Феромагнітна фаза одновісного магнетика у присутності анізотропної біквадратичної обмінної взаємодії
Дослiджено феромагнiтну фазу (ФМФ) одноосьового магнетика з одноiонною анiзотропiєю (ОА) типу “легка площина” та анiзотропною бiквадратною обмiнною взаємодiєю (БОВ). Розглянуто випадок, коли значення вузлового спiну дорiвнює одиницi S = 1. Одержано вирази для двох гiлок спектра спiнових збуджень при...
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nasplib_isofts_kiev_ua-123456789-134022025-02-09T13:27:54Z Феромагнітна фаза одновісного магнетика у присутності анізотропної біквадратичної обмінної взаємодії Ферромагнитная фаза одноосного магнетика в присутствии анизотропного биквадратичного обменного взаимодействия Ferromagnetic Phase of a Uniaxial Magnet with Anisotropic Biquadratic Exchange Шаповалов, І. Тверде тіло Дослiджено феромагнiтну фазу (ФМФ) одноосьового магнетика з одноiонною анiзотропiєю (ОА) типу “легка площина” та анiзотропною бiквадратною обмiнною взаємодiєю (БОВ). Розглянуто випадок, коли значення вузлового спiну дорiвнює одиницi S = 1. Одержано вирази для двох гiлок спектра спiнових збуджень при скiнченних температурах та визначено умови стiйкостi мод спектра. Побудовано дiаграму стiйкостi мод спектра в координатах T - h, з якої випливає, що за певних умов у системi зi зниженням температури спочатку вiдбувається порушення стiйкостi мод спектра, а потiм, з подальшим зниженням температури, стiйкiсть мод спектра вiдновлюється, тобто спостерiгається реєнтрантна поведiнка. Доведено, що температура фазового переходу (ФП) другого роду мiж ФМФ та фазою зi спонтанно порушеною симетрiєю суттєво залежить вiд константи анiзотропiї БОВ. Исследована ферромагнитная фаза (ФМФ) одноосного магнетика с одноионной анизотропией (ОА) типа легкая плоскость и анизотропным биквадратным обменным взаимодействием (БОВ). Рассмотрен случай, когда значение узельного спина равно единице. Получены выражения для двух ветвей спектра спиновых возбуждений при конечных температурах и определены условия устойчивости мод спектра. Построена диаграмма устойчивости мод спектра в координатах T - h, из которой следует, что при определенных условиях в системе с понижением температуры сначала происходит потеря устойчивости мод спектра спиновых возбуждений, а затем, при дальнейшем понижении температуры, устойчивость мод спектра восстанавливается, т. е. наблюдается реентрантное поведение. Доказано, что температура фазового перехода (ФП) второго рода между ФМФ и фазой со спонтанно нарушенной симметрией существенно зависит от константы анизотропии БОВ. The ferromagnetic phase (FMP) of a uniaxial magnet with the easy-plane single-ion anisotropy (SIA) and the anisotropic biquadratic exchange interaction (BQEI) has been studied. The case S = 1 for the site spin S has been considered. Expressions for two branches of the spin excitation spectrum at finite temperatures T have been obtained, and the conditions for spectral mode stability have been determined. The spectral mode stability diagram in the T h coordinates has been constructed. The diagram testifies that, under certain conditions, the temperature decrease is accompanied by a violation of the spectral mode stability followed, as the temperature decreases further, by its restoration; i.e. the reentrance phenomenon is observed. The temperature of the second-order phase transition (PT) from the FMP into the phase with spontaneously broken symmetry has been demonstrated to depend considerably on the BQEI anisotropy constant. 2010 Article Феромагнітна фаза одновісного магнетика у присутності анізотропної біквадратичної обмінної взаємодії / І. Шаповалов // Укр. фіз. журн. — 2010. — Т. 55, № 3. — С. 307-312. — Бібліогр.: 37 назв. — укр. 2071-0194 PACS 75.30.Kz https://nasplib.isofts.kiev.ua/handle/123456789/13402 537.61 uk application/pdf application/pdf Відділення фізики і астрономії НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
Ukrainian |
| topic |
Тверде тіло Тверде тіло |
| spellingShingle |
Тверде тіло Тверде тіло Шаповалов, І. Феромагнітна фаза одновісного магнетика у присутності анізотропної біквадратичної обмінної взаємодії |
| description |
Дослiджено феромагнiтну фазу (ФМФ) одноосьового магнетика з одноiонною анiзотропiєю (ОА) типу “легка площина” та анiзотропною бiквадратною обмiнною взаємодiєю (БОВ). Розглянуто випадок, коли значення вузлового спiну дорiвнює одиницi S = 1. Одержано вирази для двох гiлок спектра спiнових збуджень при скiнченних температурах та визначено умови стiйкостi мод спектра. Побудовано дiаграму стiйкостi мод спектра в координатах T - h, з якої випливає, що за певних умов у системi зi зниженням температури спочатку вiдбувається порушення стiйкостi мод спектра, а потiм, з подальшим зниженням температури, стiйкiсть мод спектра вiдновлюється, тобто спостерiгається реєнтрантна поведiнка. Доведено, що температура фазового переходу (ФП) другого роду мiж ФМФ та фазою зi спонтанно порушеною симетрiєю суттєво залежить вiд константи анiзотропiї БОВ. |
| format |
Article |
| author |
Шаповалов, І. |
| author_facet |
Шаповалов, І. |
| author_sort |
Шаповалов, І. |
| title |
Феромагнітна фаза одновісного магнетика у присутності анізотропної біквадратичної обмінної взаємодії |
| title_short |
Феромагнітна фаза одновісного магнетика у присутності анізотропної біквадратичної обмінної взаємодії |
| title_full |
Феромагнітна фаза одновісного магнетика у присутності анізотропної біквадратичної обмінної взаємодії |
| title_fullStr |
Феромагнітна фаза одновісного магнетика у присутності анізотропної біквадратичної обмінної взаємодії |
| title_full_unstemmed |
Феромагнітна фаза одновісного магнетика у присутності анізотропної біквадратичної обмінної взаємодії |
| title_sort |
феромагнітна фаза одновісного магнетика у присутності анізотропної біквадратичної обмінної взаємодії |
| publisher |
Відділення фізики і астрономії НАН України |
| publishDate |
2010 |
| topic_facet |
Тверде тіло |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/13402 |
| citation_txt |
Феромагнітна фаза одновісного магнетика у присутності анізотропної біквадратичної обмінної взаємодії / І. Шаповалов // Укр. фіз. журн. — 2010. — Т. 55, № 3. — С. 307-312. — Бібліогр.: 37 назв. — укр. |
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| fulltext |
I.P. SHAPOVALOV
FERROMAGNETIC PHASE OF A UNIAXIAL MAGNET
WITH ANISOTROPIC BIQUADRATIC EXCHANGE
I.P. SHAPOVALOV
I.I. Mechnikov Odesa National University
(2, Dvoryans’ka Str., Odesa 270100, Ukraine)
PACS 75.30.Kz
c©2010
The ferromagnetic phase (FMP) of a uniaxial magnet with the
easy-plane single-ion anisotropy (SIA) and the anisotropic biquadra-
tic exchange interaction (BQEI) has been studied. The case S = 1
for the site spin S has been considered. Expressions for two
branches of the spin excitation spectrum at finite temperatures T
have been obtained, and the conditions for spectral mode stability
have been determined. The spectral mode stability diagram in the
T−h coordinates has been constructed. The diagram testifies that,
under certain conditions, the temperature decrease is accompanied
by a violation of the spectral mode stability followed, as the tem-
perature decreases further, by its restoration; i.e. the reentrance
phenomenon is observed. The temperature of the second-order
phase transition (PT) from the FMP into the phase with sponta-
neously broken symmetry has been demonstrated to depend con-
siderably on the BQEI anisotropy constant.
1. Introduction
Magnets with high values of the SIA and BQEI constants
have been found in works [1–6]. In turn, it invoked the
subsequent researches of such systems [7–23]. However,
in the majority of works, where the BQEI was consid-
ered, the authors confined themselves to the isotropic
BQEI approximation.
In work [17], magnets with SIA and anisotropic BQEI
with the site spin S = 1 were studied. It was found
that, in the case where the external magnetic field is
directed along the crystal symmetry axis (the z-axis),
two phases with spontaneously broken symmetry can
be realized in the system, besides the symmetric fer-
romagnetic and quadrupole phases. One of the asym-
metric phases is the so-called Q<FMZ phase. It is
a quadrupole-ferromagnetic phase with both a ferro-
magnetic ordering axis coinciding with the z-axis and
a plane of quadrupole ordering, whose orientation de-
pends on Hamiltonian parameters. When the magnetic
field hZ grows, the fraction of the ferromagnetic com-
ponent increases and that of the quadrupole one de-
creases. At a definite hZ-value, the phase Q<FMZ con-
tinuously transforms into the FMP, i.e. there occurs
a PT of the second kind induced by the field. Unfor-
tunately, the authors of work [17] confined the consid-
eration to the low-temperature case, which made the
study of temperature-induced PTs impossible. A gen-
eralization of the ferromagnetic and Q<FMZ phase re-
searches to the finite temperature interval has been car-
ried out in work [20]. In particular, a boundary between
the ferromagnetic and Q<FMZ phases in the field ver-
sus temperature coordinates was determined. However,
the problem of spin excitation spectra remained unre-
solved.
Another asymmetric phase (the Q<FM< phase, ac-
cording to the terminology of the authors of work [17]),
is a phase, in which the magnetization is directed at an
angle to the field hZ . At T = 0, when the external field
grows, the quadrupole phase (QP) → Q<FM< phase
transition occurs at hZ = hc1. If the field grows further,
the phase transition, either Q<FM< phase → Q<FMZ
phase or Q<FM< phase → FMP, occurs at hZ = hc2.
Hence, the Q<FM< phase is realized only provided that
hc1 < hc2, with hc1- and hc2-values depending on Hamil-
tonian parameters. In work [24], expressions for hc1
and hc2 were obtained in the case where there exists
anisotropic BQEI in the system (the corresponding ex-
pressions are given below).
In the absence of BQEI, only one asymmetric phase
can be realized in an easy-plane magnet, namely, in the
case where the magnetic field is perpendicular to the
easy plane. This phase is an analog of the Q<FM< one
(an angular phase). The existence of this phase was pre-
dicted in works [25, 26] and confirmed by experiments
carried out with nickel compounds [27–30]. Further re-
searches of the angular phase have been continued till
now [32–35]. For instance, in work [33], an experimental
T − h phase diagram is presented, in which the bound-
aries of the angular phase with the quadrupole and fer-
romagnetic ones agree well with the experimental results
of work [31]. A comparison between the results obtained
in works [33] and [24] gives the following result: the
expressions for hc1 obtained in both works completely
agree with each other, whereas expressions for hc2 are
different to a certain extent.
306 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 3
FERROMAGNETIC PHASE OF A UNIAXIAL MAGNET
The spectra of spin excitations in magnets with
anisotropic BQEI at finite temperatures were the object
of researches in work [36], where a special method was
developed. The method is based on the application of
a suitable dynamic matrix, the characteristic values of
which coincide with the values for spin excitation ener-
gies.
The main purpose of this work consists in construct-
ing the spin excitation spectra for a FMP and determin-
ing the conditions of their stability. The analysis of the
problem is carried out within the method elaborated in
work [36].
2. Hamiltonian
In the most general case, the uniaxial Hamiltonian with
S = 1 which makes allowance for SIA and BQEI looks
like
H = −hZ
∑
i
SZ
i −
−
∑
i,j(i6=j)
Jij
[
SZ
i S
Z
j − 2ξS+
i S
−
j
]
+D
∑
i
O0
2i−
−
∑
i,j(i6=j)
Kij
(
3O0
2iO
O
2j − 2ηO1
2iO
−1
2j + 4ζO2
2iO
−2
2j
)
, (1)
where Jij are the exchange interaction constants, Kij
are the BQEI constants, D is the SIA constant; ξ, η,
and ζ are positive numbers; and Om
l with l = 1, 2 and
m = 0,±1, . . . ,±l are the tensor operators that form the
Lie algebra of the group SU(3). The first term in Hamil-
tonian (1) is the energy of site spins in an external mag-
netic field (the Zeeman energy). The second term is the
energy of exchange interaction which becomes isotropic
at ξ = 1. The third term is the energy of spins in the
crystalline field. The fourth term is the BQEI energy.
In the case where η = ζ = 1, the BQEI is isotropic:
HBQEI = −
∑
i,j(i6=j)
Kij (SiSj)
2
. (2)
The deviations of the parameters η and ζ from 1 char-
acterize the anisotropy degree of BQEI; i.e. those pa-
rameters are the BQEI anisotropy constants.
The operators Om
l are connected with spin operators
by the relations
O0
1 = SZ ; O1
1 ≡ S+ =
1√
2
(
SX − iSY
)
;
O−1
1 ≡ S− =
−1√
2
(
SX + iSY
)
;
O0
2 =
(
SZ
)2 − 2
3
; O± 1
2 = −
(
SZS± + S±SZ
)
;
O± 2
2 =
(
S±
)2
. (3)
The average values of these operators determine the spin
ordering in the system. In the FMP, only the diago-
nal averages 〈SZ〉 and 〈O0
2〉 are different from zero, and
the order parameter is therefore two-component. In this
work, we confine ourselves to the consideration of the
single-sublattice ordering in easy-plane magnets, which
is provided by the conditions Jij > 0, Kij > 0, and
D > 0.
In the molecular-field approximation,
H0 = −
(
hZ + 2J0〈SZ〉
)
×
×
∑
i
SZ
i +
(
D − 6K0〈O0
2〉
)∑
i
O0
2i, (4)
where J0 ≡
∑
i Jij and K0 ≡
∑
iKij .
Depending on the spin projection SZ on the z-axis
(SZ = 0,±1), the energy levels of lattice site atoms are
determined by the formulas
E0 = −2
3
D + 4K0〈O0
2〉;
E1 = −hZ − 2J0〈SZ〉+ 1
3
D − 2K0〈O0
2〉;
E−1 = hZ + 2J0〈SZ〉+ 1
3
D − 2K0〈O0
2〉. (5)
Since the condition E1 < E−1 is satisfied automati-
cally in the FMP, this phase can be realized, provided
that E1 < E0 or
hZ + 2J0〈SZ〉 > D − 6K0〈O0
2〉. (6)
At zero temperature (T = 0) in the FMP,
〈SZ〉 = 1; 〈O0
2〉 =
1
3
. (7)
At finite temperatures, the averages 〈SZ〉 and 〈O0
2〉 are
determined by the system of two equations [36]
〈SZ〉 =
2sh
hz + 2J0〈SZ〉
θ
exp
6K0〈O0
2〉 −D
θ
1 + 2ch
hz + 2J0〈SZ〉
θ
exp
6K0〈O0
2〉 −D
θ
,
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 3 307
I.P. SHAPOVALOV
〈O0
2〉 =
1
3
− 1
1 + 2ch
hz + 2J0〈SZ〉
θ
exp
6K0〈O0
2〉 −D
θ
,
(8)
where θ is the temperature expressed in energy units
(θ = kT ).
Note that system (8) looks identically for the
quadrupole and ferromagnetic phases; however, the
quantities 〈SZ〉 and 〈O0
2〉 are different in both phases.
For the identification of solutions of the system which
correspond to different phases, it is expedient to use the
passage to the limit T → 0. In this case, 〈SZ〉 → 0 and
〈O0
2〉 → − 2
3 in the QP, and 〈SZ〉 → 1 and 〈O0
2〉 → 1
3 in
the FMP.
3. Spin Excitation Spectrum
The value SZ = 1 corresponds to the ground state of site
atoms in the FMP. At finite temperatures, there arise the
spin excitations with SZ = 0 and SZ = −1. The creation
operators for these excitations are the Hubbard opera-
tors X01 and X−11, and the corresponding annihilation
operators are X10 and X1−1.
In order to find branches of the spin excitation spec-
trum, we used the method proposed in work [36] which
consists, briefly, in the following. Calculating the com-
mutators of non-diagonal Hubbard operators and the
Hamiltonian, the dynamic matrix is constructed. The
number of characteristic values of this matrix coincides
with the number of non-diagonal Hubbard operators
that were used, with every characteristic value corre-
sponding to a certain Hubbard operator. The expres-
sions for those characteristic values, which correspond
to the annihilation operators X01 and X−11, coincide in
the k-space with the expressions for branches of the spin
excitation spectrum.
To calculate the commutators, it is expedient to pass
to Hubbard operators in Hamiltonian (1). At SZ = 1,
the relations between the Om
l and Hubbard operators
are given by the formulas
SZ = X11 −X−1−1, S+ = −X10 −X0−1,
S− = X01 +X−10, O0
2 = X11 +X−1−1 − 2
3
,
O1
2 = X10 −X0−1, O−1
2 = −X01 +X−10,
O2
2 = X1−1, O−2
2 = X−11. (9)
Accordingly, Hamiltonian (1) reads
H = −hZ
∑
i
(
X11
i −X−1−1
i
)
−
−
∑
i,j
Jij
[(
X11
i −X−1−1
i
) (
X11
j −X−1−1
j
)
+
+ 2ξ
(
X10
i +X0−1
i
) (
X01
j +X−10
j
)]
+
+D
∑
i
(
X11
i +X−1−1
i − 2
3
)
−
−
∑
i,j
Kij
[
3
(
X11
i +X−1−1
i − 2
3
)
×
×
(
X11
j +X−1−1
j − 2
3
)
−
−2η
(
X10
i −X0−1
i
) (
−X01
j +X−10
j
)
+ 4ζX1−1
i X−11
j
]
.
(10)
To construct a suitable dynamic matrix, it is necessary
to calculate the commutators [X10
f , H], [X0−1
f , H], and
[X1−1
f , H] and use the approximation
XiXj = Xi〈Xj〉+ 〈Xi〉Xj ,
〈Xnm〉 = 0, (n 6= m) (11)
which was proposed in work [36]. In the k-space, the
corresponding commutators are[
X10
k , H
]
= p11 (k)X10
k + p12 (k)X0−1
k ,
[
X0−1
k , H
]
= p21 (k)X10
k + p22 (k)X0−1
k ,
[
X1−1
k , H
]
= p33 (k)X1−1
k , (12)
where the coefficients pim constitute a dynamic matrix
with the components
p11 (k) = hZ + 2J0〈SZ〉 −D + 6K0〈O0
2〉−
−
(
〈SZ〉+ 3〈O0
2〉
)
(ξJk + ηKk) ,
308 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 3
FERROMAGNETIC PHASE OF A UNIAXIAL MAGNET
p12 (k) =
(
〈SZ〉+ 3〈O0
2〉
)
(ηKk − ξJk) ,
p21 (k) =
(
〈SZ〉 − 3〈O0
2〉
)
(ηKk − ξJk) ,
p22 (k) = hZ + 2J0〈SZ〉+D − 6K0〈O0
2〉−
−
(
〈SZ〉 − 3〈O0
2〉
)
(ξJk + ηKk) ,
p33 (k) = 2hZ + 4〈SZ〉 (J0 − ζKk) . (13)
The characteristic values of the dynamic matrix are
determined by the secular equation∣∣∣∣∣∣
p11 (k)− ε (k) p12 (k) 0
p21 (k) p22 (k)− ε (k) 0
0 0 p33 (k)− ε (k)
∣∣∣∣∣∣ = 0.
(14)
Its solution gives three characteristic values which can be
expressed, by taking Eq. (13) into account, as follows:
ε1 (k) = 2hZ + 4〈SZ〉 (J0 − ζKk) , (15)
ε2,3 (k) = hZ + 〈SZ〉 (2J0 − ξJk − ηKk)∓
∓
{(
〈SZ〉
)2
(ξJk − ηKk)2 +
+
[
D − 6〈O0
2〉 (K0 − ξJk)
] [
D − 6〈O0
2〉 (K0 − ηKk)
]}1/2
.
(16)
Those characteristic values of the dynamic matrix,
which correspond to the Hubbard annihilation opera-
tors, coincide with the branches of the spin excitation
spectrum. The characteristic values ε1 (k) and ε2 (k)
correspond to the operators X10 and X1−1; therefore,
the branches of the spin excitation spectrum are
ω1 (k) = ε1 (k) , (17)
ω2 (k) = ε2 (k) . (18)
Since Jk and Kk are even functions of the wave vector
k, both branches are characterized by the square-law
dispersion in the long-wave limit,
ω1 (k) = Δ1 + α1k2;ω2 (k) = Δ2 + α2k2. (19)
In work [37], the inequality ω (k) > ω (0) was proved
to be valid at every k 6= 0 in the cases where the single-
sublattice ordering takes place in the system. Therefore,
the condition for spectrum mode stability is given by the
system of inequalities
ω1 (0) > 0,
ω2 (0) > 0, (20)
and the stability boundary is determined by two equali-
ties, ω1 (0) = 0 and ω2 (0) = 0, or
hZ + 2〈SZ〉 (J0 − ζK0) = 0, (21)
hZ + 〈SZ〉 (2J0 − ξJ0 − ηK0) =
=
{(
〈SZ〉
)2
(ξJk − ηKk)2 +
[
D − 6〈O0
2〉 (K0 − ξJk)
]
×
×
[
D − 6〈O0
2〉 (K0 − ηKk)
]}1/2
. (22)
Expression (22) coincides with the corresponding expres-
sion for the QP (see work [36]). However, since the quan-
tities 〈SZ〉 and 〈O0
2〉 are different in the ferromagnetic
and quadrupole phases, the stability boundaries in the
T −h coordinates do not coincide, generally speaking, in
both phases.
Note that curve (21) coincides with the curve corre-
sponding to the second-kind PT between the Q<FMZ
and ferromagnetic phases, which was obtained in work
[20] (see Introduction).
4. Stability Diagram
First of all, it is worth noting that we consider the case
where the phase Q<FM< is not realized, i.e. the con-
dition hc1 > hc2 is satisfied. For the critical fields hc1
and hc2, the following expressions were obtained in work
[24]:
hc1 =
√
[D + 4K0(1− η)][D + 4(K0 − ξJ0)],
hc2 = D − 2J0(1− ξ)− 2K0(1− η). (23)
In Fig. 1, the mode stability diagram for the FMP
spectrum is depicted in the θ̃ − h̃ coordinates, where
θ̃ = θ/K0 and h̃ = hZ/K0. The Hamiltonian parameters
are chosen so that the condition hc1 > hc2 is satisfied, i.e.
a unique asymmetric phase is the phase Q<FM<. Zone
1 is the range, where FMP spectrum modes are stable,
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 3 309
I.P. SHAPOVALOV
Fig. 1. Mode stability diagram for the FMP spectrum of an easy-
plane magnet with anisotropic BQEI: (1 ) region of FMP spec-
trum mode stability, (2 ) region of FMP spectrum mode instabil-
ity, (3 ) curve given by expression (21), and (4 ) curve given by
expression (22). The diagram is plotted for the parameter values
J0 = 0.8, D = 0.4, K0 = 1, ζ = 1.2, η = 0.8, and ξ = 1.25
and zone 2 is the range, where their stability is violated.
Curves 3 and 4 are given by expressions (21) and (22),
respectively. In the external magnetic field interval h̃1 <
h̃ < h̃2, the stability of spectrum modes demonstrates
a reentrant behavior. In particular, at h̃ = h̃∗, if the
temperature decreases, the stability is first violated at
the point θ̃1. But the further temperature decrease gives
rise to the restoration of the spectrum mode stability at
the point θ̃2.
Since curve (21) is not only a curve, where the mode
stability of the spin excitation spectrum is violated, but
also a PT curve, it is expedient to study the dependence
of the temperature θ̃c of the second-kind PT between
the Q<FMZ and ferromagnetic phases on the BQEI
anisotropy constant ζ at various fields h̃. Such a de-
pendence is depicted in Fig. 2. The figure demonstrates
that the transition temperature substantially depends on
the constant ζ. At the same time, for large enough ζ’s,
the temperature θ̃c is almost independent of the external
field h̃.
5. Discussion of Results
In this work, the expressions for two branches of the spin
excitation spectrum in the FMP have been obtained.
Both branches demonstrate the square law of the disper-
sion in the long-wave limit. When determining the spin
excitation spectrum branches, the condition D > 0 was
not used. Therefore, expressions (17) and (18) for the
spectrum branches and expressions (21) and (22) for the
Fig. 2. Dependences of the temperature of the second-kind PT
between the ferromagnetic and Q<FMZ phases on the constant
ζ at various hZ = 0.6 (1 ), 0.8 (2 ), and 1 (3 ). All curves were
calculated for the parameter values J0 = 0.8, D = 0.4, and K0 = 1
boundary of the spectrum mode stability range remain
valid in the case D < 0, i.e. for an easy-axis magnet.
Owing to a mismatch between the stability bound-
aries for the ferromagnetic and quadrupole phases (see
Section 3), there emerges a region in the stability dia-
gram plotted in the T −h coordinates, where the modes
of spectra of both phases are stable. The presence of
such a region brings about two essential consequences.
First, the curve of the PT between the ferromagnetic and
quadrupole phases coincides with the curve, where the
free energies in both phases are identical; in this case, the
corresponding PT is of the first kind. Second, there are
two regions of metastability. In one of them, the FMP is
metastable and the QP is stable; in the other, the situ-
ation is opposite. Hence, the stability diagram does not
coincide with the phase one. The results of researches
of phase diagrams, metastable regions, and the influence
of the BQEI anisotropy constants on the first-kind PT
between the ferromagnetic and quadrupole phases will
be reported elsewhere.
It has to be noted that the method proposed in work
[36] can be directly used only for those phases which
preserve the Hamiltonian symmetry, i.e. for the QP and
the FMP. In the case of phases with spontaneously bro-
ken symmetry, it is necessary, first, to diagonalize the
zero Hamiltonian with the help of a unitary transforma-
tion for the application of the method proposed to be
eligible. This transformation is one-parametric for the
Q<FMZ phase and two-parametric for the Q<FM< one.
The calculations of spin excitation spectra in the asym-
310 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 3
FERROMAGNETIC PHASE OF A UNIAXIAL MAGNET
metric phases at T 6= 0 will be a subject of a separate
research.
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Received 22.04.09.
Translated from Ukrainian by O.I. Voitenko
ФЕРОМАГНIТНА ФАЗА ОДНОВIСНОГО МАГНЕТИКА
У ПРИСУТНОСТI АНIЗОТРОПНОЇ БIКВАДРАТИЧНОЇ
ОБМIННОЇ ВЗАЄМОДIЇ
I. Шаповалов
Р е з ю м е
Дослiджено феромагнiтну фазу (ФМФ) одноосьового магне-
тика з одноiонною анiзотропiєю (ОА) типу “легка площина”
та анiзотропною бiквадратичною обмiнною взаємодiєю (БОВ).
Розглянуто випадок, коли значення вузлового спiну дорiвнює
одиницi S = 1. Одержано вирази для двох гiлок спектра спi-
нових збуджень при скiнченних температурах та визначено
умови стiйкостi мод спектра. Побудовано дiаграму стiйкостi
мод спектра в координатах T − h, з якої випливає, що за пев-
них умов у системi зi зниженням температури спочатку вiдбу-
вається порушення стiйкостi мод спектра, а потiм, з подаль-
шим зниженням температури, стiйкiсть мод спектра вiднов-
люється, тобто спостерiгається реєнтрантна поведiнка. Дове-
дено, що температура фазового переходу (ФП) другого роду
мiж ФМФ та фазою зi спонтанно порушеною симетрiєю суттє-
во залежить вiд константи анiзотропiї БОВ.
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