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Submitted on 1 Jan 1978
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DYNAMICS OF THE CLASSICAL PLANAR SPIN
CHAIN
H. de Raedt, B. de Raedt
To cite this version:
JOURNAL DE PHYSIQUE Colloque C6, supplhment au no 8, Tome 39, aolit 1978, page C6-708
DYNAMICS OF THE CLASSICAL PLANAR SPIN CHAIN
H. De Raedt and B. De ~aedt+
+ Department Natuurkunde
,
Universitaire Inste ZZing Antuerpen B-2610 W i Z r i jk, Be lgiwnI n s t i t u t ffir Theoretische Physik, Universitiit desSaarlandes 0-6600 Saarbrilcken,Gemany
R8sum6.- La technique Monte Carlo a 8tE utilis6e pour calculer les moments de la fonction de rela- xation pour la charne de spins classiques avec anisotropie dans le plan x-y. Ainsi on obtient des rgsultats d6taill6s pour les propri6tCs dynamiques. Comparaison est faite avec 11exp8rience et au-
tres th8ories.
Abstract.- The Monte Carlo method has been used to calculate the moments of the relaxation function for the classical planar spin chain. In this way extended results for the dynamic properties of this system are obtained. Comparison is made with experiment and other theories.
In a previous paper the authors presented a method, based on the projection operator formalism.
to obtain expressions for the relaxation function in a systematic way /l/. Ti was shown, that taking a spin component and its first two time derivatives as relevant variables, the normalized relaxation function can be written as
z2+zrj(z)-(<ak> /<a2> -<a2>
@ (2)' k k k ) (1)
k z(z2 -<W"> /<w~>~)+c~(z) k
In the simplest approximation the transport coeffi- cient Ck(z) was found to be given by / l /
<u6> -<a4>
k ;/
C,(z) =
-
(2)< w ~ > ~ - ~ < w ~ > <a4> + < u ~ > ~
' / *
z+i
[
k k5
The relaxation function is now completely determi- ned by its moments and < w ~ > ~ , and it satisfies the sum rules
identically.
The classical planar spin chain is described by the hamiltonian / 2 , 3 , /
where the spins are classical unit vectors, and we will take A = 0.21, corresponding to the anisotro- py in CsNi-F3, for which measurements were made by Steiner and a1./4/. Loveluck and al. calculated some moments using the transfer operator method /5/. But because the higher moments are very complicated ex- pressions, that give not much physical insight, we have used a Monte Carlo simulation to calculate the-
se quantities numerically. A more complete descrip- tion of the method, and extensed results will be presented in a forthcoming paper 161. Some results will now be summarized.
Fig.1 : The second, fourth and sixth moments for both the X- (left) and z-(right)components, for T = 0.5, and as a function of the wave vector k.
In figure I bare results for the moments are shown for the temperature T = 0.5. The moments of the X-component do not disappear at zero wave vec- tor, because the total spin in the X-direction is not a conserved quantity, in contrast with the to- tal spin in the z-direction. In figure 2 we give some results for the imaginary part of the relaxa- tion function for the wave vector k = 0.25 n. At lower temperatures the width of the spin wave peak of the z-component is extremely small, which means. that the excitations are almost undamped. At higher
Fig. 2 : The imaginary part of the normalized rela- xation function for the wave vector k P 0.2511 for
the X- (left) and z- (right) components.
temperatures the peak of the z-component seems to disappear more rapidly than that of the X-component. The difference between the X-and z-components at low-temperatures can be understood in terms of the correlation lengths. At zero temperature the corre- lation length
E x
diverges, because the spins order ferromagnetically in the X-y plane, whileCZ
remains finite /6/.Our results agree with the experimental observations on CsNiF3 /4/, and with the theoreti-
References
/ l / De Raedt, H. and De Raedt, E. Phys. Rev. B
2
(1977) 5379./2/ Villain, J., J.Physique
35
(1974) 27. /3/ Loveluck, J.M., Lovesey, S.W. and Aubry, S.,J. Phys. C
8
(1975) 3841./ 4 / Steiner, M., Villain, J. and Windsor C.G., Adv. Phys.
5
(1976) 87./5/ Loveluck, J.M. and Lovesey, S.W., J.Phys. C
8
(1975) 3857./6/ De Raedt,B. and De Raedt, H., to be published in Phys. Rev. B.
