Phase transitions in the Mitsui model
In this paper, phase transitions in the Mitsui model without longitudinal field but with a transverse one are investigated in the mean field approximation. The one-to-one correspondence has been established between this model and the two-sublattice Ising-type model with longitudinal and transverse f...
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irk-123456789-1199822017-06-11T03:04:18Z Phase transitions in the Mitsui model Dublenych, Yu.I. In this paper, phase transitions in the Mitsui model without longitudinal field but with a transverse one are investigated in the mean field approximation. The one-to-one correspondence has been established between this model and the two-sublattice Ising-type model with longitudinal and transverse fields. Phase diagrams and diagrams of existence of the ferroelectric phase are constructed. In the case Ω = 0 (Ω is the transverse field), a simple analytical expression for the tricritical temperature and the condition of existence of the tricritical point are obtained. For Ω ≠ 0, systems of equations for the tricritical point and for the condition of its existence are written. В роботi в наближеннi середнього поля дослiджено фазовi переходи в моделi Мiцуi без поздовжнього поля, проте з поперечним полем. Встановлено взаємооднозначну залежнiсть мiж такою моделлю та двопiдґратковою моделлю типу Iзiнга з поздовжнiм i поперечним полями. Побудовано фазовi дiаграми та дiаграми областей iснування сегнетофази. Для випадку Ω = 0 (Ω–поперечне поле) одержано простий аналiтичний вираз для трикритичної температури й умову iснування трикритичної точки. Для Ω 6= 0 записано системи рiвнянь для трикритичної точки й умови її iснування. 2011 Article Phase transitions in the Mitsui model / Yu.I. Dublenych // Condensed Matter Physics. — 2011. — Т. 14, № 2. — С. 23603:1-20. — Бібліогр.: 13 назв. — англ. 1607-324X PACS: 64.60.De, 64.60.Cn, 77.80.-e DOI:10.5488/CMP.14.23603 arXiv:1106.5139 http://dspace.nbuv.gov.ua/handle/123456789/119982 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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In this paper, phase transitions in the Mitsui model without longitudinal field but with a transverse one are investigated in the mean field approximation. The one-to-one correspondence has been established between this model and the two-sublattice Ising-type model with longitudinal and transverse fields. Phase diagrams and diagrams of existence of the ferroelectric phase are constructed. In the case Ω = 0 (Ω is the transverse field), a simple analytical expression for the tricritical temperature and the condition of existence of the tricritical point are obtained. For Ω ≠ 0, systems of equations for the tricritical point and for the condition of its existence are written. |
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Dublenych, Yu.I. |
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Dublenych, Yu.I. Phase transitions in the Mitsui model Condensed Matter Physics |
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Dublenych, Yu.I. |
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Dublenych, Yu.I. |
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Phase transitions in the Mitsui model |
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Phase transitions in the Mitsui model |
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Phase transitions in the Mitsui model |
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Phase transitions in the Mitsui model |
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Phase transitions in the Mitsui model |
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phase transitions in the mitsui model |
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Інститут фізики конденсованих систем НАН України |
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Phase transitions in the Mitsui model / Yu.I. Dublenych // Condensed Matter Physics. — 2011. — Т. 14, № 2. — С. 23603:1-20. — Бібліогр.: 13 назв. — англ. |
| series |
Condensed Matter Physics |
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AT dublenychyui phasetransitionsinthemitsuimodel |
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2025-07-08T17:01:14Z |
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2025-07-08T17:01:14Z |
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1837098941379772416 |
| fulltext |
Condensed Matter Physics, 2011, Vol. 14, No 2, 23603: 1–20
DOI: 10.5488/CMP.14.23603
http://www.icmp.lviv.ua/journal
Phase transitions in the Mitsui model
Yu.I. Dublenych
Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 79011 Lviv, Ukraine
Received April 14, 2011, in final form June 14, 2011
In this paper, phase transitions in the Mitsui model without longitudinal field but with a transverse one are
investigated in the mean field approximation. The one-to-one correspondence has been established between
this model and the two-sublattice Ising-type model with longitudinal and transverse fields. Phase diagrams and
diagrams of existence of the ferroelectric phase are constructed. In the case Ω = 0 (Ω is the transverse field),
a simple analytical expression for the tricritical temperature and the condition of existence of the tricritical point
are obtained. For Ω 6= 0, systems of equations for the tricritical point and for the condition of its existence are
written.
Key words: phase transition, ferroelectric phase, Mitsui model, tricritical point
PACS: 64.60.De, 64.60.Cn, 77.80.-e
1. Introduction
The Mitsui model was proposed in 1958 [1] to theoretically explain the ferroelectric properties of
the Rochelle salt. In 1971, Žekš, Shukla, and Blinc [2] formulated this model in terms of pseudospins
and just in this form it is known at the present time.
For better quantitative description of the Rochelle salt the conventional Mitsui model was
modified in many ways. For instance, in reference [3] an additional piezoelectric interaction was
included in the Hamiltonian of the model and in references [4] and [5] a transverse field (or tun-
neling) was included as well. To take into account a realistic structure of Rochelle salt crystal,
the four-sublattice Mitsui model was considered [6]. Besides the Rochelle salt, the Mitsui model
with transverse field was used for theoretical description of some other ferroelectric compounds,
notably, RbHSO4 [7, 8] and NH4HSO4 [8] crystals.
In reference [9] the effect of hydrostatic pressure on thermodynamic properties of the Rochelle
salt was studied. Hydrostatic pressure makes it possible to change the parameters of the model.
It turns out that the Mitsui model covers not only the ferroelectrics of the order-disorder
type but also other physical objects with two-minimum asymmetric potential. For instance, in
references [10] and [11] pseudospin-electron models based on the Mitsui model were considered.
Despite a rather wide use of the Mitsui model, there is no detailed analysis of its phase behavior.
In reference [12] the diagram of existence of the ferroelectric phase was calculated. However, this
diagram is far from being complete. More detailed though still incomplete diagram was obtained
in reference [13].
In the present paper, a rigorous and original mathematical investigation of phase transitions in
the Mitsui model is proposed. We managed to construct a complete phase diagram of the Mitsui
model in the mean field approximation, first without tunneling (or transverse field) and then with
nonzero tunneling. For all curves (surfaces) of the diagram the analytical expressions, equations or
systems of equations are given.
c© Yu.I. Dublenych, 2011 23603-1
http://dx.doi.org/10.5488/CMP.14.23603
http://www.icmp.lviv.ua/journal
Yu.I. Dublenych
2. Hamiltonian of the Mitsui model in the mean field approxima tion
The Hamiltonian of the Mitsui model without external field reads
H = −1
2
∑
ij
Jij
(
SzA
i SzA
j + SzB
i SzB
j
)
−
∑
ij
KijS
zA
i SzB
j −∆
∑
i
(
SzA
i − SzB
i
)
−Ω
∑
i
(
SxA
i + SxB
i
)
.
(2.1)
Here subscripts A and B denote two sublattices, SαA
i is α-component of the pseudospin on ith site
of sublattice A; Jij and Kij are the interactions between pseudospins of the same sublattice and of
different sublattices, respectively, ∆ is the asymmetry of the anharmonic potential; transverse field
Ω describes the tunneling between two wells of the two-minimum potential. The model considered
is a lattice one. However, since we use the mean field approximation, we do not need to specify the
type of the lattice.
The transformations SzB
i → −SzB
i , Kij → −Kij transform Hamiltonian (2.1) into the Hamil-
tonian of the two-sublattice model with longitudinal field ∆ (the same for both sublattices) and
transverse field Ω. Therefore, the two Hamiltonians are equivalent.
In the mean-field approximation, Hamiltonian (2.1) reads
H =
N
2
{
Kη
A
η
B
+
J
2
(
η2
A
+ η2
B
)
}
+
∑
i
(
HA
i +HB
i
)
, (2.2)
where the following notations are introduced:
HA
i = − (∆ +Kη
B
+ Jη
A
)SzA
i − ΩSxA
i , (2.3)
HB
i = − (−∆+Kη
A
+ Jη
B
)SzB
i − ΩSxB
i , (2.4)
K =
∑
i
Kij =
∑
j
Kij , J =
∑
i
Jij =
∑
j
Jij , (2.5)
ηA = 〈SzA
i 〉 and ηB = 〈SzB
i 〉 are the average values of pseudospin on sublattices A and B, respec-
tively, N is the total number of pseudospins.
3. Ω = 0 case
3.1. Free energy, thermodynamic equilibrium conditions an d order parameters
Let us first consider the Ω = 0 case. In this case the eigenvalues of Hamiltonians HA
i and HB
i
are as follows:
λ
A
= −1
2
(∆ +Kη
B
+ Jη
A
) , −λ
A
;
λ
B
= −1
2
(−∆+Kη
A
+ Jη
B
) , −λ
B
.
(3.1)
The partition function for one unit cell reads
Zi =
(
e−βλ
A + eβλA
) (
e−βλ
B + eβλB
)
e−β[Kη
A
η
B
+ J
2 (η
2
A
+η2
B
)] . (3.2)
The free energy per one unit cell is as follows:
F = −θ lnZi ,
F = Kη
A
η
B
+
J
2
(
η2
A
+ η2
B
)
− θ ln
(
e−βλ
A + eβλA
)
− θ ln
(
e−βλ
B + eβλB
)
, (3.3)
where θ = 1/β is thermodynamic temperature. From thermodynamic equilibrium conditions
(
∂F
∂η
A
)
T,∆
= 0,
(
∂F
∂η
B
)
T,∆
= 0
(3.4)
23603-2
Phase transitions in the Mitsui model
we obtain the following equations for η
A
and η
B
:
2η
A
= tanh (−βλ
A
),
2η
B
= tanh (−βλ
B
).
(3.5)
Taking into account equations (3.5), we can rewrite the expression for free energy in the following
form:
F = Kη
A
η
B
+
J
2
(
η2
A
+ η2
B
)
+
θ
2
ln
[(
1
4
− η2
A
)(
1
4
− η2
B
)]
. (3.6)
The system of equations (3.5) has the solutions of two kinds: for the first ones η
A
= −η
B
(then the system is reduced to one equation only), they exist for arbitrary ∆; for the second ones
η
A
6= −η
B
, they exist in a bounded region of values of ∆ and correspond to the ferroelectric phase.
Let us divide the (K, J) plane into eight segments as shown in figure 1. The plots of the free
energy as a function of ∆ at zero temperature in the centers of the unit-circle arcs for every segment
are shown in figure 2. (It is easy to obtain these plots from equations (3.5) and (3.6) setting θ → 0
or β → +∞.) As one can see from figure 2, in segments 4 and 5 there are no phase transitions,
in segments 6, 7, and 8 there are phase transitions at ∆ = 0 only. In segment 3 there are only
second-order phase transitions that are easy to investigate. The most complicated and interesting
picture of phase transitions is observed for segments 1 and 2, therefore we consider only the region
K > 0, J > 0.
Figure 1. Division of the (K, J) plane into eight segments, which correspond to different relations
between parameters K and J .
Let us introduce new variables: ferroelectric order parameter ξ = η
A
+ η
B
and antiferroelectric
order parameter σ = η
A
− η
B
. Let us also divide all energetic values by K + J and introduce the
following notations:
a =
K − J
K + J
, γ =
∆
K + J
, t =
θ
K + J
, f =
F
K + J
. (3.7)
Now equations (3.5) can be rewritten in the following form:
eξ/t =
(1 + ξ)2 − σ2
(1− ξ)2 − σ2
,
e−aσ/t = e−2γ/t (1 + σ)2 − ξ2
(1 − σ)2 − ξ2
.
(3.8)
If ξ = 0 (first type solutions, both sublattices are equivalent up to a sign of η) then the first
equation becomes an identity and the system (3.8) is reduced to a single equation. If ξ 6= 0 (second
23603-3
Yu.I. Dublenych
Figure 2. Free energy as a function of ∆ at zero temperature. Numbers over figures correspond
to the unit-circle points indicated in figure 1.
type solutions) then equations (3.8) can be rewritten in a simpler form:
σ = ±
√
1 + ξ2 + 2ξ
1 + eξ/t
1− eξ/t
,
γ =
a
2
σ +
ξ
2
+ t ln
1 + σ − ξ
1− σ + ξ
.
(3.9)
In view of the symmetry we consider only σ > 0, ξ > 0 and γ > 0.
23603-4
Phase transitions in the Mitsui model
The expression for the free energy per unit cell in terms of new variables reads
f =
1
4
(
ξ2 − aσ2
)
+
t
2
ln
{[
1− (ξ + σ)2
] [
1− (ξ − σ)2
])
− 2t ln 2. (3.10)
3.2. Second-order phase transitions
The second-order ferroelectric phase transition corresponds to the branchpoint of the curve
σ(γ) (a and t being fixed) or of the curve σ(t) (a and γ being fixed) where a solution of the first
type turns into a solution of the second type. We denote the value of σ in this point by σ̃. It is as
follows:
σ̃ = lim
ξ→0
σ =
√
1− 4t . (3.11)
One can see from this expression that the second-order phase transitions exist up to the temperature
tmax =
1
4
. (3.12)
Substituting ξ = 0 and σ from equation (3.11) in equation (3.9), we obtain the equation for the
curve γ(t) of the second-order phase transitions:
γ =
a
2
σ̃ + t ln
1 + σ̃
1− σ̃
. (3.13)
Now let us find the minimal temperature for the existence of the second-order phase transitions
at fixed a. If a = 1, then only second-order phase transitions exist. If −1 < a < 1, then there are
phase transitions of both second and first orders. The latter exist from t = 0 to a certain value of
the temperature. As one can see from figure 3, the tricritical point, i.e. the point where the order
of phase transition changes, can be determine from the following condition:
lim
ξ→0
dγ
dσ
= 0. (3.14)
Figure 3. Antiferroelectric order parameter as
a function of γ for several values of temper-
ature (numbers over the curves). a = 0.0875.
Thermodynamically stable states are depicted
by heavy lines. At t = 0.03 and t = 0.1 there are
first-order phase transitions and at t = 0.1507
and t = 0.2 there are second-order ones.
Figure 4. Antiferroelectric order parameter as
a function of γ for several values of parame-
ter a (numbers near the curves). The values
of the temperature are calculated using expres-
sion (3.15). At a = −0.1 and a = 0.25 there are
second-order phase transitions and at a = 0.35
and a = 0.45 there are first-order ones.
23603-5
Yu.I. Dublenych
We obtain for the tricritical temperature:
ttc =
1
3
+
1
6(a− 1)
. (3.15)
The tricritical point exists if, at the temperature determined by equation (3.15), the following
condition is satisfied:
lim
ξ→0
d2γ
dσ2
6 0 (γ > 0). (3.16)
This is clear from figure 4 where curves σ(γ)/2 for several values of parameter a and corresponding
values of the temperature [see equation (3.15)] are depicted. The existence of the region where the
dependence σ(γ) for ξ 6= 0 is two-valued indicates that there is a first-order phase transition. This
two-valuedness disappears with decreasing a and the order of the phase transition changes. From
equations (3.15) and (3.16) one obtains:
a 6
1
4
. (3.17)
Let us find the maximum of curve γ(t) at fixed a. The extremum condition dγ/dt = 0 and
equation (3.13) yield:
ln
1 + σ̃
1− σ̃
=
a+ 1
σ̃
, γ =
1
4
(
a+ 1
σ̃
+ (a− 1)σ̃
)
. (3.18)
Excluding σ̃, we obtain the equation for γ:
(
2γ +
√
(2γ)2 − a2 + 1
)
tanh
2γ +
√
(2γ)2 − a2 + 1
2
= a+ 1. (3.19)
3.3. First-order phase transitions
Up to here we analyzed the second-order phase transitions and found the expression for the
surface γ = γ(t, a) of these transitions as well as the tricritical point and the condition for its
existence. Now let us consider the first-order phase transitions.
A first-order phase transition corresponds to a point of self-intersection of the curve for the
free energy f(ξ, σ) as a function of γ at fixed a and t (or as a function of t at fixed a and γ). But
only this point of self-intersection gives the first-order phase transition in which the multivalued
function for the free energy takes the minimal value from all possible values at fixed γ.
If −1 6 a 6 1/4, then the first-order phase transitions exists for 0 6 t < ttc [see equation
(3.15)]. The curve for them in the (γ, t)-plain (i.e, the intersection points of the branches ξ = 0
and ξ 6= 0 of the free energy) can be found from the following system of equations:
σ =
√
1 + ξ2 + 2ξ
1 + eξ/t
1− eξ/t
,
γ =
a
2
σ +
ξ
2
+ t ln
1 + σ − ξ
1− σ + ξ
,
γ =
a
2
σ1 + t ln
1 + σ1
1− σ1
,
f(ξ, σ) = f(0, σ1).
(3.20)
If t = 0, then the first-order phase transition occurs at
γ = ±1
4
(a+ 1). (3.21)
In the t 6= 0 case the system of equations (3.20) can be solved only numerically.
23603-6
Phase transitions in the Mitsui model
The phase coexistence curves for several values of parameter a are shown in figure 5. The
region of ferroelectric phase is bounded by such a curve and by the coordinate axes. The curve of
tricritical points [more exactly, its projection on the plane (γ, t)] is also shown in figure 5 (heavy
line). From this curve another two curves branch off: the curve of minima of function γ(t) for all
possible values of a and the curve of maxima for the second-order phase transitions. The curve of
tricritical points furcates into the curve of branchpoints and the curve of critical points (dashed
line). If a & 0.1793293 (four curves on the right), then there is a region with two second-order phase
transitions (see also figure 6). At this value of a the maximum of the curve γ(t) of the second-order
phase transitions coincides with the tricritical point.
Figure 5. Coexistence curves in the (γ, t)-plane for several values of parameter a (numbers near
the curves). The curve of tricritical points (heavy line), the curve of minima for the first-order
phase transitions, the curve of maxima for the second-order phase transitions, the curve of
branchpoints and the curve of critical points (dashed line) are indicated.
If a > 1/4, then the upper part of the curve of the first-order phase transitions corresponds to
the phase transitions within the ferroelectric phase. The temperature of these transitions (at fixed
a and γ), i.e. the temperature for the self-intersection points of the free energy curve, can be found
from the following system of equations (the solutions of the type ξ = ξ1 should be rejected):
σ =
√
1 + ξ2 + 2ξ
1 + eξ/t
1− eξ/t
,
γ =
a
2
σ +
ξ
2
+ t ln
1 + σ − ξ
1− σ + ξ
,
σ1 =
√
1 + ξ21 + 2ξ1
1 + eξ1/t
1− eξ1/t
,
γ =
a
2
σ1 +
ξ1
2
+ t ln
1 + σ1 − ξ1
1− σ1 + ξ1
,
f(ξ, σ) = f(ξ1 , σ1).
(3.22)
In figures 6(b) and (c) some fragments of phase diagrams in coordinates (γ, t) for a > 1/4
are shown. The upper part of the curve of the first-order phase transitions corresponds to the
transitions within the ferroelectric phase.
If a > 1/4, then the critical temperature can be determined from the following system of
23603-7
Yu.I. Dublenych
Figure 6. Fragments of the phase coexistence curves for a) a = 0.25 (the tricritical point is
shown), b) a = 0.46 and c) a = 0.7 (see figure 5).
equations:
dγ
dσ
= 0,
d2γ
dσ2
= 0,
σ =
√
1 + ξ2 + 2ξ
1 + eξ/t
1− eξ/t
,
γ =
a
2
σ +
ξ
2
+ t ln
1 + σ − ξ
1− σ + ξ
.
(3.23)
This system of equations can be rewritten in the form:
σ2 = 1 + ξ2 − 2t+
2
a
(
t−
√
a2ξ2 + 2atξ2(1− a) + t2(1 + a)2
)
,
σ2 = 1 + aξ2 − 3t+
1
a
(
t−
√
[aξ2(1− a) + t(1 + a)]2 − 4a2ξ2(2at− 2t− a)
)
,
σ2 = 1 + ξ2 + 2ξ
1 + eξ/t
1− eξ/t
,
γ =
a
2
σ +
ξ
2
+ t ln
1 + σ − ξ
1− σ + ξ
,
(3.24)
whence it follows
ξ2 =
−q − [(3a− 1)t− a]
√
q − 2(a+ 1)t [(a− 1)2t− a2]
2a(a− 1)[2(a− 1)t− a]
, (3.25)
23603-8
Phase transitions in the Mitsui model
where q = (a−1)(3a2−14a−1)t2−2a(2a2−5a+1)t+a2(a−1), and hence the system of equations
is reduced to a single transcendental equation.
As one can see from figure 6, the curve γ(t) of the first-order phase transitions has a minimum
if a is big enough. To find it, let us differentiate the last equation of system (3.20) with respect to
t and then substitute dγ/dt = 0. After simple transformations we obtain the following equation:
ξ2 − aσ2 + 4γσ = −aσ2
1 + 4γσ1 , (3.26)
which together with equations (3.20) gives the point of minimum. At zero temperature the solutions
of equations (3.20) also satisfy it:
lim
t→0
dγ
dt
= 0. (3.27)
The branchpoint of the curve (at fixed a > 1/4) can be found from equations (3.20), substituting
t =
1− σ2
1
4
. (3.28)
3.4. Regions of existence of the ferroelectric phase
To conclude, we obtained explicit expressions or system of equations for all special points of
the phase curve at fixed a. This makes it possible to construct a diagram of the regions where the
ferroelectric phase exists. There are seven regions in the (γ, a) plane (figures 7 and 8). They are
bounded by the following curves:
(1) the curve of minima for the first-order phase transitions [equation (3.26)];
(2) the curve of tricritical points [equations (3.11), (3.13), and (3.15)];
(3) the curve of branchpoints [equations (3.20), and (3.28)];
(4) the curve of critical points [equation (3.24)];
(5) the curve of maxima for the second-order phase transitions [equation (3.19)];
(6) the straight line of the first-order phase transitions at zero temperature [equation (3.21)].
In the point (a ≈ 0.179329, γ = 0.283995) the curve of minima for the first-order phase
transitions and the curve of maxima for the second-order phase transitions branch off from the
Figure 7. Regions of existence of the ferroelec-
tric phase (see also figures 9 and 10). Dashed
line (prolongation of the line which bounds re-
gion V) is shown only for comparison with the
diagram from [12].
Figure 8. Regions of existence of the ferroelec-
tric phase (fragment). The filled circle is the
branchpoint.
23603-9
Yu.I. Dublenych
curve of tricritical points and in the point (a = 0.25, γ = 0.307041) the curve of critical points
branches out into the curve of branchpoints and the curve of critical points. All curves except for
the curve of maxima for the second-order phase transitions converge at a = 1.
Figure 9. Parameters of ferroelectric and antiferroelectric ordering for different regions where
the ferroelectric phase exists. Only thermodynamically stable states are depicted. a = 0.36.
From top to bottom: I) γ = 0.337; II) γ = 0.3375; III) γ = 0.338; and IV) γ = 0.339. For region
III the phase transition within the ferroelectric phase is indicated by filled circles.
In figures 9 and 10, the behavior of parameters σ and ξ is shown for every region. In region
I (figures 7 and 8) only one high-temperature second-order phase transition exists. In region II,
in addition to the mentioned one, there are two first-order phase transitions. In narrow region III
there are four phase transition: two of the first and two of the second order; one first-order phase
transition is within the ferroelectric phase. In region IV there are three phase transitions, one of
them being the first-order transition. In region V there are two second-order phase transitions just
like for the Rochelle salt. In region VI there is only one phase transition (of the first order) and in
23603-10
Phase transitions in the Mitsui model
Figure 10. (Continuation of figure 9.) V) a = 0.36, γ = 0.34; VI) a = −0.4, γ = 0.14;
VII) a = 0.36, γ = 0.35.
region VII there are no phase transitions at all.
4. Ω 6= 0 case
4.1. Free energy and conditions for thermodynamic equilibr ium
Now let us consider the case of nonzero tunneling. The eigenvalues of one-site Hamiltonians
(2.3) and (2.4) are as follows:
λ
A
= −1
2
√
∆2
A
+Ω2 , −λ
A
, ∆
A
= ∆+Kη
B
+ Jη
A
;
λ
B
= −1
2
√
∆2
B
+Ω2 , −λ
B
, ∆
B
= −∆+Kη
A
+ Jη
B
;
(4.1)
and the free energy per one unit cell reads
F = Kη
A
η
B
+
J
2
(
η2
A
+ η2
B
)
− 1
2
(√
∆2
A
+Ω2 +
√
∆2
B
+Ω2
)
−
− θ ln
(
1 + e−β
√
∆2
A
+Ω2
)
− θ ln
(
1 + e−β
√
∆2
B
+Ω2
)
. (4.2)
23603-11
Yu.I. Dublenych
The conditions of thermodynamic equilibrium (3.4) yield the following equations:
2η
A
=
∆
A
√
∆2
A
+Ω2
tanh (−βλ
A
),
2η
B
=
∆
B
√
∆2
B
+Ω2
tanh (−βλ
B
).
(4.3)
Like in the Ω = 0 case, let us pass to dimensionless values, introducing one more notation:
ω =
Ω
K + J
. (4.4)
Let us also introduce new variables:
x =
∆
A
+∆
B
2(K + J)
=
η
A
+ η
B
2
=
ξ
2
,
y =
∆
A
−∆
B
2(K + J)
= γ − a
η
A
− η
B
2
= γ − a
σ
2
.
(4.5)
Then the system of equations (4.3) becomes:
2x = A(x + y)−A(−x+ y),
2(γ − y)
a
= A(x+ y) +A(−x+ y),
(4.6)
where
A(z) =
z
2
√
z2 + ω2
tanh
(√
z2 + ω2
2t
)
, (4.7)
and the expression for free energy takes the form:
f = x2 − (γ − y)2
a
− 1
2
(
√
(x+ y)2 + ω2 +
√
(x− y)2 + ω2
)
−
− t ln
(
1 + e−
√
(x+y)2+ω2/t
)
− t ln
(
1 + e−
√
(x−y)2+ω2/t
)
. (4.8)
4.2. Second-order phase transitions
Like in the Ω = 0 case, the system of equations (4.6) has solutions of two types: 1) x = 0; then
the first equation becomes an identity; and 2) x 6= 0, which corresponds to the ferroelectric phase.
Letting x tend to zero, we obtain from (4.6) the system of equations for hypersurface γ = γ(ω, a, t)
of the second-order phase transitions:
B(ỹ)− 1 = 0,
γ = ỹ + aA(ỹ),
(4.9)
where B(z) = dA(z)/dz, ỹ = lim
x→0
y. Having determined B(z), we can write the system in the form:
s2 − 2tω2s
ỹ2
√
ỹ2 + ω2
+
4t
(
ỹ2 + ω2
)
ỹ2
− 1 = 0,
γ = ỹ +
asỹ
2
√
ỹ2 + ω2
,
(4.10)
where the following notation is introduced:
s = tanh
(
√
ỹ2 + ω2
2t
)
.
23603-12
Phase transitions in the Mitsui model
Setting γ equal to zero in equation (4.10), we obtain the expression for maximal temperature at
which the phase transitions exist:
tmax =
ω
ln
1 + 2ω
1− 2ω
. (4.11)
The logarithm in the latter expression makes sense if ω 6 1/2. Hence,
ωmax =
1
2
. (4.12)
Differentiating equations (4.10) with respect to t, setting dγ/dt equal to zero, and eliminating
dỹ/dt, we obtain after simple transformations:
2(1 + a)
(
ỹ2 + ω2
)
(
ỹ2s
√
ỹ2 + ω2 − t
(
ỹ2 + ω2
)
)
= a
(
3tω2s
√
ỹ2 + ω2 − 2ỹ2s
√
ỹ2 + ω2 + 6tω2
(
ỹ2 + ω2
)
− ω2ỹ2s2
)
. (4.13)
This equation together with equations (4.10) determines the point of maximum for the curve γ(t)
of the second-order phase transitions (at fixed ω and a).
The equation for tricritical point is similar to that in the Ω = 0 case:
lim
x→0
dγ
dy
= 0. (4.14)
From this equation we obtain:
(a+ 1)D (ỹ)− 3a [C (ỹ)]2 = 0, (4.15)
where the following notations are introduced:
C(z) =
d2A(z)
dz2
, D(z) =
d3A(z)
dz3
. (4.16)
The tricritical point exists if both equation (4.14) and the following condition are satisfied:
lim
x→0
d2γ
dy2
6 0 (γ > 0). (4.17)
This yields the equation
8 [D (ỹ)]
3 − 9C (ỹ)D (ỹ)E (ỹ) +
9
5
[C (ỹ)]
2
F (ỹ) = 0, (4.18)
where
E(z) =
d4A(z)
dz4
, F (z) =
d5A(z)
dz5
. (4.19)
If the second-order phase transitions exist until t = 0, then it follows from equations (4.10),
that for this zero-temperature transition γ is the following:
γ = (2ω)−2/3
(
a+ (2ω)2/3
)
ỹ, (4.20)
where
ỹ =
1
2
(2ω)2/3
(
1− (2ω)2/3
)1/2
. (4.21)
Rewriting equation (4.15) for zero temperature we obtain:
a =
5(2ω)2/3 − 4
4(2ω)2/3 − 5
. (4.22)
If a is bigger than this value (at fixed ω), then the second-order phase transitions begin at zero
temperature. It follows from equation (4.18) that equation (4.22) is satisfied under condition
ω > 2−5/2 ≈ 0.1768; (4.23)
then a 6 1/2.
23603-13
Yu.I. Dublenych
4.3. First-order phase transitions
The points of the first-order ferroelectric phase transitions can be calculated from the following
system of equations:
2x = A(x+ y)−A(−x+ y),
2(γ − y)
a
= A(x+ y) +A(−x+ y),
γ − y1
a
= A(y1),
f(x, y) = f(0, y1).
(4.24)
Differentiating the last equation with respect to t and eliminating the derivatives, we obtain the
following equation:
x2 − (γ − y)2
a
−
(
x+
γ − y
a
)
(x+ y)2 + ω2
x+ y
−
(
x− γ − y
a
)
(x− y)2 + ω2
x− y
= − (γ − y1)
2
a
− 2(γ − y1)
a
y21 + ω2
y1
, (4.25)
which, together with the system of equations (4.24), yields the extremum of the curve γ(t) of the
first-order phase transitions (at fixed ω). If t = 0, then the solutions of the system of equations (4.24)
satisfy equation (4.25) as well. Hence, for the curve γ(t) of the first-order phase transitions we have
lim
t→0
dγ
dt
= 0. (4.26)
Like in the Ω = 0 case, we can obtain the system of equations for the tricritical point setting
dγ/dy and d2γ/dy2 equal to zero:
a =
2−B− −B+
2B−B+ −B− −B+
,
C+ (1−B−)
3
+ C− (1−B+)
3
= 0,
2x = A+ −A− ,
2(γ − y)
a
= A+ +A− ,
(4.27)
where A± = A(±x+ y), B± = B(±x+ y), and C± = C(±x+ y).
Rewriting the system of equations (4.27) for zero temperature, we obtain:
z+
(
2
(
z2− + ω2
)3/2 − ω2
)3
(
z2− + ω2
)2 +
z−
(
2
(
z2+ + ω2
)3/2 − ω2
)3
(
z2+ + ω2
)2 = 0,
z+
2
√
z2+ + ω2
− z−
2
√
z2− + ω2
− z+ + z− = 0,
a =
4
(
z2+ + ω2
)
3
2
(
z2− + ω2
)
3
2 − ω4
ω2
(
ω2 −
(
z2+ + ω2
)
3
2 −
(
z2− + ω2
)
3
2
) + 1,
γ =
a
4
z+
√
z2+ + ω2
+
z−
√
z2− + ω2
+
1
2
(z+ + z−) ,
(4.28)
where z± = ±x + y. The nontrivial solution of this system at fixed ω 6 2−5/2 corresponds to the
point where the curve of the first-order zero temperature phase transitions and the curve of the
critical points end.
23603-14
Phase transitions in the Mitsui model
Figure 11. Phase coexistence curves in (γ, t)-plane for several values of a (number near curves),
curve of tricritical points (hard line), curve of extrema for the first-order phase transitions (not
hole), curve of maxima for the second-order phase transitions, curve of branchpoints and curve
of critical points (dashed line). ω = 0.1.
The phase coexistence curves in the (γ, t)-plane for several values of a as well as the curves of
tricritical, critical, and branchpoints at ω = 0.1 are depicted in figure 11. The latter two curves
do not converge at zero temperature and, hence, there is an interval of values of a where the
phase diagrams look as shown in figure 12 (b), i.e., they are composed of the curve of second-order
phase transitions and of the curve of first order phase transitions within ferroelectric phase. At
ω = 2−5/2 the curves of critical points and branch- points converge again at zero temperature, and
if 2−5/2 < ω < 0.196815 these curves converge at nonzero temperatures and pass into the curve
Figure 12. Fragments of the phase coexistence curves at ω = 0.1 (see figure 11). a) a = 0.34 (cir-
cles are the extremum points), b) a = 0.662 (dashed line on the left is the curve of branchpoints
and the line on the right is the curve of critical points).
23603-15
Yu.I. Dublenych
of tricritical points (figure 13). The latter is many-valued in low-temperature region. This leads to
the existence of a new type of phase diagram with two tricritical points (figure 14). If ω is bigger
than ≈ 0.196815, only the curve of tricritical points remains.
Figure 13. Phase coexistence curves in (γ, t)-plane for several values of a (number near curves).
ω = 0.18. The heavy line is the line of tricritical points. The curve of tricritical points and the
curve of branchpoints are indistinguishable on this scale.
Figure 14. Fragment of the phase coexistence curve at a = 0.4941, ω = 0.18 (see figure 13).
Dashed line is the curve of tricritical points.
4.4. Regions of existence of the ferroelectric phase
In figures 15–20, the diagram of the regions where the ferroelectric phase exists is shown for
ω = 0.1. The diagram is composed of the same curves as in the Ω = 0 case as well as of the
straight line of second-order zero-temperature phase transitions (4.20) and the curve of maxima
for the first-order phase transitions (4.25) which passes very closely to the curve of first-order
zero-temperature phase transitions.
Like in the ω = 0 case, two curves branch off (in the same point) from the curve of tricritical
points: the curve of minima for the first-order phase transitions and the curve of maxima for the
second-order phase transitions. If ω < 0.196815, then the curve of tricritical points bifurcates in
the curve of critical point and the curve of branchpoints which, if ω > 2−5/2, conflow again into the
curve of tricritical points. The curve of first-order zero-temperature phase transitions is composed
of two parts. The first one corresponds to the transitions from (or into) the ferroelectric phase.
It begins at the origin of coordinates and ends at an extremity of the curve of the branchpoints
(ω 6 2−5/2) or of the tricritical points (ω > 2−5/2) (this extremity is an end point of the curve for
the second-order zero-temperature phase transitions). The second part exists only at ω < 2−5/2,
corresponds to the phase transitions within the ferroelectric phase and ends at an extremity of
the curve of the critical points (see figure 16). In the ω > 2−5/2 case, the point where the curve
23603-16
Phase transitions in the Mitsui model
Figure 15. Regions of existence of the ferroelec-
tric phase. ω = 0.1.
Figure 16. Regions of existence of the ferro-
electric phase (fragment). ω = 0.1. (See also
figure 21).
of tricritical points and the curve of the first-order zero-temperature phase transitions meet is
determined by equation (4.22); in this point the latter curve smoothly turns into the curve of the
second-order zero-temperature phase transitions.
At nonzero ω new regions occur. For instance, in region VIII there are one low-temperature
first-order phase transition into the ferroelectric phase and one second-order phase transition. In
region IX three phase transitions exist: first-order one and two second-order ones (the first-order
phase transition occurs within the ferroelectric phase). In region X there are one first-order phase
transition within the ferroelectric phase and one second-order phase transition.
The curves of minima and maxima for the first-order phase transitions converge in region VIII
Figure 17. Regions of existence of the ferroelec-
tric phase (fragment). ω = 0.1. The filled circle
indicates the point where the curve of tricriti-
cal points forks into the curve of critical points
and the curve of branchpoints.
Figure 18. Regions of existence of the ferroelec-
tric phase (fragment). ω = 0.1.
23603-17
Yu.I. Dublenych
Figure 19. Regions of existence of the ferroelec-
tric phase. ω = 0.18.
Figure 20. Regions of existence of the ferroelec-
tric phase (fragment). ω = 0.18. Dotted line is
the upper part of the curve of tricritical points.
(a ≈ 0.3433, γ ≈ 0.3255, t ≈ 0.0484) cutting off a long spit from it (figure 18). This is region XI
with two low-temperature first-order phase transitions and one second-order phase transition.
Further, the curve of maxima for the first-order phase transitions cuts off narrow strips from
regions IX, V, and VII. These strips are regions XII, XIII, and XIV, respectively, where, in addition,
two close low-temperature first-order phase transitions appear. In figure 17 these regions are not
seen because they are too narrow.
As one can see, at a sufficiently small value of ω the diagram is richer than at ω = 0 but if
the value of ω exceeds some number, the regions of the diagram disappear one after another and
the diagram becomes poorer. Region III disappears firstly. In figure 17 it looks like a short curve
segment at the beginning in the branchpoint. It is region XII that disappears the next, and, at
ω = 2−5/2, region X becomes a point.
5. Conclusions
Hence, we performed a complete analysis of phase transitions in the Mitsui model (without
and with transverse field Ω) in the mean field approximation. Some results concerning the phase
diagram of the Mitsui model were obtained earlier [5, 12, 13] but they were incomplete (for the
Ω = 0 case) or partial (for the Ω 6= 0 case).
In the Ω = 0 case, we derived an analytical expression for tricritical temperature and the
condition of its existence. In this case, there are seven regions in the plane (a, γ) that correspond
to seven different types of behavior of order parameters. At sufficiently small but nonzero Ω their
number doubles. With Ω increasing these regions change their form and shift in the (a, γ)-plane.
Starting with certain value of Ω the number of regions decrease and at ω > 1/2 only the region
without phase transitions remains.
At nonzero Ω, second-order phase transitions are possible at zero temperature, which is not
possible at Ω = 0. The maximal number of phase transitions is four and five at Ω = 0 and at
Ω 6= 0, respectively.
23603-18
Phase transitions in the Mitsui model
Figure 21. Ferroelectric and antiferroelectric order parameters for the regions of existence of
ferroelectric phase. ω = 0.1. Only thermodynamically stable states are depicted. From top to
bottom: VIII) a = 0.65, γ = 0.4035; IX) a = 0.66, γ = 0.4065; X) a = 0.667, γ = 0.4085; and
XIV) a = 0.2, γ = 0.2912765. The first-order phase transition within the ferroelectric phase is
shown by filled circles.
Acknowledgements
The author is grateful to Prof. I. Stasyuk, Prof. R. Levitskii, Dr. T. Verkholyak and Dr. O. Danyliv
for useful discussions.
23603-19
Yu.I. Dublenych
References
1. Mitsui T., Phys. Rev., 1958, 111, 1529; doi:10.1103/PhysRev.111.1259.
2. Žekš B., Shukla G.G., Blinc R., Phys. Rev. B, 1971, 3, 2306; doi:10.1103/PhysRevB.3.2306.
3. Levitskii R.R., Zachek I.R., Verkholyak T.M., and Moina A.P., Phys. Rev. B, 2003, 67, 174112;
doi:10.1103/PhysRevB.67.174112.
4. Levitskii R.R., Andrusyk A.Ya., Zachek I.R., Condens. Matter Phys., 2010, 13, 13705.
5. Levitskii R.R., Zachek I.R., Andrusyk A.Ya., J. Phys. Stud., 2010, 14, 3701.
6. Stasyuk I.V., Velychko O.V., Ferroelectrics, 2005, 316, 51l; doi:10.1080/00150190590963138.
7. Levitskii R.R., Andrusyk A.Ya., Preprint of the Institute for Condensed Matter Physics, ICMP–11–
03U, Lviv, 2011 (in Ukrainian).
8. Levitskii R.R., Andrusyk A.Ya., Preprint of the Institute for Condensed Matter Physics, ICMP–05–
13U, Lviv, 2005 (in Ukrainian).
9. Levitskii R.R., Moina A.P., Andrusyk A.Ya., Slivka A.G., Kedyulich V.M., J. Phys. Stud., 2008, 12,
2603.
10. Danyliv O.D., Physica C, 1998, 309, 303; doi:10.1016/S0921-4534(98)00597-8.
11. Dublenych Yu.I., Preprint of the Institute for Condensed Matter Physics, ICMP–01–01U, Lviv, 2001
(in Ukrainian).
12. Vaks V.G., Introduction to the Microscopic Theory of Ferroelectrics. Nauka, Moscow, 1973 (in Rus-
sian).
13. Levitskii R.R., Verkholyak T.M., Kutnii I.V., Hil I.G., Preprint of the Institute for Condensed Matter
Physics, ICMP–01–11U, Lviv, 2001 (in Ukrainian); Preprint arXiv:cond-mat/0106351, 2001.
Фазовi переходи в моделi Мiцуi
Ю.I. Дубленич
Iнститут фiзики конденсованих систем НАН України, вул. I. Свєнцiцького, 1, 79011 Львiв, Україна
В роботi в наближеннi середнього поля дослiджено фазовi переходи в моделi Мiцуi без поздовжньо-
го поля, проте з поперечним полем. Встановлено взаємооднозначну залежнiсть мiж такою модел-
лю та двопiдґратковою моделлю типу Iзiнга з поздовжнiм i поперечним полями. Побудовано фазовi
дiаграми та дiаграми областей iснування сегнетофази. Для випадку Ω = 0 (Ω – поперечне поле)
одержано простий аналiтичний вираз для трикритичної температури й умову iснування трикрити-
чної точки. Для Ω 6= 0 записано системи рiвнянь для трикритичної точки й умови її iснування.
Ключовi слова: фазовий перехiд, сегнетофаза, модель Мiцуi, трикритична точка
23603-20
http://dx.doi.org/10.1103/PhysRev.111.1259
http://dx.doi.org/10.1103/PhysRevB.3.2306
http://dx.doi.org/10.1103/PhysRevB.67.174112
http://dx.doi.org/10.1080/00150190590963138
http://dx.doi.org/10.1016/S0921-4534(98)00597-8
Introduction
Hamiltonian of the Mitsui model in the mean field approximation
=0 case
Free energy, thermodynamic equilibrium conditions and order parameters
Second-order phase transitions
First-order phase transitions
Regions of existence of the ferroelectric phase
=0 case
Free energy and conditions for thermodynamic equilibrium
Second-order phase transitions
First-order phase transitions
Regions of existence of the ferroelectric phase
Conclusions
|