Discrete breathers in an one-dimensional array of magnetic dots
The dynamics of the one-dimensional array of magnetic particles (dots) with the easy-plane anisotropy is
 investigated. The particles interact with each other via the magnetic dipole interaction and the whole system is
 governed by the set of Landau–Lifshitz equations. The spatially...
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| Cite this: | Discrete breathers in an one-dimensional array of magnetic dots / R.L. Pylypchuk, Y. Zolotaryuk // Физика низких температур. — 2015. — Т. 41, № 9. — С. 942–948
 . — Бібліогр.: 46 назв. — англ. |
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| author | Pylypchuk, R.L. Zolotaryuk, Y. |
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| citation_txt | Discrete breathers in an one-dimensional array of magnetic dots / R.L. Pylypchuk, Y. Zolotaryuk // Физика низких температур. — 2015. — Т. 41, № 9. — С. 942–948
 . — Бібліогр.: 46 назв. — англ. |
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| container_title | Физика низких температур |
| description | The dynamics of the one-dimensional array of magnetic particles (dots) with the easy-plane anisotropy is
investigated. The particles interact with each other via the magnetic dipole interaction and the whole system is
governed by the set of Landau–Lifshitz equations. The spatially localized and time-periodic solutions known as
discrete breathers (or intrinsic localized modes) are identified. These solutions have no analogue in the continuum
limit and consist of the core where the magnetization vectors precess around the hard axis and the tails
where the magnetization vectors oscillate around the equilibrium position.
|
| first_indexed | 2025-12-07T17:34:27Z |
| format | Article |
| fulltext |
© Roman L. Pylypchuk and Yaroslav Zolotaryuk, 2015
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 9, pp. 942–948
Discrete breathers in an one-dimensional array
of magnetic dots
Roman L. Pylypchuk
1
and Yaroslav Zolotaryuk
2
1
Physics Department, Ludwig-Maximilians-Universität, Theresienstrasse 37, 80333 München, Germany
2
Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, Kyiv, 03680, Ukraine
E-mail: yzolo@bitp.kiev.ua
Received April 16, 2015, published online July 24, 2015
The dynamics of the one-dimensional array of magnetic particles (dots) with the easy-plane anisotropy is
investigated. The particles interact with each other via the magnetic dipole interaction and the whole system is
governed by the set of Landau–Lifshitz equations. The spatially localized and time-periodic solutions known as
discrete breathers (or intrinsic localized modes) are identified. These solutions have no analogue in the con-
tinuum limit and consist of the core where the magnetization vectors precess around the hard axis and the tails
where the magnetization vectors oscillate around the equilibrium position.
PACS: 63.20.Pw Localized modes;
63.20.Ry Anharmonic lattice modes;
75.10.Hk Classical spin models.
Keywords: magnetic dots, antiferromagnets, Landau–Lifshitz equations.
1. Introduction
The artificially manufactured periodic arrays of mag-
netic particles have received much attention in the litera-
ture during the last two decades [1–8]. Apart from their
technological importance as candidates for the high-
density magnetic storage media [1,2,5], these arrays appear
to be a good testing ground for studying various nonlinear
magnetic wave phenomena [9–11].
Many of these phenomena, such as magnetic solitons,
domain walls, vortices are well studied and documented in
scientific literature [12,13]. Owing to the fact that they are
well described by the continuum version of the Landau-
Lifshitz (LL) equation, these solutions can be treated ana-
lytically. In some cases, if the underlying system is
integrable, the inverse scattering method has been applied
[14]. The anharmonic localization in lattices occupies a
special place among other nonlinear wave phenomena. The
discrete breathers (DBs) (also know as intrinsic localized
modes) [15–21] are time-periodic and spatially localized
excitations. Unlike their continuum counterparts that nor-
mally exist only in the integrable systems [22], discrete
breathers can exist in discrete media that are not necessari-
ly described by the integrable equations. Discrete breathers
owe their existence to the fact that the spectrum of the lin-
ear waves is bounded and as a result all the resonances
with the linear excitations can be avoided if the breather
frequency and/or the system parameters are chosen appro-
priately [23,24]. In magnetic systems DBs have been stud-
ied rather extensively [20,25–33]. In particular, experi-
mental observation of breathers in antiferromagnets has
been reported [34,35].
The bulk of these studies was performed for the Hei-
senberg magnets, where the interaction between the neigh-
bouring spins occurs via the exchange interaction. For the
Heisenberg models the exchange interaction usually domi-
nates over the single-ion anisotropies, so the breather solu-
tions in the underlying models can be treated as the weakly
discrete modes. Also, interesting solutions that have no
analogue in the continuum limit has been discussed
[29,30,36,37]. The weakness of the interspin interaction
comparing to the anisotropy (or they should be at least of
the same order) is necessary for these excitations. In the
magnetic dots arrays the interaction between the dots is of
the dipole-dipole kind, which decays as 3r (with r being
the distance between the interacting magnets). Thus, if the
array is manufactured with the spacing large enough to
guarantee the fulfilment of the necessary non-resonance
conditions, it is highly probable that DBs will exist in such
a system. To investigate the DB existence in the magnetic
dot array is the main aim of this article.
Discrete breathers in an one-dimensional array of magnetic dots
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 9 943
The paper is organized as follows. In the first section
the model of interacting magnetic dipoles is presented. The
next section is devoted to the numerical studies of discrete
breathers. Discussion and conclusions are given in the last
section.
2. The model
2.1. The Hamiltonian and equations of motion
We consider the one-dimensional array of N immobile
equidistant magnetic particles that interact as magnetic
dipoles.
The Hamiltonian of this model consists of the dipole-
dipole interaction energy between all dots and the magnet-
ic anisotropy term for each dot [38]:
2
3 3
=1
( , ) 3( , )( , )1
= .
2 | |
N N
zn m mn n mn m
n
m n n
H D
a m n
M M ν M ν M
M
(1)
Here D is the anisotropy constant, a is the distance be-
tween the adjacent dots, ,mn xν e and =nM
( , , )x y z T
n n nM M M 2 2(| | = )n MM is the magnetic dipole
momentum of the nth particle. In this article the easy-
plane anisotropy is considered with the plane of the array
(xy) being the easy plane, thus, < 0.D
It is convenient to introduce the new dimensionless var-
iables in which the total magnetic dipole momentum is
normalized to unity:
2= / , = / | | , 2 | | .n n M H H D M t D Mtμ M (2)
In the new dimensional variables the dynamics of the
magnetic moment of the nth dot is described by the dis-
crete version of the LL equation [38]:
= [ ] , n nn
Hμμ μ
( ) ( ) ( )
= ,x y zx y zn
n n n
μ e e e (3)
where the dot denotes differentiation with respect to time.
Now the system of coupled dipoles has only one parame-
ter: 3=1/ (2 | | ).D a This parameter appears as a
prefactor in the dipole-dipole term of the dimensionless
energy .H It can be treated either as a measure of the dis-
creteness of the system or as the ratio of the dipole-dipole
and exchange energies.
2.2. Dispersion law
Before embarking on studies of the nonlinear vibrations
of the array it is useful to recall the dispersion law of the
linear waves (magnons). The magnon spectrum can be
found when Eq. (3) is linearized around the obvious
ground state, where the dipoles are lying in the easy plane
and are oriented tail-to-tail: ( ) ( ) ( )=1, = = 0.x y z
n n n
Consider first the infinite array. Then the dispersion law is
well-known and reads [7,11]
2
3 3
=1 =1
cos( ) 2 cos( ) 2
( ) = 2 2 1 .L
n n
nq nq
q
n n
(4)
Typical curves for the dispersion law for the different val-
ues of the coupling constant are given in Fig. 1(a). If the
array is finite the magnon band becomes discrete. It con-
sists of the set of modes
( )
,
n
L = 1, 2, , .n N The de-
pendence of these modes as a function of the coupling con-
stant is given in Fig. 1(b). Strictly speaking, the discrete
translational invariance is lost for the finite linear array and
we can speak about it only in the approximate sense if is
small. As a result, there is a mode that is placed below the
band (see Fig. 1(b)). If the periodic boundary conditions
are applied the translational invariance of the array holds
exactly.
Fig. 1. (Color online) Dispersion law (4) for = 0.01 (curve 1),
= 0.03 (curve 2) and = 0.06 (curve 3) (a). Frequencies of the
magnon modes as a function of for the finite chain with N = 11
(blue) and N = 31 (red). Thick black lines demonstrates L(0) and
L( ) from Eq. (4) (b).
Roman L. Pylypchuk and Yaroslav Zolotaryuk
944 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 9
3. Discrete breathers and their properties
In this section we report the results of the studies of dis-
crete breathers with the help of numerical simulations.
3.1. Spontaneous localization
Discrete breathers are spatially localized excitations
that are periodic in time, i.e. =1 =1{ ( )} = { ( )}N N
n n n nt t Tμ μ ,
= 2 /T , where is the breather frequency. The con-
cept of the anti-continuum limit [17,18] is important for
constructing the DB solutions. In the current model it can
be implemented by setting = 0 . Thus, the dipole-dipole
interaction between the dots is absent and each of them can
be excited independently. If a particular dot with the num-
ber 0=n n is excited, the magnetization vector will per-
form precession around the hard axis with the frequency
( )
0
=
z
n
. Projection on the xy plane demonstrates the
following dynamics:
0 0
( ) ( )
=
x y
n n
i 2 ( )1 e .i t
Similarly, several dots located in the arbitrary places of the
array, can be excited.
In this article we will restrict ourselves to the configura-
tions that consist of the precessing core of rN dipoles.
Such an initial state can be represented as follows:
2 2
(0) 2 2
= , ,0 0
1 cos 1 cos
1 1 1 1 1 1
= 0 , 0 , , 0 , 1 sin , , 1 sin 0 , 0 , 0
0 0 0 0 0 0
n n n Nr
m . (5)
_______________________________________________
First of all we report on the simple numerical experi-
ment that demonstrates the phenomenon of dynamical lo-
calization of the magnetic dot magnetization. We take the
anti-continuum configuration (0)m from Eq. (5) as the
initial condition and integrate the LL equations numerical-
ly. The fourth order Runge–Kutta method was used. The
precision of the method was tested by monitoring the con-
servation of the total energy and the dipole moment. We
have observed different results that depend on the value of
. If this constant is sufficiently small, the localized state
Fig. 2. (Color online) Contour plots of the temporal evolution of the ( )z
n (a), (c) and ( )1 x
n (b), (d) components of the magnetization
in the array of N = 100 magnetic dots with the initially excited one [(a), (b), = 0.039] and three [(c), (d), = 0.0045] dipoles.
Discrete breathers in an one-dimensional array of magnetic dots
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 9 945
persists for rather long times. Otherwise, for larger values
of the initially excited dipoles fall into the easy plane
and localization disappears. In Fig. 2 the contour plots of
the dynamical evolution of the array magnetization are
demonstrated for = 1rN [panels (a), (b)] and = 3rN
[panels (c), (d)] initially excited dots. Here the dipoles
were initially excited with ( ) = 0.9z
n . As one can see,
energy stays with the initially excited central dots remain
in the excited state for rather long time, with the respective
magnetization vectors precessing around the hard axis. The
lifetime of the localized excitation exceeds the period of
one rotation 2 /0.9 7T by several orders of magni-
tude. Thus, the phenomenon of dynamical localization is
established. At this point we wish to know whether a local-
ized mode is an exact periodic solution that can be attribut-
ed to the excitations known as discrete breathers [19]. Be-
low we investigate these excitations in more detail.
3.2. Breather periodic orbits
In this subsection we show that time-periodic localized
modes are indeed exact solutions of the LL equation. Nu-
merically this task can be performed in the following way.
Define the evolution operator
0 0 1 2
ˆ : ( ) ( ), = col( , , , ) ,T NI t t T μ μ μm m m (6)
which stands for the integration of the LL equations (3)
along the time interval 0 0[ , ].t t T The fixed points of the
3N-dimensional map
2
0 0 0( ) ( ) | ( ) | 1= 0, =1,2, , ,nt T t t T n Nn nμ μ μ
(7)
will be the periodic solution with the period T. This map is
complemented by the term 2
0| ( ) | 1,n t Tμ which is
necessary to ensure that the normalization condition holds
after each iteration step. We start from the anti-continuum
Fig. 3. (Color online) Dynamics of the central out-of-plane and neighbouring magnetization vectors of the DB orbit on the unit sphere
for: (a) blue curve (central dot, n = 16), red curve (n = 15), other parameters = 0.022, = 0.5, N = 31; (b) same as (a) but for =
0.048, = 0.75; (c) same as (a) but for N = 20 dots with Nr = 2, blue curves correspond to the dots n = 10 and n = 11, red curve : n = 9;
(d) N = 21 dots with Nr = 3, = 0.5, = 0.016, pink curve corresponds to n = 11, blue curves n = 10 and n = 12, red curve: n = 9.
Roman L. Pylypchuk and Yaroslav Zolotaryuk
946 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 9
limit (5), “turn on” the dipole-dipole interaction by setting
> 0 and show that the spatially localized excitation per-
sist. Then can be increased gradually and the breather
periodic orbit can be followed until it ceases to exist. In
case of successful choice of the initial configuration 0( )tm
convergence with the desired precision takes place after
several iteration steps. The numerical scheme based on the
Newton method [39] has been designed [40] for finding
DB periodic oribits. It has been shown to work successful-
ly in the number of models [41], including magnetic lattic-
es [29,30]. Below we report the main results.
Starting from the anti-continuum approximation (5) and
using the method, described above, we have managed to
detect the breather periodic orbits for the different values
of and , number of dots in the array N, and number of
the precessing dipoles, .rN The phase in the initial
condition (0)m does not seem to play any role as we have
achieved conversion to the same solutions for the different
values of . The dynamics of the magnetization vector
nμ of these periodic orbits is shown in Fig. 3. The struc-
ture of DB's in all these figures is the same: it consists of
the core of few dipoles that precess around the hard axis
(although due to the interaction the precession trajectory is
tilted towards the x axis) and the weakly oscillating tails.
If the coupling constant is increased, the precessing tra-
jectory is tilted stronger toward the x axis as the in-plane
dipoles interact stronger with the precessing dipole (com-
pare Figs. 3(a) and 3(b)). Note, that the oscillations beyond
the precessing core appear to be rather weak (shown by the
red trajectories).
Next we estimate the existence area of DBs on the param-
eter plane ( , ). We remind that in order to exist, the
breather frequency together with its multiples should not
resonate with the linear waves of the system. In the anti-
continuum limit ( = 0) the allowed range of the breather
frequencies is 0 < < 1. Since the precession frequency
coincides with the z component of the magnetization vector,
cannot exceed 1. If the coupling is on, the allowed breath-
er frequencies lie in the range [0, ] ( ) < < 1maxq L q .
Thus, the existence area of DBs in the ( , ) parameter
plane coincides approximately with the upper left triangle in
Fig. 2(b) with the edges, given by = 1 , = 0 and
( )
=
N
L . We have managed to track numerically the DB
periodic orbit starting from = 0 up to the critical values
when the Newton method ceases to exist. We have found the
orbits to persist into the magnon spectrum. In that case the
breather tails do not decay asymptotically as 0, .n N In-
stead, we observe a bound breather-magnon state. However,
these solutions appear to be unstable.
The asymptotic behaviour of the breather tails is given in
Fig. 4. The decay law is close to the power law if we are not
far from the anti-continuum limit. Indeed, we observe almost
power law decay for = 0.018 with ( ) 6
01 | |x
n n n
and ( , ) 3
0| | .y z
n n n The power-law decay is in ac-
cord with other models that possess long-range interac-
tion [42–44]. As increases and the dipole-dipole inter-
action becomes more prominent. At this point we notice
that the decay law becomes faster than the power law [see
Fig. 4(b)]. This can be attributed to the fact that the array
does not possess the discrete translational invariance.
Although the absence of this symmetry is felt rather
weakly, it becomes more and more pronounced as the
coupling constant increases. Moreover, it should mani-
fest itself in the strongest way at the edges of the array, since
Fig. 4. (Color online) Spatial decay of the breather profile [1 –
| n|
(x)
(+), ( )y
n (), ( )z
n ()] on the log-log scale for the array
of N = 31 dots with = 0.75 and = 0.018 (a) and = 0.048 (b).
The solid lines in the panel (a) approximate the decay of the
magnetization (see text for details).
Discrete breathers in an one-dimensional array of magnetic dots
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 9 947
the dipoles at the edges interact only with the dipoles to the
left (to the right), while the dipoles in the middle of the array
interact symmetrically with all their neighbours. We remind,
that the discrete translational invariance is possible only
when the periodic boundary conditions are imposed.
Suppose we are not looking for the breather periodic
orbit. Instead, we are simply interested in the details of the
time evolution of the initial configuration (5) on the large
time scale. Then we obtain the quasiperiodic localized so-
lution. Its spatial structure will be the same as for the
breathers, discussed in the previous paragraphs. The time
evolution of the magnetization components appears to be
quasiperiodic, as shown in Fig. 5. Here we have excited
initially = 5rN dipoles with the precession frequency
= 0.75. As the course of evolution the localized struc-
ture persisted, but the temporal evolution exhibits two fre-
quencies: the precession frequency 0.75 and the much
lower envelope frequency. Within of one modulation peri-
od the magnetic moment can encompass the hard axis ap-
proximately ten times. It is not possible to trace the
quasiperiodic breather solution with the method used in
this section for the periodic breathers. The problem of the
quasiperiodic breather existence is an interesting problem
on its own [45,46] and will be pursued independently.
4. Discussion and conclusions
Discrete breathers (intrinsic localized modes) have been
demonstrated to exist in the one-dimensional array of
magnetic dots that interact as magnetic dipoles. We have
focused on the arrays with the easy plane anisotropy. DBs
are time periodic and spatially localized solutions of the
Landau–Lifshitz equation. The structure of the breather
solution is as follows: several dipoles in the core of the
breather rotate around the hard axis and the rest perform
small amplitude oscillations while lying in the easy plane.
It should be noted that this type of breathers has no ana-
logue in the continuum limit. The breather frequency
should not resonate with the linear modes of the array
(magnons). It appears that the area of breather existence is
limited from below by the maximal frequency of the
magnon band and by the value = 1 (in the dimensionless
units) from above.
In terms of the structure and the existence conditions
the solutions obtained in this article are similar to the DBs
in classical ferromagnetic Heisenberg chains with the easy-
plane anisotropy, obtained earlier [29,30]. The difference
is that in the Heisenberg chains the ground state is degen-
erate while for the array of magnetic dipoles it is con-
strained to the state ( ) = 1.x
n Another difference appears
due to the long-range dipole-dipole interaction, and it man-
ifests itself in the asymptotic behaviour of the magnetiza-
tion away from the breather core. Also, we believe that
experimental observation of DBs in the arrays of magnetic
particles seems to be much more easier as compared to the
previously studied Heisenberg models. While in both mod-
els (Heisenberg and magnetic dots) the breathers exist if
the interaction is considerably weaker than the anisotropy,
such a situation is rather rare for the Heisenberg lattices,
where the exchange interaction usually dominates over the
anisotropy energy. In the case of magnetic dots the interac-
tion can be chosen sufficiently weak by increasing the dis-
tance between the particles.
As far as the further research is concerned, we believe
that the following directions are of considerable interest:
(i) the influence of dissipation and external magnetic fields
(both constant and periodic) on the breather existence area;
(ii) DB existence and properties in arrays with easy-axis
anisotropy; (iii) breather existence in the two-dimensional
arrays of magnetic particles.
We thank V.P. Kravchuk for useful discussions. One of
the authors (Y.Z.) acknowledges the financial support from
the Ukrainian State Grant for Fundamental Research
No. 0112U000056.
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| id | nasplib_isofts_kiev_ua-123456789-128076 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0132-6414 |
| language | English |
| last_indexed | 2025-12-07T17:34:27Z |
| publishDate | 2015 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Pylypchuk, R.L. Zolotaryuk, Y. 2018-01-05T17:25:26Z 2018-01-05T17:25:26Z 2015 Discrete breathers in an one-dimensional array of magnetic dots / R.L. Pylypchuk, Y. Zolotaryuk // Физика низких температур. — 2015. — Т. 41, № 9. — С. 942–948
 . — Бібліогр.: 46 назв. — англ. 0132-6414 PACS: 63.20.Pw, 63.20.Ry, 75.10.Hk https://nasplib.isofts.kiev.ua/handle/123456789/128076 The dynamics of the one-dimensional array of magnetic particles (dots) with the easy-plane anisotropy is
 investigated. The particles interact with each other via the magnetic dipole interaction and the whole system is
 governed by the set of Landau–Lifshitz equations. The spatially localized and time-periodic solutions known as
 discrete breathers (or intrinsic localized modes) are identified. These solutions have no analogue in the continuum
 limit and consist of the core where the magnetization vectors precess around the hard axis and the tails
 where the magnetization vectors oscillate around the equilibrium position. We thank V.P. Kravchuk for useful discussions. One of
 the authors (Y.Z.) acknowledges the financial support from
 the Ukrainian State Grant for Fundamental Research
 No. 0112U000056. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур К 80-летию уравнения Ландау–Лифшица Discrete breathers in an one-dimensional array of magnetic dots Article published earlier |
| spellingShingle | Discrete breathers in an one-dimensional array of magnetic dots Pylypchuk, R.L. Zolotaryuk, Y. К 80-летию уравнения Ландау–Лифшица |
| title | Discrete breathers in an one-dimensional array of magnetic dots |
| title_full | Discrete breathers in an one-dimensional array of magnetic dots |
| title_fullStr | Discrete breathers in an one-dimensional array of magnetic dots |
| title_full_unstemmed | Discrete breathers in an one-dimensional array of magnetic dots |
| title_short | Discrete breathers in an one-dimensional array of magnetic dots |
| title_sort | discrete breathers in an one-dimensional array of magnetic dots |
| topic | К 80-летию уравнения Ландау–Лифшица |
| topic_facet | К 80-летию уравнения Ландау–Лифшица |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/128076 |
| work_keys_str_mv | AT pylypchukrl discretebreathersinanonedimensionalarrayofmagneticdots AT zolotaryuky discretebreathersinanonedimensionalarrayofmagneticdots |