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Solitons in a modulated antiferromagnetic chain
0038-1098/91 $3.00 + .00 Pergamon Press plc
Solid State Communications, Vol. 77, No. 6, pp. 461-463, 1991. Printed in Great Britain.
SOLITONS IN A M O D U L A T E D A N T I F E R R O M A G N E T I C CHAIN A.S.T. Pires Departamento de Fisica, Universidade Federal de Minas Gerais, C.P. 702, Belo Horizonte, MG, 30161, Brazil (Received 24 September 1990 by P. Burlet)
In this paper we study soliton excitations in a modulated onedimensional classical easy plane antiferromagnetic model.
THERE ARE several examples of modulated structures in magnetic crystals [1]. For instance the onedimensional classical easy-plane antiferromagnetic model with nearest- and next-nearest-neighbour interaction, for certain values of the ratio of the nearest and next-nearest-neighbour interaction, leads to helical spin structures at zero temperature, in connection with domain walls between opposed chiralities of the helical state [2, 3]. Although the helical order disappears for T ¢ 0, helical short-range order will be present at low temperatures. In this paper we will present a theoretical investigation of soliton excitations in a classical onedimensional antiferromagnet with a modulated structure. Let us start from the following one-dimensional model =
2 J ~ [S.S.+, + d(S~)2],
(1)
n
where J > 0, and d > 0 is an easy-plane anisotropy. Following the usual procedure [4] we will assume that two neighbouring spins are almost antiparallel to each other at low temperatures, and write S. =
the continuum approximation, in *
=
2J I dz ~(t3O~2 (c3q~2 1 [ \ O z J + sin20 \ O z J + (4JS) 2
F(a0)2
x LkOt J + sin 2 0 \ o t j j -
+ 2dcos z0
2a sin20 c~xJ
(3)
where z is the coordinate along the chain and we have taken the lattice constant as the unity. We have also added the term ( - 2a sin200~b/Oz) which will be responsible for the modulated structure. One way to obtain this term will be, for instance, by the addition to (1) of a Lifshitz term of the form (SYOSX/Oz-Sx~SV/Oz). The ground state of Hamiltonian (3) is a spiral magnetic structure 0 = re/2, 4~ = az. The equations of motion can be obtained directly from the Hamiltonian (3). We obtain 020 Oa2
020 sin 0 cos 0 F ( ~ ) 2 l (~)~ C2 Ot2 = Lk Oz J - -~ \ ot J 1
(-- 1)" {sin (0, + (-- l)nq,) -
× cos (q~, + (-- 1)"ct,) sin (0° + (-- 1)"r/,) x sin (~b, + ( - 1 ) % t , ) c o s (0, + (-1)"r/,)}. (2) Where 0 and 4> are the angles giving the sublattice magnetization, whereas r/ and ~ describe the deviations from perfect anti-alignment, and can be assumed to be small at low temperatures. Substituting (2) into the Hamiltonian (1) we can keep only the terms up to the second order in the small quantities q, ct and the spatial variation of 0 and q~. The variables r/ and ~ are then eliminated with ~O/dt and O(o/Ot, which can be derived from the equation of motion of the spins. This reduction from (1) finally results, using 461
c~2q~
102q~
Oz:
c~ at ~
2d-
2 a - ~ -z
,
(4)
_ 2 cot 0 [(~?O)(c~q5) =
~
1 -
7 \~-TJ\-~-JJ
so + 2a sin 0 cos 0 ~zz'
(5)
where c = 4JS. In the linear approximation the above equations have spin wave solutions with frequencies ~o~(q) --- cq for the in plane mode and 0922(q) = 2d + a 2 q- q2)c for the out of plane mode.
462
SOLITONS
Now we consider chain. We set 4(z, t)
=
az -
where s = z -
+ a $
’
vt. Rewriting -
(sin’@
and taking
solitary waves moving
ot + 5(s),
:l-(sin’B$)
IN A MODULATED
=
ecz, t) = equation
ANTIFERROMAGNETIC
along the
e(sj,
(6)
(5) as
$(sin’Og)
therefore, from (9) leading to the result A = o = 0. Thus the soliton solution, above the equilibrium spiral structure, is given by cos d(z, t)
=
&z, t)
az,
=
sech [yJm
0,
(7)
E,
=
(15)
4JS’ydm.
a$ -= o2 _
_
0,
aZ
!52
(16)
_-
leading
d28 -= ds2
0 ds
x
&co.
(17)
-2,,&$?!!!$?+&$? i (10)
a
(18)
de 2
to the following =
Its solution sin 0 cos 8,
(9) as
sin’ 19’
ds=
where A is the first integral of equation (7) and y = (1 - v~/c~))“~. Finally, from (8) and (9) we obtain y’
z+
In order to satisfy (17) we write equation
d5
case
=
expression
(2d + a2)y2 cos’6’ -
for equation
a2 cot’0.
(11) (19)
yields sech [my(z
(2d :d,‘y’)
which gives
-
vt)],
(20)
and
d0 ’ -
D’ cos’ 6’ + y4A2cot2 0
=
B2,
(11) &z, t)
=
tan-’
where =
(2d + u2 + 02/c2 + y202v2/c4)y2,
and 8’ is the first integral obtain 0 s ,/B2 B,
of (10). Integrating
s -
sg.
(13)
y4A2 cot’ 9
cos d(s)
=
dm
sech (sJ_). (14)
z+*CO,
vt)]
=
4JS2,j%(2d
+ a2)y ’
(22)
Acknowledgement -This work was partially supported by Conselho National de Desenvolvimento Cientifico e Tecnologico.
REFERENCES 1.
’
-
We note that, in contrast to the behaviour of the energy given by equation (16), the energy given by (22) tends to zero in the limit ZJ -+ c. The soliton solutions for the easy-axis anisotropy -2d + a2 < 0 can be obtained following the same procedure as we have used here.
The requirement that the energy of a soliton excited from the spiral ground state, should be finite leads to the following boundary condition for the angle 4 84 -_=a
[my(z
The energy is now given by
(11) we
a2y2+2d =
cotanh
(21)
Es
d6’ + D2 cos’ 8 -
a my
(12)
Now taking 8 = H/2, de/& = 0 as the boundary conditions at infinity (s = f co) for a soliton (localized wave) equation (13) can be easily integrated with solution
aZ
vt)],
Let us now study a soliton against the homogeneous distribution of spins. The boundary condition for the angle 4 becomes here
(9)
0’
(z -
sin 8 cos 8
-2d
0 ds
Vol. 77, No. 6
with energy
(6) into (4) and (7) we find
-z d2e -= ds2
CHAIN
V.G. Bar’yakhtar, A.J. Zhukov & D.A. Yablonskii, Sov. Phys. Solid State 21, 454 (1979); T.K. Soboleva, E.P. Stafanovskii & V.V.
Vol'. 77, No. 6
SOLITONS IN A MODULATED ANTIFERROMAGNETIC CHAIN
Tarasenko, Sov. Phys. Solid State 22, 1370 2. (1980); I.M. Vitebskii, Sov. Phys. JETP 55, 390 3. (1982); V.G. Bar'yakhtar & D.A. Yablonskii, Soy. Phys. Solid State 24, 1435 (1982); Soy. 4. Phys. Solid State 27, 1454 (1985); J. von Baehm & P. Bak, Phys. Rev. Lett. 42, 122 (1979).
463
I. Harada, J. Phys. Soc. Jpn 53, 1943 (1984). T. Horiguchi & T. Morita, J. Phys. Soc. Jpn. 59, 888 (1990). H.J. Mikeska, J. Phys. C13, 2913 (1980).
Solid State Communications, Vol. 77, No. 6, pp. 461-463, 1991. Printed in Great Britain.
SOLITONS IN A M O D U L A T E D A N T I F E R R O M A G N E T I C CHAIN A.S.T. Pires Departamento de Fisica, Universidade Federal de Minas Gerais, C.P. 702, Belo Horizonte, MG, 30161, Brazil (Received 24 September 1990 by P. Burlet)
In this paper we study soliton excitations in a modulated onedimensional classical easy plane antiferromagnetic model.
THERE ARE several examples of modulated structures in magnetic crystals [1]. For instance the onedimensional classical easy-plane antiferromagnetic model with nearest- and next-nearest-neighbour interaction, for certain values of the ratio of the nearest and next-nearest-neighbour interaction, leads to helical spin structures at zero temperature, in connection with domain walls between opposed chiralities of the helical state [2, 3]. Although the helical order disappears for T ¢ 0, helical short-range order will be present at low temperatures. In this paper we will present a theoretical investigation of soliton excitations in a classical onedimensional antiferromagnet with a modulated structure. Let us start from the following one-dimensional model =
2 J ~ [S.S.+, + d(S~)2],
(1)
n
where J > 0, and d > 0 is an easy-plane anisotropy. Following the usual procedure [4] we will assume that two neighbouring spins are almost antiparallel to each other at low temperatures, and write S. =
the continuum approximation, in *
=
2J I dz ~(t3O~2 (c3q~2 1 [ \ O z J + sin20 \ O z J + (4JS) 2
F(a0)2
x LkOt J + sin 2 0 \ o t j j -
+ 2dcos z0
2a sin20 c~xJ
(3)
where z is the coordinate along the chain and we have taken the lattice constant as the unity. We have also added the term ( - 2a sin200~b/Oz) which will be responsible for the modulated structure. One way to obtain this term will be, for instance, by the addition to (1) of a Lifshitz term of the form (SYOSX/Oz-Sx~SV/Oz). The ground state of Hamiltonian (3) is a spiral magnetic structure 0 = re/2, 4~ = az. The equations of motion can be obtained directly from the Hamiltonian (3). We obtain 020 Oa2
020 sin 0 cos 0 F ( ~ ) 2 l (~)~ C2 Ot2 = Lk Oz J - -~ \ ot J 1
(-- 1)" {sin (0, + (-- l)nq,) -
× cos (q~, + (-- 1)"ct,) sin (0° + (-- 1)"r/,) x sin (~b, + ( - 1 ) % t , ) c o s (0, + (-1)"r/,)}. (2) Where 0 and 4> are the angles giving the sublattice magnetization, whereas r/ and ~ describe the deviations from perfect anti-alignment, and can be assumed to be small at low temperatures. Substituting (2) into the Hamiltonian (1) we can keep only the terms up to the second order in the small quantities q, ct and the spatial variation of 0 and q~. The variables r/ and ~ are then eliminated with ~O/dt and O(o/Ot, which can be derived from the equation of motion of the spins. This reduction from (1) finally results, using 461
c~2q~
102q~
Oz:
c~ at ~
2d-
2 a - ~ -z
,
(4)
_ 2 cot 0 [(~?O)(c~q5) =
~
1 -
7 \~-TJ\-~-JJ
so + 2a sin 0 cos 0 ~zz'
(5)
where c = 4JS. In the linear approximation the above equations have spin wave solutions with frequencies ~o~(q) --- cq for the in plane mode and 0922(q) = 2d + a 2 q- q2)c for the out of plane mode.
462
SOLITONS
Now we consider chain. We set 4(z, t)
=
az -
where s = z -
+ a $
’
vt. Rewriting -
(sin’@
and taking
solitary waves moving
ot + 5(s),
:l-(sin’B$)
IN A MODULATED
=
ecz, t) = equation
ANTIFERROMAGNETIC
along the
e(sj,
(6)
(5) as
$(sin’Og)
therefore, from (9) leading to the result A = o = 0. Thus the soliton solution, above the equilibrium spiral structure, is given by cos d(z, t)
=
&z, t)
az,
=
sech [yJm
0,
(7)
E,
=
(15)
4JS’ydm.
a$ -= o2 _
_
0,
aZ
!52
(16)
_-
leading
d28 -= ds2
0 ds
x
&co.
(17)
-2,,&$?!!!$?+&$? i (10)
a
(18)
de 2
to the following =
Its solution sin 0 cos 8,
(9) as
sin’ 19’
ds=
where A is the first integral of equation (7) and y = (1 - v~/c~))“~. Finally, from (8) and (9) we obtain y’
z+
In order to satisfy (17) we write equation
d5
case
=
expression
(2d + a2)y2 cos’6’ -
for equation
a2 cot’0.
(11) (19)
yields sech [my(z
(2d :d,‘y’)
which gives
-
vt)],
(20)
and
d0 ’ -
D’ cos’ 6’ + y4A2cot2 0
=
B2,
(11) &z, t)
=
tan-’
where =
(2d + u2 + 02/c2 + y202v2/c4)y2,
and 8’ is the first integral obtain 0 s ,/B2 B,
of (10). Integrating
s -
sg.
(13)
y4A2 cot’ 9
cos d(s)
=
dm
sech (sJ_). (14)
z+*CO,
vt)]
=
4JS2,j%(2d
+ a2)y ’
(22)
Acknowledgement -This work was partially supported by Conselho National de Desenvolvimento Cientifico e Tecnologico.
REFERENCES 1.
’
-
We note that, in contrast to the behaviour of the energy given by equation (16), the energy given by (22) tends to zero in the limit ZJ -+ c. The soliton solutions for the easy-axis anisotropy -2d + a2 < 0 can be obtained following the same procedure as we have used here.
The requirement that the energy of a soliton excited from the spiral ground state, should be finite leads to the following boundary condition for the angle 4 84 -_=a
[my(z
The energy is now given by
(11) we
a2y2+2d =
cotanh
(21)
Es
d6’ + D2 cos’ 8 -
a my
(12)
Now taking 8 = H/2, de/& = 0 as the boundary conditions at infinity (s = f co) for a soliton (localized wave) equation (13) can be easily integrated with solution
aZ
vt)],
Let us now study a soliton against the homogeneous distribution of spins. The boundary condition for the angle 4 becomes here
(9)
0’
(z -
sin 8 cos 8
-2d
0 ds
Vol. 77, No. 6
with energy
(6) into (4) and (7) we find
-z d2e -= ds2
CHAIN
V.G. Bar’yakhtar, A.J. Zhukov & D.A. Yablonskii, Sov. Phys. Solid State 21, 454 (1979); T.K. Soboleva, E.P. Stafanovskii & V.V.
Vol'. 77, No. 6
SOLITONS IN A MODULATED ANTIFERROMAGNETIC CHAIN
Tarasenko, Sov. Phys. Solid State 22, 1370 2. (1980); I.M. Vitebskii, Sov. Phys. JETP 55, 390 3. (1982); V.G. Bar'yakhtar & D.A. Yablonskii, Soy. Phys. Solid State 24, 1435 (1982); Soy. 4. Phys. Solid State 27, 1454 (1985); J. von Baehm & P. Bak, Phys. Rev. Lett. 42, 122 (1979).
463
I. Harada, J. Phys. Soc. Jpn 53, 1943 (1984). T. Horiguchi & T. Morita, J. Phys. Soc. Jpn. 59, 888 (1990). H.J. Mikeska, J. Phys. C13, 2913 (1980).