On the cumulant expansion peculiarity for partition function functional of the cluster de Gennes model
The problem of the functional representation for cluster de Gennes model partition function within the collective variables method is discussed. Contrary to the usual Ising model case the coefficients of the obtained partition function functional (cluster cumulants) depend on temperature T and tra...
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irk-123456789-1209912017-06-14T03:03:52Z On the cumulant expansion peculiarity for partition function functional of the cluster de Gennes model Korynevskii, N.A. The problem of the functional representation for cluster de Gennes model partition function within the collective variables method is discussed. Contrary to the usual Ising model case the coefficients of the obtained partition function functional (cluster cumulants) depend on temperature T and transverse field Γ . Therefore there exists a rigorous limitation on the value of Γ parameter at low temperatures. The equation for maximum value Γl , temperature and short-range intracluster interaction V is obtained. The solutions of this equation have been found. Досліджується проблема функціонального зображення функціонала статистичної суми кластерної моделі де Жена в методі колективних змінних. На противагу до звичайної моделі Ізінга коефіцієнти отриманого функціонала статистичної суми (кластерні кумулянти) залежать від температури T і поперечного поля Γ . Внаслідок цього при низьких тмпературах виникає строге обмеження на величину параметра Γ . Отримано рівняння для максимального значення Γl , температури і величини короткосяжних внутрікластерних взаємодій V . Знайдено розв’язки цього рівняння. 2000 Article On the cumulant expansion peculiarity for partition function functional of the cluster de Gennes model / N.A. Korynevskii // Condensed Matter Physics. — 2000. — Т. 3, № 4(24). — С. 737-747. — Бібліогр.: 7 назв. — англ. 1607-324X DOI:10.5488/CMP.3.4.737 PACS: 05.60.+W, 05.70.Ln, 05.20.Dd, 52.25.Dg, 52.25.Fi http://dspace.nbuv.gov.ua/handle/123456789/120991 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| description |
The problem of the functional representation for cluster de Gennes model
partition function within the collective variables method is discussed. Contrary to the usual Ising model case the coefficients of the obtained partition function functional (cluster cumulants) depend on temperature T and
transverse field Γ . Therefore there exists a rigorous limitation on the value
of Γ parameter at low temperatures. The equation for maximum value Γl ,
temperature and short-range intracluster interaction V is obtained. The
solutions of this equation have been found. |
| format |
Article |
| author |
Korynevskii, N.A. |
| spellingShingle |
Korynevskii, N.A. On the cumulant expansion peculiarity for partition function functional of the cluster de Gennes model Condensed Matter Physics |
| author_facet |
Korynevskii, N.A. |
| author_sort |
Korynevskii, N.A. |
| title |
On the cumulant expansion peculiarity for partition function functional of the cluster de Gennes model |
| title_short |
On the cumulant expansion peculiarity for partition function functional of the cluster de Gennes model |
| title_full |
On the cumulant expansion peculiarity for partition function functional of the cluster de Gennes model |
| title_fullStr |
On the cumulant expansion peculiarity for partition function functional of the cluster de Gennes model |
| title_full_unstemmed |
On the cumulant expansion peculiarity for partition function functional of the cluster de Gennes model |
| title_sort |
on the cumulant expansion peculiarity for partition function functional of the cluster de gennes model |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2000 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120991 |
| citation_txt |
On the cumulant expansion peculiarity for partition function functional of the cluster de Gennes model / N.A. Korynevskii // Condensed Matter Physics. — 2000. — Т. 3, № 4(24). — С. 737-747. — Бібліогр.: 7 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT korynevskiina onthecumulantexpansionpeculiarityforpartitionfunctionfunctionaloftheclusterdegennesmodel |
| first_indexed |
2025-07-08T18:58:58Z |
| last_indexed |
2025-07-08T18:58:58Z |
| _version_ |
1837106347764613120 |
| fulltext |
Condensed Matter Physics, 2000, Vol. 3, No. 4(24), pp. 737–747
On the cumulant expansion peculiarity
for partition function functional of the
cluster de Gennes model
N.A.Korynevskii
Institute for Condensed Matter Physics
of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 79011 Lviv, Ukraine
Received February 16, 2000
The problem of the functional representation for cluster de Gennes model
partition function within the collective variables method is discussed. Con-
trary to the usual Ising model case the coefficients of the obtained parti-
tion function functional (cluster cumulants) depend on temperature T and
transverse field Γ . Therefore there exists a rigorous limitation on the value
of Γ parameter at low temperatures. The equation for maximum value Γl ,
temperature and short-range intracluster interaction V is obtained. The
solutions of this equation have been found.
Key words: cluster de Gennes model, partition function functional, phase
transitions
PACS: 05.60.+W, 05.70.Ln, 05.20.Dd, 52.25.Dg, 52.25.Fi
1. Introduction
The rapid progress in the second order phase transition theory which has taken
place in the last third of the twentieth century was initiated by a revolutionary appli-
cation of quantum field theory methods into this problem. The traditional statistic
physics methods of investigation were, in some sense, relegated to the background.
The idea has arised regarding the limited possibilities of statistical description of
subtle peculiarities of the phase phenomenon. I.Yukhnovskii was the first who put
modern second order phase transition theory on the rigorous base of statistical
physics. There are two most important points in this theory. The first one: a state-
ment about the basic distribution of variables connected with the order parameter
in the phase transition point vicinity. And the second one: a method of layer by
layer integration in the partition function functional which makes it possible to ob-
tain in an explicit form all universal and non-universal characteristics of the system
investigated [1].
c© N.A.Korynevskii 737
N.A.Korynevskii
The starting point of the theory is the expression for a partition function func-
tional. For a usual Ising model which is characterized by one kind of interparticle
interaction such a functional may be represented as an unlimited order exponential
form with the constant coefficients. These coefficients are average values of the usual
cumulants calculated with respect to the distribution of the external field Hamil-
tonian hSz(~Ri), the so-called “reference system”. When a reference system is not
so simple, for example for cluster ferroelectrics it contains short-range intracluster
interactions and a transverse field, the partition function functional coefficients de-
pend on these parameters, temperature and Matsubara’s frequencies [2]. Naturally,
a problem of their behaviour, namely a possibility to change its own sign, arrises.
It must be noted that the sign of a higher order coefficient in the exponential form
for a partition function functional determines the convergency of all functional in-
tegrals, that is the finity of the theory in a wide range of temperature and energy
parameters. It seems very probable that the so-called anti-Curie points which occur
in cluster ferroelectrics at low temperatures [3], arise due to the sign change in the
higher order coefficient (fourth order) of partition function functional exponential
form.
Therefore, the problem of correct calculation of cumulants of the cluster de
Gennes model and their analysis from the point of convergency of total partition
function functional is actual. For the systems described by long-range and short-
range potential and by the transverse field this task is not trivial. The aim of the
present paper is to construct a total partition function functional for the cluster de
Gennes model and to establish the conditions of its finity. The specific calculations
will be done for a cluster system containing two particles in each cell (f 0 = 2).
2. Two-particle cluster system cumulants
Let’s consider the cluster de Gennes model Hamiltonian [4, 5]
H = Γ
∑
q,f
Sx
f (
~Rg) + h
∑
q,f
Sz
f (
~Rq) + V
∑
q,f,f ′
Sz
f (
~Rq)S
z
f ′(~Rq)
+
∑
q,q′,f,f ′
Jff ′(~Rq, ~Rq′)S
z
f (
~Rq)S
z
f ′(~Rq′). (2.1)
Here Γ and h are transverse and longitudional external fields, respectively; V is a
pair intracluster interaction of particles; Jff ′(~Rq, ~Rq′) is a pair long-range potential;
Sα are Pauli matricies; 1 6 q 6 N is a cell-cluster number; 1 6 f 6 f0 is the number
of a particle in the cluster.
In the generalized transition operators representation
Yλ(~Rq) =
∑
m
UλmX
m(~Rq), (2.2)
where Xm(~Rq) are Hubbard-Stasyuk operators, Uλm are eigenfunctions of the total
738
On the cumulant expansion peculiarity. . .
interaction matrix, (1.1) takes such a convenient form:
H =
22f0∑
λ=1
{
Λλ
∑
q
Yλ(~Rq)−
1
2
∑
q,q′
Φλ(~Rq, ~Rq′)Yλ(~Rq)Yλ(~Rq′)
}
. (2.3)
Here Λλ are energy parameters of an isolated cluster (reference system), Φλ(~Rq, ~Rq′)
are eigenvalues of long-range interaction matrix. For a two-particle (f0 = 2) inter-
acting cluster system, for example, there are:
Λ1 = −2h, Φ1(~Rq, ~Rq′) = J11(~Rq, ~Rq′) + J12(~Rq, ~Rq′),
Λ9 = E1 =
√
V 2 + 4Γ2, Φ5(~Rq, ~Rq′) = J11(~Rq, ~Rq′)− J12(~Rq, ~Rq′);
Λ12 = E2 = −
√
V 2 + 4Γ2,
Λ15 = E3 = −V,
Λ16 = E4 = V, (2.4)
all other coefficients Λλ and Φλ(~Rq, ~Rq′) in this case are equal to zero.
Taking a “non-interacting” part of the (1.3) Hamiltonian
H0 =
∑
λ6=1
∑
q
ΛλYλ(~Rq), (2.5)
as a reference system, for partition function functional in the collective variables
representation one may obtain [6]:
Z = ZN
0
∫
(dρλ(~k, ν))
N exp
{22f0∑
λ=1
∑
k6B
∞∑
ν=0
[
β
2
Φλ(~k)ρλ(~k, ν)ρλ(−~k,−ν)
− 2β
√
N hρλ(~k, ν)δλ1δ(~k)δ(ν)
]}
×
∫
(dωλ(~k, ν))
N exp
{
i2π
22f0∑
λ=1
∑
k,ν
ωλ(~k, ν)ρλ(~k, ν)
}
× exp
{ ∞∑
n=1
(−i2π)n
n!
22f0∑
λ1...λn 6=1
∑
k1,...,kn6B
∞∑
ν1,...νn=0
Mλ1...λn
(~k1, ν1, ...~kn, νn)
× ωλ1
(~k1, ν1)...ωλn
(~kn, νn)
}
. (2.6)
Coefficients Mλ1...λn
(~k1, ν1, ...~kn, νn) are usual cumulant average values of the
generalized transition operators Yλ(~Rg) products in the frequency-momentum rep-
resentation
ρ̂λ(~k, ν) =
1
β
∫ β
0
dβ ′e−iβ′ν
N∑
q=1
e−βH0Yλ(~Rq)e
βH0ei
~k ~Rq . (2.7)
739
N.A.Korynevskii
In the (1.6), ρλ(~k, ν) are collective variables, corresponding to ρ̂λ(~k, ν); ν = 2π/βn
(n = 0,±1,±2, . . .) are Matsubara’s frequencies; β = 1/kT , T is the absolute tem-
perature; k is the Boltzman constant; B is the Brillouin zone boundary; Φλ(k) is a
Fourier transform of Φλ(~Rq, ~Rq′); Z0 = Sp {exp(−βH0)}.
Since collective variables ρλ(~k, ν) decribe only a long-range part of the total
Hamiltonian (short-range interactions are taken into account by Z0) among different
cumulants in (2.6), it is enough to take only the ones with λ, for which Φλ(~Rg, ~Rg′) 6=
0. In the two-particle cluster system such indeceis are λ = 1 and λ = 5. Thus, all
cluster cumulants in this case may be expressed by cumulant average values
〈RYλ(~Rq1 , β1)Yλ(~Rq2, β2)...Yλ(~Rqm, βm)...〉c0 (2.8)
in which
Y1(~Rq, β) =
0 0 beβE13 0
0 0 aeβE23 0
be−βE13 ae−βE23 0 0
0 0 0 0
,
Y5(~Rq, β) =
0 0 0 −aeβE14
0 0 0 beβE24
0 0 0 0
−ae−βE14 be−βE24 0 0
, (2.9)
where
a2 =
1
4
(
1 +
V√
V 2 + 4Γ2
)
, b2 =
1
4
(
1− V√
V 2 + 4Γ2
)
,
Eij = Ei −Ej . (2.10)
R in (2.9) is a symbol for “time” arrangement with respect to the inverse tem-
perature β.
〈...〉0 = Sp
{
...e−βH0
} [
Sp
{
e−βH0
}]−1
. (2.11)
It may be easily tested that the non-equal, identically to zero expressions (2.8),
contain only an even number of operators Yλ(~Rq, β).
The calculation of (2.8) is based on the method similar to the Vikh-Blokch-
Dominisis theorem [7] with the commutation relation
[Yλ(~Rq), Yλ′(~Rq′)] =
∑
µ
W µ
λλ′Yµ(~Rq)δ(~Rq − ~Rq′),
W µ
λλ′ =
∑
r,s,t
(UrsλUstλ′ − UstλUrsλ′)Urtµ, (2.12)
where r, s, t are ordinary indicies and λ, λ′, µ are double indices.
740
On the cumulant expansion peculiarity. . .
For the n-th order cluster cumulant we have a formula:
M(λ)
2n (
~k1, ν1, ..., ~k2n, ν2n) = 〈RYλ(~k1, ν1)...Yλ(~k2n, ν2n)〉c0
=
1
(β2N)n
∑
q1,...,q2n
e−i(~k1Rq1
+...+~k2n ~Rq2n
)
∫ β
0
dβ1...
∫ β
0
dβ2ne
−i(β1ν1+...+β2nν2n)
× 〈RYλ(~Rq1 , β1)...Yλ(~Rq2n , β2n)〉c0. (2.13)
The general form of the (2n)-th order cumulant is:
M2n = M̃2n −
[
(2n)!
2!(2n− 2)!
M̃2M̃2n−2 +
(2n)!
4!(2n− 4)!
M̃4M̃2n−4
+
(2n)!
6!(2n− 6)!
M̃6M̃2n−6 + ...
+
(2n)!
(2p)!(2n− 2p)!
(
1− δn−2p
2
)
2p6n
M̃2pM̃2n−2p
]
+ 2!
[
(2n)!
2!(2!)2(2m− 4)!
M̃2
2M̃2n−4 +
(2n)!
2!4!(2n− 6)!
M̃2M̃4M̃2n−6
+
(2n)!
2!(4!)2(2n− 8)!
M̃2
4M̃2n−8 +
(2n)!
2!6!(2n− 8)!
M̃2M̃6M̃2n−8
+
(2n)!
4!6!(2n− 10)!
M̃4M̃6M̃2n−10 +
(2n)!
2!8!(2n− 10)!
M̃2M̃8M̃2n−10
+
(2n)!
2!(6!)2(2n− 12)!
M̃2
6M̃2n−12 +
(2n)!
4!8!(2n− 12)!
M̃4M̃8M̃2n−12
+
(2n)!
2!10!(2n− 12)!
M̃2M̃10M̃2n−12 + ...
]
− 3!
[
(2n)!
3!(2!)3(2n− 6)!
M̃3
2M̃2n−6 +
(2n)!
2!(2!)24!(2n− 8)!
M̃2
2M̃4M̃2n−8
+
(2n)!
2!(2!)26!(2n− 10)!
M̃2
2M̃6M̃2n−10
+
(2n)!
2!2!(4!)2(2n− 10)!
M̃2M̃2
4M̃2n−10
+
(2n)!
2!(2!)28!(2n− 12)!
M̃2
2M̃8M̃2n−12
+
(2n)!
2!4!6!(2n− 12)!
M̃2M̃4M̃6M̃2n−12
+
(2n)!
3!(4!)3(2n− 12)!
M̃3
4M̃2n−12 + ...
]
+ 4!
[
(2n)!
4!(2!)4(2n− 8)!
M̃4
2M̃2n−8
+
(2n)!
3!(2!)34!(2n− 10)!
M̃3
2M̃4M̃2n−10 + ...
]
741
N.A.Korynevskii
+ . . .+ (−1)n−1(n− 1)!
(2n)!
(2!)nn!
M̃n
2 , (2.14)
where M̃2n are redusible parts of (2.13). After a precise calculation of (2.13), (2.14)
we obtain a final formula for the (2n)-th order cumulants
M(λ)
2n (
~k1, ν1, ...~k2n, ν2n) =
[
A(λ)(E2
1 −E2
3)
2βE1Z0
]n
Cn−1(xλ)
× δ(~k1 + ... + ~k2n)δ(ν1 + ... + ν2n). (2.15)
Here Cn−1(xλ) is a (n− 1)-order polynomos
Cn−1(xλ) =
n∑
p=1
(−1)p−1apx
n−p
λ
= a1x
n−1
λ − a2x
n−2
λ + ... + (−1)p−1apx
n−p
λ + ...+ (−1)n−1an,
xλ =
B(λ)Z0
βA(λ)(E2
1 − E2
3)
, xλ > 0,
A(λ) =
e−βE2 − e∓βE3
(E2 ∓E3)2 + ν2
+
e∓βE3 − e−βE1
(E1 ∓ E3)2 + ν2
,
B(λ) =
(E1 ∓E3)(cosh β(E2 ∓E3)− 1)
(E2 ∓E3)2 + ν2
− (E2 ∓E3)(cosh β(E1 ∓E3)− 1)
(E1 ∓E3)2 + ν2
,
Z0 =
∑
λ
e−βEλ = 2 coshβV + 2 cosh β
√
V 2 + 4Γ2. (2.16)
ap coefficients are equal to the simple sum of corresponding coefficients of the p-th
level separation in (2.14). So, for example
m = 1, c0 = 1,
m = 2, c1 = x− 3,
m = 3, c2 = x2 − 15x+ 30,
m = 4, c3 = x3 − 63x2 + 420x− 630,
m = 5, c4 = x4 − 255x3 + 4410x2 − 18900x+ 22680,
m = 6, c5 = x5 − 1023x4 + 42240x3 − 395010x2 + 1247400x− 1247400,
m = 7, c6 = x6 − 4095x5 + 390390x4 − 7207200x3 + 45405360x2
− 113513400x− 97297200,
m = 8, c7 = x7 − 16383x6 + 3554460x5 − 123513390x4 + 1394593200x3
− 6583777200x2 + 13621608000x− 10216206000. (2.17)
Taking into account the factors connected with the normalization of Yλ(~Rq) op-
erators (
√
2)2m and the cluster structure of the Hamiltonian 2 at V = 0; Γ = 0 case,
for example
MIsing
λλλλ =
1
8
(
√
2)2·2 · 2− 3
16
(
√
2)2·2 · 22 = −2,
742
On the cumulant expansion peculiarity. . .
one obtains all values of the classic Ising model cumulants:
M2 = 1, M10 = 7936,
M4 = −2, M12 = −353792,
M6 = 16, M14 = 22368256,
M8 = −272, M16 = −1903757312 (2.18)
and so on. In the cluster case at V = 0, Γ = 0 we have:
M2 =
1
4
= 0.25, M10 =
31
64
≈ 0.4844,
M4 = − 1
16
= −0.0625, M12 = −691
256
≈ −2.6992,
M6 =
1
16
= 0.0625, M14 =
87375
4096
≈ 21.3318, (2.19)
M8 = − 17
128
≈ −0.1328, M16 = −798569
4096
≈ −194.9631
and so on. (2.19) demonstrate that at V = 0, Γ = 0 all cumulants possess a good
sign and the (2.6) functional is signified correctely.
3. Role of V and Γ parameters in the cluster cumulants be-
haviour
It may be easily seen that cumulants M2n(~k1, ν1, ...~k2n, ν2n) behaviour with re-
spect to intracluster interaction V and transverse field Γ are absolutely different.
For instance let’s consider λ = 1.
V = 0
M2n =
(βΓ)n sinhn 2βΓ
β2n(4Γ2 + ν2)n(cosh 2βΓ + 1)n
Cn−1(x),
x =
sinh 2βΓ
βΓ
> 2; (3.1)
Γ = 0
M2n =
(
eβV
4 coshβV
)m
Cn−1(y),
y =
2 cosh βV
eβV
6 2. (3.2)
At large Γ all cumulants starting from n = 2 are divergent. But at large V they
are finite. So, these two types of interactions play different role in the cluster system
stability formation.
The cumulants increasing at Γ → ∞ is not a unique danger for (2.6) functional.
Because Cn−1(x) is the (n − 1)-order polynomos, the higher order cumulants (n >
743
N.A.Korynevskii
Table 1. The first real root of polynomos Cn−1(x) and the limited values of
transverse field intensity (for short range interaction absence case) which satisfy
the investigated system stability.
m 1 2 3 4 5 6 7 8
x – 3.0 2.3765 2.2017 2.1264 2.0868 2.0634 2.0482
βΓl ∞ 0.8111 0.5175 0.3833 0.3049 0.2534 0.2171 0.1895
2) become a non-monotonous function of the x-parameter (2.16), so Cn−1(x) may
change its sign at certain values of V and Γ. As a result the functional (2.6) becomes
infinite at certain values of these parameters and temperature. Evidently, the first
root of Cn−1(x) (for fixed T ) determine a region of physically possible values of those
parameters.
Introducing A(λ) and B(λ) into x (2.16) one may test that x depends on Mat-
subara frequency very slightly. So, for the analytic regard of some first polynomos
Cn−1(x) it is possible to put ν = 0. In this approximation:
x =
cosh βV + cosh β
√
V 2 + 4Γ2
βΓ2
×
[√
V 2 + 4Γ2(V 2 + Γ2)(cosh βV cosh β
√
V 2 + 4Γ2 − 1)
− V (V 2 + 3Γ2) sinhβV sinh β
√
V 2 + 4Γ2
]
×
[
(V 2 + 2Γ2) sinhβ
√
V 2 + 4Γ2 + V
√
V 2 + 4Γ2(cosh β
√
V 2 + 4Γ2 − eβV )
]−1
.
(3.3)
The first real root (x > 0) of polynomos (2.16) for m 6 8 and the corresponding
values of βΓl (at V = 0), calculated from the second equation in (3.1), are presented
in table 1. table 1
The numerical calculations for m > 8 are rather cumbersome, but it may be
prooved that
lim
m→∞
Cm−1(x) = 0, at x = 2. (3.4)
The limiting value βΓl corresponds to the situation when cluster cumulants
change their sign and the partition function functional becomes infinite, so the sys-
tem investigated becomes unstable. The limiting value Γ l nonlineary grows when
temperature falls down. At large m βΓl → 0. It means when V = 0 the total func-
tional (1.6) is convergable only for Γ = 0. Transverse field, as is well-known, destroys
the system at low temperatures.
Absolutely different situation takes place when Γ = 0. In this case polynomos
Cm−1(x) have no roots in the range 0 < V < ∞, so all cluster cumulants possess a
good sign and the cluster system is stable at every temperature.
The situation when both V 6= 0 and Γ 6= 0 is, naturally, more complicated.
The dependence of Γl on V at different temperatures is presented in figure 1 and
744
On the cumulant expansion peculiarity. . .
0
20
40
60
80
100
120
140
160
180
200
5 10 15 20 25 30 35 40 45 50
Γ
V
β=0.1
β=0.05
β=0.02
β=0.01
β=0.005
β=0.0025
β=0.002
β=0.001
Figure 1. The limit value of the transverse field Γl dependence on the intracluster
interaction V at different values of inverse temperature β.
0
10
20
30
40
50
60
70
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
Γ
β
V=2
V=5
V=10
V=20
Figure 2. The limit value of the transverse field Γl dependence on the inverse
temperature β at different values of the intracluster interaction V .
745
N.A.Korynevskii
the dependence of Γl on β at different V is presented in figure 2. One can see that
there is a limited value Γl which divides Γ, V (or Γ, T ) diagram of state into two
regions at every value of temperature and intracluster interactions. In the first one
(Γ < Γl) the functional (2.6) is defined correctly and the system remains in the stable
state. Upper of the line Γ = Γl cluster cumulants change their sign, functional (2.6)
becomes divergent and the system investigated loses its stability. Correspondingly,
calculated on the base of (2.6) functional thermodynamic functions will demonstrate
unphysical behaviour. It must be noted that the situation discussed here takes place
only for non-Gaussian approximation in (2.6) (when m > 2).
In practice, the calculation on the base of (2.6) functional envisages the usage
of a limited (i.e. ussually not high) power of ωλ(~k, ν) variables. It is reasonable that
even the first non-Gaussian approximation leads to a certain nonphysical behaviour
of thermodynamic function of de Gennes model at low temperatures. Temperature
points of unphysical behaviour (anomaly of polarization, dielectric susceptibility,
heat capacity etc.) are called anti-curie points [3]. It must be noted that minimization
procedure, used in [3] in the framework of self-consistent cluster approximation in
its origin is close to taking into account some higher order correlation effects like
non-Gaussian fourth order distribution in (2.6).
References
1. Yukhnovskii I.R. Phase Transition of the Second Order. Collective Variables Method.
Singapore, World. Scientific, 1987.
2. Korynevskii N.A. // TMPh, 1983, vol. 55, p. 291 (in Russian).
3. Vaks V.G. Introduction into Microscopic Theory of Ferroelectricity. Moskow, Nauka,
1973 (in Russian).
4. Stasyuk I.V., Levitskii R.R., Korynevskii N.A. // Phys. Stat. Sol. (b), 1979, vol. 91,
p. 541.
5. Yukhnovskii I.R., Korynevskii N.A. // Phys. Stat. Sol. (b), 1989, vol. 153, p. 583.
6. Korynevskii N.A. On the Maximum Transverse Field Magnitude in the Quantum
Cluster Model. Preprint of the Institute for Condensed Matter Physics, ICMP–99–
09U, Lviv, 1999, 20 p. (in Ukrainian).
7. Abrikosov A.A., Gor’kov L.P., Dzialoshynskii I.E. Methods of Quantum Field Theory
in Statistic Physics. Moskow, Izd. Phys.-Math., 1962 (in Russian).
746
On the cumulant expansion peculiarity. . .
Про особливості кумулянтного розкладу для
функціоналу статистичної суми кластерної моделі
де Жена
М.А.Кориневський
Інститут фізики конденсованих систем НАН Укpаїни,
79011 Львів, вул. Свєнціцького, 1
Отримано 16 лютого 2000 р.
Досліджується проблема функціонального зображення функціонала
статистичної суми кластерної моделі де Жена в методі колективних
змінних. На противагу до звичайної моделі Ізінга коефіцієнти отри-
маного функціонала статистичної суми (кластерні кумулянти) зале-
жать від температури T і поперечного поля Γ . Внаслідок цього при
низьких тмпературах виникає строге обмеження на величину пара-
метра Γ . Отримано рівняння для максимального значення Γl , тем-
ператури і величини короткосяжних внутрікластерних взаємодій V .
Знайдено розв’язки цього рівняння.
Ключові слова: кластерна модель де Жена, функціонал
статистичної суми, фазові перетворення.
PACS: 05.60.+W, 05.70.Ln, 05.20.Dd, 52.25.Dg, 52.25.Fi
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