IC/95/314

INTERNATIONAL CENTRE FOR

THEORETICAL PHYSICS

SOLITON IN AN INHOMOGENEOUS WEAKFERROMAGNET WITH DZIALOSHINSKI-MORIYA

INTERACTION

M. Daniel

and

R. Amuda

INTERNATIONALATOMIC ENERGY

AGENCY

UNITED NATIONSEDUCATIONAL,

SCIENTIFICAND CULTURALORGANIZATION

Ml RAM ARE-TRIESTE

IC/95/314

International Atomic Energy Agencyand

United Nations Educational Scientific and Cultural Organization

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

SOLITON IN AN INHOMOGENEOUS WEAK FERROMAGNETWITH DZIALOSHINSKI-MORIYA INTERACTION1

M. Daniel2

International Centre for Theoretical Physics, Trieste, Italy

and

R. AmudaDepartment of Physics, Centre for Nonlinear Dynamics,

Bharathidasan University,Tiruchirappalli - 620 024, Tamil Nadu, India.

ABSTRACT

We have identified an integrable model of the inhomogeneous radially symmetric weak

Heisenberg ferromagnet in arbitrary (n-) dimensions with Dzialoshinski-Moriya antisym-

metric spin coupling. The elementary spin excitations of the magnet are found to be

governed by soliton modes under small angle oscillation of the antisymmetric spin cou-

pling.

MIRAMARE - TRIESTE

September 1995

1 Submitted to Physical Review B.2Permanent address: Department of Physics, Centre for Nonlinear Dynamics, Bharathi-

dasan University, Tiruchirappalli - 620 024 Tamil Nadu, India.

In ferromagnets, the exchange interaction

(spin-spin coupling), the single ion anisotropy due to the crystal

field effect and Zeeman energies have been treated as common but

simple magnetic spin couplings. When the symmetry around the

magnetic ions is not high enough an unfamiliar but important

antisymmetrical coupling results due to the combined effect of

spin orbit coupling and the exchange interaction leading to the

mechanism of weak feromagnetism. Though the microscopic theory on

the above mechanism was proposed by Dzialoshinski and developed

2by Moriya more than three decades ago, not much is known about

the macroscopic properties of these ferromagnets including the

elementary spin excitations. However, very recently there is loti

of interest in the studies of the weak ferromagnets with

Dzialoshinski- Moriya (D-M) interaction because of its important

role in insulators, spin glasses and low temperature phase of

copper oxcide super conductors and also in the phase transition

3 4studies , in the quantum aspects and in the statistical

5mechanics . In a different platform, in the recent years the quasi

one dimensional isotropic ferromagnets of Heisenberg model and few

higher dimensional ferromagnets have been identified as integrable

nonlinear dynamical models exhibiting an interesting class of

localized nonlinear elementary spin excitations in the form of

solitons, domain walls etc . In this context, very recently

soliton spin excitations have also been identified in the one

dimensional classical continuum isotropic and anisotropic weak

12ferromagnetic spin chains with D-M interaction . Motivated by

this in the present paper, we investigate the nature of nonlinear

spin excitations in an inhomogdneous radially symmetric Heisenberg

ferromagnet with D-M interaction in arbitrary (n-) dimensions in

the classical continuum limit and identify soliton modes.

In order to understand the underlying nonlinear

dynamics of the above weak ferromagnetic system, we formulate the

dynamical equations starting from the Hamiltonian expressed

interms of the ionic spin operators. The Heisenberg model of the

spin Hamiltonian for an inhomogeneous weak ferromagnet with

nearest neighbour exchange spin coupling and D-M interaction can

be written interms of the spin angular momentum operators as

u - _ r r f ij(s S )+D (S ' ̂ S )t (1). x j 1 J 3+6 " j 3+& ! "

In eq.(l), the first term represents the exchange interaction

between adjacent spins with j, the exchange integral and the

second term corresponds to the D-M interaction due to

antisymmetrical spin coupling and the vector £ represents all

possible nearest neighbours. This antisymmetrical coupling acts to

cant the spins because the coupling energy is minimized when the

two spins are perpendicular to each other. The direction of the

constant vector D can be related to the symmetry of the

ferromagnetic crystal and when a centre of inversion is located at

13the point half way between the spins coupled, then D = 0. We

choose the vector D parallel to m =(1,1,1), (i.e) D = Dm . In

Hamiltonian (1), f., the site dependent function introduces

inhomogeneity in the ferromagnetic lattice which can occur either

when the distance between neighbouring ions varies or when the

wave function of the ion varies from site to site. In case of

ferromagnets when the spin angular momentum value is large, S.

x y zrepresents the classical three dimensional vector (S.,S.,S.) in

J <J J

spin space at the site j of an n-dimensional lattice and S. -arej+6

its nearest neighbours. In the low temperature, long wavelength

limit the spin vector S. and the site dependent function f . vary

very slowly over the lattice distance 'a' and we can go to the

continuum limit by introducing Taylor expansions for S. c and for

2 >f.,- upto Ota ) (a:lattice parameter) and replace S. and f .j-o 3 3respectively by the continuous vector function S (,r,t) and by the

continuous scalar function f(r,t) where r=(r ,r ,..., r ). Also~" " " 1 2 n

in order to reduce the mathematical complexity in the case of D-M

interaction we consider only those contributions during spin

7. -ST" t -KSS^ttSiilJtl 1, | : f i -H ' I jfiua*:

evolution that lies only within a small angle (9) cone whose axis

lies parallel to m (m .S^*l) (See Fig.l). In view of the above,

J4the equation of motion representing the spin dynamics (after

suitable rescaling of time and redefinition of parameter

D (d=D/Ja)) can be written as

f f C ~ ' t > = f < £ ) S > A ^ 2 S > + 7 f ( r ) ( S ^ V S ^ - d [ 2 f ( r ) - a v f ( r ) ] v i > ( 2 )

-> 2 2S rr,t)=l,V = V * y = n-dimensional Laplacian.

We choose to investigate the nonlinear spin excitations in the

radially symmetric case in arbitrary(n-) dimensions in which case2

V and y are replaced by { — + ) and T- respectively, where»« A 2 r or

or2 2 2 2

r =r +r +...+ r' ,0 <. r < «o .Eq.(2) contains several integrablej- A n

spin models exhibiting soliton excitations for different 'n1 and

'f in the absence of D-M interaction (d=0)(see for e.g refs.

6,10,11). Now it is natural to see whether the weak ferromagnetic

system (2) in the radially symmetric case with d̂ s 0 is also an

integrable model and can admit soliton spin excitations for

arbitrary 'n' and 'f and if not for specific choices of 'n' and

'f1. The answer for the question of integrability can be obtained

by carrying out the Painleve~singularity structure analysis

on eq.(2). As it is a well established systematic procedure widely

5

used in the literature ,we desist from giving details about the

procedure here.

However, due to the presence of the length

constraint S =1 in eq.(2), the Painleve"analysis of it in its

present form becomes difficult and hence we try to construct an

equivalent representation which will be amenable to the analysis.

Experience shows that spin equations for different ferromagnetic

models can be mapped to the1 family of nonlinear Schro'dinger

equations through differential geometric and gauge equivalence

10,11 L 14,18

formalisms . Thus, we map the lnhomogeneous weak

ferromagnetic system onto a moving helical space curve in E (see

Fig.2) by identifying the spin vector with the tangent vector e

— y _ ̂ 1/?of the space curve with curvature *{r,t)=(e .e, ) ' and torsion

Ir Ir

7<r,t)=s e1-^

e1 A ei '• Defining two unit normal vectors e> and

e in the usual wayr the change of orientation of the orthogonal

trihedral e., i=l,2,3 which defines the space curve uniquely

within rigid motion is determined by the Serret-Frenet equation

given by

e. = d A e., i=i,2,3, (3a)

where-> -> ->d = T&^X e3- (3b)

Now using the radially symmetric version of eq.(2) (after

replacing S by e ) and the Serret-Frenet equation (3) the time

evolution of the orthogonal trihedral e. can be written as

e ^ = en-

A & ' i=l,2,3, (4a)

It 1

where

-> -> -> ->and

Vs-

| | r L 2r ! (4c)

, - = f*T+d{2ff-af ) K (4d,e)

Here suffices with respect to r represent partial derivatives. The

compatibility conditions (e. ) =(e ) , i= 1,2,3 of eqs.(3) and

(4) lead to a set of coupled nonlinear evolution equations for the

curvature and torsion of the space curve given by

« = -MffcT) +r(fs) + -~Z--£HT+ d{u [2f- af ] I 1 J (5a)t [ r r r r rj '

r r JTt- fi |f

...(5b)

Upon suitable identification ,of the curvature and torsion of the

space curve with the energy and current densities of the

ferromagnetic system , eq.(5) can be equivalently written as a set

of coupled nonlinear evolution equations for the energy and

current densities of the ferromagnet. But in order to identify

eqs.(5) with more standard nonlinear partial differential

equations possesing soliton solutions, we rewrite eqs.(5) as the

following generalised inhomogeneous nonlinear Schrodinger equation

by making the complex transformation q=;f exp (iJr (r' , t) dr' I .

iq^+2fjqj2q+|(fq) +( -----fq )+id(2f-af )q

/if |q!2+2(n-l)-|q{2ldrl=0. (6)+ 2

Thus eq.(6) equivalently represents the spin dynamics of an

inhomogeneous radially symmetric (in arbitrary n- dimension) weak

ferromagnet. The generalised inhomogeneous nonlinear Schrodinger

equation (6) contains integrable models and soliton solutions

corresponding to the same values of n,f when d=0 as for eq.(2).

Now in order to carry out the singularity structure

analysis we rewrite eq.(6) (within an tyrbitrary function of 't'

which can be removed by a simple time dependent gauge

transformation) as

iq +2Wq+ | (fq) + { - - ~ ) f q ) + id(2f-af )qj =0, (7a)

W -f(|qf 2) -2(f -^ - " ^ f ! | q j 2 = 0. (7b)r ' ' r r r ' •

— * . - . >" ••.-.-••>*"'».•

Further, it is required to rewrite eqs.{7) and the c.c of (7a) by

replacing q(r,t) by P(r,t) and q (r,t) by Q(r,t) and for isolating

those cases for which the system of eqs.{7) are free from movable

critical manifolds so that the general solution will be single

valued around the noncharacteristic movable singular manifold

$ (r,t)=0, we express the functions P,Q and W locally in the form

of the Laurent series with atleast one of the leading function

coefficients being different from zero. An analysis on the leading

order behaviour of the Laurent series solutions in eqs.{7) (after

rewriting it interms of P and Q) shows that P *-̂ P d> ,Q**-"1Q '£ ,

_2 2 2Wv-»W d> with P Q =-0 and W =-f# and the resonances, namely the

0' 0 0 r O r

powers at which the arbitrary functions can enter into the Laurent

series are found to be -1,0,2,3,4. The resonances '-1' and '0'

correspond to the arbitrariness of the singular manifold and the

coefficient P or Q respectively. For proving the existence of

sufficient number of arbitrary functions at the resonance values

without introduction of movable critical manifolds, we substitute

the full Laurent series solutions in eqs.(7) and collect the

coefficients of different powers of -p. A detailed analysis of the

resultant equations shows that at resonance values 2,3 and 4 it is

not required to introduce any movable critical manifold when the

inhomogeneous function f(r) assumes the specific form

f(r). a r-(n-a)+/} r-a<n-l)

(oi ,fti arbitrary constants) and thus confirms the existence of

sufficient number of arbitrary functions and hence proves the

possibility of integrability. Thus we conclude that the

inhomogeneous radially symmetric Heisenberg weak ferromagnet in

arbitrary (n-) dimensions is expected to be an integr]7able

nonlinear dynamical model and hence the elementary spin

excitations can be expressed interms of solitons only when the

inhomogeneity is of the form (8).

Having identified the integrable model in eq.{6)and

hence in eq.(2), we now try to obtain the Lax pair of operators or

the associated linear eigen value problem for the purpose of

constructing soliton solutions. This can be achieved by using the

Laurent series solution of (7) truncated at the constant level

16term . Thus we substitute the truncated Laurent series solutions

P= P0 ^ 1 + P i ' Q = Qo' f V > 1 + Q l a n d W= w

0 > ) 2 + w1 ^ 1 + W 2 i n eqs .<7)

_2collect the coefficients of different powers of -p namely d> and

rp to obtain a overdetermined system of equations which upon

2 2 *Q i ^ , Q Q = x i ^ , P ^ q * Q a = < I a n d a ^ i + - */•2

2 2identification of PQ= i ^ , QQ= xi^, P^q* Qa=<I and a^=-i+-• */•2-

give the following linear eigen value problem.

V = Uf, ¥'t= Vy, (9a,b)

10

where u- =(->p ys ) and the Lax pair of operators U and V take theX A

form

U=

., n-1lA.r q

n-1 v= -B -A(9c,d)

where

A- iW-2iX2fr2(n-1J,

B= -2Xfqrn"1+if(fq) - d(2f-af

<9e)

(9f)

Here X. is the spectral parameter obtained as a constant of

integration and f takes the specific form given in eq.(8) The

consistency of the linear eigen value problem (9a) and (9b) ( with

(9c-f)) leads to the evolution equation (7), provided the spectral

parameter evolves as

= p(t)+ i>) (t)= (2nat+v)-1 (10)

Here y is the free parameter. The linear eigen value problem (9)

can also be constructed from eqs.(3) and (4).

Knowing the Lax pair of operators, the explicit

form of soliton solution to eq.(6) can now be obtained byr

constructing the Backlund transformation from the linear eigen

11 19,20value problem (9). Following the standard procedure ' ' we

obtain the Backlund transformation connecting the N and N-1

solitons as

11

-~* . (ii)2

where v = (w />« ) satisfying the Riccati equations,y = 2ir

* 2 * 2

+ct+<3 >: and y = B+2A;r+B y with A and B as given in eqs.(9e,f).

The one soliton solution q(l) can be constructed using the

Backlund transformation starting from the zero soliton solution^(0)

q(0}= 0 and with the knowledge of y (0) ------- which can be obtained

V2<0)

by solving the linear eigen value problem (9) for q(0)=0. Thus we

obtain the one soliton solution to eq.(6) as1 rn

q= 2r}(t)r sech [ 2i>(t)I- -4{ip(t)t-6 I]n 1

exp [2i(----- -2>j(p2(t)-n2(t)) t+6 }] , (12)

^^^»- [p(O)+i72(O)+2n*( P2{0)t+>72(0)t)] f

[(l+2(0)) +4n o. >7 (0)t ]

phase constants. Knowing q and hence the curvature « and torsion r

of the space curve the tangent vector e and hence the spin vector

~~y 18 21

S can be constructed using the known geometric procedure.

Thus we obtain the elementary spin excitations of the n-

dimensional radially symmetric inhomogeneous weak ferromagnet in

the form of one soliton solution corresponding to (12) as

12

. nc - _ x _ • y_ 4>7 _ . r

2 2 •" ' ' n ! - • ' ( - • 2'P +V)

n

sech[2>7l - - 4 6 ' p t - i l ) e x p [ ± 2 i { p - -2/9 (p -r . ) t+6 I ] , (13a)n 1 n 2

2 nSZ= 1~( - I 2 - - ) s e e n [ 2f>f- - 4 / ? p t - 6 , I ] . (13b)

It may be noted that the structure of the one

soliton solutions are similar to the case when d=0. However, by

inspection we notice that the structure of multi solitons will

differ. In Fig. (3) we have plotted the evolution of the one

soliton (}<j|) in the physically important circularly (n=2) and

spherically (n=3) symmetric cases for the following specifc values

of the parameters: p(0)=0.2, ̂ {0) = 0.2,-6 = 0.3, ot= 0.5, /*= 0.1. On

comparing the evolution of solitons in Figs.(3a) and (3b), we

notice that as the dimension of the ferromagne£ at a given time

increases the soliton becomes sharp and more localized with

increase of amplitude and decrease of width. Further it is

observed that the soliton in both the circularly and spherically

symmetric cases spreads as time passes on which is due to the form

of the amplitude function ?)(t). Though energy is not a constant

here( since r>(t) is not a constant) there excits an infinite

number of constants of motion each of which are summed over the

infinite number of constants of motion of the standard cubic

nonlinear Schrodinger equation multiplied by time dependent

13

coefficients (for details see refs. 11, 17 and 21 of ref.ll here).

Also finally, the complete iptegrability nature of the system

ensures the stability of solitons under collisions.

In conclusion the radially symmetric inhomogeneous

weak Heisenberg ferromagnet with D-M interaction in arbitrary

n-dimensions in the classical continuum limit is found to be an

integrable nonlinear dynamical model under small angle spin

oscillation due to antisymmetrical spin coupling and when the

inhomogeneity is of the form f(r)= ar + fir . The

elementary spin excitations of this integrable ferromagnetic

system is found to be governed by solitons.

ACKNOWLEDGEMENTS

The work of M.D forms part of a- major research

project supported by the department of Science and Technology,

Government of India and he thanks the International centre for

Theoretical Physics, Trieste, Italy for hospitality where part of

the work was done, R.A thanks the Bharathidasan university for the

award of the research fellowship.

14

REFERENCES

I.Dzialoshinski, Phys. and Chem. Solids 4_, 241 (1958) .2T.Moriya, Phys.Rev. 120, 91{1960).

J.R.de Sousa, D.F. de Albuquerque and I.P.Fittipaldi, Phys.Lett.

191A, 275(1994) and references therein.4Y.S.Xiong, L.Yi, K.L. Yao and. Z.G.Li, Z.Phvs.B96,261(1994).5O.Derzhkp and A.Monia, Ferrqelectrics 153, 49(1994).6M.Lakshmanan, Phys. Lett. 61A, 53(1977).

K.Nakamura and T.Sasada, J.Phys.CIS, L915 (1982).8H.J.Mikeska and M.Steiner, Adv. Phys.4^, 191 (1990).9A.M.Rosevich, B.A.Ivanov and A.S.Rovalov, Phys.Rep.195,

117(1990) .10

K.Porsezian, M.Daniel and M.Lakshmanan, J.Math.Phys.33,

1807(1992).

M.Daniel, K.Porsezian and M.Lakshmanan, J.Math. Phys.35,6498(1994)

12

M.Daniel and R.Amuda, J.Phys. A28(1995) To Appear.

T.Moriya in Magnetism Vol.1 (Eds.) G.T.Rado and H.Shul (Academic

Press, New York,1963).14M.Lakshmanan, Th.W.Ruijgrok and C.J. Thompson, Physica8_4A, 577 (1976) .

15M. J.Ablowitz, A.Ramani and H.Segur, J. Math. Phys. 2_1, 715(1980).

16J.Weiss, M.Tabor and T.Carnevale, J.Math.Phys.24., 522(1983).

17M.Lakshmanan and R.Sahadevan, Phys.Rep.224, 1(1993).

18L.P.Eisenhart, A Treatise on the Differential Geometry of Curvesand Surfaces (Dover, New York, 1960).

19R.M.Miura, Backlund Transformations, Lecture Notes inMathematics (Springer-Verlag, Berlin, 1976).

20A.A.Rangwala and J.A.Rao, Phys.Lett.A112, 188(1985).

21M.Daniel and M.Lakshmanan, Physica A120, 125(1983).

15

Fig. I Small angle cone (e) representing allowed region of spin oscillationsdue to D-M. interaction

F i g . 2 A helical space curve i n - E 3 [ R ! Position vector]

Co cr

att=2.0 —at t=3.0at t=4.0att=5.0 —

0 1 2 3 4 5 6 7 8 9 10r

Fig.3a Evolution of soliton ( I q l ) i n the circularly symmetric case

att=1.0att=2.0at t=3.0at t=4.0at t=5.0

1 2 3 4 5 6 7 8 9r

3b Evolution of soliton ( Iq l ) in the spherically symmetric case