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Chelyabinsk Physical and Mathematical Journal. 2020. Vol. 5, iss. 2. P. 194-201.
DOI: 10.24411/2500-0101-2020-15206
DISCRETE MAGNETIC BREATHERS IN MONOAXIAL CHIRAL HELIMAGNET
I.G. Bostrem1, E.G. Ekomasov2,3a, J. Kishine4, A.S. Ovchinnikov1, Vl.E. Sinitsyn1
1 Ural Federal University named after the first President of Russia B.N. Yeltsin, Yekaterinburg, Russia 2Bashkir State University, Ufa, Russia
3South Ural State University (National Research University), Che2yabin.sk, Russia 4 The Open University of Japan, Chiba, Japan a ekomasoveg@gmail.com
Intrinsic localized spin modes, or discrete breathers, are investigated in the forced ferromagnetic state of a monoaxial chiral helimagnet. The approximate solution of these excitations are obtained with the aid of discrete equations of spin dynamics. Conditions on the frequency of the breathers and on the easy-plane anisotropy are established under which the breathers are possible. In the presence of Dzyaloshinskii — Moryia interaction the localized spin modes become spatially modulated and, as a consequence, acquire the chirality. Energy of these excitations, including a pinning potential, is calculated.
Keywords: discrete breather, chiral helimagnet.
Introduction
According to generally accepted definition, a breather is a localized in space and periodic in time solution of either continous media equations or discrete lattice equations. The breather solutions were found, for example, for the exactly solvable sine-Gordon equation [1-3]. The discrete nonlinear lattices also reveal spatially localized oscillating modes. It was found out that the nonlinearity and the discretness are two pivotal ingredients supporting these excitations, which were named as discrete breathers or intrinsic localized modes [4; 5]. Most of the previous studies were focused on the discrete breathers in lattice models or crystals that constitutes nonlinear discrete systems [6-8]. However, there remains a need to find these excitations in magnetic models. Several studies were undertaken to verify discrete breathers in ferromagnets and antiferromagnets with an easy-plane anisotropy [9-11]. It has been confirmed that these intrinsic localized modes have high frequencies, above the maximum frequency of the spin-wave spectrum.
The purpose of this paper is to discuss localized spin modes in the model of monoaxial chiral helimagnet. Folllowing the idea suggested in [9] we address the so-called phase of forced ferromagnetism, where in the ground state all spins align with an external
The work is supported by the RFBR, grant no. 07-01-96002; by the Government of the Russian Federation, Act 211 of 16.03.2013, contracts no. 02.A03.21.00ll and no. 02.A03.21.0006; by the Foundation for the Advancement of Theoretical Physics and Mathematics "Basis", grant no. 17-11107.
Discrete magnetic breathers in monoaxial chiral helimagnet
195
magnetic field applied along the chiral axis [12]. Our analysis shows that the intrinsic localized modes exist and the Dzyaloshinskii — Moryia (DM) interaction does not hinder their emergence.
1. The model
We explore the spin Hamiltonian
H = -2 J ^ S„ ■ Sra+1 + A ^ (Sn)2 - Ho ^ Szn + D ^ [Sn x Sra+1]z ,
n n n n
where the first term corresponds to the exchange coupling of the strength J > 0 along the chiral axis (the z-axis), the Sn is the spin vector of the nth site. The second term means the easy-plane anisotropy with the constant A > 0, so the xy-plane being the easy-plane of magnetization. The third term describes the Zeeman coupling with the external magnetic field H0 directed along the z-axis. The last term stands for Dzyaloshinskii — Moryia interaction along the chiral axis with the strength D. It is supposed that the field H0 exceeds the treshold value Hcr = 2S (\J4J2 + D2 — J + A), thus the phase of forced ferromagnetism is stabilized [12].
The equation of motion for the operator S+ = Sn + iSn
ih-§f = [S+.H ]
has the explicit form
dS+
ih dSn dt
HoS+ — A (S+Sn + SnS+ +
+2J [S+ (Sn+1 + SZ_i) — Sn (S+_1 + S++i)] + iDSn (S+_1
By introducing the normalized classical variables, s± obtain
ih d +
------s+
2 JS dt n
Ho +
----s
2JSSn
2Bs+sn + s+
= S±/S and sn
Sn+1 + Sn_0 —
s++0 .
= V1 — s+s_, we
s
z
n
s
+
n_ 1
+ s++1
D
+ i 2JS
z
n
s
+
n_ 1
s
+
n+1
(1)
where B = A/2J.
We look for solutions in the form s+(t) = sn(t) exp (—iOt + ikna) and szn = \Jl — Sn, where a is a lattice constant, k is a wave number that will be specified later, O is the frequency. Notice, we admit that sn depends on time in contrast to the analysis of discrete breathers in ferromagnets [9].
The real and imaginary parts of Eq. (1) give, respectively,
Osn
sn
1 — sn+1+
1s
2
n_ 1
— \fl~Sn (Sn_1 + Sn+1) cos ka —
and
2 BsnVT—4
D
+ 2J A
sn (sn_1 + Sn+1) sin ka
(2)
h dsn
2JS^T
D
л/1 — sn (sn_1 — Sn+1) sin ka + —у/1 — s2n (sn_1 — Sn+1) cos ka.
2J
(3)
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I.G. Bostrem, E.G. Ekomasov, J. Kishine, A.S. Ovchinnikov, Vl.E. Sinitsyn
By introducing the phase 5 conditioned by
cos 5
1
sin 5
1 + D2
1 + 4 J2
D
2 J
1 +
---->
D2 4 J 2
we recast the system (2), (3) as
0sn —B sn\J1 + sn
1 - sn+1 + v 1 - sn-1
— л/1-Sn (Sn— 1 + sn+l)\j1 +
D2
4J2
cos (ka + 5),
ddTn = л/1 - sn (sn—1 - sn+1) у 1 + 4J sin(ka + 5).
In the last line the dimensionless time is determined, т = t/t0, where t0 = h/(2JS).
2. Discrete breathers
To find breather solutions we assume dsn/dr = 0. Firstly, we note that 5 = 0 at D = 0. Then, as it follows from Eq. (5), sin(ka) = 0 or k = 0. Eq. (4) is reduced to
0sn = — 2Bsn\/1 — sn + sn ( у 1 — sn+1 + у 1 — sn—И — л/1 — sn (sn—1 + sn+1) (6)
that has been investigated in the work [9].
For non-zero D a solution exists provided sin(ka + 5) = 0, i.e.
k
— tan a
1
D
2J
(7)
This relationship relates the wavenumber of the discrete breather with the pitch of spiral order in the conical phase, which is stable at H < Hcr.
As it has been demonstrated in the previous studies of ferromagnets, the discrete breathers are featured by an alternating sign of the sn arrangement, i. e. by staggered magnetization. To introduce a continuum description of the breather excitations, it is convenient to define the smoothly varying envelope function ф(г) = (—1)nsn, where z = na.
Taking into account the expressions
sn = (-1)n^(z), sn±1 = (-1)
1
n1
d^ a2 d2ф
m ±
/1- M 1- 2ф2И. Ф
s
1
dф(z)
n± 1 ~ 1 - -ф2^) T аф(г)~d~
and substituting them into Eq. (4) we obtain eventually the classical Duffing equation
d^2 - аф + вф3 = 0,
dz2
if to ignore nonlinear terms involving dф/dz and d2ф/dz2. Here, the dimensionless coordinate z = z/a is introduced and the notions
а
2B + 0
2 — 2\J 1 + 4J2 cos(ka + 5) 1 + 4J2 cos(ka + 5)
(8)
Discrete magnetic breathers in monoaxial chiral helimagnet
197
в
B — 1 — \J 1 + 4J2 cos(ka + S) ■\Jl + 4J2 cos (ka + S)
are adopted. We note that the results of [9], a = 2B + Q — 4 and в = B — 2, are reproduced from Eqs. (8), (9), if to take D = 0 and ka + S = 0 .
Given the imposed boundary conditions ф(±х>) = 0 and (d^/dz)|z=±^ = 0 valid for localized breather solutions we obtain
d^
\Jai\)2 — 1вф4
± (Z — Zo),
where Z0 is a constant of integration. Making use the change y2 = a — 1 вф2, we obtain
2
dV
--± (Z — Zo).
J V2 — a
The condition a > 0 must be fulfilled to get a localized solution. Eventually, we find
ф(z) =
Фт
cosh [т2^у/2в (z — zo)]
:ю)
where фт = \]2а/в, and, apparently, there arises в > 0.
In the presence of the Dzyaloshinskii — Moryia interaction the solution exists provided sin(ka + S) = 0, or cos(ka + S) = ±1. The upper sign yields conditions for the frequency
I D2
Q > 2 + 2y 1 + 4J2 — 2B,
and the easy-plane anisotropy
B > 1 +
1 +
D2 4J2.
We note that the opposite sign brings about
I D2 I D2
Q < 2 — fiA + iJ — 2B' B< 1 W1 + 4J5-
but the latter is inconsistent to the requirement of the easy-plane anisotropy, B > 0, thus, fixing cos(ka + S) = 1.
Notice that the breather solution (10) accumulates not only magnon density but a topological charge in contrast to the case of usual ferromagnet. Its value depends on the balance between the scale of breather localization, l = 4a/(фт\ф2ф) = 2a/^/a, and the pitch of the spiral, k, given by Eq. (7). By definition, the topological charge protected by the DM interaction is
^ До kl 1 л (D
Q = —- = — =-----== tan-1 —
2n 2n n^a \2J
(11)
where До is the total spin rotation angle along the path l. The spin arrangement of the discrete breather is presented in Fig. 1.
198
I.G. Bostrem, E.G. Ekomasov, J. Kishine, A.S. Ovchinnikov, Vl.E. Sinitsyn
Fig. 1. Snapshot of discrete breather of the chiral helimagnet in the forced ferromagnetic state
3. Energy of the discrete breather
By using the envelope function sn = (—1)nф (z) the energy stored in the breather amounts to
E — Eo = /4J2 + D2S2 ^ ф(па)'ф(na+a) + 2JS2 + 1hqS — AS2 ^ ф2(иа), (12)
nn
where E0 = —2JS2N — H0SN + AS2N is the energy of the ferromagnetic background. The direct calculation of the sum in the continuum limit gives
/“ dz
Ф (z)
а
Ф (Z) + а1гФ (Z)
dz /2, . 4/a
N(z) = —
“ dz
2 4\fa
N(z) =
Therefore, the energy of the chiral breather is
AE(1) = E- Eo
4/a
~T
/4J2 + D2 S2 + 2 JS2 + 1 Ho S — AS
2
(13)
This expression does not take into account discreteness of the lattice giving rise to a pinning energy [13]. To calculate this contribution we use the continuum solution as
f (z — zo) =
Фт
cosh [la V2e (z — zo)] ’ where dependence on the central coordinate zo is clearly indicated, and rewrite (12)
E — Eo = dz
S (z — na)
x
/ 4J2 + D2 S2 f (z — zo )f (z — zo + a) + 2 J S2 + ^ Ho S — AS2 f2 (z)
By using the identity
12
2nnz
S(z — na) = 1 + a cos T
n n= 1
(14)
(15)
and substituting it into (14), one can see that the first term of Eq. (15) leads to AE(1) while the second is related with the pinning energy
AE(2) = 2/4J2 + D2S2 ^
n=1 -™
™ dz 2nn
cos [z + zo ] f (z )f (z + a) +
a
a
Discrete magnetic breathers in monoaxial chiral helimagnet
199
+2 (2 JS2 + 1 HoS - ASAjr /°
' ' n=1 ^-c
dz f 2 nn
— cos ------
a \ a
[z + zo] f 2(z).
Calculation of the integrals
4n
dz cosf2^ [z + zo]) f (z)f (z + a)
a
4n2n
в sinh^ П2П
dz
cos
a
( 2 nnz0 a
+ n sin
( 2 nnz0
dz f 2nn r , \ 2
I2n = — cos 1 ~ ' ~ 1 ' * '
aa
[z + zo] f 2(z)
a
4n2n cos (^f0 в sinh f -7n%)
yields the result
AE(2) = 2 ^
n=1
where the term with n =1 dominates.
4. Conclusions
In summary, we demonstrate in our study that there are intrinsic nonlinear spin excitations, discrete breathers, in the state of forced ferromagnetism of the monoaxial chiral helimagnet.
The authors wish to acknowledge Professor V.D. Buchelnikov for support and attention to the work of the Ekaterinburg and Ufa schools of theoretical physicists in the field of statics and dynamics of magnetic inhomogeneities.
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V4 J2 + D2 S2hn + ( 2 JS2 + 1 HoS - AS2) I,
2
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I.G. Bostrem, E.G. Ekomasov, J. Kishine, A.S. Ovchinnikov, Vl.E. Sinitsyn
10. Rakhmanova S., Mills D.L. Nonlinear spin excitations in finite Heisenberg chain. Physical Review B, 1996, vol. 54, pp. 9225-9231.
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Article received 06.05.2020 Corrections received 20.05.2020
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Челябинский физико-математический журнал. 2020. Т. 5, вып. 2. С. 194-201.
УДК 530.182.1 DOI: 10.24411/2500-0101-2020-15206
ДИСКРЕТНЫЕ МАГНИТНЫЕ БРИЗЕРЫ В ОДНООСНЫХ ХИРАЛЬНЫХ ГЕЛИМАГНЕТИКАХ
И. Г. Бострем1, Е. Г. Екомасов2,3", Дж. Кишине4, А. С. Овчинников1, Вл. Е. Синицын1
1 Уральский федеральный университет имени первого Президента России Б. Н. Ельцина, Екатеринбург, Россия
2 Башкирский государственный университет, Уфа, Россия 3Южно-Уральский государственный университет (национальный исследовательский университет), Челябинск, Россия 4 Открытый университет Японии, Чиба, Япония
" ekomasoveg@gmail.com
В форсированном ферромагнитном состоянии моноаксиального хирального гелимаг-нетика исследуются собственные локализованные спиновые моды, или дискретные бризеры. Получено приближённое решение этих возбуждений с помощью дискретных уравнений спиновой динамики. Устанавливаются условия для частоты бризеров и для анизотропии типа легкая плоскость, при которых бризеры возможны. При наличии взаимодействия Дзялошинского — Мории локализованные спиновые моды становятся пространственно модулированными и, как следствие, приобретают ки-ральность. Вычислена энергия этих возбуждений, включая потенциал пиннинга.
Ключевые слова: дискретный бризер, хиральный гелимагнетик.
Поступила в редакцию 06.05.2020 После переработки 20.05.2020
Сведения об авторах
Бострем Ирина Геннадьевна, кандидат физико-математических наук, доцент, доцент кафедры теоретической и математической физики, Институт естественных наук и математики, Уральский федеральный университет имени первого Президента России Б. Н. Ельцина, Екатеринбург, Россия; irina.bostrem@yandex.ru.
Екомасов Евгений Григорьевич, доктор физико-математических наук, профессор, профессор кафедры теоретической физики, Башкирский государственный университет, Уфа, Россия; научный сотрудник лаборатории функциональных материалов, ЮжноУральский государственный университет (национальный исследовательский университет), Челябинск, Россия; ekomasoveg@gmail.com.
Кишине Джан-ичиро, доктор наук, профессор, Отделение естественных наук и наук об окружающей среде, Открытый университет Японии, Чиба, Япония; kishine@ouj.ac.jp. Овчинников Александр Сергеевич, доктор физико-математических наук, доцент, профессор кафедры теоретической и математической физики, Институт естественных наук и математики, Уральский федеральный университет имени первого Президента России Б. Н. Ельцина, Екатеринбург, Россия; alexander.ovchinnikov@urfu.ru.
Синицын Владимир Евгеньевич, кандидат физико-математических наук, доцент кафедры теоретической и математической физики, Институт естественных наук и математики, Уральский федеральный университет имени первого Президента России Б. Н. Ельцина, Екатеринбург, Россия; sinitsyn_ve@mail.ru.
Работа поддержана грантом РФФИ, проект № 07-01-96002; Правительством РФ, Постановление № 211, контракты № 02.A03.21.0011 и № 02.A03.21.0006; Фондом развития теоретической физики и математики «Базис», грант № 17-11-107.
