SPECTRAL EXCITATIONS OF LINEAR MAGNETIC CHAINS

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SPECTRAL EXCITATIONS OF LINEAR MAGNETIC

CHAINS

J. Bonner, H. Blöte, J. Johnson

To cite this version:

JOURNAL D E PHYSIQUE

Colloque C6, supplhment au no

8 ,

Tome 39, a021 1978, page C6-710

J.C. Bonner, H.W.J. ~1gt:and

J.D. Johnson

++

L l$ziversity o f

Rhode

Island, Kingston, R. I., 02881, U.S.A.

++I(amerlingh Ones Laboratoriwn, University of Leiden, Netherlands Los A l m s S c i e n t i f i c Laboratory, Los A l m s , New Mexico, 87545, U.S.A.

R6sumd.- Le spectre d'exitations dldmentaires de la chaTne magndtique est remarquablement complexe

et interessant. Nous pr6sentons ici un r6sum6 des recherches r6centes qui ont dt6 inspirges par (a)

un projet qui a pour but d'apporter une solution analytique et complPte, et (b) les ddcouvertes ex-

pdrimentales r6centes concernant la transition de type "spin-Peierls", les groupes localisds des

spins dans les systSmes ferromagndtiques lindaires et anisotropes, et les caract6ristiques non-clas-

siques de la dynamique des spins d'une charne lingaire

1

basse tempgrature.

Abstract.- Although the linear magnetic chain may be regarded as the simplest magnetic model, the

spectrum of elementary excitations is remarkably complex and interesting. This paper summarises some

recent investigations inspired by (a) a continuing project directed towards obtaining a complete,

analytic solution for a general linear Ising-Heisenberg-XY chain, and (b) by recent experimental dis-

coveries of the spin-Peierls transition, bound spin complexes in anisotropic linear ferromagnets,

and non-classical features of low temperature linear chain spin dynamics.

INTRODUCTION.- Recent theoretical investigations of

the spectral excitations of various linear magnetic

chains will be summarised. First of all, we consider

bound spin complex (localized) and spin-wave-type

(delocalized) excitations in the linear Ising-Heisen-

berg ferromagnet. The results are interesting because

they are not in accordance with generalized smooth-

ness-universality considerations. As the spin aniso-

tropy decreases, from the Ising limit to the Heisen-

berg limit, a cross-over anisotropy value is reached

where the character of the dominant low-lying ther-

mal excitations changes from localized to delocali-

zed. In the uniform Heisenberg ferromagnetic limit

there is an unresolved question whether the linear

chain behaves in accordance with spin wave theory.

The doubt arises because of the prominence of low-

lying bound states of two (and more) reversed spins

/ l / .

Our calculations, which are essentially exact,

have not been extended to the Heisenberg limit be-

cause of convergence problems. However, the strong

implication is that spin-wave-type thermal excita-

tions will prevail there. The theoretical situation

is also very interesting in the light of current

work /2/ which has revealed the presence of bound

and delocalized solitons in continuum, linear, fer-

romagnetic chains. Our predictions in the region

close to the Ising limit agree with theoretical and

experimental studies of bound spin complexes

1 3 1 .

*

Work supported in part by the U.S.D.O.E.,

by Nato,

and the U.S.N.S.F.

It would be very interesting, therefore, to perform

similar experiments close ot the Heisenberg limit.

Unfortunately, a good l-D ferromagnet with low spin

anisotropy does not yet seem to be available.

It is also an interesting question how well

spin wave theory describes low spin (spin 112) li-

near Heisenberg antiferromegnets (AF'

S) ;

for exam-

ple, the low temperature spin dynamics. Classical

spin wave theory describes high spin systems (e.g.

spin 512

TMMC)

rather well. Significant deviations

have, however, been predicted theoretically for

spin

112

AF's. These studies indicate that the spec-

tral weight should not be entirely located at the

des Cloizeaux and Pearson (des C/P) triplet excita-

tion energies, as a spin wave picture would suggest.

An appreciable tail should persist above the des

C/P energies (frequencies) even at T

0

/4,5,6/.

Very recent neutron scattering experiments have

confirmed deviations from a simple des C/P scheme

/ 7 / .

However, the nature of the contributing exci-

tations higher in energy than the des C/P states is

not yet clear.

The linear

AF

alternating chain has been a fo-

cus of interest recently, as a result of the disco-

very of experimental spin-Peierls systems

/ B / .

Al-

ternating AF chains also interest physical chemists

in the area of spin exciton theory

191.

Crucial

t6

both spin-Peierls theory and spin exciton theory is

the existence of

an

energy gap between the singlet

ground state and the lowest-lying excitations which

form a band of t r i p l e t s t a t e s . Approximate t r e a t - ments have c o n s i s t e n t l y p r e d i c t e d t h e occurrence of such a gap, b u t r e c e n t work based on t h e i s o t r o p i c c h a r a c t e r of t h e Heisenberg Hamiltonian has sugges- t e d t h a t a g a p l e s s a c o u s t i c branch should be pre- s e n t , i n a d d i t i o n t o an o p t i c branch s p l i t o f f from t h e ground s t a t e by t h e energy gap /10/. Hence t h e e x c i t a t i o n spectrum h a s been v e r y car'efully s t u d i e d f o r t h e S = 112 a l t e r n a t i n g c h a i n by e x a c t c a l c u l a - t i o n s on f i n i t e c h a i n s .

LINEAR ISING-HEISENBERG FERR0MAGNET.- This system i s

d e s c r i b e d by Hamiltonian

where J (>O) i s the exchange c o n s t a n t and y i s a parameter which measures t h e s p i n a n i s o t r o p y i n t h e system. The l i m i t y = 0 g i v e s t h e I s i n g model, and y = 1 t h e Heisenberg model. An a n a l y t i c s o l u t i o n f o r the elementary e x c i t a t i o n s of t h i s system i n e e r o magnetic f i e l d has been o b t a i n e d which appears t o be e x a c t . The r e s u l t i s d i s p l a y e d i n f i g u r e 1 .

An e x c i t a t i o n energy gap appears f o r a l l y < I . The gap e x p r e s s i o n i s continuous, b u t a cross-over i n behavior appears a t y = 2

-

fi

=

0.586.

where again a thermal e x c i t a t i o n gap appears f o r a l l Y

<

I , a s shown i n f i g u r e I . However, t h e AFgap

i s a s i n g l e e x p r e s s i o n without any cross-over.

EXCITATION GAPS I N ANTIFERROMAGNETIC ALTERNATING CHAINS.- These systems a r e d e s c r i b e d by t h e Heisen- b e r g Hamiltonian

where

a

p l a y s t h e r o l e of an a l t e r n a t i n g parameter, such t h a t

a

= 0 corresponds t o a completely dimeri- zed system, and

a

= 1 g i v e s t h e uniform l i m i t . F i g u r e 2 shows t h e r e s u l t s of e x t e n s i v e computations and e x t r a p o l a t i o n s on

AF

a l t e r n a t i n g c h a i n s of up t o 12 s p i n s I l l / . No evidence can be found f o r t h e presence of an a c o u s t i c ( g a p l e s s ) d i s p e r s i o n branch p r e d i c t e d by r e c e n t theory /10/. 4 / A €

-

-

IJI

I -

"d

O!I 012 d3 0:4

is

d.6

d.e

04

to

.

DINER a- UNIFORM

F i g . I : Thermal e x c i t a t i o n gap of t h e Ising-Hei- senberg ferromagnetic c h a i n i n comparison w i t h t h e gap of t h e corresponding a n t i f e r r o m a g n e t . For y < 0.586, t h e ferromagnetic thermal gap i s given by

1

AE, = J ( l

-

Ty2), and f o r y > 0.586 by AE2=2J(1-y)

For y < 0.586 (approaching the I s i n g l i m i t ) t h e do- minant e x c i t a t i o n s a r e bound s p i n complexes. For y > 0.586 (approaching t h e Heisenberg l i m i t ) spin- wave-like e x c i t a t i o n s dominate t h e spectrum. What i s i n t e r e s t i n g i s t h a t a t y = 0.586 a s i g n i f i c a n t change i n t h e p h y s i c a l c h a r a c t e r of t h e problem i s

not accompanied by any change i n t h e symmetry of t h e Hamiltonian. This c o n t r a s t s w i t h t h e AF l i m i t ,

F j g

.

2 : E x t r a p o l a t e d energy gap of t h e a l t e r n a - t i n g , s p i n 112, Heisenberg a n t i f e r r o m a g n e t i c chain. The gaps f o r f i n i t e N c h a i n s a r e shown a l s o . The c r o s s e s give t h e a l t e r n a t i o n energy gap of Bulaev- k i i .

I n t e r e s t i n g l y , t h e n u m e r i c a l l y e x t r a p o l a t e d gap curve l i e s very c l o s e t o t h e approximate Bulaevskii Hartree-Fock r e s u l t 1121, and v a n i s h e s only i n t h e uniform l i m i t a = l . This r e s u l t s u p p l i e s f u r t h e r s t r i k i n g evidence f o r b e h a v i o r a l d i f f e r e n c e s a t low temperatures between S = 112 and S = m systems.

A c l a s s i c a l s p i n wave (S = m) c a l c u l a t i o n shows

m.

The a l t e r n a t i n g

XY

magnetic chain can be s o l - ved e x a c t l y i n both t h e S = 112 and l i m i t s .

R e f e r e n c e s

/ I / Wortis,

M.,

Phys. Rev.

132

(1963) 85. Bonner, J . C .

T h e s i s , U. of London.

/ 2 / L u t h e r , A . , Phys. Rev.

B14

(1976) 2153.

1 3 1 T o r r a n c e , J.B., J r . , and Tinkham, M., Phys. Rev. 187 (1969) 587 & 595.

/ 4 / Hohenberg, P.C., and Brinkman, W.E., Phys. Rev.

E

(1974) 128.

/ 5 / Bonner, J.C., e t a l . , AIP Conf. Proc.

2

(1974) 335

161 Miiller, G., and Beck, H., J . Phys. C c (1978) 483,

and p r i v . comm.

171 S h i r a n e , G., p r i v a t e conrmunicatbon.

/ 8 / J a c o b s , I.S., e t a l , Phye. Rev. (1976) 3036. 1 9 1 Soos, Z.G., J. Chem. Phys.

66

(1967) 4284.

/10/ Drawid,

M.,

and H a l l e y , J . W . , AIP Conf. P r o c .

2

(1976) 208.

/ I ] / Duffy, W., J r . , i s thanked f o r d e t a i l s of h i s e a r - l i e r c a l c u l a t i o n s .