- Email: [email protected]
Structural states in the 2D XY discrete model with dipole and exchange interactions
__ _. __ t!B
1 September
2s
1997
PHYSICS
Physics Letters A 233 (1997)
ELSEYIER
LETTERS
A
471-475
Structural states in the 2D XY discrete model with dipole and exchange interactions Valery V. Beloshapkin
‘,
Valery V. Muhin
Kirensky Institure of Physics, 660036 Krasnoyarsk. Russian Federarion Received 20 November
1996; revised manuscript received 27 May 1997; accepted Communicated by A.R. Bishop
for publication
12 June 1997
Abstract The structural states of the discrete 2D XY model with dipole and exchange interactions are studied numerically. It is shown that the ground state passes through a sequence of transitions of staircase type with increasing exchange constant. For small values of the exchange the constant ground state is continually degenerate. Metastable chaotic structures are shown to exist. @ 1997 Published by Elsevier Science B.V. PACS: 75.10.Hk;
61.43.Bn
Considerable progress has been made recently in the study of the structural states of discrete onedimensional models. This progress was stimulated by Aubry who reduced the task of finding all structural states to the investigation of the phase trajectories of equivalent dynamical systems [ 11. Further development of this method has led to a complete classification of all structural states for one-dimensional discrete systems [l-5]. In particular, it was shown that the discreteness of the models can give rise to metastable stochastic structure states [ l-3,5]. These structures correspond to stochastic trajectories in the phase space of the equivalent dynamical system. The nature of these structures was investigated in Ref. [ 61. As it was shown in Ref. [ 61 these structures are stable to small perturbations and consist of regular regions with different integer periods with solitons between these regions. The sizes of the regions are
random. The competition of interactions between structural elements leads to the appearance of commensurate and incommensurate ground states and transitions between them through the devil’s staircase kind of phase diagram with changing parameters of the models [ l-3,5-7]. An extension of the discrete map method to twodimensional systems is possible; however, for this case the equivalent dynamical system turned out to be infinite dimensional and the investigation of these systems leads to fundamental mathematical problems. Hence numerical experiments play an important role in the investigation of two-dimensional systems. In the present paper we report the results of a numerical analysis of the discrete classical two-dimensional spin model (XY model) with exchange and dipole interactions at zero temperature. A one-dimensional XY model with exchange and dipole interactions was studied in Ref. [ 41. It was shown numerically that an equivalent dynamical system of this model is integrable. This means, in
’ E-mail: [email protected]. 0375.9601/97/$17.00 @ 1997 Published PII SO375-9601(97)00491-X
by Elsevier Science B.V. All rights reserved.
472
VV Beloshapkin, VI! MuhidPhysics
Letters A 233 (1997) 471-475
particular, that there are no me&stable chaotic structures. The exact proof of this result was given in Ref. [ 81. It was proved that an equivalent dynamical system of the one-dimension XYZ-model is integrable [ 91. Because, in the one-dimensional case, ferromagnetic exchange and dipole interactions do not compete, the ground state is ferromagnetic [4]. In the two-dimensional case the situation is completely different. Dipole interactions form a micro-vortex structure in the ground state [ lo]. Only exchange interactions would give rise to a homogeneous spin structure. However, because of competition between them and dipole interactions, the structural states of the twodimensional XY model are of much interest. In this paper we show that in the two-dimensional case, competition of interactions leads to a sequence of structural transitions between the ground states with increasing exchange constant. Also metastable chaotic structures may exist. In our model all XY spins are spaced at the nodes of the square lattice and are described by the angle 4 between spin direction and the axis x. The Hamiltonian of the model is
c
H = -;
J(s,,, - s,14)
n’“Pq
+E
s,,, . sPq - 3 ( s,,~
- enln.Pq)
( sPq
- fhpq
)
ILw13
n’nPq
(1) m,n,p,q=
l,...,N,
where J is the constant of the exchange interaction, N is the numerical size of the square lattice, IRnn,,,qlis the distance between nodes (m, n) and (p, q), en,,,pq is the unit vector from node (m, n) to node (p, q), s,,, is the unit vector defining the orientation of the spin at the site (m, n), snm= (~0s dh,
sin Am 1.
The exchange interactions in the first sum ( 1) are assumed to exist between nearest neighbours only. The ground state of model ( 1) for J = 0 was found in Refs. [ IO,1 11. In particular it was shown that the ground state of the dipole system on the square lattice is a square lattice of micro-vortices. The size of a micro-vortex equals one period of the lattice. Furthermore, the authors of Refs. [ IO,1 11 found a continual
Fig. I. (a) Elementary cells corresponding to the ground state of system ( I ) at J = 0. (b) Elementarycell forming the ground state of the two-dimensional system ( 1) at J = 0.1455. (c) Elementary cell forming the ground state of the two-dimensional system ( I ) at J = 0.146.
degeneration of the ground state. In Fig. la we show the ground state structure for system ( 1) at J = 0. In the present paper we use the relaxation method [ 12,6,13] to determine the structural states of system ( 1) . According to this method all equilibrium structures correspond to a local minimum of the energy and are found from the solution of the equations
Wnn -_=--
6’z-f
dt
adh ’
m,n=
l,...,
N.
(2)
As I + 03, the functions c#+,,,(t) describe the equilibrium configuration of the system ( 1). The integration of Eqs. (2) is terminated when the quantity
KY Beloshapkin, VVI MuhidPhysics
becomes sufficiently small (in most cases w 10-15). The specification of various sets of initial conditions ( &,n (0) ) determines the various metastable configurations. We sought stable ground-state structures on a class of periodic solutions (periodic boundary conditions) with N the size of one elementary cell which was chosen by the global minimum of the energy. To find the ground state of system (1) we integrated Eqs. (2) for a single cell with a radius of dipole interactions Rn,,,,,* - 200~ - 300~2, which corresponds to the linear dimension of the system L N 400~ - 600~. This radius of interactions gives an accuracy 10P510P6 of the calculations of energy per particle. A further increase of the size of the system does not change the results presented below. For each specific N and J, we found that the square lattice of the vortices has the lowest energy. However, the size of the elementary cell is changed with variation of the exchange constant J. In Figs. lb and lc we show two elementary cells of the ground-state structure for J = 0.1455 and J = 0.146 respectively. Thus, we found that the ground state of the model ( 1) is the square lattice of the micro-vortices in which the size of the elementary cell (micro-vortices) increases with increasing exchange constant J. As J -+ 0 the size of the elementary cell is reduced to one period of the lattice, as found in Refs. [ 10,111. We found also that the ground state is continually degenerate at J < 0.25 (as for the pure dipole case [ lo] ) . In Fig. 2 the elementary cell is shown for the energy equal to the energy of the elementary cell shown in Fig. lb. All structures which can be obtained by continual transformation from the structure of Fig. lb to the structure of Fig. 2, have the same energies. If the exchange coupling exceeds 0.25, the continual degeneration disappears and the ground state becomes vertical. The phase diagram of the ground state is given in Fig. 3, where also the dependence of the vortex size on the exchange coupling is given. That diagram is similar to the devil’s staircase one of one-dimensional discrete systems [ 1,2]. As mentioned above, there are no metastable chaotic structures in model ( 1) for the one-dimensional case. This fact is a consequence of the integrability of the equivalent dynamical system [ 4,8]. The expression “chaotic structures” here means that they correspond
Letters A 233 (1997) 471-475
473
t
t
t Fig. 2. An example of the continual degeneration of the ground state for J = 0.1455.
The structure consisting of these cells has
an energy equal to the energy of the structure shown in Fig. I b.
N 14 12 -
108642-
0.1445
0.1455
0.1465
Fig. 3. Ground-state phase diagram of the system (
I)
J The depen-
dence of the vortex size N on the exchange coupling is shown.
to chaotic trajectories in an equivalent dynamical system. We consider the possibility of metastable chaotic structures in the two-dimensional case. As we pointed in Ref. [ 63, the existence of stochastic structures in the one-dimensional case is related to a set of dynamical stable regular structures. Any stochastic structure consists of regions of regular structures. The sizes of the regions are random and have no upper limits. However, there are inferior limits for the sizes of the regions since solitons attract each other more strongly than they are pinned to the lattice for a sufficiently small distance between the solitons dividing the regions. As shown above, in the two-dimensional model ( 1)) there are an infinite number of stable structures, consisting
474
V1! Beloshapkin, c(V MuhidPhysics
Letters A 233 (1997) 471-475
-__-__-._
,-v--
;\
-
,._~-__-_.r-\l\,.-__-_.,\-__-_r.\ll .___-__. \
1
,;‘,‘,j-..-.,pf,
, , \
;
!
,,‘,.,..;.,;:
’ ‘
, I’
: :
:
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I
I
,JJ ‘, I
:,‘,‘,,,,,,.,,..c______
! :;:
: 1;;,\“\ 1
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;;x;;~~~~:::::~::;\L
,‘~..---_-~~.~,.
,~,‘,‘~
,: .__.,
‘\
,..,..,r-_--r-.r.r._,~
I”
1
\\~l,,~,rr..r,--,-..-_~..,, ’t;;,r::/ ’ ’ \’ ,’ ‘\ . ;.‘.‘,-.-.~~-~~.-~-.~~-~-_-_=-~-_-, : ’*;;,, ~‘/,:,l\,x ,,,,,,,,,,~~_-‘__._-..-,~~-~~~~~ T,
’f ; 1, I I d ‘ /‘,/, , ,‘, . :.-,-,~.‘.-~~_-.-.~_-:.~_-.-:.~, ?’ ; ; : ,’ ,t ,: ,‘,~,~;,:,,;; : ’I’ ’ ‘*‘,8,d,‘/.8, .‘, z._ < _ _ ,‘, , , , ,‘/,f,
,
Y
1Hi.j - 6i,.iw2/ = 0,
i,j = 1,. . ., N2,
(3)
where H,,,, = s,
I
.I
be negative, which means an instability of the sample. The corresponding eigenvectors are localized in the boundary between the regions. Further we used this structure as a starting point for the relaxation procedure (2) at free boundary conditions. The result of the relaxation is shown in Fig. 4b. Because of the absence of negative squares of eigenfrequencies for the new annealed structure, we may conclude that the structure is dynamically stable. As one can see, this structure consists of two regular regions, if we neglect distortions at the boundary. Thus, the numerical experiment shows that the regions of different regular structures can coexist in one sample. This result gives rise to the possibility of the existence of inhomay
r
_
,
,\\,
‘,./
\,’
‘..,‘\
,,,t;; , J 6 ’ (, ,;;,,, ‘,,,f;;
J ‘
,‘,‘,‘I
\,\, ::’ , ;i ,\ ,,,‘..“-‘--“..‘,“, ,_..___...,_.,\.,\ I”’ ,;‘,‘,;:-~~==~:~~:~::;;, ’ ,;\ .;:::; : : ; ,’ ,’,,,.::~_._-___.~/._____.. _._.._._. . . . . c.r_-,‘\,,‘,‘,‘!,;; :
of vortices with different sizes, for every value of the exchange constant J. Although each structure with a global minimum of energy is formed by vortices with a fixed size, vortices with a different size can contribute to inhomogeneous metastable structures, both regular and stochastic. With the aim to prove the existence of metastable chaotic structures, we performed two numerical experiments. First we investigated the possibility of the coexistence of the regions consisting of different regular structures within one sample. With this aim we prepared a sample with the size 16 x 16 consisting of two regular regions as shown in Fig. 4a. Every vertical region forms the ground state of model (1) at different values of the exchange constant J. However, some of the calculated squares of the frequencies of small oscillation defined by the equation
r
‘Y“ .--‘,,~.\~r.rr~rr..~,~~,~,,,~,,~~I ,‘,‘.--‘..‘..“.--_--..~,~~,~ t IX -,- ‘,,’, , \ ,~.~,-,-,~_-~-_-_-~-~~_~~-; ‘\ ‘\ .,,.. _..__ ..,,., : : : ,,; t,;;. ,,--_-----,,-,-~,-,,,,
,“,
:
\:
;.‘,
,
\
-_---_-_
I
.,,, r(-__._.__._,
!
,
,
t t
I
f
,
,
1::
~,,~,~.‘_-_-_-_-_-_~~-:,‘:;:~~~~~,~~; ,,-_--._._~\\__._.-_
)
;;,,t,),
I
:: 1, ..*.;.
I
: ; ; , ,t
:
Fig. 4. (a) Initial unstable structure on the 16 x I6 lattice consisting of two regular structures which anneals to the stable structure consisting of two regular clusters. (b) Dynamical stable structure consisting of two regular structures at J = 0.1457.
;
‘, , ,
/I::
‘,,:;,“,“,_‘_~-_..,.,.,~
I,”
: :
::;
, ‘,
:
;,
1,.,/t ,r___-l
Fig. 5. Metastable stochastic structure for the two-dimensional model ( 1) on the 36 x 36 lattice at J = 0.1457. structural states for model ( 1) formed of clusters with random sizes. In Fig. 5 an example of stochastic structures for the lattice with size 36 x 36 is shown for .I = 0.1457. This structure was obtained by integration of the relaxation equations (2) with random initial conditions (&,,,, (0) ) at free boundary conditions. The calculation shows that the structure is dynamically stable. As one can see, this structure consists of domains of regular structures shown in Figs. 1 and 2 with different sizes of the domains randomly arranged about each other. Therefore, the nature of the chaotic structures in the two-dimensional case is identical to that of chaotic structures in one-dimensional systems, where the chaotic structures consist of regular domains with different length periods and solitons as intermediate regions between them [ 61. In conclusion we discuss two problems which concern the relaxation method to find the ground states. As we said earlier, the choice of the initial conditions (+,,, (0) ) for Eqs. (2) defines the stable structures which have to be found by the relaxation procedure (2). However, there is no certainty that this structure corresponds to the global minimum. We sought structures for each value of the constant J by randomly changing initial conditions. We did not find structures with energy lower than the energy of the vertical structure. Furthermore, we verified our results by a random search method (an analog of the Monte Carlo method at finite temperature). With the help of this method we did not find structures with energy lower than the energies of the vortex ones found earlier. mogeneous
,
KV Beloshapkin.
VV MuhidPhysics
The second problem is the periodic boundary conditions used to find the ground state of the system ( 1) . To verify the effect of the periodic boundary conditions on our results we investigated the ground state for a 24 x 24 lattice with free boundary conditions at J = 0.146. The lowest energy is realized as the square lattice of 6 x 6 vortices. This result coincides with the result for periodic boundary conditions (see Fig. lc) . Thus, we can conclude that the ground state of 2D XY models with exchange and dipole interactions is the square lattice of vortices. The sizes of the vortices increase with increasing exchange constant. Also metastable chaotic structures are shown to exist. The nature of the chaotic structures in the two-dimensional case is identical to that of the chaotic structures in one-dimensional systems. When we prepared this article, we learned of the investigation of model (1) at finite temperature by a Monte Carlo simulation [ 141. The authors of this article found a phase transition to the ordered phase characterized by a circular arrangement of the spins, which corresponds to the results of our investigation.
Letters A 233 (1997) 471-475
41s
References 1I] S. Aubry and PY. Le Dacron, Physica D 8 ( 1983) 381. [21 P Bak, Rep. Progr. Phys. 45 (1982) 587. 131 P.I. Belobrov, A.G. Tretyakov and GM. Zaslavsky. Phys. Len. A 97 (1983) 409. 141 P.I. Belobrov, V.V. Beloshapkin, A.G. Tretyakov and G.M. Zaslavsky, Sov. Phys. JETP 60 ( 1984) 180. [S] V.V. Beloshapkin and A.G. Tretyakov. Phys. Lett. A 106 (1984) 47. [6] V.V. Beloshapkin, A.G. Tretyakov and G.M. Zaslavsky. Fiz. Nizk. Temp. I2 (1986) 733 [Sov. J. Low Temp. Phys. 12 (1986) 4151. 171 P.I. Belobrov, V.V. Beloshapkin, A.G. Tretyakov and G.M. Zaslavsky, Phys. Len. A 122 ( 1987) 323. [Sl Ya.1. Granovsky and A.S. Gedanov, Zh. Eksp. Tear. Fiz. (USSR) 44 ( 1986) 237. 191 Ya.1. Granovsky and AS. Gedanov, Theor. Math. Phys. (USSR) 71 (1987) 143. [IO] PI. Belobrov, R.S. Gekht and V.A. Ignatchenko, Sov. Phys. JETP 57 (1983) 636. I I II P.1. Belobrov, V.A. Voevodin and V.A. Ignatchenko. Sov. Phys. JETP 61 (1985) 522. [ 121M. Peyrard and S. Aubry, J. Phys. C 16 (1983) 1593. [ 131 V.V. Beloshapkin, G.P. Berman, E.V. Shtukkert and A.G. Tretyakov, Sov. Phys. JETP, 73 (1991) 683. [ 14) F. Matsubara and J. Sasaki, e-journal Condensed Matter, article 9604190 ( 1996).
1 September
2s
1997
PHYSICS
Physics Letters A 233 (1997)
ELSEYIER
LETTERS
A
471-475
Structural states in the 2D XY discrete model with dipole and exchange interactions Valery V. Beloshapkin
‘,
Valery V. Muhin
Kirensky Institure of Physics, 660036 Krasnoyarsk. Russian Federarion Received 20 November
1996; revised manuscript received 27 May 1997; accepted Communicated by A.R. Bishop
for publication
12 June 1997
Abstract The structural states of the discrete 2D XY model with dipole and exchange interactions are studied numerically. It is shown that the ground state passes through a sequence of transitions of staircase type with increasing exchange constant. For small values of the exchange the constant ground state is continually degenerate. Metastable chaotic structures are shown to exist. @ 1997 Published by Elsevier Science B.V. PACS: 75.10.Hk;
61.43.Bn
Considerable progress has been made recently in the study of the structural states of discrete onedimensional models. This progress was stimulated by Aubry who reduced the task of finding all structural states to the investigation of the phase trajectories of equivalent dynamical systems [ 11. Further development of this method has led to a complete classification of all structural states for one-dimensional discrete systems [l-5]. In particular, it was shown that the discreteness of the models can give rise to metastable stochastic structure states [ l-3,5]. These structures correspond to stochastic trajectories in the phase space of the equivalent dynamical system. The nature of these structures was investigated in Ref. [ 61. As it was shown in Ref. [ 61 these structures are stable to small perturbations and consist of regular regions with different integer periods with solitons between these regions. The sizes of the regions are
random. The competition of interactions between structural elements leads to the appearance of commensurate and incommensurate ground states and transitions between them through the devil’s staircase kind of phase diagram with changing parameters of the models [ l-3,5-7]. An extension of the discrete map method to twodimensional systems is possible; however, for this case the equivalent dynamical system turned out to be infinite dimensional and the investigation of these systems leads to fundamental mathematical problems. Hence numerical experiments play an important role in the investigation of two-dimensional systems. In the present paper we report the results of a numerical analysis of the discrete classical two-dimensional spin model (XY model) with exchange and dipole interactions at zero temperature. A one-dimensional XY model with exchange and dipole interactions was studied in Ref. [ 41. It was shown numerically that an equivalent dynamical system of this model is integrable. This means, in
’ E-mail: [email protected]. 0375.9601/97/$17.00 @ 1997 Published PII SO375-9601(97)00491-X
by Elsevier Science B.V. All rights reserved.
472
VV Beloshapkin, VI! MuhidPhysics
Letters A 233 (1997) 471-475
particular, that there are no me&stable chaotic structures. The exact proof of this result was given in Ref. [ 81. It was proved that an equivalent dynamical system of the one-dimension XYZ-model is integrable [ 91. Because, in the one-dimensional case, ferromagnetic exchange and dipole interactions do not compete, the ground state is ferromagnetic [4]. In the two-dimensional case the situation is completely different. Dipole interactions form a micro-vortex structure in the ground state [ lo]. Only exchange interactions would give rise to a homogeneous spin structure. However, because of competition between them and dipole interactions, the structural states of the twodimensional XY model are of much interest. In this paper we show that in the two-dimensional case, competition of interactions leads to a sequence of structural transitions between the ground states with increasing exchange constant. Also metastable chaotic structures may exist. In our model all XY spins are spaced at the nodes of the square lattice and are described by the angle 4 between spin direction and the axis x. The Hamiltonian of the model is
c
H = -;
J(s,,, - s,14)
n’“Pq
+E
s,,, . sPq - 3 ( s,,~
- enln.Pq)
( sPq
- fhpq
)
ILw13
n’nPq
(1) m,n,p,q=
l,...,N,
where J is the constant of the exchange interaction, N is the numerical size of the square lattice, IRnn,,,qlis the distance between nodes (m, n) and (p, q), en,,,pq is the unit vector from node (m, n) to node (p, q), s,,, is the unit vector defining the orientation of the spin at the site (m, n), snm= (~0s dh,
sin Am 1.
The exchange interactions in the first sum ( 1) are assumed to exist between nearest neighbours only. The ground state of model ( 1) for J = 0 was found in Refs. [ IO,1 11. In particular it was shown that the ground state of the dipole system on the square lattice is a square lattice of micro-vortices. The size of a micro-vortex equals one period of the lattice. Furthermore, the authors of Refs. [ IO,1 11 found a continual
Fig. I. (a) Elementary cells corresponding to the ground state of system ( I ) at J = 0. (b) Elementarycell forming the ground state of the two-dimensional system ( 1) at J = 0.1455. (c) Elementary cell forming the ground state of the two-dimensional system ( I ) at J = 0.146.
degeneration of the ground state. In Fig. la we show the ground state structure for system ( 1) at J = 0. In the present paper we use the relaxation method [ 12,6,13] to determine the structural states of system ( 1) . According to this method all equilibrium structures correspond to a local minimum of the energy and are found from the solution of the equations
Wnn -_=--
6’z-f
dt
adh ’
m,n=
l,...,
N.
(2)
As I + 03, the functions c#+,,,(t) describe the equilibrium configuration of the system ( 1). The integration of Eqs. (2) is terminated when the quantity
KY Beloshapkin, VVI MuhidPhysics
becomes sufficiently small (in most cases w 10-15). The specification of various sets of initial conditions ( &,n (0) ) determines the various metastable configurations. We sought stable ground-state structures on a class of periodic solutions (periodic boundary conditions) with N the size of one elementary cell which was chosen by the global minimum of the energy. To find the ground state of system (1) we integrated Eqs. (2) for a single cell with a radius of dipole interactions Rn,,,,,* - 200~ - 300~2, which corresponds to the linear dimension of the system L N 400~ - 600~. This radius of interactions gives an accuracy 10P510P6 of the calculations of energy per particle. A further increase of the size of the system does not change the results presented below. For each specific N and J, we found that the square lattice of the vortices has the lowest energy. However, the size of the elementary cell is changed with variation of the exchange constant J. In Figs. lb and lc we show two elementary cells of the ground-state structure for J = 0.1455 and J = 0.146 respectively. Thus, we found that the ground state of the model ( 1) is the square lattice of the micro-vortices in which the size of the elementary cell (micro-vortices) increases with increasing exchange constant J. As J -+ 0 the size of the elementary cell is reduced to one period of the lattice, as found in Refs. [ 10,111. We found also that the ground state is continually degenerate at J < 0.25 (as for the pure dipole case [ lo] ) . In Fig. 2 the elementary cell is shown for the energy equal to the energy of the elementary cell shown in Fig. lb. All structures which can be obtained by continual transformation from the structure of Fig. lb to the structure of Fig. 2, have the same energies. If the exchange coupling exceeds 0.25, the continual degeneration disappears and the ground state becomes vertical. The phase diagram of the ground state is given in Fig. 3, where also the dependence of the vortex size on the exchange coupling is given. That diagram is similar to the devil’s staircase one of one-dimensional discrete systems [ 1,2]. As mentioned above, there are no metastable chaotic structures in model ( 1) for the one-dimensional case. This fact is a consequence of the integrability of the equivalent dynamical system [ 4,8]. The expression “chaotic structures” here means that they correspond
Letters A 233 (1997) 471-475
473
t
t
t Fig. 2. An example of the continual degeneration of the ground state for J = 0.1455.
The structure consisting of these cells has
an energy equal to the energy of the structure shown in Fig. I b.
N 14 12 -
108642-
0.1445
0.1455
0.1465
Fig. 3. Ground-state phase diagram of the system (
I)
J The depen-
dence of the vortex size N on the exchange coupling is shown.
to chaotic trajectories in an equivalent dynamical system. We consider the possibility of metastable chaotic structures in the two-dimensional case. As we pointed in Ref. [ 63, the existence of stochastic structures in the one-dimensional case is related to a set of dynamical stable regular structures. Any stochastic structure consists of regions of regular structures. The sizes of the regions are random and have no upper limits. However, there are inferior limits for the sizes of the regions since solitons attract each other more strongly than they are pinned to the lattice for a sufficiently small distance between the solitons dividing the regions. As shown above, in the two-dimensional model ( 1)) there are an infinite number of stable structures, consisting
474
V1! Beloshapkin, c(V MuhidPhysics
Letters A 233 (1997) 471-475
-__-__-._
,-v--
;\
-
,._~-__-_.r-\l\,.-__-_.,\-__-_r.\ll .___-__. \
1
,;‘,‘,j-..-.,pf,
, , \
;
!
,,‘,.,..;.,;:
’ ‘
, I’
: :
:
t’,‘,
I
I
,JJ ‘, I
:,‘,‘,,,,,,.,,..c______
! :;:
: 1;;,\“\ 1
’
I
_-_____~\,________.~
;;x;;~~~~:::::~::;\L
,‘~..---_-~~.~,.
,~,‘,‘~
,: .__.,
‘\
,..,..,r-_--r-.r.r._,~
I”
1
\\~l,,~,rr..r,--,-..-_~..,, ’t;;,r::/ ’ ’ \’ ,’ ‘\ . ;.‘.‘,-.-.~~-~~.-~-.~~-~-_-_=-~-_-, : ’*;;,, ~‘/,:,l\,x ,,,,,,,,,,~~_-‘__._-..-,~~-~~~~~ T,
’f ; 1, I I d ‘ /‘,/, , ,‘, . :.-,-,~.‘.-~~_-.-.~_-:.~_-.-:.~, ?’ ; ; : ,’ ,t ,: ,‘,~,~;,:,,;; : ’I’ ’ ‘*‘,8,d,‘/.8, .‘, z._ < _ _ ,‘, , , , ,‘/,f,
,
Y
1Hi.j - 6i,.iw2/ = 0,
i,j = 1,. . ., N2,
(3)
where H,,,, = s,
I
.I
be negative, which means an instability of the sample. The corresponding eigenvectors are localized in the boundary between the regions. Further we used this structure as a starting point for the relaxation procedure (2) at free boundary conditions. The result of the relaxation is shown in Fig. 4b. Because of the absence of negative squares of eigenfrequencies for the new annealed structure, we may conclude that the structure is dynamically stable. As one can see, this structure consists of two regular regions, if we neglect distortions at the boundary. Thus, the numerical experiment shows that the regions of different regular structures can coexist in one sample. This result gives rise to the possibility of the existence of inhomay
r
_
,
,\\,
‘,./
\,’
‘..,‘\
,,,t;; , J 6 ’ (, ,;;,,, ‘,,,f;;
J ‘
,‘,‘,‘I
\,\, ::’ , ;i ,\ ,,,‘..“-‘--“..‘,“, ,_..___...,_.,\.,\ I”’ ,;‘,‘,;:-~~==~:~~:~::;;, ’ ,;\ .;:::; : : ; ,’ ,’,,,.::~_._-___.~/._____.. _._.._._. . . . . c.r_-,‘\,,‘,‘,‘!,;; :
of vortices with different sizes, for every value of the exchange constant J. Although each structure with a global minimum of energy is formed by vortices with a fixed size, vortices with a different size can contribute to inhomogeneous metastable structures, both regular and stochastic. With the aim to prove the existence of metastable chaotic structures, we performed two numerical experiments. First we investigated the possibility of the coexistence of the regions consisting of different regular structures within one sample. With this aim we prepared a sample with the size 16 x 16 consisting of two regular regions as shown in Fig. 4a. Every vertical region forms the ground state of model (1) at different values of the exchange constant J. However, some of the calculated squares of the frequencies of small oscillation defined by the equation
r
‘Y“ .--‘,,~.\~r.rr~rr..~,~~,~,,,~,,~~I ,‘,‘.--‘..‘..“.--_--..~,~~,~ t IX -,- ‘,,’, , \ ,~.~,-,-,~_-~-_-_-~-~~_~~-; ‘\ ‘\ .,,.. _..__ ..,,., : : : ,,; t,;;. ,,--_-----,,-,-~,-,,,,
,“,
:
\:
;.‘,
,
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Fig. 5. Metastable stochastic structure for the two-dimensional model ( 1) on the 36 x 36 lattice at J = 0.1457. structural states for model ( 1) formed of clusters with random sizes. In Fig. 5 an example of stochastic structures for the lattice with size 36 x 36 is shown for .I = 0.1457. This structure was obtained by integration of the relaxation equations (2) with random initial conditions (&,,,, (0) ) at free boundary conditions. The calculation shows that the structure is dynamically stable. As one can see, this structure consists of domains of regular structures shown in Figs. 1 and 2 with different sizes of the domains randomly arranged about each other. Therefore, the nature of the chaotic structures in the two-dimensional case is identical to that of chaotic structures in one-dimensional systems, where the chaotic structures consist of regular domains with different length periods and solitons as intermediate regions between them [ 61. In conclusion we discuss two problems which concern the relaxation method to find the ground states. As we said earlier, the choice of the initial conditions (+,,, (0) ) for Eqs. (2) defines the stable structures which have to be found by the relaxation procedure (2). However, there is no certainty that this structure corresponds to the global minimum. We sought structures for each value of the constant J by randomly changing initial conditions. We did not find structures with energy lower than the energy of the vertical structure. Furthermore, we verified our results by a random search method (an analog of the Monte Carlo method at finite temperature). With the help of this method we did not find structures with energy lower than the energies of the vortex ones found earlier. mogeneous
,
KV Beloshapkin.
VV MuhidPhysics
The second problem is the periodic boundary conditions used to find the ground state of the system ( 1) . To verify the effect of the periodic boundary conditions on our results we investigated the ground state for a 24 x 24 lattice with free boundary conditions at J = 0.146. The lowest energy is realized as the square lattice of 6 x 6 vortices. This result coincides with the result for periodic boundary conditions (see Fig. lc) . Thus, we can conclude that the ground state of 2D XY models with exchange and dipole interactions is the square lattice of vortices. The sizes of the vortices increase with increasing exchange constant. Also metastable chaotic structures are shown to exist. The nature of the chaotic structures in the two-dimensional case is identical to that of the chaotic structures in one-dimensional systems. When we prepared this article, we learned of the investigation of model (1) at finite temperature by a Monte Carlo simulation [ 141. The authors of this article found a phase transition to the ordered phase characterized by a circular arrangement of the spins, which corresponds to the results of our investigation.
Letters A 233 (1997) 471-475
41s
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