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Singular optics of spin waves in a two-sublattice antiferromagnet with uniaxial magnetic
anisotropy
Yu. I. Gorobets, and O. Yu. Gorobets
Citation: Low Temperature Physics 43, 564 (2017);
View online: https://doi.org/10.1063/1.4985212
View Table of Contents: http://aip.scitation.org/toc/ltp/43/5
Published by the American Institute of Physics
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LOW TEMPERATURE PHYSICS
VOLUME 43, NUMBER 5
MAY 2017
Singular optics of spin waves in a two-sublattice antiferromagnet with uniaxial magnetic
anisotropy
Yu. I. Gorobetsa)
Institute of Magnetism, National Academy of Sciences and Ministry of Education and Science of Ukraine,
Kiev 03142, Ukraine
O. Yu. Gorobetsb)
Institute of Magnetism, National Academy of Sciences and Ministry of Education and Science of Ukraine,
Kiev 03142, Ukraine and National Technical University of Ukraine “KPI,” Kiev 03056, Ukraine
(Submitted September 30, 2016)
Fiz. Nizk. Temp. 43, 707–713 (May 2017)
Based on exact 3D solutions of the Landau-Lifshitz equations in a two-sublattice antiferromagnet
with uniaxial magnetic anisotropy, the existence of nonlinear spin waves with singular points on
the wavefront is predicted. These waves are spin-wave analogs of optical singularities. Published
by AIP Publishing. [http://dx.doi.org/10.1063/1.4985212]
recently been experimentally demonstrated by creating radiofrequency vortices with two different states of orbital angular
momentum.13 This latest technology, in principle, allows the
transmission of an unlimited number of channels within a
given fixed frequency band.13 In view of new technical possibilities for detecting the phase and observing the spin-wave
front, this paper theoretically shows the possibility of propagation of nonlinear spin waves with modulated fronts and threedimensional solitons in a two-sublattice antiferromagnet with
uniaxial magnetic anisotropy based on three-dimensional nonlinear solutions of the Landau-Lifshitz equations.
In particular, the modulation of the front of such nonlinear spin excitations can contain wavefront singularities of
the type of optical screw dislocations of different orders,14,15
as well as the plane singular wavefield points of the circulation, source, and sink types.16
Introduction
Recently, high resolution, both spatially and temporally,
techniques for measuring the intensity distribution of spin
waves have been successfully developed.1–4 Using these techniques, the behavior of one- and two-dimensional spin-wave
packets—envelope and bullet spin-wave solitons—has been
studied, including their free propagation, collisions, parametric
excitation and wavefront reversal.5–7 Lately, experimental
approaches have been developed that allow to obtain information about the phase of wave packets, therefore enabling the
observation of a spin-wave front with a total two-dimensional
phase resolution and detection of the phase of linear and nonlinear spin-wave packets.8 At present, the inversion of the
wavefront of spin-wave pulses is effectively used for processing ultra-high frequency signals.9 The use of peculiarities of
the wavefront of spin waves for the transmission of information has mainly been investigated in ferromagnetic materials
and ferrites. However, new promising materials for spin-wave
electronics—antiferromagnets—have also been considered
since they allow operating at higher frequencies.10
In addition, in connection with the technical capabilities of
recording the phase and observing the wavefront of spin waves,
it seems relevant to apply the methods developed and realized
in singular optics (i.e., in the wave optics of helical fields)11 to
create singularities on the front of linear and nonlinear spin
waves for recording and transmitting information. To date, the
ideas of singular optics have been successfully used for a wide
range of electromagnetic wave lengths, as well as for wave
fields of a different nature. For instance, such examples include
singular electron optics12 and vorticity of radio waves.13 In the
latter case, the possibility of coding and simultaneous transmission of several radio channels at the same frequency has
Main section
Let us consider a two-sublattice antiferromagnet with uniaxial magnetic anisotropy and sublattice magnetizations M1
¼ M2, jM1j ¼ jM2j ¼ M0, where the magnetization modulus
of both sublattices is M0 ¼ const. Taking into account the conditions for the constancy of the modulus of the antiferromagnetism vector jLj ¼ L0 ¼ const, we select a parametrization
Lx ¼ 2M0 sin h cos u;
Lz ¼ 2M0 cos h;
Ly ¼ 2M0 sin h sin u;
where h and u are the polar and azimuth angles for the antiferromagnetism vector and Lx, Ly, and Lz are the Cartesian
coordinates of the antiferromagnetism vector. Then the
Landau-Lifshitz equations describing the dynamics of the
vector L have the form17
8 >
@
@u
>
2
>
c2 div ðruÞ sin2 h ¼ 0
>
< @t sin h @t xH
"
2 #
2
>
@
h
@u
>
2
2 2
2
2
>
>
sin h cos h ¼ 0;
: @t2 c r h þ x0 sgnðb1 Þ þ c ðruÞ @t xH
1063-777X/2017/43(5)/6/$32.00
(1)
564
(2)
Published by AIP Publishing.
Low Temp. Phys. 43 (5), May 2017
Yu. I. Gorobets and O. Yu. Gorobets
565
where xH ¼ gH0, g ¼ 2lh 0 (l0 is the Bohr magneton, ¯ is the Planck constant, and H0 is the external magnetic field strength),
(
sgnðb1 Þ ¼
1; b1 > 0;
1; b1 < 0;
c¼
4l0 M0 pffiffiffiffiffiffiffiffiffi
4l M0 pffiffiffiffiffiffiffiffiffiffiffi
Aa1 ; x0 ¼ 0
Ajb1 j;
h
h
(3)
where A is the homogeneous exchange energy constant, a1 is the inhomogeneous exchange constant, and b1 is the uniaxial
magnetic anisotropy constant. Equation (2) has the following particular three-dimensional nonlinear solutions18–21 [the derivation of Eq. (2) and the method of obtaining particular solutions are considered in Ref. 22]:
8
z vP t
x y
>
>
;
þf
>
< h ¼ 2arctanfH Pð x; y; zÞ g; Pð x; y; zÞ ¼ p
l0
l0 l0
Ð
>
z vQ t
x y
>
>
þ xH ðtÞdt;
;
þg
:u ¼ q
l0
l0 l0
where x, y, and z are the Cartesian coordinates of the radius
vector to an arbitrary point in the antiferromagnet, v is the
qffiffiffiffiffiffi
velocity of a nonlinear spin wave, l0 ¼ jba1 j ¼ xc00 and the
1
parameters p and q are determined by the expressions
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sgnðb1 Þ
p ¼ 0; q ¼ 6
1 v2Q =c2
(5)
or
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sgnðb1 Þ
p¼6
; q ¼ 0;
1 v2P =c2
(6)
and the function H(P) has the form
H ð PÞ ¼
b0
pffiffiffiffiffiffiffiffi
;
dn c0 jC1 jP; k1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffi
þ 1þ4C1
1þ2C1 1þ4C1
¼
where c0 ¼ 1þ2C12jC
,
b
,
0
2jC1 j
1j
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffi
2 1þ4C
1
pffiffiffiffiffiffiffiffiffiffi
¼ 1þ2C
, 14 < C1 < 0, and 0 < k1 1, or
1 þ 1þ4C1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 snðP; k2 Þ
H ð PÞ ¼
;
1 þ snðP; k2 Þ
(7)
k1
(8)
1
, C1 > 0 and 0 < k2 1. The functions
where k2 ¼ pffiffiffiffiffiffiffiffiffiffi
1þ4C
1
f(X, Y) and g(X, Y) have the form
8
X
X
2
>
>
n~~ lnðjr r0i jÞ þ k1;2 K ðk1;2 Þ
n~i ai þ C2
f ð X; Y Þ ¼
>
>
>
p
>
i
i
>
>
>
>
X X AðiÞ ð Þ
>
ð iÞ
>
i
n
>
þ
B
cos
na
þ
C
sin
na
;
>
i
i
n
<
jr r0i jn n
n
i
X
X
2
>
>
>
ai n~~i þ C3
n~i lnðjr r0i jÞ þ
gð X; Y Þ ¼ k1;2 K ðk1;2 Þ
>
>
p
>
i
i
>
>
>
>
X X AðiÞ ð Þ
>
>
ð iÞ
i
n
>
þ
C
cos
na
þ
B
sin
na
;
>
i
i
n
n
n
:
jr r0i j
n
i
(9)
where the following notation is introduced:
X¼
(4)
x
y
; Y¼ ;
l0
l0
(10)
where r is a two-dimensional vector with the coordinates
r ¼ (X, Y) in the XOY plane, r0i is a two-dimensional vector
with the coordinates in the XOY plane perpendicular to the
propagation direction of the spin wave—r0i ¼ (X0i, Y0i)
where X0i and Y0i are some dimensionless constants,
YY0i
, i, n, n~i , n~~ i are integers, and
ai ¼ arctan XX
0i
8
>
< 0;
HðnÞ ¼
>
: 1;
p
n0
n>0
ð2
;
dn
K ðkÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi :
1 k2 sin2 n
(11)
0
Note that the expression for the function f(X, Y) in Eq.
(9) is an expansion in powers of jr r0i j of an arbitrary harmonic function of two variables X and Y, and the expression
for the function g(X, Y) in Eq. (9) represents an expansion in
powers of jr r0i j of a harmonic function of the same two
variables X and Y, which is the conjugate function with
respect to the function g(X, Y). This means that the functions
f(X, Y) and g(X, Y) are connected by the Cauchy-Riemann
conditions22 and are the eigenfunctions of the twodimensional Laplace operator.
It should be noted that in the static case (i.e., for @h
@t ¼ 0
¼
0),
Eq.
(2)
describing
the
coordinate
dependences
and @u
@t
of the polar and azimuth angles of the antiferromagnetism
vector in a two-sublattice antiferromagnet with uniaxial
magnetic anisotropy coincide with the corresponding equations for the spatial distribution of the polar and azimuth
angles of the magnetization vector in a ferromagnet with uniaxial magnetic anisotropy in the exchange approximation17
(i.e., in the case when the magnetostatic energy of the ferromagnet can be neglected17) Therefore, the functional form
of all the solutions of Eq. (2) obtained in the present paper
for vP ¼ 0 and vQ ¼ 0 also extends to the case of static magnetization distributions of a ferromagnet with an easy-axis
anisotropy or “easy plane.” In this sense of the word, in analyzing the particular cases of solutions (4), we will also discuss the known analogous solutions in a ferromagnet.
Let us analyze the obtained three-dimensional nonlinear
solutions (4) of the Landau-Lifshitz equations for a
566
Low Temp. Phys. 43 (5), May 2017
Yu. I. Gorobets and O. Yu. Gorobets
two-sublattice antiferromagnet with uniaxial magnetic
anisotropy (2). For example, let us choose the function H(P)
in the form (8) and consider the external magnetic field
H0 ¼ 0. Also, for definiteness, we consider the case of
magnetic anisotropy of the easy-axis type with the parameters p and q determined by Eq. (6). Then the projections of
the antiferromagnetism vector on the axes of the Cartesian
coordinate system take the form
8
!
>
1
z vP t
x y
x y
>
>
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
>
;
k
;
¼
2M
cn
;
;
L
þ
f
cos
g
x
0
2
>
>
l0
l0 l0
l0 l0
1 v2P =c2
>
>
>
>
>
>
! <
1
z vP t
x y
x y
ffi
þ
f
;
k
;
Ly ¼ 2M0 cn pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
;
;
2 sin g
l0
l0 l0
l0 l0
>
1 v2P =c2
>
>
>
>
>
!
>
>
1
z vP t
x y
>
>
ffi
L ¼ 2M0 sn pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
;
; k2 :
þf
>
>
: z
l0
l0 l0
1 v2P =c2
Expression (12) represents a nonlinear
propa
spin wave
gating along the Oz axis. For f
x y
l0 ; l0
¼g
x y
l0 ; l0
¼ 0, solu-
tion (12) is a nonlinear plane wave. It is known that the
wavefront of the wave beams that are similar in properties to
a plane wave looks like a family of disjoint surfaces. The
distance between the adjacent surfaces
wave is equal to the x y
x y
length. Furthermore, the functions f l0 ; l0 and g l0 ; l0 represent the modulation of the plane wavefront. In optics, the
deviations of wavefronts from planarity occurring in real
beams are called optical aberrations. However, all the aberrations considered in the classical theory deform a wavefront
without changing its topology. A different picture is
observed in the presence of optical vortices in a monochromatic wave.11–15 If such vortices appear, then singular points
arise on the wavefront surface, which are in many respects
analogous to crystal lattice defects known in solid state physics—screw dislocations—and have the same name.11–15 At
the very singular point, the amplitude of light oscillations is
zero, and the phase value is not defined. In its vicinity, there
are sharp collapsing phase changes. Because of this singularity, the phase distribution function belongs to the class of
singular functions, which is the reason for the term “singular
optics” mentioned above. As will be shown below, a spin
wave
of
the type
(12)
with a specific choice of the functions
x y
x y
f l0 ; l0 and g l0 ; l0 can contain singular points, including
screw dislocations, on the surface of the wavefront [if we
use the terminology adopted for electromagnetic waves in
the optical range for the spin waves of the type (4)]. Thus,
in optics, wavefront points such that the phase of light
oscillations along the path traced around them in a plane
perpendicular to the propagation direction changes by 2pl,
where l is a non-zero integer are called screw dislocations.
The quantity l is called the dislocation order or the topological charge of the wavefront surface. The amplitude of
the electric field E(r, a) near optical screw dislocations r
! 0 of the order l is described by the expression E(r, a)
¼ rlexp(6ila).14
Depending on the direction of the twist of the
wave surface “screw,” screw dislocations are classified
(12)
as left (negative) and right (positive) dislocations. On
the surface of the wavefront, there can be both a single
screw dislocation and an entire system of dislocations.
The appearance of screw dislocations radically changes
the topology of the wave front. The equiphase surface
ceases to be multi-sheeted, and a transition to a single
surface with a specific helical structure takes place
(Fig. 1).11–15
To describe analogous singularities on the front of a spin
wave in Eq. (12), we consider the oscillations of the antiferromagnetism vector in the vicinity of a singular point in the
plane perpendicular to the propagation direction of the spin
wave z ¼ vpt. Here and below, the presence of the index i in
the respective coefficients actually means the possibility of
describing the system of singular points with radius vectors
r ¼ r0i in a plane perpendicular to the propagation direction
of the spin wave.
In this case, we take the asymptotes for jr r0ij ! 0 in
Eq.
(9) for the full expansion of the harmonic function
f
x y
l0 ; l0
and its conjugate harmonic function g
x y
l0 ; l0
in
powers of jr r0ij. Let us consider such a perturbation of the
wave front, for which the coefficients at ln(jr r0ij) and negative integer powers of jr r0ij vanish. Substituting this
asymptotes into Eq. (12), we obtain:
Fig. 1. Wavefront structure in the absence (a) and in the presence of a screw
dislocation (b); k is the wavelength.11–15
Low Temp. Phys. 43 (5), May 2017
Yu. I. Gorobets and O. Yu. Gorobets
8
ðiÞ
ðiÞ
ðiÞ
ðiÞ
ðiÞ
ðiÞ
>
>
L ¼ 2M0 cnðAl jr r0i jl ðBl cos nai þ Cl sin lai Þ; k2 Þ cos ðAl jr r0i jl ðCl cos lai Bl sin lai ÞÞ;
>
< x
ðiÞ
ðiÞ
ðiÞ
ðiÞ
ðiÞ
ðiÞ
Ly ¼ 2M0 cnðAl jr r0i jl ðBl cos lai þ Cl sin lai Þ; k2 Þ sin ðAl jr r0i jl ðCl cos lai þ Bl sin lai ÞÞ;
>
>
>
: L ¼ 2M cnðAðiÞ jr r jl ðBðiÞ cos la þ CðiÞ sin la Þ; k Þ:
z
0
0i
l
i
l
l
In Eq. (13), from the whole series (9) of the expansion
of a harmonic function in powers of jr r0ij for jr r0ij !
0, only the term with n ¼ l remains, where l > 0 is the minimum positive power of jr r0ij in the respective series.
Also, taking into account the condition jr r0ij ! 0, in Eq.
(13) the trigonometric as well as elliptic sine and cosine can
be expanded in a Taylor series up to linear terms in the argument of these functions
8
>
>
< Lx 2M0 ;
ðiÞ
ðiÞ
ðiÞ
Ly ¼ 2M0 Al jr r0i jl ðCl cos lai Bl sin lai Þ; (14)
>
>
: Lz ¼ 2M0 AðiÞ jr r0i jl ðBðiÞ cos lai þ CðiÞ sin lai Þ:
l
l
l
The expansion (14) is valid if
ðiÞ
ðiÞ
ðiÞ
jAl jr r0i jl ðBl cos lai þ Cl sin lai Þj < jKðk20 Þj
ðiÞ
jCl
and
ðiÞ
Bl
cos lai sin lai j < 1;
pffiffiffiffiffiffiffiffiffiffiffiffiffi
where k0 ¼ 1 k2 .
If we choose a wavefront perturbation for which the
ðiÞ
ðiÞ ðiÞ
coefficient Bl ¼ 0 and denote CðiÞ ¼ Al Cl , then it follows
from Eq. (14):
8
>
< Lx 2M0 ;
Ly 2M0 CðiÞ jr r0i jl cos lai ;
(15)
>
:
Lz 2M0 CðiÞ jr r0i jl cos lai :
It is evident from Eq. (15) that in the vicinity of a singular point with a radius vector r r0i in the plane perpendicular to the propagation direction of the spin wave, the
magnitude of the deviation of the antiferromagnetism vector
from the uniform direction tends to zero as jr r0ij ! 0,
and for a path enclosing the singular point, the phase
increases by 2pl. This allows us to treat Eq. (15) as a spinwave analogy of an optical screw dislocation. Depending on
the rotation direction of the antiferromagnetism vector,
which is determined by the sign of the topological charge l,
Eq. (15) describes left (negative) or right (positive) screw
dislocations (Fig. 2).
If, in solution (12), we do not assume vanishing coefficients at ln(jr r0ij) or at integer negative powers of jr r0ij, singularities with a nonzero amplitude on the front of a
spin wave can be obtained. The following example of the
spin-wave singularity illustrates the case for which the amplitude of the deviation of the antiferromagnetism vector from
the uniform direction is finite for jr r0ij ! 0, and for a
path enclosing the singular point, the phase increases by 2pl:
8
>
~
~
>
< Lx ¼ 2M0 cnðn~ i lnðjr r0i jÞ; k2 Þ cos ðai n~ i þ C3 Þ;
~
Ly ¼ 2M0 cnðAn~ i lnðjr r0i jÞ; k2 Þ sin ðai n~~ i þ C3 Þ; (16)
>
>
: L ¼ 2M snðn~~ lnðjr r jÞ; k Þ:
z
0
i
0i
2
i
567
(13)
2
Despite the fact that, except for special cases, the superposition principle is not valid for solutions of nonlinear
equations, in solutions (4) the superposition principle for
harmonic functions f ðlx0 ; ly0 Þ and gðlx0 ; ly0 Þ under the sign of the
function H(P) is realized. Specifically, the sum of harmonic
functions is also harmonic, and their expansions into series
(9) are possible due to the linearity of the Laplace operator,
the eigenfunctions of which they are. Therefore, for a nonlinear spin wave in an antiferromagnet considered in this paper,
both a single singular point as well as an entire system of
singularities with an arbitrary arrangement on the wavefront
surface can appear.
Several interesting particular cases of solutions of the
type (4) should also be mentioned. It is well known that for
the modulus of the elliptic function equal to one, the elliptic
functions degenerate into hyperbolic functions, and Eq. (8)
for the function H(P) is considerably simplified:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 tanhðPÞ
:
(17)
H ð PÞ ¼
1 þ tanhðPÞ
In this case, in the parametrization (1), the usual trigonometric sine and the cosine of the polar angle for the antiferromagnetism vector are given by the expressions
8
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
>
>
>
¼
2M
sin
h
cos
u
¼
2M
1 tanh2 ðPÞ cos u;
L
0
0
< x
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
(18)
Ly ¼ 2M0 sin h sin u ¼ 2M0 1 tanh2 ðPÞ sin u;
>
>
>
: L ¼ 2M cos h ¼ 2M thðPÞ;
z
0
0
where P and u are given by Eq. (4).
The expression Lz ¼ 2M0 tanhðPÞ in the last equation
resembles a kink-like soliton (monopole).24 Formally, a kink
can be introduced as a solution of the Korteweg-de Vries,25
the nonlinear Schr€odinger,26 or the sin-Gordon27 equations,
described by a hyperbolic tangent. Inverting the sign of the
kink-type solution produces an anti-kink. In the general case,
Fig. 2. Two types of singularities for the rotating components of the antiferromagnetism vector: circulation23 according to Eq. (16) for C3 ¼ 6(p/2) and
n~~ i ¼ 1 (a); source or sink23 according to Eq. (15) for l ¼1 or Eq. (16) for
n~~ j ¼ 1 and C3 ¼ 0 or p (b).
568
Low Temp. Phys. 43 (5), May 2017
Yu. I. Gorobets and O. Yu. Gorobets
Lz ¼ 2M
0 tanhðPÞ is a three-dimensional kink, and for
x y
f l0 ; l0 ¼ 0 and g lx0 ; ly0 ¼ 0 it becomes one-dimensional.
Furthermore, Eq. (8) for the function allows a significant
simplification when the modulus of the elliptic function is zero
since in this case the elliptic functions degenerate into ordinary
trigonometric sine and cosine. Taking into account the above
limiting case, Eq. (8) for the function H(P) takes the form
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 sin ðPÞ
:
(19)
H ð PÞ ¼
1 þ sin ðPÞ
At the same time, in the parametrization (1), the usual
trigonometric sine and cosine of the polar angle for the antiferromagnetism vector are also expressed in terms of trigonometric sine and cosine:
8
>
< Lx ¼ 2M0 sin h cos u ¼ 2M0 cosðPÞ cos u;
(20)
Ly ¼ 2M0 sin h sin u ¼ 2M0 cosðPÞ sin u;
>
:
Lz ¼ 2M0 cos h ¼ 2M0 sin ðPÞ;
where P and u are also given by Eq. (4).
From the above correspondence between the solutions
(4) and analogous solutions for the magnetization distributions in a ferromagnet with uniaxial magnetic anisotropy, it
is worth noting
that for vP ¼0 inthe one-dimensional case
(i.e., for f
x y
l0 ; l0
¼ 0 and g
x y
l0 ; l0
¼ 0, P ¼ PðzÞ ¼
z
l0 ),
the
substitution of Eq. (18) into Eq. (1) yields a flat domain
boundary.28,29 In the same sense, the substitution of Eq. (12)
into Eq. (1) in the one-dimensional case for vP ¼ 0 describes
the magnetization distribution in a ferromagnet, obtained for
the first time by Shirobokov30
8
>
z
>
>
Lx ¼ 2M0 cn
; k2 ;
>
>
l0
<
Ly ¼ 0;
(21)
>
>
z
>
>
>
: Lz ¼ 2M0 sn l0 ; k2 :
This and the following equations can be used to draw an
analogy between the solutions (4) and static distributions of
the magnetization of a ferromagnet if we set L ¼ M—the
magnetization vector of a ferromagnet, and MS ¼ 2M0 —the
saturation magnetization of a ferromagnet. However, these
equations simultaneously describe nonlinear static solutions
of Eq. (2) in a two-sublattice antiferromagnet.
Let us also consider the above correspondence between
the solutions (4) and analogous solutions for the magnetization distributions in a ferromagnet with uniaxial magnetic
anisotropy in the two-dimensional case (i.e., in the absence
of the dependence of the angles h and u on the coordinate z,
which occurs in a magnet without anisotropy, i.e., for
b1 ¼ 0). Then for vP ¼ 0 and for the following choice of the
functions f(X, Y) and g(X, Y)
(
f ðX; YÞ ¼ n~~ lnðrÞ;
(22)
gðX; YÞ ¼ an~~ ;
the substitution of Eq. (18) into Eq. (1) yields the
well-known two-dimensional Belavin-Polyakov soliton in
an isotropic ferromagnet as a special case of the solutions (4)31
8 n~~
>
<
h
1
¼
tan
;
(23)
2
r
>
:
~
u ¼ n~ a;
or, in a different notation
8
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
!2
u
>
>
u
n~~
n~~
>
r
r
>
t
>
>
Lx ¼ 2M0 1 ~
cos an~~ ;
>
>
n~ þ r n~~
>
r
>
>
>
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
>
!2
<
u
~
~
u
r n~ r n~
t
~
>
> Ly ¼ 2M0 1 r n~~ þ r n~~ sin a n~ ;
>
>
>
>
!
>
>
>
n~~
n~~
>
r
r
>
>
:
>
: Lz ¼ 2M0 n~~
r þ rn~~
(24)
Thus, for particular soliton cases, the superposition of
individual terms in the expansion of the functions f(X, Y) and
g(X, Y) in powers of jr r0ij actually represents a
“superposition” of the shape modulations of threedimensional moving solitons.
Conclusion
Special cases of the considered type of the solutions of
the Landau-Lifshitz equations for an antiferromagnet with
uniaxial magnetic anisotropy, when put side by side with the
static solutions for the magnetization distributions of a uniaxial ferromagnet, include well-known solutions of the type
of the two-dimensional Belavin-Polyakov solitons,31 onedimensional Shirobokov solitons,30 Bloch domain boundary,28 Khodenkov solitons,32 target soliton,33 as well as
some other known nonlinear solutions. Furthermore, this
class of solutions of the Landay-Lifshitz equations for an
antiferromagnet with uniaxial magnetic anisotropy demonstrates the fundamental possibility of realizing vortex optics
based on nonlinear spin waves.34
This project has received funding from the European
Union’s Horizon 2020 research and innovation program
under the Marie Skłodowska-Curie Grant Agreement No.
644 348 (MagIC).
a)
Email: [email protected]
Email: [email protected]
b)
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Translated by L. Gardt
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