m. meister, f.g. mertens and a. sanchez- soliton diffusion on the classical, isotropic heisenberg...

8/3/2019 M. Meister, F.G. Mertens and A. Sanchez- Soliton diffusion on the classical, isotropic Heisenberg chain

1/13

Eur. Phys. J. B 20, 405417 (2001) THE EUROPEANPHYSICAL JOURNAL B

cEDP SciencesSocieta Italiana di FisicaSpringer-Verlag 2001

Soliton diffusion on the classical, isotropic Heisenberg chain

M. Meister1,a, F.G. Mertens1, and A. Sanchez2

1 Physikalisches Institut, Universitat Bayreuth, 95440 Bayreuth, Germany2 Grupo Interdisciplinar de Sistemas Complicados (GISC), Departamento de Matematicas, Universidad Carlos III de Madrid,

Edificio Sabatini, Avenida de la Universidad 30, 28911 Leganes, Madrid, Spain

Received 23 November 2000 and Received in final form 31 January 2001

Abstract. We investigate the diffusive motion of a solitary wave on a classical, isotropic, ferromagneticHeisenberg spin chain with nearest-neighbour exchange interaction. The spins are coupled magnetically

to Gaussian white noise and are subject to Gilbert damping. The noise induces a collective, stochastictime evolution of the solitary wave. Within a continuum version of the model we employ implicit collectivevariables to describe this stochastic behaviour. Thermally excited magnons are disregarded. We derivestochastic equations of motion for the collective variables and solve them numerically, in particular to obtaintheir variances as functions of time. These results are compared to data from spin dynamics simulationsof a discrete chain. For some of the collective variables we find good agreement with respect to the longtime behaviour, whereas for other variables the agreement is only qualitative; reasons for this are given.For shorter times we derive analytical expressions for the variances of the collective variables, which alsoagree well with spin dynamics.

PACS. 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion 75.10.HkClassical spin models 05.45.Yv Solitons

1 Introduction

Throughout physics the importance of coherent excita-tions in nonlinear systems solitons and solitary wavesare the most widely known is well recognized. As it isimpossible to list but the most important publications inthis field, we only give a few sample references. An intro-ductory discussion of solitary excitations and a summa-rizing overview of typical problems can be found in [1]. In[2], an exhaustive review of solitons in magnetic systems isgiven. More recent publications include [38]. A strongeremphasis on particle physics is put in [9].

Amongst all coherent excitations, solitons are very spe-cial. These excitations of localized energy density arise assolutions of integrable wave equations. Integrable physicalsystems usually are strong idealizations, because in mostmore realistic systems perturbations which spoil integra-bility will be present. There are many cases however, inwhich the perturbed, non-integrable system still exhibitscoherent excitations of localized energy density, now nolonger solitons but solitary waves. The soliton of the un-perturbed system can often serve as a good approxima-tion to these solitary waves and as a starting point forthe treatment of the perturbed system; see [1013] for ex-ample. Of course one can also start with a non-integrablesystem that exhibits solitary waves and study the effects

a e-mail: [email protected]

of perturbations there. The most prominent example ofthis type is the 4 system [3,4,1416].

Due to their omnipresence, finite temperatures are aparticularly important type of perturbation. The effects oftemperature are often modelled by (white) noise coupledto the system as a stochastic perturbation. In these sys-tems, the diffusion of the solitary excitations is one of theproblems investigated; publications on this issue include[1719] and references therein.

The dynamics of coherent excitations is often investi-gated in terms of collective variables, a technique whichtakes advantage of the particle-like nature of the excita-tions. If extended modes (linear waves, phonons, magnons)are neglected, as will be done in this work, this techniqueaffords a reduction of the large number of degrees of free-dom to a small number of collective variables describingthe coherent excitation [20,21]. Moreover, investigationswere carried out [22,23] in which the total number of de-grees of freedom was not reduced, but where collectivevariables appeared as a subset of a set of variables whichhad been specially tailored for describing a system in thepresence of a particle-like excitation.

When using collective variables to study coherent ex-citations, there is, in general, the choice [24] between ex-plicit collective variables, which are defined directly, i.e.explicitly, as functions of the degrees of freedom of thesystem, and implicit collective variables, which appear asvariables in shape-functions describing the configuration


2/13

406 The European Physical Journal B

of the system. Explicit variables have the advantage thatthey are clearly defined in terms of the degrees of free-dom of the system; the definitions can be used unambigu-ously in calculations as well as in simulations. However,

the structure of solitons and solitary excitations is usuallyparametrized by implicit collective variables. These vari-ables appear within a function describing the excitation inquestion and have to be isolated from this function in oneway or another before their dynamics can be investigated;it is not always possible to reexpress these variables in anexplicit way.

In this article implicit collective variables are in-troduced to study the stochastic dynamics of solitaryexcitations on the classical Heisenberg spin chain withferromagnetic, isotropic, nearest neighbour exchange in-teraction; in [25] an explicit collective variable, the posi-tion X of the excitation, was used towards the same end.We will employ both a discrete version of the model and a

continuum one and will alternate freely between these twoversions throughout the paper. Spin dynamics simulationswill be carried out for the discrete model, while analyti-cal calculations will be done in the continuum chain. Werestrict our investigations to solitary waves of a width con-siderably larger than the lattice constant (set equal to 1)of the discrete model, so a continuum approximation foranalytical calculations of solitary wave dynamics is justi-fied. As we neglect magnons altogether, we need not worryabout differences in magnon dispersion relations betweenthe discrete and the continuum case. Our aim is to derivestochastic equations of motion for the collective variablesdescribing a solitary excitation, to use these equations for

predicting the variances of the collective variables and tocheck the predictions by spin dynamics simulations.

The Hamiltonian of the unperturbed model in the dis-crete case is

H = JN1n=1

Sn Sn+1 (J > 0), (1)

which in the continuum version becomes

H =J

2

rS rSdr, (2)

where r denotes the coordinate along the chain. We areusing dimensionless quantities: J gives the exchange cou-pling in units of a positive scale J0 and S measures thespinlength in units of a scale S0. As a result, J0S20 is theunit of energy and /(J0S

20 ) the unit of time. In (2) a

constant ground state energy has been subtracted, so thisexpression gives the excitation energy above the groundstate. Furthermore, we have moved from a discrete chainof N spins to a continuum one of infinite length. We as-sume that the spin field in the continuum case tends to-wards the same ground state configuration (there is aninfinite degeneracy for the case of an isotropic chain) aswe approach infinity, on either side of the excitation. Forthe discrete chain we use free boundary conditions.

The perturbed spin dynamics of the chain is governedby the Landau-Lifshitz equation with Gilbert damping of

strength and magnetically coupled white noise b:

d

dtS+ S

d

dtS= SB S b. (3)

This equation has to be considered in Stratonovich in-terpretation [26,27]; the question of the correct interpre-tation is important here because the magnetic couplingleads to multiplicative couplings between the componentsof the spins and of the noise. For the continuum model,S denotes the spin field; in the discrete case, (3) gives theequation of motion for each individual spin of the chain.The quantity B is the effective magnetic field, arising fromthe interaction amongst the spins. It is given by

Bjn =H

Sjnand B = J2rS 2JS, (4)

respectively; Bjn denotes the component j at the site n.The noise b is Gaussian and white, with properties

bi(r, t) = 0

bi(r1, t1)bj(r2, t2) =

2ij(r1 r2)(t1 t2). (5)

Analogous relations obtain in the discrete case. The per-turbations are regarded as the effects of coupling thespins to a heat bath of temperature T. This requires2 = 2kBT, kB being Boltzmanns constant. A moredetailed discussion of these perturbations can be foundin [25,26]. For the unperturbed, isotropic, continuumHeisenberg chain (3) reduces to

tS= JS 2rS. (6)

This equation has solitons amongst its solutions [2831],which in the representation

S= S

1 2 cos ,

1 2 sin ,

(7)

for the Cartesian spin components can be written as

(r, t) = 1 A

sech

r X

2,

(r, t) = 0 + 2

A 1

r X

+ arctan

A

2 Atanh

r X

, (8)

where 0 = 0 + JS2

A2t and X = X0 + JS

2

2A 1 t.

A is the amplitude, is the width, and X is the posi-tion of the soliton. The excitation energy of the soliton,as calculated from (2), is 4JS2/.

In Section 2 we neglect damping and derive stochas-tic equations of motion for the collective variables X, Aand . We perturbatively find an analytic expression forthe time dependence of the variances of these variables.Gilbert damping is taken into account in Section 3.Stochastic equations of motion for collective variables de-scribing general coherent excitations in d-dimensional spin


3/13

M. Meister et al.: Soliton diffusion on the classical, isotropic Heisenberg chain 407

systems are found, and then specialized to the solitary ex-citations given by (8). This yields equations of motion forthe collective variables X, 0, A and . These equationsare not solved analytically, but numerical solutions are

presented in Section 4. There we also compare the resultsobtained in this way with spin dynamics simulations andwith the results derived in Section 2. The short time be-haviour of our spin dynamics data turns out to be welldescribed by the analytical results of Section 2, while thenumerical solutions of the equations obtained in Section 3show a different behaviour. For long times, the latter re-sults agree very well with spin dynamics in the case ofthe variables X and 0, but for A and the agreementis not satisfactory. These findings are discussed. Finally,Section 5 gives our conclusions.

2 Stochastic equations of motion withoutdamping

In this section we calculate the variance of the collectivevariables X, A and . The approach chosen here corre-sponds to the way the position X is determined from spindynamics simulations (see the appendices). We do not takeinto account the damping term for reasons that will be-come evident below. This restricts the results of this ap-proach to time scales for which the effects of damping canbe neglected. Thus, equation (3) takes the form

d

dtS= JS 2rS S b. (9)

Here we use only the z-component of this equation and anansatz for the spin field Swhich is given by (8), with timedependent collective variables X,A,and 0. Noting thatSz = S is independent of 0, we find

Sz

XX+

Sz

AA +

Sz

= J

S 2rS

z [S b]

z.

(10)

A Gilbert damping term would produce an additionalterm proportional to 0 on the left, which would subse-quently cause problems, see below. Now we multiply thisequation with S

z

X, S

z

Aand S

z

, respectively, and inte-

grate over the chain, obtaining the system of equations

S2

1615

A2

0 0

0 43

23

A

0 23

A 42

45A2

X

A

=

JS3 3215

A3

2

2A

2

0

0

Sz

X(S b)z dr

Sz

A(S b)

zdr

Sz

(S b)z dr

. (11)

As Sz does not depend on 0, we cannot multiply by thecorresponding derivative. Had we taken into account the

damping term in (3), we would have three linear equations

for four unknowns, namely X, 0, A and .After inversion of system (11) we obtain:

X = 2JS

2A 1 + X ,

A = A,

= . (12)

Here the quantities are effective stochastic forces, which,on the level of collective variables, reflect the effects ofthe noise coupled to the spins. Assuming that the collec-tive variables are stochastically independent of the noiseb (we postpone a detailed discussion of this assumptionto Sect. 3), the can be found to satisfy U = 0 forU = X, A, and

X (t1)X(t2) = 2(t1 t2) 514

3 A

A

, (13)

A(t1)A(t2) = 2(t1 t2)

1

(42 15)2

A

96

54

312

72 180 +

A

7

288

54 + 962 + 620

, (14)

(t1)(t2) = 2(t1 t2)

8

15

45

42

15

2

A

1 +

42

21

A

63

42

62

3

, (15)

A(t1)(t2) = 2(t1 t2)

9

(42 15)2

80

352

212 +

A

21

1522

2050

3

, (16)

X(t1)A(t2) = 0, X(t1)(t2) = 0. (17)

In Section 3 we will again calculate correlations of stochas-tic forces, and there we will give a more detailed deriva-tion, which is parallel to the steps not shown here. Intro-ducing quantities UV by

U(t1)V(t2) = 2(t1 t2)UV(A, ) (18)

and mutually independent Wiener processes WX , W1 andW2, we can express the equations of motion (12) as follows:

dX =2JS

2

A 1dt + (A, )dWX ,

dA = (A, )dW1,

d = 1(A, )dW1 + 2(A, )dW2, (19)


4/13


where 2 = XX , 2 = AA, 1 = A and 21 + 22 =

. The set of equations (19) is problematic, however:The stochastic quantities A and may assume unphysicalvalues, i.e. the amplitude A may grow beyond its maxi-

mum 2 or assume negative values, and likewise the widthcan become negative. In any event, the probability for thisto occur is very small, if we consider only weak noise (lowtemperatures) and short times. Here it is important torecall that the set (19) has been derived from an ansatzdescribing a solitary wave, where the effects of the pertur-bations have been taken into account only by allowing fora time dependence of the collective variables; all extendedexcitations (magnons) have been neglected. Furthermore,we have neglected some correlations in the derivation ofthe set (19) by assuming that the noise is stochasticallyindependent of the collective variables. These assumptionsappear justified as long as the noise is weak.

Since the derivation of (19) included approximations

anyway (weak noise and short times), we do not solve theequations (19) exactly. Instead, we resort to a perturbativetechnique. A small noise expansion [27] X = X0 + X1 +2X2 + . . . , and similarly for A and , yields for the firstnontrivial order (1):

dX1 = 2JS

20

2

A0 11dt

2JS

A200

2

A0 1

12

A1dt + (A0, 0)dWX ,

dA1 = (A0, 0)dW1,

d1 = 1(A0, 0)dW1 + 2(A0, 0)dW2. (20)

From this we find

A1(t) = (A0, 0) [W1(t) W1(0)] , (21)

1(t) = 1(A0, 0) [W1(t) W1(0)]

+ 2(A0, 0) [W2(t) W2(0)] , (22)

and

X1(t) = 2JS

20

2

A0 1

t0

{1(A0, 0) [W1(t)

W1(0)] + 2(A0, 0) [W2(t) W2(0)]}dt

2J S

A200

2

A0 1

12

t0

(A0, 0) [W1(t) W1(0)] dt

+ (A0, 0) [WX (t) WX(0)] , (23)

where A0 and 0 denote the constant initial values. Tothis order we have Var[X(t)] = 2X1(t)2 and thus wefinally obtain, with UV evaluated at U0, V0:

Var[X(t)] = 2XX t + 4J2S2

2

320

1

20

2

A0 1

+

AA

A40

2A0

1

+

2AA200

t

3

. (24)

The term linear in t derives from the direct effect of thenoise on the position X, expressed by the stochastic forceX . The cubic term is the result of an indirect effect: Thenoise also changes A and stochastically, but these two

variables determine the velocity of the excitation. The po-sition, in turn, is, up to a constant, given by the time in-tegral of the velocity; therefore differences in the velocity(e.g. those induced by the noise) will lead to differencesin the position that grow with time. Thus the time de-pendence of the variance of the position can be expectedto include higher than linear powers of time. Within thefirst order approximation applied above, the velocity isdescribed by a Wiener process, so the position contains atime integral of a Wiener process. The cubic term in (24)straightforwardly derives from this integral. Similar su-perdiffusive behaviour was found in [32]. There the effectsof noise on the position of a vortex in a two-dimensionaleasy-plane Heisenberg model were studied; the indirect

effect of the noise on the position originated from an ex-ternal force acting on the vortex. The force was such thatit enlarged differences between trajectories. Consequently,some of the variances of interest in that case were foundto grow faster than linear in time.

For A and we obtain

Var [A(t)] = 2AA(A0, 0)t,

Var [(t)] = 2(A0, 0)t. (25)

As we will see in Section 4, the predictions for A and agree with the spin dynamics results only for a short time(t 100), whereas the prediction for the position is goodup to several hundreds of time units. Results describingthe long time behaviour will be obtained in the followingsection.

3 Stochastic equations of motionwith damping

In this section we derive equations of motion that ac-count for the Gilbert damping term in (3) and hence canbe expected to apply to longer times than the results ofSection 2. As a first step, we derive general equations of

motion for collective variables describing coherent excita-tions in a d-dimensional spin system, and afterwards wespecialize to the Heisenberg chain. We set out from theansatz

S(r, t) = Sc (r , . . . , U i(t), . . . ) , (26)

where r denotes a point in d-dimensional space, and Uiis one of finitely, say N, many collective coordinates. Thefunction Sc describes a particular type of coherent excita-tion in terms of r and the U1, . . . , U N. From (26) we findimmediately

tS=

Ni=1

S

cUi

Ui, (27)


5/13


and using this in the perturbed Landau-Lifshitz equa-tion (3), we arrive at

N

i=1

ScUi

+ Sc Sc

Ui Ui = Sc (B + b) . (28)

The next step is to isolate from these equations the timederivatives of the Ui. In analogy to [33], along the linesof [26,32], we first take the vector product of (28) withScUk

, then form the inner product of the result with Scand integrate over all space, obtaining

Ni=1

Sc

ScUk

ScUi

+ ScUk

Sc

ScUi

ddrUi

=

Sc

ScUk

(Sc (B + b))ddr, (29)

for each of the collective variables Uk. We introduce avector C := Cin +Cex with

Cink :=

Sc

ScUk

(Sc B)

ddr,

Cexk :=

Sc

ScUk

(Sc b)

ddr, (30)

where Cin is the intrinsic part, containing the effectivemagnetic field B that arises from the interaction amongstthe spins, and Cex is the external part, proportional to the

external noise b. As ScScUk

ScUi =

ScUk

Sc ScUi

and ScUk

Sc ScUi

= Sc

ScUk

ScUi

, because

Sc ScUk

= 12

Uk

(Sc Sc) vanishes due to the invariance

of the spinlength S, (29) can be written as

Ni=1

M2,ki + S

2M1,ki

Ui = Ck, (31)

where

M1,ki :=

ScUk

ScUi

ddr, and

M2,ki := ScUk

Sc ScUi

ddr. (32)Introducing a matrix L() by L()ki := M2,ki+S

2M1,kiwe arrive at

Ui =Nk=1

L()1

ik

Ck. (33)

Defining Fi :=

L()1Cini

and fi :=

L()1Cexi

we

finally obtain from (33):

Ui = Fi + fi. (34)

On the level of collective coordinates the interaction ofthe constituents of the system is thus reflected by in-trinsic forces Fi, whereas the quantities fi are effective

stochastic forces, analogous to the s of Section 2. Thecorrelations of the fi are given by (the -dependence ofthe matrix L()1 is not written explicitly):

fi(t1)fj(t2) =

Nm=1

Nk=1

L1ik (t1)Cexk (t1)L1jm(t2)Cexm (t2).

(35)

Note that, by making the ansatz (26), we have already ne-glected magnons; however, the function Sc depends on thecollective variables, and these are stochastic quantities.Therefore the expectation values in (35) cannot be eval-uated straightforwardly, because they involve products offunctions of the collective variables and noise terms b. As(3) is considered a Stratonovich equation, there could becorrelations between the noise and the collective variableswhich may prevent an exact evaluation of (35).

On the other hand we can derive the Ito-version of(33). In this formulation, the noise terms are stochasti-cally independent of the collective variables, but there willappear, in general, correction terms to Fi. These correc-tion terms are of second order in the noise ( i.e. 2),whereas the fi are of first order in . For small valuesof (weak noise) we can therefore neglect the correctionterms in Fi and treat the collective variables as stochasti-cally independent of the noise. This allows us to split theexpectation values of the products in (35) into a productof expectation values:

fi(t1)fj(t2) =

Nk=1

Nm=1

L

1

ik (t1)L1

jm(t2)S4

3h=1

3g=1

ShcUk

(t1, r1)SgcUm

(t2, r2)

bh(r1, t1)bg(r2, t2)

ddr1d

dr2. (36)

Using the properties ofb, equation (5), we find

fi(t1)fj(t2) = 2S4

Nk=1

Nm=1

L1ik (t1)L

1jm(t1)

ScUk

Sc

Um(t1, r)d

dr(t1 t2)= 2S4

L1M1

L1

Tij

(t1 t2)

=: Dij(t1 t2), (37)

where the superscript T denotes the transpose. In a similarway it can be seen that fi = 0. Thus, within the aboveapproximations, the fi represent additive and white (i.e.the correlation time is zero) noise on the level of collec-tive variables. Without the approximations, the expres-sions (35) could contain non-vanishing correlation timescales (i.e. the noise may be coloured). However, as theresult (37) shows, these time scales become negligible forweak noise. Analogous statements apply to the results forthe correlations of the s in Section 2.


6/13


In order to evaluate the components of the matricesM and thence also of L and its inverse, we use the repre-sentation (7) and obtain after some algebra

ScUk

ScUi

= S2

112c

cUk

cUi

+

12c

cUk

cUi

,

(38)

and

ScUk

Sc

ScUi

= S3

cUk

cUi

cUi

cUk

. (39)

Now we specialize to the isotropic, one-dimensionalcase and consider the soliton solutions (8) given above.For this type of excitation we use four collective variables,namely X, 0, A and . For an isotropic chain with near-est neighbour exchange, B is given by (4), so

Ck = J

r

Sc

rSc

Sc

ScUk

dr

S2b

ScUk

dr. (40)

Using the soliton solutions (8), the intrinsic parts can beevaluated further, eventually leading to the equations forthe collective variables:

X = 2JS

2A 1

9

9+

2 +6

2S2

[9(9+2S2 (22 +12))]+4S42 (2 +12)+f1 (41)

0 =2JS

A29

9 + 2S22+ f2 (42)

A =36JS2

2S2 (2 + 12) + 9

A 2

2+ f3 (43)

= 12JS2

(2 6)2S2 +9

A3

22S2 +9

(22S2 + 9)((2 + 12) 2S2 + 9) A

+ f4.

(44)

For the correlations (37) of the stochastic forces it turnsout that

D =

DXX DX0 0 0

DX0 D00 0 0

0 0 DAA DA0 0 DA D

. (45)

Here the components depend on and A, which in turndepend on time:

DXX = 2

4A

92 (3A)+54A+2S22

2 +12

(3A)

(9+2S22)[9+2S2 (2 +12)],

DX0 = 2 3

4

2

A 1

2

A (9 + 2S22),

D00 = 2 3

2

2

A2 (9 + 2S22),

DAA = 182 A (2 A)

[9 + 2S2 (2 + 12)],

DA = 92 A 2

9 + 2S2 (2 + 12),

D = 32

A

9 (3 A) + 2S2

2 (3 A) + 6A

(9 + 2S22) [9 + 2S2 (2 + 12)]

(46)

When solving the stochastic collective variable equationsnumerically, we will evaluate the components of the abovematrix subject to the approximation

D(A, ) = D(A, ). (47)

Expanding D in a Taylor-series around the expectationsof its arguments, we find:

D(A, ) = D(A, ) +

AD(A, )(A A)

+

D(A, )( ) + . . . (48)

Taking into account that A A = 0 and the samefor , we see that the error in the equations of motionis of at least second order in the noise; thus the approxi-

mation (47) is consistent with the above approximations.The time dependence of A and is assumed to be de-termined by the effects of damping alone. From (4144) itis clear that this assumption involves approximations like(47) and hence, like (47), is justified for weak noise.

4 Numerical results

We have solved numerically the set of equations (4144)and have obtained the results shown below by averagingover 10 000 realizations. For comparison the spin dynam-ics of a chain of 1000 spins has been simulated. The spinshave been subject to equation (3), which has been solvedusing the Heun algorithm [34], which provides an ade-quate treatment of the multiplicative couplings between


7/13


0 500 1000 1500 2000 2500

t

0

0.005

0.01

0.015

relativedeviation

A=0.5

A=1.0

A=1.5

0 1000 2000 3000 4000 5000

t

0

0.002

0.004

0.006

0.008

0.01

relativedeviation

A=0.5

A=1.0

A=1.5

Fig. 1. Deviations of the simulated spin configuration from theansatz with fitted collective variables. The damping parameterwas = 0.01. The initial values of A are given in the insets.The initial values of were Top: = 10, Bottom: = 20.Some of the graphs had to be cut before t = 5000, because thesolitary wave reached the end of the chain before that time.

the components of the spins and of the noise, arising dueto the vector product S b in (3). The spin dynamicssimulations have been extended up to 5000 time units andover 1000 realizations; we have made the choice J = 1 andS = 1. The collective variables X, 0, A, have been de-termined from spin dynamics by a fitting procedure that

is described in detail in Appendix A.First we check numerically how well the spin config-uration is approximated by the ansatz. We take the val-ues for the collective variables X, 0, A and determinedfrom the fit, plug them into the exact soliton solutions(8) and calculate the total squared deviation from thespin configuration obtained in a simulation with damp-ing, but without noise. This quantity is then divided by2A, which gives the z-component of the total magneti-zation of an unperturbed soliton and which can thus beconsidered a measure for the size of the excitation. Thequantity obtained in this fashion is shown in Figure 1 as afunction of time. The results of Figure 1 have led us to re-strict ourselves to values A 1; our approach starts froman ansatz and we can only expect good results as long asthe ansatz describes the spin configuration well. Figure 1

shows that this is the case for A = 1 and A = 1.5, whereasthe relative deviations are much larger for A = 0.5. Fig-ure 2 compares results for the variance of the position Xof the solitary wave, obtained from spin dynamics simu-

lations, numerical solution of the collective variable equa-tions (4144) and analytically (24). As can be seen fromthe left column, the analytical result (24) describes wellthe behaviour for short and intermediate times, whereasthe numerical solution of (4144) clearly deviates fromthe spin dynamics result. For long times, as shown in theright column, the situation is vice versa. This is due tothe fact that the analytical prediction is in harmonywith the way the position is measured in spin dynamics,whereas the collective variable equations obtained in theprevious section are not (see Appendix B). For long timesthis is not relevant, as was surmised in [25] for the case ofan explicit collective variable. The same argument giventhere applies to the implicit collective variables here. The

noise produces fluctuations, amongst which are distortionsof the soliton shape that cause a shift of the soliton po-sition (translation modes). For a fixed distortion, differ-ent methods of determining the position from a simulatedspin configuration can produce different results. However,these differences are limited in magnitude, as they are dueto localized fluctuations. Therefore they become negligi-ble when the standard deviation of the soliton positionis comparable to the width of the excitation. Comparingthe long time result for A = 1.5, = 20 (Fig. 2, top)with the long time result for A = 1.5, = 40 (Fig. 2,bottom) we see that for the case = 40 we have, att 4500, a standard deviation of about 5, whereas for the

case = 20 it is about 13. Consistently, the relative devi-ations between spin dynamics and the solutions of (4144)are much smaller in the case = 20. Naturally, the first-order result of Section 2 does not hold for arbitrarily longtimes, as has been anticipated. The results of Figure 2 havebeen obtained at kBT = 0.001. We get the same qualita-tive picture, the same agreement, for kBT = 0.002 andkBT = 0.003, but do not show the figures to save space.At kBT = 0.01 we have observed that the code we areusing runs into difficulties determining the parameters ofthe solitary wave, as thermal fluctuations distort the exci-tation strongly. Therefore we have restricted our investi-gations to the values of kBT given above. In this context

the reader should note that for kBT 1 the mean ther-mal energy stored in one link of the chain in equilibriumis approximately kBT. If we consider 2 as the number ofspins within the solitary wave, i.e. in that region of thechain where the structure of the excitation clearly domi-nates, then for = 20 we have that on average a thermalenergy of 0.04 is stored within the soliton at kBT = 0.001.This is 20% of the soliton energy 0.2, obtained as given af-ter (8); at T = 0.003 the thermal energy even amounts to60% of the soliton energy. Hence the temperatures consid-ered here are not small. Indeed, at T = 0.01 the thermalenergy is double the soliton energy and the distortionsare so strong that the parameters of the solitary wave canno longer be determined. For the variable 0 we have ob-tained good agreement as well, see Figure 3. The peaks oc-curing in the graph of the variance determined from spin


8/13


0 250 500

t

0

0.1

0.2

0.3

Var(X)

0 1000 2000 3000 4000 5000

t

0

100

200

Var(X)

spin dynamics

numerical

analytical

0 250 500

t

0

0.1

0.2

0.3

0.4

0.5

Var(X)

0 1000 2000 3000 4000 5000

t

0

100

200

300

Var(X)

0 250 500

t

0

0.05

0.1

0.15

0.2

0.25

ar

0 1000 2000 3000 4000 5000

t

0

10

20

30

Var(X)

Fig. 2. The variance of the position as a function of time. The spin dynamics result (solid line) is an average over 1000realizations, the result from the collective variable equations of Section 3 (dashed line) is an average over 10 000 realizations.The dot-dashed line represents the analytical result (24). The plots on the left are zooms of the early stages of the plots onthe right. For all cases the damping was = 0.01 and the temperature kBT = 0.001. Top: Initially A = 1.5, = 20. Middle:Initially A = 1, = 20. Bottom: Initially A = 1.5, = 40.

dynamics are artifacts of the measuring procedure. Thelatter involves FORTRANs ATAN2 function, which hasa limited range, (, ). The peaks occur when the meanvalue of 0 approaches a boundary of this range, so thatfor some realizations the value of 0 has already been

shifted back to the opposite end of the range, whereas thisback shifting has not yet occurred for other realizations.

Thus near the boundaries of the range, large differences ofthe order 2 can occur between the values of 0 from dif-ferent realizations. The smoothing of the jumps from to in the spin dynamics graph for 0 has the same rea-son. On the other hand, when solving the collective vari-

able equations, 0 is treated as a variable of unboundedrange, the average over the realizations is calculated


9/13


0 1000 2000 3000 4000 5000

t

4

2

0

2

4

0

0 1000 2000 3000 4000 5000

t

0

0.1

0.2

0.3

Var(

)

0

Fig. 3. Results for the collective variable 0. Initially: A =1.5, = 20. The temperature was kBT = 0.001, the damp-ing = 0.01. Shown are the mean value of 0 (top) and thevariance of 0 (bottom). The solid lines give spin dynamicsdata (1000 realizations), the dashed line the results from nu-merically solving the collective variable equations of Section 3(10 000 realizations). The peaks in the variance have been cutoff, because they are artifacts of the measuring procedure, asare the smoothed jumps from to in the spin dynamicsresults for 0. See main text for further explanation.

and then this single value is reshuffled into the (, )range for better comparison with spin dynamics. There-fore we have sharp jumps for the dashed line.

In Figure 4 we show the mean values ofA and . Thetiny deviations of the spin dynamics result for A from thesolution of the collective variable equations are caused bythe measuring procedure: the amplitude of these devia-tions increases, if the threshold used in the fit (see Ap-pendix A) is increased. Apart from this, both graphs agreevery well. For the spin dynamics result is slightly abovethe result of the collective variable equations. This dif-ference increases with increasing temperature. The prob-able reason for this systematic deviation is a dressingof the soliton with fluctuations. Especially spike-like dis-tortions appearing on the sides of the solitary wave canbe suspected to increase the width determined from spin

0 1000 2000 3000 4000 5000

t

1.2

1.3

1.4

1.5

A

0 1000 2000 3000 4000 5000

t

20

21

22

23

24

25

Fig. 4. Mean values for A (top) and (bottom) at kBT =0.001. The damping was = 0.01. Shown are results from spindynamics (solid line, 1000 realizations) and from the solutionof the collective variable equations of Section 3 (dashed line,10 000 realizations).

dynamics. The fact that there is agreement here roughlyduring the first 1000 time units is quite consistent: At first,no fluctuations at all are present on the chain; after thattime, as we have observed from spin dynamics, the systemhas approximately saturated energetically. But this also

means that the fluctuations have fully developed then. Theagreement for the variances ofA and is less satisfactorythan for X and 0, as can be seen from Figures 5 and 6.This is due to the fact that the standard deviation ofA and is, even after 5000 time units, still small compared to theabsolute values of A and (< 10%). Therefore it can beconcluded that for these two quantities 5000 time units isstill a short time. Thus deviations between spin dynamicsand the numerical solution of the collective variable equa-tions (4144) are due to a disagreement between the wayA and are determined from spin dynamics and the waythey enter the calculations. As the analytical calculationsof Section 2 correspond to the fitting procedure employedin simulations (see Appendix B), the good agreement be-tween the results of Section 2 and spin dynamics for shorttimes supports this view. The mechanisms responsible for


10/13


0 50 100

t

0

0.005

0.01

0.015

0.02

0.025

0.03

Var(

)

0 1000 2000 3000 4000 5000

t

0

1

2

3

4

Var(

)

Fig. 5. Variance of at kBT = 0.001 and = 0.01. Initially

the parameters wereA

= 1.5

, = 20. The solid line is theresult of spin dynamics (1000 realizations), the dashed line

has been obtained by solving the collective variable equationsnumerically (10 000 realizations), and the dot-dashed line givesthe analytical result (24). The top plot is a zoom of the earlystages of the bottom one.

the discrepancies in the mean values of A and also con-tribute to the deviations showing up in the variance.

5 Conclusion

On a classical, isotropic, ferromagnetic Heisenberg spinchain subject to Gilbert damping and Gaussian whitenoise coupled magnetically to the spins, solitary wavesshow diffusive motion, which has been investigated in thisarticle by both analytical and numerical methods. Re-stricting ourselves to times short enough for damping tobe neglected, we have derived analytical results for thetime dependence of the variances of X, A and . Further-more, we have obtained equations of motion for arbitrarycollective variables, parametrizing the spin configurationof a general coherent excitation in a d-dimensional spinsystem, subject to Gilbert damping and magnetically cou-pled Gaussian white noise. The equations of motion arefirst-order in time and include a force composed of twoparts: an intrinsic part, reflecting the interaction amongst

0 50 100

t

0

0.0001

0.0002

Var(A)

0 1000 2000 3000 4000 5000

t

0

0.005

0.01

0.015

Var(A)

Fig. 6. As Figure 5, but for the collective variable A.

the spins on the collective level, and an external, additivewhite noise part. The latter derives from the noise coupledmagnetically to the spins. The equations of motion havebeen specialized to the case of the Heisenberg chain, withcollective variables X, 0, A and , and have been solvednumerically. Both the analytical results and the numericalsolutions of the equations of motion have been comparedwith data from spin dynamics simulations. For all fourof the above collective variables we have observed thatfor short times there is a disagreement between data ob-tained from spin dynamics simulations and the numerical

solutions of the equations of motion for the four collectivevariables given above. This is due to a lack of correspon-dence between the way these variables are measured inspin dynamics and the way they enter the calculations.As the method used to calculate the analytical results forthe variances of X, A and is more suited to the pro-cedures employed in spin dynamics, we obtain agreementbetween spin dynamics and the analytical results for shorttimes. There is no corresponding analytical result for 0,because, for reasons given in Section 2, this variable hasnot been involved in this approach. For long times, theresults obtained by numerically solving the equations ofmotion in Section 3 match the spin dynamics data for Xand 0 very well. Within the time span of 5000 time unitsconsidered in this work, there is a systematic deviation be-tween spin dynamics and numerical results from (4144)


11/13


for the variances of A and . This has been discussed indetail in Section 4, and hence we do not repeat this dis-cussion here.

For the position X, which can be considered the most

important of the four collective variables, because it pos-sibly is the easiest to be accessed experimentally, we havefound the following: There is a short time regime, in whichthe stochastic behaviour of the position is dominated bythe direct effect of the noise on the position. This givesa linear dependence of the variance on time. The damp-ing term, which is always taken to be weak, can safelybe neglected. For intermediate times it is still possible toneglect a weak damping term, however, the stochastic fluc-tuations ofA and lead to stochastic fluctuations of thevelocity (which is a function of A and ) of the solitarywave. This in turn causes additional stochastic changesof the position. A term of cubic time dependence appearsin the variance. In the long time regime damping can no

longer be neglected. Together with the effects mentionedfor short and intermediate times it affects the time de-pendence of the variance of X. The way the position ismeasured in spin dynamics is no longer important.

Fruitful discussions with A.R. Bishop (Los Alamos), Yu.Gaididei (Kiev), A. Kovalev (Kharkov) and G. Lythe (Madrid)are gratefully acknowledged. Work at Leganes was supportedby DGESIC (Spain) through grant PB96-0119.

Appendix A: Determining the collective

variables from spin dynamics simulationsHere we will explain in detail how the collective variablesX, 0, A and are determined from spin dynamics sim-ulations. At first we discuss, for the sake of clarity, thehypothetical situation that there are no magnons in thesystem. Afterwards we turn to the real case.

If there are no magnons in the system, then in sim-ulations we (hypothetically!) only have a solitary wavedescribed by (8), where the collective variables X, 0, Aand depend stochastically on time. In particular, thez-component of the spin field depends on the collectivevariables as Sz (r X, A, ), with X, A, unknown. Wedefine

FX,A,

Y; A,

:=Sz (r X,A,)

Szcr

r Y, A,

dr. (A.1)

Here we use the index c to distinguish the configurationof the system, Sz(r X,A,), from the analytical solu-tion (8), Szc (r Y, A, ). In the simulations S

z will bereplaced with the simulated spin configuration in (A.1);hence the distinction in notation. As long as we work un-der the hypothesis that there are no magnons, Sz and Szcare the same functions, of course. Due to the symmetry of

the soliton (8), FX,A,(X, A, ) = 0. A partial integrationshows that

dFX,A,dY

X, A,

> 0. Both relations obtain

independently of the values chosen for A and . It seemsthat a similar approach was followed in [3]. In our sim-ulations we evaluate (a discretized version of) (A.1) for

several trial values of Y, and fixed A and . The trial val-

ues for Y are distributed symmetrically around the valueof the position determined the previous time (for each re-alization and starting from a common, known initial po-sition) in an interval of a total length equal to 40; thedistance between two consecutive values is 0.25 (the lat-tice constant is unity). The values of FX,A, that we haveobtained in this way are then inspected for a change ofsign from negative to positive as Y is increased. The loca-tion of the zero on the Y-axis is estimated from a linearinterpolation and considered as the position of the solitarywave.

In reality magnons are present in our simulations andunfortunately it turns out that magnons lead to a depen-

dence of the position, determined as described above, andits variance, on the value of . The position is still in-dependent of A, because this quantity only appears as apositive, multiplicative factor of FX,A, and thus changesneither the position of the zeros of this function on theY-axis nor the sign of the slope of the function at sucha zero. Because of the dependence on , we modify thealgorithm as follows:Choosing a fixed value for A, e.g. A = 1, we calculate avalue Xi for the position as described above, for each el-ement of a set of 21 trial values i, which are distributedequidistantly in the range from 0.80 to 1.80, where 0is the initial width. As the width increases due to thedamping, the range of trial values for has been chosenasymmetrically. Next, we calculate a value Ai of the am-plitude for each of these pairs (Xi, i), yielding a collec-tion of triplets (Xi, Ai, i). The values Ai are determinedaccording to the formula

Ai =

1 S

z

S

sech

rXii

2dr

sechrXii

4dr

=3

4i

1

Sz

S

sech

r Xi

i

2dr, (A.2)

which can be derived in the following way: The quantity[Sz Szc (r Xi, A , i)]

2 dr (A.3)

obviously gives the squared deviation of the function Szcfrom an arbitrary spin configuration Sz. Considering Xiand i as fixed, differentiating (A.3) with respect to A,requiring that the result vanishes and solving for A yields(A.2). So the Ai represent the best values, as determinedby a least square fit, for the amplitude, for given valuesofXi and i. In simulations, of course, the simulated spinconfiguration will be used for Sz in (A.2).

As is obvious from the above, the quantities Xi and Aidepend on i. We therefore define, for a fixed simulated


12/13


spin configuration,

() :=

[Sz Szc (r X(), A(), )]

2dr, (A.4)

the squared deviation of the simulated spin configurationSz from the analytic solution Szc , in which the parametersX and A are considered functions of . The value of for which this function has a minimum should be consid-ered the correct value of the width. In order to find the(approximate) location of this minimum on the -axis, weevaluate from the simulated configuration, for each triplet(Xi, Ai, i), the quantity i as a discretized version of(i), where X(i) = Xi, A(i) = Ai. In calculating thei we only take into account sites where S

z deviates morethan a certain threshold from its groundstate value; oth-erwise the result would be dominated by thermal fluctua-tions in regions of the chain far from the solitary wave andtherefore would no longer provide information about theparameters of the excitation. The threshold values havebeen 0.05, 0.1 and 0.2 for kBT equal to 0.001, 0.002 and0.003, respectively. We choose the minimum value of thei, and denote it iM . Next, we interpolate the three val-ues iM , iM1 and iM+1 with a parabola, which, like, has to be considered a function of . The position ofthe minimum of the parabola on the -axis is accepted asthe definitive value of the width. With this value of thewidth we calculate again a value for the position and thenfor the amplitude, precisely as detailed above, and acceptthis triplet of numbers as values for position, amplitudeand width of the solitary excitation in the simulated spinconfiguration.

What remains is to determine the angle 0. From (8)we see that the angle is composed of the global part,0, and a part that depends on r and X. The latter willbe denoted X in the following. In simulations the to-tal angle can be measured easily for each of the spinsof the chain, as = ATAN2 (Sy, Sx), ATAN2 denotingthe FORTRAN function. Far from the soliton, Sy and Sx

are practically zero. This would cause problems with theATAN2 function. For this reason we restrict the calcula-tion of the values to the same spins as the calculation ofthe i. We have determined X, A and , and hence cancalculate X and subtract it from . So we obtain a pre-liminary value for 0. As the range of the ATAN2 function

is bounded to (, ), whereas X , as given by (8), is un-bounded, we carry out the subtraction in this way: We de-fine wx := cos(X ) = cos() cos(X)+sin()sin(X)and wy := sin()cos(X) cos()sin(X ) to obtain asa preliminary value 0 = ATAN2(wy,wx). Due to localthermal fluctuations, this preliminary value will not be thesame for all spins. Therefore the final value of the globalangle 0 is determined by averaging over all preliminaryvalues.

Appendix B: Correspondence of calculationsand simulations

In this section we explain why the calculations of Section 2correspond to the way the collective variables have been

determined in simulations of the spin dynamics. Basically,the latter is a nonlinear least-square fit. We define, forpurposes of this appendix, the function

F(U, U) :=1

2

Sz

(U) Szc (U)

2dr. (B.1)

Here Sz represents (a continuum version of) the simulatedspin configuration, depending on a set of time dependentcollective variables U. The function Szc denotes the shape-function which is to be fitted to the simulated spin con-figuration by varying the set of parameters U. Note thatthere is a one-to-one correspondence between U and U,each element of the latter being just a trial value for thecorresponding element of the former. The necessary con-dition for the least-square fit of a given simulated spinconfiguration (i.e. with fixed U) is a vanishing of the gra-

dient of F with respect to U:

F

Ui=

Sz

SzcUi

dr

Szc

SzcUi

dr = 0, (B.2)

for each Ui. As the fit parameters U appear nonlinearly inthe set of equations (B.2), solving these equations is notstraightforward. Therefore we have instead used the pro-cedure detailed in Appendix A, which can, however, stillbe considered a least-square fit. Taking the time derivativeof the middle part of (B.2), we find

jSz

Uj

SzcUi

drUj ; (B.3)

it should be noted that the Ui are independent of time.Only the values obtained as solutions of (B.2) change withtime as the spin configuration changes. Moreover, if weneglect all extended excitations and consider only the timedependence of Sz caused by the time dependence of thecollective variables U, we can identify Sz and Szc and also

Sz/Ui with Szc /Ui, because the dependence ofSz and

Szc on U and U, respectively, is identical. We can find outabout the Uj in (B.3) as follows: We insert an ansatz like(26) in equation (9) and consider the z-component. Thisproduces

j

Sz

UjUj (B.4)

on the LHS, just as in (10) for the case of the soliton (8).Proceeding analogously to Section 2 and multiplying (B.4)with Sz/Ui, for each of the Ui, and integrating overr gives a system of equations analogous to (11); again,the analogy is an identity for the soliton (8). Using theabove identifications, we see that the LHS of this systemis precisely the same as (B.3). Therefore the variables Uiand in particular their time derivatives, appearing in thissystem of equations, are precisely those involved in theleast-square fit. In this sense, the calculations of Section 2correspond to the fit performed in spin dynamics, whilethe calculations of Section 3 do not.


13/13


References

1. A.R. Bishop, J.A. Krumhansl, S.E. Trullinger, Physica D1, 1 (1980).

2. A.M. Kosevich, B.A. Ivanov, A.S. Kovalev, Physics Re-ports 194, 117 (1990).

3. J. Dziarmaga, W. Zakrzewski, Phys. Lett. A 242, 227(1998).

4. J. Dziarmaga, W. Zakrzewski, Phys. Lett. A 251, 193(1999).

5. A.M. Kosevich, V.V. Gann, A.I. Zhukov, V.P. Voronov, J.Exp. Theor. Phys. 87, 401 (1998).

6. B.A. Ivanov, H.J. Schnitzer, F.G. Mertens, G.M. Wysin,Phys. Rev. B 58, 8464 (1998).

7. B.A. Ivanov, V.M. Muravev, D.D. Sheka, J. Exp. Theor.Phys. 89, 583 (1999).

8. P.G. Kevrekidis, M.I. Weinstein, Physica D 142, 113(2000).

9. C. Rebbi, G. Soliani, Solitons and Particles (World Scien-tific, Singapore, 1984).

10. D.W. McLaughlin, A.C. Scott, Phys. Rev. A 18, 1652(1978).

11. E. Mann, J. Math. Phys. 38, 3772 (1997).12. E. Mann, J. Phys. A 30, 1227 (1997).13. N.R. Quintero, A. Sanchez, F.G. Mertens, Phys. Rev. E

62, R60 (2000).14. M.J. Rodriguez-Plaza, L. Vazquez, Phys. Rev. B 41, 11437

(1990).15. N.R. Quintero, A. Sanchez, F.G. Mertens, Phys. Rev. Lett.

84, 871 (2000).16. N.R. Quintero, A. Sanchez, F.G. Mertens, Phys. Rev. E

62, 5695 (2000).

17. Y. Wada, Prog. Theor. Phys. Supp. 113, 1 (1993).18. N.R. Quintero, A. Sanchez, F.G. Mertens, Phys. Rev. E

60, 222 (1999).19. N.R. Quintero, A. Sanchez, F.G. Mertens, Eur. Phys. J. B

16, 361 (2000).20. F.G. Mertens, H.J. Schnitzer, A.R. Bishop, Phys. Rev. B

56, 2510 (1997).21. A. Sanchez, A.R. Bishop, SIAM Rev. 40, 579 (1998).22. R. Boesch, P. Stancioff, C.R. Willis, Phys. Rev. B 38, 6713

(1988).23. H.J. Schnitzer, F.G. Mertens, A.R. Bishop, Physica D 141,

261 (2000).24. R. Rajaraman, Solitons and Instantons (North Holland,

Amsterdam, 1998).25. M. Meister, F.G. Mertens, J. Phys. A 33, 2195 (2000).26. T. Kamppeter, F.G. Mertens, E. Moro, A. Sanchez, A.R.

Bishop, Phys. Rev. B 59, 11349 (1999).27. C.W. Gardiner, Handbook of Stochastic Methods for

Physics, Chemistry and the Natural Sciences (Springer,Berlin, 1983).

28. K. Nakamura, T. Sasada, Phys. Lett. A 48, 321 (1974).29. M. Lakshmanan, T.W. Ruijgrok, C.J. Thompson, Physica

A 84, 577 (1976).30. J. Tjon, J. Wright, Phys. Rev. B 15, 3470 (1977).31. H.C. Fogedby, J. Phys. A 13, 1467 (1980).32. T. Kamppeter, F.G. Mertens, A. Sanchez, A.R. Bishop, F.

Dominguez-Adame, N. Grnbech-Jensen, Eur. Phys. J. B7, 607 (1999).

33. A.A. Thiele, Phys. Rev. Lett. 30, 230 (1973).34. M. San Miguel, R. Toral, in Nonequilibrium Structures VI,

edited by E. Tirapegui (Kluwer, Dordrecht, 1998).

m. meister, f.g. mertens and a. sanchez- soliton diffusion on the classical, isotropic heisenberg...

Documents