field.This continuum approximation can be recoveredfrom our model in the limit h g&1. With the use ofEq. (35), it is possible to obtain an expression forthe longitudinal correlation length g~~. By using the 1.6

(a) ~ h=020

2.2

In Fig. 8 we compare these results of the varioustheories with the transfer-matrix results. Of course,this comparison is justified only for temperatureslower than k~ T=hg~

~

at which a maximum occursfor g~~. At this temperature the Zeeman energy hg~~

of the correlated spins equals the thermal energy

kz T; for higher temperatures the boson approach issurely inadequate. The free-boson result for

g~~is

K ', independent of temperature. In the limitT~~O this is correct and expresses the decrease ofthe correlation associated with the ordering effectof the field. As the temperature increases, Fig. 8(a)shows the striking differences between the behaviorspredicted by a simple inclusion of HF renormaliza-tion and by Eq. (37). In the first case g~~ =Kr ' is adecreasing function of T* in complete disagreementwith the transfer-matrix result. The correctbehavior is instead reproduced very well by Eq. (37).The thermally induced crossover toward a nearly

1.2 1.8

- 1.40.5

0 0I

0.50.5I

1 0 1T"

FIG. 7. Temperature dependence of the longitudinal

(a) and transverse (b) correlation lengths for differentvalues of h, as calculated by means of the transfer-matrixtechnique.

040

I

0.2 0.404I 1

0.2 0T"

FIG. 8. Temperature dependence of the longitudinal(a) and transverse (b) correlation lengths for the two dif-ferent values of h. The dashed and the dashed-dottedlines represent the predictions of the free-boson theoryand of the Hartree-Fock approximation, respectively.The solid line is the result of the present theory. The tri-angles represent the analytical results of the continuummodel (Ref. 16) valid for T~ &&V h . The circles are theexact transfer-matrix results.


isotropic situation appears also in this case. Thelongitudinal correlation length firstly increases toapproach its isotropic value and Eq. (37) shows thatphysical origin of the increase of g~~ is due to thethermally activated coupling term which overcomesthe effect of the renormalization in this range oftemperature. Furthermore, we note that at lowfield, in the limit of low temperatures (T* &&~h )

our result recovers the analytical form of the con-tinuum model'

0.8

06

04.

0.2

1 T~& ll pI 1/2 4g1/2

1+ (38) 0 0.2 0.4 0,6

Our interpretation about the role played by thespin-energy coupling is corroborated by the compar-ison between the transfer-matrix results and a bosontheory for the transverse correlation length (j, asshown in Fig. 8(b). In this case the behavior of g~is well reproduced by the HF prediction which isthe only leading temperature effect:

Ei=hr—1/2

At low temperatures it gives a linear decrease ac-cording to Eqs. (12) and (13).

Another quantity which can be calculated bymeans of the theory is the magnetization. This isgiven by

(S') =S —J„dh'I (h', T') (40)

with

X (h, T*)= lim (ST')-'(S' „S„'(0)) .k o

The free-boson limit [Eq. (32)] is in good agreementwith the numerical data only at very low tempera-tures or high fields (see Fig. 9). For increasing tem-peratures an analytical expression for (S') can befound: expanding Eq. (35), a decrease of the mag-netization stronger than the linear behavior is ob-tained:

(42)

When the complete expression (35) is used, the mag-netization must be evaluated in a numerical wayand an excellent agreement is found for a widerrange of the parameters. Obviously, our theorycannot account for the asymptotic approach to zeroof the magnetization for higher temperatures.

V. CONCLUSIONS

In this paper we have presented a spin-wave ap-proach for describing both static and dynamical

FIG. 9. Temperature dependence of the magnetizationfor different values of h. The dashed and the solid linerepresent the prediction of the free-boson theory and ofthe present theory, respectively. The asterisks representthe exact transfer-matrix results.

longitudinal properties of a ferromagnetic Heisen-berg classical chain in an external magnetic field.In the framework of interacting bosons all thesequantities can be analytically calculated in a ratherstraightforward way. The crucial quantity whichrules the behavior of the system has been found tobe the coupling between energy and magnetizationfluctuations, as previously guessed. ' However, wehave found that the mechanism is thermally in-duced and therefore the spectra are strongly affect-ed by temperature, which opposes the field-inducedanisotropy trying to restore the isotropic situation.The boson treatment, in the large range of tempera-ture and fields where the magnetization is substan-tially different from zero, gives a very physical andsimple explanation of the observed phenomena: Inparticular, a theoretical interpretation is given tothe transition between two regimes observed in thesimulations. Moreover, the static quantities are inexcellent agreement with the transfer-matrix results.In the authors' opinion, in the aforementioned rangeof temperatures, fields, and wave vectors where thecomputer experiments show peaks at intermediatefrequencies, the interacting-boson approach appearsto be completely exhaustive. Other approachesbased on nonlinear-mode representation' ' (soli-tons) have been thought to give a possible explana-tion of the observed features in the spectra. ' How-ever, in our opinion, the assumptions involved inthese methods are outside the range of tempera-tures, field, and wave vectors probed in the simula-tion.


The possibility of observing the so-called second-

magnon features in real 1D systems has been dis-cussed' by Loveluck and Balcar. The conclusionsare not particularly optimistic, since at the momentno good reahzations of high-spin (nearly classical)ferromagnetic chains have been found. Moreover, it

is not easy to separate the weak longitudinal partfrom the transverse one. An attempt in this direc-tion has been made by Steiner et al. ' who used aneasy-plane system, CsNiF3 with S=1. No effecthas been observed in this ferromagnet, which, how-

ever, does not meet the required criteria.

APPENDIX A

Solving the linear system of Eqs. (16) we find for ga(E) the following expression:

ga(E}= [ga (E)Aa(E)+aha '(E)Ba(E) ata '(—E)Ca(E)]

X( I 1+aya[ga '(E) ha '(E—)j+a sinkta '(E) jAa(E)

+ [a ya[hP'(E) —ca '(E)]+a sinkra '(E) jBa(E)

aCa(E—)Iya[ta (E) ra (E)J+—sink [ga (E)—ca '(E)]j)where

(A 1)

Aa.(E)= 1 —a [ha '(E)—ga '(E)]

+a Ir' '(E)ta '(E)—[rp'(E)] —ha '(E)ga (E)+ha (E)ca '(E) +gp (E)ca '(E) [ca '(E)—] j,

(A2)

Ba(E)= [ga' '(E)—hP'(E)]

+a I [g(0)(E)]2 (0)(E) (o)(E) (o)(E)I (0)(E)+h (0)(E) (o)(E) [t(0)(E)]2+t(o)(E) (o)(E) j

(A3)

Ca(E)=ta '(E) a[ ta '(E)ha—'(E) ta '(E)ca '(E—) ga '(E)ra '(E—) +ha '(E)ra '(E)] . (A4)

(A5)

(A6)

The final expressions for the numerator and the denominator of the Green's function ga(E) are, respectively,

~=gp'«}+a t [ga '(E)]'—[hp'(E)]' —[ta"'(E)]'j

+a'I [ ga'( E)]' c'"a( E) +[ h'"a( E)]' c'"a(E—} [t,"'(E}j'c"(E}+2ha '(E)ra '(E)ta (E} [ra '(E)] ga (—E) [ca '(E)]2gp'(—E)—[ha(o)(E)]2ga(0)(E) j

~= 1+a Iya[ga '(E)—ha '(E)]+sinkta '(E)—hp'(E)+gp '(E) j

+a'«a"(E)ta"(E) —[ra"(E)]'+ha '(E)[cp'(E) —gg '(E)]+ga '(E)cp '(E)

[ca (E}l +3 a I [ga (E}] ga (E)ha (E) ca (E)ga '(E)+—ca '(E)hp'(E)

—[tP (E)] +tP'(E)ra '(E) j

+»nk [ra"«)ga"(E)—rp (E)hp (E}—tp'(E)ha '(E)+c' '(E)tp'(E) j ) .

Letting a =0 in Eqs. (A5) and (A6) we find the un-

perturbed result

ga(E}=ga(o)(E) .

Retaining only the linear terms in the couplingparameter a, we obtain expression (19}.

APPENDIX 8In this appendix we report the explicit expres-

sions for the quantities (17a}—(17e) and for

p (E)1 (ani+e —(an)-~

cos(12vrN E +4JSas sinaq (131)


Using conventional complex variables techniqueswe find for x & 4bk.

(p) 1 kg T bk br+2Regk '(x)=

2~r (JSa)' D(k, x)

that

Imt [gk"(x}1'—[hk '(x)]'—[tk '(x)]']

,g'(k, x)2 Reg,'"(x) Img„'"(x)4bk2

X [x bk(KT+bk }l

k, T bkImg,"'(x)=

2n(JS.a)' D(k, ) g(k, }

X [x —bk(Kr+8 —bk)],

(82)

(83)

—2RePk(x) ImPk(x),

Re[[gk '(x)] —[hk '(x)] —[tk '(x)] I

2g (kx)1

4bk

(810)

1 ka T cxbj 1RePk(x) = 2' (JSa)~ D(k, x) Kz

X[x (Kz+2) 2bk(K—r+bk)], (84)

where sk=sink/2 and ck ——cosk/2. For x &4bkboth quantities are real and do not contribute to thespectrum. All the other quantities (17b)—(17e) canbe expressed in terms of gk '(E) and Pk(E) in thefollowing way:

hk(x) =ckPk(x) —gk '(x), — (86)

tk (x)= skPk(x) .— gk (x—), —(o) x 1+cosk (o)

4 sink

(87)

Xck '(x)= —,gk '(x) (1+cosk)—

4 1 —cosk

x sink 1 'Vk 4Pkx+4 sk 2n 4 (JSa )2

hp+2X (88)

Xrk '(x) = —,gk '(x) sink —1

4(1 )k)-

ImPk(x) = kg T ckbk(h z +2)xg(k, x),

2n (Jsa)(85)

X I [ Regk (x)]~—[ Imgk~o~(x)]~I

—I [ RePk(x)] —[ ImPk(x)] I . (811)

From Eqs. (84) and (85) we observe that pk(x) hasno singularities. Taking irito account the expressionof gk '(x) in Eqs. (23a) and (23b), it turns out thatthe correction terms at the numerator in Eq. (19}donot present any divergence.

APPENDIX C

The static correlation function of the classical 1Dferromagnetic Heisenberg chain in a magnetic fieldcan be numerically evaluated using the transfer-matrix technique. This method has been extensive-ly discussed by Loveluck et al. , Blume et al. ,and De Raedt and De Raedt. ' We follow the nota-tions of the last authors and refer to their paper fora detailed description of the method. Here we onlydiscuss the definitions adopted for the inverse longi-tudinal and transverse correlation lengths and givetheir explicit expressions in terms of the eigenvaluesand eigenfunctions of the transfer matrix.

For ferromagnetic coupling, the inverse correla-tion length can be calculated using the definition[Loveluck et al. ; 13=(kii T) ']

X,=0/P

—,'

X I'[&s:st &-&s:&']

h~+2 —1Ez-

(89)

1 sink ka TxPk(x)

4sk 2n4( JSa }2' [&sost &—&so& ]

I = —oo

g I'[&s,st &—&s, &']

l=o

R =X,P,Z

From the explicit expressions (86), (87) it followswhere the wave-vector-dependent susceptibility Pqis expressed in terms of the classical two-spin corre-

4986 BALUCANI, PINI, TOGNETTI, AND RETTORI

lation functions by means of the fluctuation-dissipation theorem:

x,-xp= g e'&'((s,s, ) —(s, )') .

The definition {Cl) seems to be the most con-venient, since it can be applied whatever thebehavior of the two-spin correlation function, in

particular even when it is not exponentially decay-ing.

For gq and Xq =7q" we have

21 —Alp

x, /p= g 2 «ooio)' (C2)l~p 1 —2Alpcosq+Al

xx 1 —Ali2

&e ip=2 X p «oori»l@p 1 —2Al i cosq +Al i

(C3)

where the I summations run over Xl values, Xl isthe number of points used in the numerical Gauss-ian integration of the transfer-integral equation, and

the quantities Alp, Al i and Cpplp, cppl i are related tothe eigenvalues and eigenvectors of the transfer ma-

trix, according to Eqs. (4.15) and (4.16) of the paperby De Raedt and De Raedt. ' The denominator ofEq. (Cl) can easily be written in terms of

2Kx—

1+Alp

l@p lp

+Alp3 Alo(COOIO)

I~o (1—Aio)

1+Al i(Cooi& )

l@0 l 1

1+Al )

, A„(C+„)2i~o (1—AI |)'

(C4)

(C5)

The accuracy of these expressions, as well as of anyother static quantity, is determined by the number

EI of points used for the Gaussian integration. Atlow temperatures this number needs to be increased—PH,due to a less smooth behavior of the kernel e

II, =s, s, +,——(s,'+s,'+, )2 1 l+ 2l

of the integral equation. In practice, N&=16 wasfound to give good convergence for values of T~down to 0.1 and for all values of h.

+q

dq' p, ,=o

and the following expressions for the longitudinaland transverse inverse correlation lengths are found:

'J. M. Loveluck and E. Balcar, Phys. Rev. Lett. 42, 1563(1979).

E. Balcar and J. M. Loveluck, J. Magn. Magn. Mater.15-18, 365 (1980).

S. W. Lovesey and J. M. Loveluck, J. Phys. C 12, 4015(1979).

4S. %. Lovesey and J. M. Loveluck, J. Phys. C 9, 3659(1976).

~S. W. Lovesey, Z. Phys. B 37, 307 (1980).M. Blume, P. Heller, and N. A. Laurie, Phys. Rev. B 11,

4483 (1975).7S. W. Lovesey and J. M. Loveluck, J. Phys. C 9, 3639

(1976).E. Balcar and J. M. Loveluck, J. Phys. C 15, 4533

(1982)~

G. Reiter, P. Heller, M. Blume, and A. Sjolander, SolidState Commun. 36, 365 (1980).B. De Raedt and H. De Raedt, Phys. Rev. B 21, 4108(1980).

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' G. Reiter and A. Sjolander, J. Phys. C 13, 3027 (1980).'5A. R. McGurn and D. J. Scalapino, Phys. Rev. 8 11,

2552 (1975).K. Nakamura and T. Sasada, J. Phys. C 11, L171{1978).

~7A. R. Bishop, Z. Phys. B 37, 357 (1980).A. R. Bishop, Z. Phys. B 44, 185 (1981).M. Steiner, K. Hirakama, G. Reiter, and G. Shirane,Solid State Commun. 40, 65 (1981).J. M. Loveluck, S. W. Lovesey, and S. Aubry, J. Phys.C 8, 3841 (1975).H. De Raedt and B. De Raedt, Phys. Rev. B 21, 304(1980).

classical one-dimensional ferromagnets in a magnetic field: static and dynamical properties

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