Excitation spectrum of the ferromagnetic Ising-Heisenberg chain at ...

PHYSICAL REVIEW B

VOLUME 2S, NUMBER 7

Excitation spectrum of the ferromagnetic

1 APRIL 1982

Ising-Heisenberg

T. Schneider and E. Stoll IBM Zurich Research Laboratory, 8803 Ruschlikon-ZH, (Received 23 October 1981) The excitation spectrum of the ferromagnetic

s

=2

chain at zero field

Switzerland

Ising-Heisenberg

chain as probed by

dynamic form factors is studied. Exact results are presented for zero temperature, and the modifications introduced by thermally induced transitions are elucidated by finite-chain calculations. A particularly interesting result is the occurrence of a thermally induced resonance around zero frequency for an Ising-like model. Its origin is traced back to transitions involving magnon bound states. The resonance structure of the dynamic form factors associated with a Heisenberg-like system turns out to be less spectacular, because the contributions of different induced transitions nearly overlap on the frequency scale; so that rather broad resonance struc-

tures result. Our findings have an important bearing on related models with s & 2, such as CsNiF3, and on the soliton aspect of the classically treated systems.

I.

INTRODUCTION

In this paper we investigate the excitation spectrum, probed by dynamic form factors, of the ferromagnetic Ising-Heisenberg chain, defined by the Hamiltonian,

H = —J

"

x Sl*sf+J + g1 (SI"Sl+, +—Sl sf+I ) l

= —J g

sl'Sf+I+

(S~+S&+I

+St

St+I )

where

S,+=S,"+ISg, S,

the diagonalization of other one-dimensional manyparticle systems, ' ' including the xyz, xxz (Refs. 7— 9) models and bosons interacting with a two-body 5-function potential. ' A different application of the method was discovered by Lieb" to diagonalize the transfer matrix of classical two-dimensional models. The subsequent development in soluble classical lattice models was unified and extended in the very important work of Baxter, leading to a solution of the eight-vertex model. Moreover, Baxter showed that the eigenvalues of the xyz-spin chain could be obtained from the eight-vertex-model transfer matrix. Another and more recent line of development was initiated by the extension of the inverse-scattering transform method for solving certain classical soliton equations"' to quantum systems. In fact, the relationship between the inverse scattering and Bethe's ansatz was recently fully realized by the development of the quantum inverse method. 4' Despite this flurry of activity, devoted to the diagonalization of one-dimensional many-body systems, our understanding of thermodyarnic and dynamic properties is less advanced. Exceptions include the s = z xy chain introduced by Lieb, Schultz, and Mattis" "and the Bose gas interacting with a twobody 5-function potential. ' Otherwise, and even in the s = —, ferromagnetic Ising-Heisenberg chain

=St &S(. —'

Periodic boundary conditions are assumed, and g varies between the Ising limit g = and isotropic Heisenberg model g =1. Hamiltonian (1) is a special case of the xyz Heisenberg model,

~

H =—

xJ,S('S(pl

corresponding to J, = J, J„=J» =1/gJ and, accordingly, also called the xxz model. The other special case, with J„— Jy Jz is the xxx model or isotropic Heisenberg model. These quantum-mechanical-model systems for magnetism were proposed by Heisenberg in 192S,' and the diagonalization of the Hamiltonian, the thermodynamic and dynamic properties have been studied by many authors, beginning with the pioneering work of Bethe. In fact, Bethe diagonalized the xxx model with s = —,. This exact solution was accomplished in terms of the famous Bethe ansatz for the eigenvectors, which has been applied successively to 25

(xxz), where the ground state is trivial and the conservation law

[Hsf)

0,

Sj= xsl*

(4)

l

leads to considerable simplifications: The mathematical complexity of the eigenstates and eigenvalues is such that it has not proved possible so far to evaluate correlation functions of spin operators or dynamic 4721

T. SCHNEIDER AND E. STOLL

4722

form factors at finite temperatures. Considerable progress has been made, however, for the thermoand the zero-temperature dynamic properties' behavior of correlation functions and dynamic form factors Finite-temperature results are of considerable interest because the excitation spectrum probed by dynamic form factors will be modified due to thermally induced transitions. Moreover, these properand light scattering ties can be measured by neutron experiments on materials '2 which are well represented by the xyz Heisenberg chain and its special cases. Dynamic form factors are defined by"

the sum rule,

'

S»(q,

X

Q

( Qj —Qj + Qj g)

(5)

g

where

Z

= Tre t'", P = I/T,

A, =

1

N

(6)

Xer'At . l

(kI denotes the eigenvectors, and

co~

the eigenvalues

of 0; q is the wave number, and A( stands for operators of interest. Examples are $Z

A(=

Sf ~( ~(+1

+ ~(

~(+1

The associated dynamic form factors are listed in Table I, where S (q, co) =S~(q, c«) and S (q, co) are accessible by neutron scattering. $~~(q, co) is related Static correlato the light scattering cross section. tion functions are obtained from Eq. (5) in terms of

"

TABLE I. Labeling of the dynamic form factors fEq. (5) j and the static correlation functions fEq. (9)] associated with the operators listed in Eq. (8).

gZ

S~(q, o)) (q)

~( ~(+i

des

S»(q, «)) = (A, (0)A,t(0) ) (9)

The complexity resulting from the double sums appearing in Eqs. (5) and (9) is considerably reduced at

T =0, where the eigenstates (h. include the ground state only. Here, Eqs. (5) and (7) simplify to I

S»(q, co) = QI(OIA

IX')I g(«o

+~(

S~(q, a))

S (q, o))

S)q(q, o))

S (q)

S (q)

S»(q)

~(+1

—«o„, +~o) . (10)

and

S»(q) = XI&OIA, I}, )

dt e ~(Aq(t)Aq"(0))

)

cu)

25

I

.

Thus, the excitation spectrum probed by dynamic form factors becomes much richer for finite T, because it not only involves transitions between the ground state and excited states, but also between excited states. Clearly the occurrence of a transition is determined by the matrix elements (GAIA«I h. '). The program of this paper is to investigate the excitation spectrum of the xxz chain probed by dynamic form factors. In Sec. II, we summarize the classification of the eigenvectors and the results for the eigen1 value spectrum for s = —,. Exact results for dynamic form factors at T =0 are derived in Sec. II for general s values by means of the equation-of-motion technique for Green's functions. Here, we also derive approximate finite-temperature results on the basis of linear spin-wave theory. In Sec. III, we present zero and finite-temperature results for finite chains and s = —,, to explore the thermally induced modifications of the excitation spectrum. A particularly interesting result is the occurrence of a thermally induced resonance centered around zero frequency, a so-called central peak, in S (q, o&), S (q, co), and S~~(q, «o) for Ising-like systems. At low but finite temperatures, its origin is traced back to transitions involving magnon bound states. Accordingly, its occurrence cannot be explained in terms of finiteorder magnon perturbation theory, because bound states represent an infinite-order phenomenon. The interpretation of the finite-chain results for a Heisenberg-like system turns out to be more complicated, because the frequency regimes, where the various transitions contribute to the dynamic form factors, merge due to the more pronounced dispersion of the eigenvalues and their small gaps. As a consequence, close to the Heisenberg limit and at finite temperatures, the dynamic form factors are composed of various transitions which overlap on the frequency scale for fixed q. This overlap makes it impossible to explain the rather broad resonance structure in simple terms, such as magnon, two-magnon

EXCITATION SPECTRUM OF THE FERROMAGNETIC ISING-.

25

bound-state, two-magnon-continuum resonances, etc. Presumably, these difficulties can be alleviated by applying an external field, which introduces gaps in the eigenvalue spectrum, and thus separates contributions of different transitions on the frequency scale, In Sec. VI, we briefly indicate the implications of our results for s & —, and for the soliton problem. %e hope that our results will stimulate experimental studies of the excitation spectrum as probed by dynamic form factors in Ising-Heisenberg chains. The discovery of good s —, ferromagnetic IsingHeisenberg chains, in principle, permits such studies. Moreover, the aforementioned evidence for the failure of finite-order magnon perturbation theory at finite temperatures also poses challenging and interesting theoretical problems.

tonian

(1), namely, (20)

where TS$

T

k =2m

S&+i

Noting that Sr [Eq. (4)1 commutes with Hamiltonian (1), the eigenvectors and eigenvalues might be classified according to the eigenvalues m of

=e

is the totally

aligned ground state with energy

m

Calculation of the eigenvalue spectrum is then reduced to a diagonalization of the Hamiltonian in the subspaces

=Si,

L,

L, and

the eigenvec-

0. =x,y, z

yields, according to Gochev,

1 cosk k= J 1 ——

1

Am=

For

&

s= ~, we 1

mop4m

l]

e

XI 4, ~

L, yielding

—

Bc

= J 1 —1 cos— g2

(16)

~ 2, k ~ Tc

(27)

1 k sc =2J 1 ——cos—

(28)

1 k =2J 1+ — cos—

(29)

g

2, 2

In fact, ~q, q is a two-magnon there is a binding energy

have a)g k

™iim

(26)

2

where

~wc

SI, . . . SI 10) .

I

2, the two-magnon

and the two-magnon continuum, bounded by the top of conc and the bottom conc and given by

the Bethe

o

bound state, because

—cos pk—= — o)gc, 4J

2 k —J —o)gc = —J + — J cos —

g

2

g~

1

2

(30)

where

m=0, 1, . . . , N

(18)

The sum in Eq. (16) goes over all (g) basis vectors of m reversed spins. The 2" & 2 matrix then splits into N +1 blocks of order 1pNp

for

(25)

g

g

can be expanded in a basis on ansatz9

(24)

1,

o)g, g

(14)

then HP E Now, if Q 6 tor of the Schrodinger equation

(22)

(23)

the magnon energies; for m bound states

(13)

SI 10) )

is then also classified by

4~, k

the diagonalization

co&,

a)p= JNs~

eo

(21)

~

n —, n =0, 1, . . . , N —1

(12)

i0),

xpypz

Imposing periodic boundary conditions,

AND EIGENVALUE SPECTRUM

by

cx

p

where

II. CLASSIFICATION OF THE STATES

The state with

SI j

In fact, each eigenvalue a wave number

2 4m, k

+ Srr . m =0, denoted

4723

for N even. For closed chains, a further reduction can be obtained by using the translational invariance of Hamil-

"'

m, p = —Ãs

..

Q

f

~

~

4f

N ]

f

~

~

~

fNp1

vanishing only for g =1 and k =0. The m th magnon bound state is conveniently written in the form Q)yak

=

sinh@ —, (cosh1ll $ —cosk) g sinhm@

(31)

where

cosh/ =g

(32)


4724

Clearly, also for m & 2, there are continuum states in addition. To summarize, the eigenvalues and eigenvector might be classified according to the quantum numbers m [Eq. (18)] and k [Eq. (22)], and the eigenvalue spectrum consists of magnons, magnon bound states, and magnon continua.

FACTORS AND CORRELATION FUNCTION AT T 0

= —iO(t) ([A (t),At])

(33)

The equation of motion is then

= 2m

([A, A t]) + (([A,H];A t)), (34)

t+DO

i

„—

e(t) exp[i(a)+i x

e) t]

(36)

To solve Eq. (33) for particular operators A, we use the usual relations,

1

QP+Mi

&+ I c

where 1

2 Js 1

(42)

cosQ

S 5(a) —a)(, ) S~(q, m) = —

of Hamiltonian

Moreover, the symmetry

(43)

(1) implies

S (q, ru) =S~(q, a)

(44)

Accordingly, the spectra of the dynamic form factors S (q, co) and S~(q, 0() are exhausted by magnon 5function resonances. This result might also be understood in terms of the Schrodinger representation of S (q, 0() given in Eq. (10), by evaluating the matrix elements

'&ols'(q)+s-(q)l)'& &ols"(q)l(') = —

= y, , , =

with eigenvalue

.

(4s)

(37)

N

ra( k

Xe'"'s, +Io),

(46)

[Eq. (25)], yielding

(47) so that according to Eq.

(10),

S (q, a) = ~5(o) —~(,, ) 1

(48)

= —, agrees 1

with Eq. (43). From Eqs. (33) and (36) —(39), we also obtain, adopting the notation of Table I, with the exception

which for s

[S-, I' S',I' ] = —25,« S;

of g=1,

and srlo&

—Mi, &+ I c

(41)

(35)

G~~(~) = —,(1 —e s")S~~(~) .

'

1

0)

(

I), ')

([A(t), 8])dt .

The imaginary part of G~~(0() and the dynamic form factor S~~(co) are related by"

= +5, [S+ S,+«'I' I' I'-S;]

I,

In fact, the only nonvanishing contributions come from the I((.') associated with one reversed spin. The eigenvectors are

where

((A;B)) =

s 4ff

=

in Table

and by invoking Eq. (36) at T =0,

The simple ground state of the ferromagnetic xxz model with g «1 [Eq. (1)], allows the explicit calculation of certain Green's functions and associated dynamic form factors at T =0. This could be achieved with the aid of the eigenvectors [Eq. (7)] and the definition of the dynamic form factor [Eq. (10)]. Here, we follow Wortis, '0 by using the equation of motion for Green's functions, defined by

0(Ggg(0()

G (q, cu)

ct)] q

III. EXACT RESULTS FOR DYNAMIC FORM

Ggg(t)

the notation summarized

=-s lo), s'lo)

S(+S,+I0) = (2s)

= (2s)'i'li)

— 1+5„1—5„,«2s

S (q, o))=0 . Considerably

lll')

.

(39)

we obtain from Eqs.

),-

(33), (37), and (38)

more complicated is the calculation of

S(((q, 0() (Table 1), probing the fluctuations of two adjacent spin deviations. In fact, the matrix elements appearing in Eq. (10) are in this case of the form

Ill') denotes a state in which the spins at sites i and I' are flipped away from the ground state. Noting that [Eq. (2)]

S("= 2(s(++ S(

(49)

N

X(s( s(+(+s,+s,++, )e-'"ll ') = (01

by adopting

N

$(s(-s,—,() e'&'I ~'), (

(50)


25

..

4725

so that ~lt') must belong to the subspace with two-flipped spin and eigenvalues corresponding to the two-magnon and two-magnon continuum states. The associated Green's function G~t(q, ru) may be written as

=

G~t(q, ro)

—g QG

(q, co) coskcosk',

(51)

k

G„„(q, co)

where

of

is the solution

Using Eqs. (37) and

(38), the equation of motion reduces to the r

G, (q, a&) =4s'

G

(q, ~)8, —

2Ns

Gkk(q,

integral equation

gcosk" +2J~Gkk(q, co) —

~)

k

II

'1

—cosq/2

—cosk

g

G

(q, m)

(53) where

(54) The solution is 1

G ~(q,

«&)

=4s G„„g„i —

—cosq/2

GIk+

0 0 0 — GkkG„I„I cosk —GlkB~ r

k k

given by Eq.

where 1 — "k XGkok(q, —cosq/2 —cosk cosk =— Gkk(q, N $

B„(q, ra) =

co) cos

o&)

1

p

mJ

(57) Substitution

1

—2

JR(,

)

8 1

B2+

(58)

(36) and (58),

=4s

[1 —2m'JR'(q,

The integrals B2(q, co) and R (q, cu) are easiest to calculate for s = 2. In this case, we obtain for 'C caisc

1 = 2J 1 —— cosq/2

S~~(q, Ol)

=

8(QI

QI2, &)

where co2, is the two-magnon

bound-state

(61) (62)

0.8—

J

(65)

-

I

q

0.6 — y

77/p

= o~—

0

0

FIG. 1. Frequency dependence of S»(q, eo) for some fixed q values for 1/g and T =0, corresponding to the xxx chain.

-1

frequency

a)

cog~ and mTc denote the bottom and top of the twomagnon continuum, respectively, and are given by

B,"(q, cu) o)) ]2+ [2m JR "(q, o)) ]2 (60)

2+

(64) 1 B=— [(cv —ruse) (~rc —a&) ] &/z

(59)

we finally obtain from Eqs.

2

cosq/2 g

2

J

B2B(' —B2'B]' =0

Ql

cosq/2 g

where

Noting that

td)

(63)

of the solution (55) into Eq. (51) yields B2(q, «))

Sii(q,

(26), and for

CHIC

2

g

k

(55)

2Pfs

S]](q, cd)

1

' cu)

0) ~»

(56)

k

R (q,

cosk

T. SCHNEIDER AND E. STOI.I.

Oa—

I

3

where

I

g

ln

=1 —12

S]~ (q)

7l/p

g

Il ~

-O4-

S[) (q) =

CA

ll ll li

Eqs. (28) and (29). From Eqs. (62) and (64), it follows that St~(q, ru) probes the two-magnon bound states and the two-magnon continuum. The frequency dependence of S~~(q, ao) is given in Figs. I, 2, and 3 for fixed q values and 1/g = 1, 0.8, and 0.13, respectively. Apart from the exceptional case, 1/g =1 and q =0, the spectrum consists of a 8function two-magnon bound-state resonance, separated by a gap from the broad two-magnon continuum structure, which is bounded by the bottom and top of the continuum. This gap corresponds to the binding energy of the bound state [Eq. (30)]. In the isotronic case, 1/g =1 and q =0; however, this gap vanishes and a square-root singularity occurs for eu ~0 of the form 2 =— ~

1

[mcoTc(q

(66)

-0))' '

Such square-root singularities are otherwise an artifact of the linear spin-wave approximation, where the magnon interaction, appearing in the present case in terms of the 8 term in the denominator of Eq. (58), is neglected. The corresponding correlation functions or static form factors are now readily obtained from the sum rule (9) and Eqs. (43), (44), (49), (62), and (64). The results are for g & I

s (q) =s~(q) --,' s (q)=0, S»(q) =Ss, (q)+S[, (q}

V)

I

0.06—

I l

0.04—

=I,

I

I

g

Extension of the exact expressions for dynamic form factors derived in Sec. III to finite temperatures is hampered by the difficult problem of taking the bound states properly into account. In fact, in the framework of perturbation theory, starting from a magnon basis, the bound states represent an In this view, it is clear infinite-order phenomenon. that finite-order calculations of the magnon self-energy for G (q, co) will fail in the sense that the bound-state effects will be lacking. From Eqs. (43), (44), and (49), it is seen that these effects are of thermal origin in the dynamic form factors S (q, co) S~(q, co) and S (q, co), which are accessible in neutron scattering experiments. In fact, at T =0, S (q, co) S~(q, ru) probes the magnons only, and S (q, ro) vanishes. As a consequence, observation of bound-state effects seems to rely on a quantitative theory, predicting the thermally induced bound-state structure in these dynamic form factors. Progress in this direction was made in the xxx model, by using the Dyson-Maleev bosonization, and a low magnon-density expansion. Here, we only consider the predictions of linear magnon theory, to compare with the finite-chain calculations, where the thermally induced effects are fully taken into account. For this purpose, we intro' of the spin duce the Dyson-Maleev bosonization' operators

-

S+Qg

S, =(2s)'~

7F/p

Ql

0.5

(72)

2s

a~

[a;,a, 1 = &~p [a,, a, ]

I

1.0

t

0

1 I

0

IV. APPROXIMATE TREATMENT FOR DYNAMIC PORN FACTORS AT FINITE T

g S]=

0.02—

0

c— os'q/2 .

S;t= (2s)'~'a, 1—

0.08— U'

2

S~~ is the two-magnon bound state, and Sft the twomagnon continuum contribution. From Eqs. (69) —(71), it is seen that the two-magnon bound state exhausts the spectrum of S~~(q, cu) for q = ~.

FIG. 2. Frequency dependence of SIq(q, ~) for some fixed q values for 1/g 0.8 and T =0 for the xxz chain.

S)t(0, a))

g

(70)

cos q/2

2.0

FIG. 3. Frequency dependence of S~~(q, 0)) for some fixed q values for I/g 0.13 and T =0 for the xxz chain.

Using

&I=

~

vN

~ g,

&k

- [a,t a, )=0 . ,

EXCITATION SPECTRUM OF THE FERROMAGNETIC ISING-. we obtain for Hamiltonian

H = t»o+ xt»t, ktta

+

— & $

(1) the bosonized form

ttk

V(kt, k2, q) as/t+k,

k, , k, , ~

04

as/2 k,

J cosk2 coskt V(kt, k2, q) = —

(75)

1

N

N

t»s

g)-

FIG. S. Frequency dependence of $~(tr, ru) for 1/g =0.8, T/J 1/3 according to linear magnon theory [Eq. (76)l. The arrows mark the locations of the square-root singulari———q ties lEq. (80)l. q -m/2,

-a.

)t») g P/2+ //

e+p/2

e-p/2 "

t

4JS g

slnq

—t»

/2

+~

I/2

(77)

t

QI

ki

2

k2

= arcsin

(78)

0)

4Js

sinq/2

= m' —k)

~k =2 Js

(79)

1 —— cosk,

I

I /

3 U

/

0)

r

N N CA

0

l

-0.2

—0.&

0

0.5

=—

at

(80)

sinq/2

The frequency dependence associated with these approximate expressions for S (q t») are glvell lll Figs. 4 and 5 for T/J = —,, 1/g =0.13, 1/g = 0.8, q sr/2 and %. One expects, however, that the square-root singularities in S (q, t») appearing at t» values given by Eq. (80) will be removed by taking the magnon interaction, given by the third term in Eq. (75), into account. Iri fact, in the hydrodynamic regime, the dynamic form factor, S (q, t»), will exhibit diffusion poles and possibly a second-magnon pole. 32 ' The second magnon is expected to occur only in a temperature windo~, bounded from below by a temperature still guaranteeing the establishment of local thermal equilibrium, and from above by umklapp processes tending to overdamp the mode. 3 In this window, the magnon momentum will also be approximately conserved. It should be noted, however, that the discussions of the diffusion and second-magnon modes have so far been restricted to threedimensional ferromagnets. While the underlying physics, such as the conservation laws, is not particularly restricted to three dimensions, certain features might be different in the one-dimensional case. In fact, molecular-dynainics studies of one-dimensional lattice dynamic models, such as the sine-Gordon chain35 and the Toda lattice36 demonstrated that the occurrence of second sound is not restricted to the hydrodynamic regime. Similar features were also found in the Heisenberg chain. '

-

l/

I

—=2Js I —— cos+ coski g 2

/

This expression exhibits square-root singularities

2

= QS'(q, t»),

2m e~

I

I —— cosq/2

Xna(l +ns-k)8(t»+c»t

=1 S'(q, t») =

I

0.2—

/

corresponds to the magnon interaction. Neglecting this interaction leads to the linear magnon theory, On this basis, S (q, t») (Table 1) is now readily calculated. We obtain for q WO,

=

I

I

U

x +q/2+k2&q/2-k2

S~(q, t»)

..

0.2

FIG. 4. Frequency dependence of S (q, ~) for 1/g =0.13, T/J =1/3, according to linear magnon theory fEq. (76)]. The arrows mark the locations of the square~. root singularities faq. (80)l. q = ~/2, ———q

-

V. FINITE-CHAIN RESULTS FOR DYNAMIC FORM FACTORS

In view of the aforementioned problems associated with an analytic calculation of dynamic form factors


4728

and correlation functions at finite temperature, we calculated the dynamic form factors listed in Table I numerically for finite chains subjected to periodic boundary conditions. Such finite-chain calculations have provided quantitative results for static proper' ' in various speties' and dynamic properties' cial cases of the XYZ-Heisenberg chain, including the antiferromagnets. Here, we consider the ferromagnetic Ising-Heisenberg chain and, in particular, an Ising-like (I/g =0.13) and a nearly isotropic

(I/g =0.8) model.

The original 2" 2" matrix associated with H can be reduced to considerably smaller blocks, as discussed in Sec. II. Once then the eigenvalues and eigenvectors are calculated numerically, the partition function [Eq. (6)] and the dynamic form factors of interest [Eq. (5) and Table I] are readily obtained. For finite chains, these form factors are best represented, for fixed q values, as histograms with an interval b ta/J. Some insight into the limitations of this method are obtained by considering the eigenvalue spectrum. Fig. 6 shows a comparison of the eigenvalues resulting from an N =8 chain for I/g =0.13 and the corresponding results for the infinite chain. First of all, in a finite chain there are only N k values [Eq. (22)]. For the ground state and the N-magnon states, the eigenvalues are exact, nevertheless. In the magnon case, this is simply due to the cyclic property of the one-spin deviation block [Eq. (46)]. For all other eigenvalues, however, finite-chain results represent approximations only to the spectrum of the infinite chain. This is easily seen in the case of the twomagnon bound-state energies, which according to Eq. (58) are the solutions of &&

1 =2m''(q,

2

(81)

re)

In the infinite system, the sum in 8'(q, &a) over the wave numbers k is replaced by an integral, and the bound-state energies become the solutions of an integral equation. In finite systems, however, the sum includes k values only, and one of the solutions will be an approximation for the bound state of the infinite system at q. Another limitation of finite chains becomes manifest in the continua, as illustrated in Fig. 6, for the two-magnon continuum, where in the shaded area and fixed q, only a very limited number of continuum states occurs. These limitations will certainly affect the accuracy of the calculated dynamic form factors and, in particular, the continuum resonances. For a quantitative discussion of these limitations, it appears to be more useful, however, to compare the finite-chain results for dynamic form factors with the corresponding exact results for infinite systems, as derived in Sec. III for T =0. Let us then turn to the finite-chain results for N = 8. The presentation and discussion will be subdivided into two main parts, namely, in the results for the Ising-like systems with I/g =0.13, and those for the Heisenberg-like systems with I/g =0.8. Moreover, we shall treat S (q, ta) = S~(q, &a), Stt(q, ra), and S (q, &a) (Table I) in separate subsections. To illustrate the wave number and temperature dependence of the resonance structure, we shall consider = m and various temperatures between q = m/2, q T/J=0 and T/J=1. From Eq. (5), it is seen that for any dynamic form factor the contributions in a given frequency interval can be identified in terms of transitions between eigenstates. This information allows an unambiguous identification of the dominant contributions to a dynamic factor in a given frequency interval an. d, accordingly, the elucidation of its physical origin. To provide this information, the contributions of the dominant transitions will be listed for S»(m/2, re) in tables.

S

I

T=O

T/3

=

1/3 gF

3.

0

I I

45

II

0

0

I

I

0.25

0.50

T/& =

1.0

0.75

FIG. 6. Eigenvalue spectrum for I/g =0.13. k, O, and ~ denote magnon, two-magnon bound states, and two-magnon are the continuum states for N=8, respectively. 4- and magnon and magnon bound states of the infinite system, jAfter and the shaded area is the two-magnon continuum. T. Schneider and E. Stoll, Phys, Rev. Lett. 47, 377 (1981}.]

~

Z/3

&C

CO

0

FIG, 7. S~(m/2, various temperatures

ao)

for a finite chain of eight spins at =0.13 and Aeo =0.1 J.

1/g


25

TABLE II. Dominant relative contributions to S~(e/2. T/J =1/3. ec cot. ground-state — magnon transition; rot transitions;

co~

magnon

co2.

—two-magnon

CtJp

S~(a/2,

—1.1 —1.0 —0.9 —0.1

continuum

+—OJ ]

1.0 1.1

S (~/2)

—CsJ3

~0J2

OJ2 +

0J~

—QJ4

~

0.03 0.16 0.04

0.67 0, 66 0. 11 0.67

0.68

0.248 75

0.40

-0.13

$~(q, ra)

At zero temperature, this dynamic form factor probes the magnons [Eq. (43)]. From Fig. 7, it is seen, however, that the associated 8-function resonance at tat, = J(1 —/g cosq) is dramatically modified for finite temperatures and q =m/2, due to the thermally induced transitions. The magnon resonance adopts a finite width, the corresponding destruction peak appears, and more importantly, there is a resonance centered around zero frequency, a socalled central peak (CP). The origin of the thermally induced structure is readily identified in terms of the

0.84

0.04 0. 17 0.03

0.10

0.15

0.07

eigenvectors and eigenvalues contributing to S (q, ru) in a particular frequency interval =0.1 J. The dominant contributions are listed in Table II, and reveal that the thermally induced CP is two-magnon bound-state and dominated by magnon — two-magnon — three-magnon bound-state transitions. Moreover, the broadening of the magnon resonances two-magnon — continuum is dominated by magnon — resonances. These findings lead to the important conclusion that finite-order magnon perturbation theory cannot account for the thermally induced CP, because bound states are involved, representing an infinite-order phenomenon. Qualitatively similar results are obtained for q = m, as illustrated in Fig. 8.

u) 3 I

I

T/3

=

1/3 ///

///

//

0

I/ I

~

0

3.

(0

I

=

2/3

I

8

2

0~

I

—1

0

FIG. 8. S~(m, ous temperatures,

T/3 =1

-0.

T/3

=

2/3

--0.04 --0.02

--0.04 --0.02 0

--0.06 8

T/3 =

1

--0.04 --0.02

g

0

for a finite chain of eight spins at vari13 and Aced =0.1 J. 1/g

co)

1/5

--0.06 V)

T/3

=

0

bl

//

2

--0.06 T/3

-- 0.02

//

I

'I

—0.06 --0.04

I

T=O

CsJ2

(sJ3 +

0.137 56 1.411 34 0.169 55

A. Ising-like xxs chain, 1/g

1. $~(q, ru)

and $~(n/2) for 1/g =0.13 and two-magnon bound-state magnon — transitions. ca2.

0J~

0.086 30 0.452 64 0.116 54

0, 9

4729

s&)

s&)

0.006 57 0.07076 0.00931

0.0 0.1

..

FIG. 9. S~~(n/2, eo) for a finite chain of eight spins at various temperatures, 1/g =0.13 and bee=0. 1 J. ———Exact result for the infinite system [Eq. (71)]. Note the different scale for ao =2.


4730

25

to S~~(rr/2, ru) and St~(a /2) for 1/g =0.13 and bound-state transitions; ~~ ~3'. magnon threeco4'. two-magnon — four-magnon bound-state transitions; continuum transitions.

TABLE III. Dominant relative contributions

T/J =

1

$Q

3

$)

ground-state

—two-magnon

magnon bound-state transitions; m~ ground-state — iwo-magnon

Sii( m/2,

—1.2 —1.1 —1.0 —0.9 —0.8 —0.3 —0.2 —0.1

o))

(do

0.001 64 0.013 64 0.25228 0.025 96 0.00360

—GJp

+

0.85 0.14 0.&6

0.0 0.1 0.2 0.3

0, 402 92 0.019 92 0.000 84

0.8 0.9 1.0 1.1 1.2

0.037 92 0.37068 4.93900 0.366 12 0.060 56

1.8

0.007 56 0.026 80 0.00492 0.82643

S„(rr/2)

2.

Stt(q,

0.48

At zero temperature, this dynamic form factor probes the two-magnon bound states and the twotaker 3). These features are magnon continuum (Figs. 1 — again illustrated in Fig. 9 for N =8, and compared with the exact T 0 behavior for an infinite system

T/3

=

1/3 ~I

g

3 0

I

n

I

D

I

M

8

T/3

0

I I

a

—O)4

0.55

0.77

ra)

T=0

Q)p

0.77

0.00036 0.011 88 0.297 04 1.415 96

2.0 2.2

—C03

M~ +

I

0, 09

0, 10

[Eqs. (62) and (64)]. Due to the I/gr prefactor in Eq. (71), this contribution is very small. As noted above, the finite-size effect is manifest only in the continuum resonance. At finite T, the resonance structure is again dramatically modified due to the thermally induced transitions, in terms of a CP, the destruction bound-state peak (not shown), and the broadening of the two-magnon bound-state resonance. The domiant contributions to St~(rr/2, ca) listed in Table III reveal that the thermally induced boundthree-magnon — CP is dominated by magnon — — four-magnon bound-state state and two-magnon transitions. The resonance structure at q = m is seen to be qualitatively similar (Fig. 10), with the important difference, however, that there is no twomagnon continuum contribution [Eqs. (70) and (71)].

I I

=

2/3

3. 0

-1

0.91 0.79 0.87 0.004

I

0

l

-1

0

FIG. 10. S~~(m, ao) for a finite chain of eight sp'ins at various temperatures, 1/g =0.13 and Iso =0.1 J.

$„(q, ca)

This dynamic form factor vanishes at T =0 [Eq. in Figs. 11 and 12 are due to thermally induced transitions only. They are dominated by a CP which exhibits side bands at q = m. Moreover, there is a weak resonance

(49)]. Accordingly, the spectra seen


I,

cu/3

0

I I

T/3

=

1

0.2

0--

--0.003

05--

I

ic

I

-1

II

I I

I

T/3 = 3/$

T/3 =

Qy

--0.003

0

0 I I

I I

0.2

Q5-PI

BL.. I

0

--0.001 IS

0

1IIs.. s

-0.

1. From Table IV, it tO

TABLE IV. Dominant relative contributions

T/J

~ . &

continuum ao2.

—1.1

-1.0

-0.9 -4.8 -0.3

C91

~

.

Cell'

lllaglloll-Inaglloll

0.0 0.1 0.2 0.3

0.09448 0.716 39 0.12906 0.045 82 0.00002

1,0

1.1 1.2

I&

=1.

the two-magnon continuum and the two-magnon bound states. At q = Ir, the CP is seen to exhibit side bands (Fig. 12), because the magnon (Fig. 4) and two-magnon contributions are now separated from the sharp CP around eo 0, coming from difference processes between two-magnon and three-magnon bound states. Another interesting feature is that the CP around co 0 becomes weaker with increasing wave

to $~(cr/2. c») and $~(cl/2) for I/g tuCJ»

Cell:

tWO-lllaglloll

0.27 0.07 0.01 0.07 0.26

0.00038 0.003 19 0.00285 0.00262 0.00018

0.10220

I

FIG. 12. S~(~, ~) for a finite chain of eight spins at vari0.13 and neo 0. 1 J. Note the dif-

COIltilllllllll

=0.13 and —tWO-nlaglloll

0.20 0.43 0.36 0.67

0.68 0.42 0.07 0.46 0.68

II

ous temperatures, 1/g ferent scale for m

0.00010 0.00014 0.00022 0.00003 0.00001 0.02652

0.8 0.9

~I..

I

two-magnon continuum transitions; transitions; eccl cud: two-magnon bound-state — two-magnon bound-state transitions; eo3 e3.'three-magnon bound-state transitions.

—0.2

-0.1

'trallaltlollS;

--0.005

--0.005

0.2--

is seen that the the CP in $ (Ir/2, Rc) COmeS from difference processes between two-magnon, as well as between three-magnon bound states. Its broadening stems from two-magnon difference processes as expected from Fig. 4, superimposed by a weaker contribution from two-magnon continuum transitions. Finally, the weak resonance around 1 is seen to originate from transitions between Rl/J ccc/J

T/3=2/3

9I

FIG. 11. $ (cr/2, co) for a finite chain of eight spina at 13 and ctRc-0. 1 J. Note the various temperatures, I/g different scale for co

I

1

I

I N), I 0

COntributiOn

Q l

ff)

gI

close to

--0.005

3

Q

"-0.003

gi9

=1.

I I

0.2--

10---0.003 05---0.001

T/0 =2/5

"

0 q„„T/g = 5/5

--0.005

N N

0

"«

Q5--

-- 0.00'I

cu

I

I

..

0.53

0.84 0.34 0.43 0.22 0.89

0.04

0.003

0.27


4732 —2

p

I

I

I

I

T/3

0

2

I

I

I

I

I

I

P~ ~yI

T=O

1/5

=

—2

I

I I

I

I

0I

I

I

I

T/3

=1/5

0.5--

05--

p~/ lg~/

3

0

CU

I I

I I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

T/3 X u&

0 ~1—

I I

=

I

3

2/5

T/3

05--

0.5--

I

0

I

-

I

I

I

I

I

I

I

I

I I

2/5

=

05--

05-pg~

0 FIG. 13. S~(e/2,

B. Heisenberg-like

xxz chain, 1/g

-0.8

An important difference from the Ising-like systems treated above, is the more pronounced disperTABLE V. Dominant relative contributions

S~(m/2, o))

1.2

—09 —0.6 —0.3 0.0 0.3 0.6 0.9

1.2 1.5 1.8 2, 1

2.4

S (~/2)

0.003 25 0.01641 0.619 05 0.619 68 0.058 34 0.045 63 0.10096 6.279 27 0. 12008 6.010 74 0.00608 6.001 10 0.00050 0.204 22

I

0

I

I

2

various temperatures,

number, while the peak height of the weak resonance around co/J =1 increases by approaching the zone boundary (q = m). These results then reveal that the thermally induced resonance structure is considerably more complex than linear magnon theory predicts (Fig. 4). On the basis of magnon hydrodynamics, one expects a CP and possibly low-frequency side peaks, representSuch features are ing the second-magnon mode. seen in Figs. 11 and 12 but at wave numbers where hydrodynamics is no longer valid. In addition, we found a weak high-frequency resonance around ~/J =1, involving two-magnon continuum and twomagnon bound-state transitions.

"'

I

FIG. 14. S~(m, ~) for a finite chain of eight

for a finite chain of eight spina at 1/g =0.8 and Ace =0.3 J.

ru}


I

-2

=0.8 and

1/g

Ace

spins at

=0.3 J.

bound-state frequension of the magnon, magnon — cies and of the top and bottom of the two-magnon continuum [Eqs. (25), (26), (28), and (29)]. More0 and g 1, the gaps in the eigenvalue over, for q spectrum vanish. Consequently, we expect the thermally induced and rather sharp resonance structure found in the Ising-like case to become much broader for I/g =0.8.

1. S ( q, ca)

- S~(q,

sa)

The resulting smearing effect, is clearly seen by comparing Figs. 7 and 13, as well as Figs. 8 and 14. In fact, for 1/g =0.8, the CP is no longer as clearly resolved from the magnon creation resonance. Nevertheless, from Tables II and V, we see that the dominant contributions to the spectrum do indeed have the same origin as in the Ising-like model. An important difference is the considerably larger contri-

to S~(m/2,

eu)

and S~(m/2) for 1/g

Gl2

o

M3

~

cal3

4

CV4

=0.8 and

T'/J

= 3.

—

0.05

0.04

6.36 6.22 6.66 6.01 0.04 0.01 0.00 6.23 0.29

0.07

0.04 6.09 6.24 6.16

0.35 0.35 0.54 0.33 0, 29

6.68

EXCITATION SPECTRUM OF THE FERROMAGNETIC ISING-. . .

I I

—2 I.

0

—2

4

2

I I I '~ g&I

I

I I

I

I

I

I

I

6-- T=0

I I

=

T/3

1.

~

0 I

I

I I

I

I

I

I

I I

I

I

T=0

'I/5

0

—2

2

I

I I

I I

T/j

I/$

=

Q8--

0 M

M

I I I I

16--

0

I

I

T/3 =

I

I

-2

I

I I

I

z~k»

0

aL

I

I

I

I

!

I

i

I I

I

I

I

I

I

I

0,

I

I I

I

I

I I

I

I

T/3

I

I

0

I

I

2

0 (u/3

I

I

I

—2

I

0

I

I

I

2

I I

I

I

-2

4

=

0

I

I

I

2/5

I

I

I

I

~~~gg ~ I

I

lQ I

I

I

I

I i

I I

I I

I

I

I

I

I

2/5

0

2

FIG. 15. SII(rr/2, ru) for a finite chain of eight spina at various temperatures, 1/g =0.8 and heo =0.3 J. ———Exact result for the infinite system [Eq. (71)].

FIG. 16. S~~(m, eo) for a finite chain of eight spins at various temperatures, 1/g =0.8 and hm =0.3 J.

two-magnon continuum tranbution of the magnon — sitions, leading to the broadening of the magnon res-

[Egs (62) and (64) and Fig. 2] As ill the Ising-jlke case, Fig. 9, the finite-chain effects become particularly manifest in the two-magnon continuum resonance (Fig. 15) which, as expected [Eqs. (26) and

onances. 2. SII(if, rti)

In the finite-chain results shown in Figs. 15 and

16, we have again included the zero-temperature case to illustrate the limitations of the method, by comparing with the exact behavior for the infinite system

TABLE VI. Relative dominant contributions

~

(28)], almost merges with the two-magnon boundstate resonance. The associated small binding energy of the bound state and the aforementioned consequences of the more pronounced dispersion of the eigenvalues partially explains why, at finite temperature, the spectrum collapses to a rather broad reso-

to S&&(m/2, 0)) and S~~(m/2) for 1/g

=0,8

and

'r/J= 3. Si] (m/2, 2. 1 1.8

1.5 1.2 0.9 0.6 0.3 0.0

0.3 0.6 0.9 1.2 1,5

1.8 2. 1 2.4 2.7 3.0

S„( /2)

o))

0.00024 0.00016 0.00048 0.006 12 0.02208 0.087 68 0.152 32 0.348 08 0.346 20 0.556 28 0.34204 0.215 28 0.037 32 0.034 20 0.12908 0.018 32 0.011 68 0.01436 0.697 03

M~

—c02

+- Ql3

QJp +

0.50 0.03 0.03 0.28 0.34 0.11 0.36 0.18 0.04 0.02

0.61 0.21 0.46 0. 18

~

~

0.54 0.54

0.17

0. 11

0.64 0.09

E. STOLL

T'. SCHNEIDER AND

-2

I

I

T/3

I

I

0.)

=

-2

—2

0

I

I I

I

I

T/

I

P

—2

0 I I

I

T/3

I

0.1

=

I

T/3

=

2 Pl

I

I

3/5

0.3--

prg~P ~ rp~ ~ryy

p/~

0

I

I I

P/

~~e~y~z

Py~g/1

~g~ ~8yP p g~

~Z~~gZ

0

3

c I

I

I

I

N

V)

I

I

2 5

O'I--

L. I

OJ I

I

I I

3

I

0 --,'

I

(J)

p~

&~i&F

F Fr/~ ~~y ~~( F ~~~~z~z

~g&~i

c~pr~

PF p 8 pPp~8Y

~y~

I

I

T/3

p

p/p g~~

i

I

=

I

FIG. 17. S~(cr/2,

~rz

f»yZ P

F

r&gq«

4~~~~~""""~~~~%

PirZi

ous temperatures,

0

2

e) for I/g

Comparing Fig. 5 to Figs. 17 and 18, it is seen that at T/J =0.1 and —,, the shape of the fully resonance is rather close to that temperature-induced predicted by the linear magnon approximation [Eq. (76) l. An important difference is the evidence for the removal of the square-root singularity in an infin-

0.0 0.3 0.6 0.9 1.2 1.5 1.8 2. 1 2.4 2.7 S~(m/2)

0.24 0.27

to

S (cr/2,

0.05 0.03 0.00 0,05 0.00 0.04 0.00 0 04 0.06

0.16

gyp'~

2

a finite chain of eight spina at variItcu =0.3 J.

co) and

S~(cr/2) for 1/g

-0.8 and

1

T/J =-, .

—oJ3

0.002 59 0.013 50 0.00976 0.031 99 0.074 37 0.079 89 0.053 23 0.153 70 0.082 89 0.00674 0.00623 0.00102 0.00051 0.00008 0.15499

~ ~g~~g ~y~gp

=0.8 and

033 +

1 t2

YY~y

jpgled

the magnon bound states and the two-magnon continuum, as well as between the two-magnon bound states are comparable to those of the magnon transitions. Compared to the Ising-like case (Figs. 11 and 12), there is much more overlap in the present case between the various contributions, giving rise to a rather broad resonance. Consequently, it is no longer possible to resolve a CP from transitions between three-magnon bound states and weak peaks from two-magnon continuum iwo-magnon bound states — transitions.

3. S (cJ, cu)

—09 —0.6 —0.3

p~

ite system in terms of the smeared low- and highfrequency cutoffs. In any case, Table VII reveals that the qualitative agreement with the linear magnon approximation is misleading. In fact, at least for T/J = —,, the contributions of the transitions between

nance. From Table VI, where the dominant contributions to Sit(e'/2, oi) are listed, it is seen that this collapse to a broad resonance is due to the small binding energy of the two-magnon bound state and the more pronounced dispersion, leading to a rather broad background of the magnon-threemagnon bound-state transltlons.

TABLE VII. Dominant relative contributions

P Z jP~ yPyF ygy g&~

4lK

0

FIG. 18. $~(m,

for a finite chain of eight spina at 1/g =0.8 and isaac =0.3 J.

co)

I I

I

p~~g

-2


I

I

2/5

p pi~~&

2

I

O'I--

Ppg

g~p~zg~8ytg~~zpp~ N4~~~~i~w~i~P&m

I

I I

0.03

0.25 0.25 0, 04 0.18 0.03 0.16 0.05

0.25 0.24 0.18 0.02 0.55 0.27 0.75 0.17

0.34 0.23 0.40

0, 07 0.27 0.05

0.12

0.05

EXCITATION SPECTRUM OF THE FERROMAGNETIC ISING-. . .

VI. IMPLICATIONS FOR s &

&

AND

THE SOLITQN ASPECT

The zero-temperature results for the dynamic form factors derived in Sec. III are not restricted to s = —, hut hold for all s values. As a consequence, S~(q, ~) = S~(q, co) probes the magnons [Eqs. (43) and (44)], and Stt(q, ru) [Eqs. (36) and (58)l the two-magnon bound states and the two-magnon continuum. In fact, this main physical content simply follows from the classification of states according to the eigenvalues of the spin deviation operator [Eq. (12)] and the eigenvalues of the translation operator [Eq. (23)). Consequently, even the finite T behavior will involve a thermally induced structure, similar to that found here for 8 = 2. This conclusion can be further substantiated by referring to a recent molecular-dynamics study of the classical IsingHeisenberg chain, 46 yielding, as far as the essential features are concerned, a similar resonance structure for S (q, cu) and St~(q, ~). These findings then also raise new aspects for the interpretation of the recent neutron scattering experiments on CsNiF3, which is an easy-plane s ferromagnetic chain. In fact, a indicates that the system molecular-dynamics study might behave almost like an s = I isotropic fexromag-

-1

'%. Heisenberg,

Z. Phys. 49, 619 (1928).

2H. Bethe, Z. Phys. 71, 205 (1931). Mathematical Physics in One Dimensions, edited by E. H. Lieb and D. C. Mathis (Academic, Neer York, 1966). 4L. A. Takhtadzhan and L. D. Faddeev, Usp. Mat. Nauk

34:5, 13 (1979) fRuss. Math. Surv. 34;5, 11 (1979}]. 5H. B. Thacker, Rev. Mod. Phys. 53, 253 (1981). 6R. J. Baster, Ann. Phys. (N. Y.) 70, 323 (1972}. 7R. Orbach, Phys. Rev. 112, 309 (1958). sC. N. Yang and C. P. Yang, Phys. Rev. 150, 321 (1966); 150, 327 (1966); 151, 258 (1966). I. G. Gochev, Zh. Eksp. Teor. Fiz. 61, 1674 (1972} |Sov. Phys. -JETP 34, 892 (1972)]. ~OE. H. Lieb and %. Liniger, Phys. Rev. 130, 1605 (1963). ' E. H. Lieb, Phys. Rev. 162, 162 (1967). ' R. J. Baxter, Ann. Phys. (N. Y.) 70, 173 (1972); 76, 1, 25, 48 (19733. ~3C. S, Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, Phys. Rev. Lett. 19, 1095 (1967). Solitons, edited by R. K. Bullough and P. J. Caudrey (Springer-Verlag, Heidelberg, 1980). ~5E. Lieb, T. D. Schultz, and D. Mattis, Ann. Phys. (N, Y.) 16, 407 (1961). 6Th. Niemejer, Physica (Utrecht) 36, 377 (1967).

T. Horiguchi, and M. Suzuki, Physica (Utrecht) 46, 67 (1970). M. Gaudin, Phys. Rev. Lett. 26, 1301 (1971). ~~J. D. Johnson and J. C. Bonner, Phys. Rev. B 22, 251 (1980). ~7S. Katsura,

netic chain in a field, so that the results of the present work for 1/g =0.8 are qualitatively applicable. Finally, let us turn to the soliton aspect. In a classical treatment and by taking the continuum limit of the ferromagnetic xxz chain, one obtains an exactly integrable system exhibiting envelope solitons. 49 Recently, it has been sho~n that a semiclassical quantization of these envelope solitons leads to the correct bound-state energies for 1/g & 1 and large quantum numbers m. '~ Moreover, it was found that the bound-state resonance in a dynamic form factor related to Stt(q, co) survives the classical limit and, accordingly, may be interpreted as an envelope-soliton feature. 50 In view of this insight and the close similarities in the dynamic form factors of the s = 2 and the classical ferromagnetic xxz chain, 6 it becomes clear that envelope solitons and the associated phenomena represent merely the classical limit of magnon bound states and their effect on properties of interest. ACKNOWLEDGEMENTS

%e have benefitted from illuminating discussions R. Jauslin, J. M. Loveluck, S. %. Lovesey, K. A. Muller, and R. H. Swendsen. with H.

20M. %ortis, Phys. Rev. 132, 85 (1963). 'G. Muller, H. Beck, and J. C. Bonner, Phys. Rev. Lett.

43, 75 (1979). M. Steiner, in Physics in One Dimension, edited by J. Bernasconi and T. Schneider (Springer-Verlag, Heidelberg, 1980), p. 140. 23J, B. Torrance, Jr. , and M. Tinkam, Phys. Rev. 187, 595

(1970). 4J. C. Bonner, H. %. J. Blote, H. Beck, and G. Muller, Physics in One Dimension (Springer-Verlag, Heidelberg, 1981), p. 115. 25%. Marshall and S. %. Lovesey, Theory of Thermal Ãeutron Scattering (Clarendon, Oxford, 1971). 26C. P. Landee and R. D. %illett, Phys. Rev. Lett. 43, 463

(1979). D. Johnson and B. M. McCoy, Phys. Rev. A 6, 1613 {1972). 28J. D. Johnson, Phys. Rev. A 9, 1743 (1974). 29R. Silberglitt and A. B. Harris, Phys. Rev. 174, 640 (1968). 3 F. D. Dyson, Phys. Rev. 102, 1217, 1230 (1956). 3'S. V. Maleev, Zh. Eksp. Teor. Fiz. 33, 1010 (1957) [Sov.Phys, JETP 6, 776 (1958)]. ~J.

32D. Forster, Hydrodynamic F/uctuations, Broken Symmetry and Correlation Functions (Benjamin, Reading, Mass. , 1975). 336. F. Reiter, Phys. Rev. 175, 631 (1968). 4F. Schwabl and K. H. Michel, Phys. Rev. B 2, 189 {1970). 35T. Schneider and E. Stoll, Phys. Rev. B 23, 4631 (1981). 36T. Schneider and E. Stoll, Phys. Rev. Lett. 45, 997 (1980).


4736

J. M. Loveluck

and E. Balcar, Phys. Rev. Lett. 42, 1563 (1979). J. C. Bonner and M. E. Fisher, Phys. Rev. 135, A540 (1964). H. %. Blote, Physica (Utrecht) 79B, 427 (197S). J. D. Johnson and J. C. Bonner, Phys. Rev. 22, 251 (1980).

'F. Carboni 42P.

and H. Shiba, Prog. Theor. Phys. (Kyoto) 63, 743 (1980). 45J. P. Groen, T. O. Klaassen, N. J. Paulin, G. Muller, H. Thomas, and H. Beck, Phys. Rev. B 22, 5369 (1980). 4 H. R. Jauslin, T. Schneider, E. Stoll, and J. M, Loveluck, J. Phys. C 15, L69 (1982). ~7J. K. Kjems and M. Steiner, Phys. Rev. Lett. 41, 1137 44N. Ishimura

and P. M. Richards, Phys. Rev. 177, 889

(1969). M. Richards and F. Carboni, Phys. Rev. B 5, 2014

(1972). 436. Muller and H. Beck, J. Phys. C 11, 483 (1978).

25

(1978). ~

J. M. Lovelucg, T. Schneider, E. Stoll,

Phys. Rev. Lett. 45, 1SOS (1980), " E. K. Sklyanin {unpublished). T. Schneider, Phys. Rev. B 24, 5327

and H.

(1981).

R. Jauslin,