Low temperature dynamics of a linear chain of classical spins with Dzyaloshinski–Moriya interaction

SSC 4462

PERGAMON

Solid State Communications 109 (1999) 87±91

Low temperature dynamics of a linear chain of classical spins with Dzyaloshinski±Moriya interaction A.S.T. Pires Departamento de FõÂsica, Universidade Federal de Minas Gerais, CP 702, Belo Horizonte, 30123-970, MG, Brazil Received 25 August 1998; accepted 19 October 1998 by P. Burlet

Abstract We calculate the low temperature dynamic relaxation function for the one-dimensional classical planar antiferromagnet with a Dzyaloshinski±Moriya interaction normal to the xy-plane using a projection operator technique. At absolute zero temperature the interaction gives rise to a canted structure similar to the weak ferromagnetic arrangement observed in some three-dimensional compounds. We obtain an analytical expression for the time dependent memory function and its Laplace transform. q 1998 Published by Elsevier Science Ltd. Keywords: D. Spin dynamics; D. Thermodynamic properties

1. Introduction The study of the dynamics of one-dimensional magnetic systems has been the subject of much research activity. The experimental information about these systems is provided, mainly, by techniques such as neutron scattering which can be interpreted by dynamic correlation functions. However, the evaluation of these functions is far from trivial. In this paper we calculate the transverse dynamical correlation function for the classical one-dimensional XY antiferromagnet with Dzyaloshinski±Moriya interaction. As far as we know this is the ®rst calculation, of this kind, performed in such a model. We use a projection operator technique, proposed by Reiter and co-workers [1±6]. This method has proven successful in the study of the classical and quantum Heisenberg model in one and two dimensions, and in a previous paper [7] we have applied it to the quantum XY model. The theory is exact in the large S limit, in that the leading term in the frequency shift and

damping is given exactly, to ®rst order in 1/S, in the limit T ! 0. Terms of higher order will involve the decay of a spin wave into three or more models, and these corrections are thought to be small. The Hamiltonian for our model is X ÿ x x
J Sn Sn11 1 Syn Syn11 Hˆ n

ÿ
1 …21†n11 D Sxn Syn11 2 Syn Sxn11 Š:

…1:1†

Nearest neighbour spins are coupled by an antiferromagnetic interaction J. D is the common amplitude for all the Dzyaloshinski vectors, and the factor (21) n11 takes into account their alternation from one pair to another of nearest neighbours. The spins are classical vectors length S and three ÿ with ®xed
components S n ˆ Sxn ; Syn ; Szn . The XY model should be distinguished from the plane rotator where the spins are constrained within the xy plane and therefore has no dynamics. In most weak ferromagnets with Dzialoshinski-type coupling D is small compared to

0038-1098/99/$ - see front matter q 1998 Published by Elsevier Science Ltd. PII: S 0038-109 8(98)00532-8

88

A.S.T. Pires / Solid State Communications 109 (1999) 87±91

J. There are some evidences that the above Hamiltonian may provide an appropriate description for the spin-canting in RbCoCl3´2H2 and for certain types of weak ferromagnetism in BaVF5 and BaFeF5 [8±10]. Exact expressions for all the thermodynamic properties of this model are known [8,9]. For instance the two spins static correlation function is given by kSx0 Sx2n l ˆ kSy0 Sy2n l ˆ

1 j2nj u ; 2

kSx0 Sx2n11 l ˆ kSy0 Sy2n11 l ˆ

…1:2†

cos a j2n11j u ; 2

kSx0 Sy2n l ˆ 2kSy0 Sx2n l ˆ 0; kSx0 Sy2n11 l ˆ 2kSy0 Sx2n11 l ˆ

…1:3† …1:4†

1 sin auj2n11j ; 2

…1:5†

with

ÿ
ÿ
u ˆ 2I1 bJ~ =I0 bJ~ ;

…1:6†

where In is the ®rst-kind modi®ed Bessel function, and p ~ J~ ˆ J 2 1 D2 …1:7† cos a ˆ J=J: At low temperatures we have ~ 2; u < 21 1 T=2JS

which can also be written as  2 2 2 ÿ
S2 4 1 2 u cos …a=2† xq ˆ 2T 1 1 u2 2 2ucosq 

1 2 u sin …a=2†  : 1 1 1 u2 1 2ucos q 2

q1 ;q2 ;q3

  ÿ
ÿ
 G1 q1 ; q2 ; q3 :Szq1 Szq2 Sxq3 1 Syq3 1 G5 q1 ; q2 ; q3 n

h i ÿ
 Sxq1 Sxq2 Sxq3 1 Syq1 Syq2 Syq3 1 q1 ; q2 ; q3 Syq1 Syq2 Syq3 X ÿ
1 G2 q3 ; q2 ; q1 Syq1 Sxq2 Sxq3 g 1 4 dq1 1q2 1q3 ;q2p n

q1 ;q2 ;q3

i ÿ
h 2 G3 q1 ; q2 ; q3 : 2Sxq1 Sxq2 Sxq3 1 Syq1 Syq2 Syq3 o ÿ
ÿ
2G4 q3 ; q2 ; q1 Syq1 Syq2 Sxq3 1 G4 q1 ; q2 ; q3 Syq1 Sxq2 Sxq3 …1:11†

…1:8†

where we have taken kB ˆ 1. The q-dependent susceptibility is   4 2 ÿ
S2 1 2 u 1 2u 1 2 u cos a cos q xq ˆ ; …1:9† 2T 1 2 2u2 cos2 q2 1 u4



of the spins causes a ferromagnetic momentum of amplitude sin(a /2) to appear on each site. The general theory for the dynamics is presented in [7] and the reader is referred to Ref. [7] where he will ®nd expressions for the relaxation function S(q,v ) and the time dependent memory function Mq(t). We give  only the ®nal expression for the term L2 Sxq 1 Syq , where L is the Liouville operator. From the equation of motion de®ning the Liouville operator (see [7]) and the Hamilttonian (1.1), written in q-space, we ®nd after a straightforward calculation   X dq1 2q2 2q L2 Sxq 1 Syq ˆ 4

where  ÿ
ÿ
 G1 q1 ; q2 ; q3 ˆ J 2 1 D2 cos q1 2 q cos q3 ;

…1:12†

 ÿ
 G2 q1 ; q2 ; q3 ˆ D2 2 J 2 cos q1 cos q3 1 J 2 cos q1 cos q2 ; ÿ
G3 q1 ; q2 ; q3 ˆ JDcos q3 cos q1 ;

…1:13† …1:14†

ÿ
ÿ
G4 q1 ; q2 ; q3 ˆ 2JDcos q3 cos q2 2 2cos q1 ; …1:15†

2

…1:10†

In this expression the ®rst term is the magnetic susceptibility of a pure antiferromagnetic chain with nearest-neighbour interaction J~ instead of J. Besides this modi®cation of the effective exchange constant, the Dzyaloshinski±Moriya coupling introduces an extra term which is at the origin of the speci®c properties of the weak ferromagnetic chain [8]. The canting

ÿ
G5 q1 ; q2 ; q3 ˆ D2 cos q3 cos q1 : …1:16†   For LS 00q ; LS 00q we have     ~ 2 u: LS 00q ; LS 00q ˆ 2T JkSx0 Sx1 l 1 DkSx0 Sy1 l ˆ 2JTS …1:17† In Section 2 we describe the calculation of the static correlations, that are need for the dynamics, by means

A.S.T. Pires / Solid State Communications 109 (1999) 87±91

of the spin wave theory. We then evaluate and obtain the dynamical correlation function. 2. Spin wave theory and evaluation of the memory function The Hamiltonian is rewritten in terms of the variables Szn and fn de®ned by the transformation h h ÿ
i1=2 ÿ
i1=2 Sn ˆ S 1 2 Sz =S 2 cos fn ; 1 2 Sz =S 2 

sin fn ; Szn :

…2:1†

The angle variable fn is then rewritten in terms of its deviation from its equilibrium value through

fn ˆ …21†n ‰p=2 1 a=2Š 1 cn

q

Hamiltonian (2.3) can be diagonalized using the following canonical transformation: h ÿ 
i21=4  1 cq ˆ 4S2 1 2 cos q aq 1 a2q ; …2:4†  h ÿ
i21=4  1 aq 2 a2q ; Szq ˆ i 4S2 1 2 cos q =4

the normal coordinates. This is " Sxq

ˆ S 2 dq;0 sin…a=2† 2 cos…a=2†cq1p 3   sin…a=2† X 1 z z 1 Sq Sq2q1 1 cq1 cq2q1 5; 2 S2 1 q1 …2:8†

 cos…a=2† Syq ˆ S 2dq;p cos…a=2† 2 sin…a=2†cq 2 2   X 1 z z Sq Sq2q1 1p 1 cq1 cq2q1 2p :  S2 1 q1 …2:9†

…2:2†

where a is the canting angle between neighbouing spins. This canting angle is chosen such that the terms linear in cn vanish from the Hamiltonian, that is it is found from the equation tan a ˆ D/J. The harmonic Hamiltonian can then be written as i X h 2ÿ
S 1 2 cos q cq c2q 1 Szq Sz2q : …2:3† H0 ˆ J~

…2:5†

where a1 q and aq are the boson creation and annihilation operators, respectively. We remark that it is easier at this point (following standard procedure) to use magnon operators to perform the calculations and then take the classical limit at the end of the calculation. We obtain  X  1 vq aq aq 1 1=2 ; …2:6† H0 ˆ

This choice of coordinates seems to presuppose that the spin vectors lie along the y axis, but in fact the axis in the xy plane that we choose for the quantization is arbitrary. As long as we evaluate only averages that are invariant to rotations about the z axis (such as kS 00q S 00q l) and involve spins separated by distances less than the coherence length j (i.e. for wave vectors q . k, where k ˆ j21 ) we can calculate rotationally invariant correlation functions using this procedure. The memory function Sq should be reliably calculated in this form since it is determined by short-distance correlations. Using now (2.4), (2.5), (2.8) and (2.9) in the expression for the memory function given in [7], and the classical relation ka1 q aq l ˆ 1=bvq , we ®nd n ÿ
X  ÿ
ÿ
~ 2 sin2 …a=2† F1 q; k cos V1 q; k t Mq …t† ˆ 2T JS k

ÿ
ÿ
1 F2 q; k cos V2 q; k tŠ 1 cos2 …a=2†  ÿ
ÿ
 F1 q 2 p; k cos V1 q 2 p; k ÿ
ÿ
1 F2 q 2 p; k cos V2 q 2 p; k Šg …2:10†

q

where

where

p ~ 1 2 cos q: vq ˆ 2JS

89

…2:7†

Now to evaluate the memory function we expand the spin operators Sxq and Syq in (1.11) to second order in

ÿ
F1 q; k ˆ c1 1 c2 sin2 …k=2† 1 c3 sin4 …k=2†; ÿ
F2 q; k ˆ d1 1 d2 cos k 1 d3 cos 2k;

…2:11†

90

A.S.T. Pires / Solid State Communications 109 (1999) 87±91 2ÿ


ÿ
c1 ˆ cos q=2 c3 ; c2 ˆ cos q=2 c3 ; h ÿ
ÿ
i c3 ˆ 4 1 2 2cos q=2 1 cos2 q=2 ;

Performing the Laplace transform we obtain  21=2 X 2 ~ …z† ˆ 22JTsin …a=2†{c1 z2 1 x2 q

2 c2

ÿ
ÿ
ÿ
d1 ˆ 3=2 1 cos q=2 2 …3=2†cos2 q=2 1 cos4 q=2 ;

d2 ˆ 2sin

  q q 1 2 cos ; 2 2

  1 q q 1 1 2cos 1 cos2 ; d3 ˆ 2 2 2

   3=2 2 c3 3zx2 2 2 z2 1 x2 12z3 =2x4 

1 d3

 2 21=2

q 2 2 2 ‰d1 1 d2 z 1 y 2 z =y2

q 4 2 ~ z2 1 y2 2 z =y4 Š} 2 2JTcos …a=2†

 fthe same terms with the replacement q ! q~g …2:15†

p  ÿ
ÿ
ˆ 2 2J S sin q=4 1 k=2 ^ sin q=4 2 k=2 jg p  ÿ
ÿ
ˆ 2 2J S sin q=4 1 k=2 ^ sin q=4 2 k=2 jg: …2:13† As we can see, the memory function has two contributions one ferromagnetic and one antiferromagnetic. Substituting the sum in k by an integral we ®nd for the classical memory function the analytical expression

q

2

1 z 1y

…2:12†

ÿ
ÿ
ÿ
V^ q; k ˆ v q=2 1 k ^ v q=2 2 k

X

p  z2 1 x2 1 z =x2

2 ~ …t† ˆ 22JT{sin …a=2†‰c1 J0 … xt† 1 c2 J1 … xt†=xt

ÿ
ÿ
1 3c3 J2 … xt†=… xt†2 1d1 J0 yt 1 d2 J2 yt ÿ
1 d3 J4 yt Š 1 cos2 …a=2† £ ‰the same terms with the replacement q

where z ˆ v 1 i1: In the low temperature limit and for q~ ! 0 (i.e. for q ! p we ®nd for the second moment   c2 k2 1 q~ 2 ; …2:16† kv2q l ˆ 2 cos …a=2† p ~ 2 and c ˆ JS ~ 2. At q ˆ p and for where k ˆ T=2JS small frequencies we can write the imaginary part of the Laplace transform of the memory function as h i X 00 p …v† ˆ 2ck cos2 …a=2† 1 2 2sin2 …a=2† : …2:17† q

P This equation holds for small q~ as well. q 00 …v† vanishes at zero temperature, where the coherence length is in®nite, and the excitations are perfectly well-de®ned at all wavelengths. The real part of the memory function is very small for q~ ! 0 and can therefore be neglected. Acknowledgements This work was partially supported by Conselho Nacional de Desenvolvimento Cienti®co e Tecnologico (Brazil) and FundaCËaÄ de Amparo a Pesquisa do Estado de Minas Gerais (FAPEMIG).

! q ˆ q 2 pŠ}; …2:14† p p ÿ
ÿ
~ ~ where x ˆ 4 2JSsin q=4 and y ˆ 4 2JScos q=4 :

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