3D Ising system in an external field. Recurrence relations for the asymmetric ρ⁶ model
The 3D one-component spin system in an external magnetic field is studied using the collective variables method. The integration of the partition function of the system over the phase space layers is performed in the approximation of the sextic measure density including the even and the odd powers o...
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| Опубліковано в: : | Condensed Matter Physics |
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Інститут фізики конденсованих систем НАН України
2001
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| Цитувати: | 3D Ising system in an external field. Recurrence relations for the asymmetric ρ⁶ model / I.V. Pylyuk, M.P. Kozlovskii // Condensed Matter Physics. — 2001. — Т. 4, № 1(25). — С. 15-24. — Бібліогр.: 18 назв. — англ. |
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Pylyuk, I.V. Kozlovskii, M.P. 2017-06-08T11:14:23Z 2017-06-08T11:14:23Z 2001 3D Ising system in an external field. Recurrence relations for the asymmetric ρ⁶ model / I.V. Pylyuk, M.P. Kozlovskii // Condensed Matter Physics. — 2001. — Т. 4, № 1(25). — С. 15-24. — Бібліогр.: 18 назв. — англ. DOI:10.5488/CMP.4.1.15 https://nasplib.isofts.kiev.ua/handle/123456789/119752 The 3D one-component spin system in an external magnetic field is studied using the collective variables method. The integration of the partition function of the system over the phase space layers is performed in the approximation of the sextic measure density including the even and the odd powers of the variable (the asymmetric ρ⁶ model). The general recurrence relations between the coefficients of the effective measure densities are obtained. The new functions appearing in these recurrence relations are given in the form of a convergent series. Тривимірна однокомпонентна спінова система в зовнішньому магнітному полі досліджується з допомогою методу колективних змінних. Інтегрування статистичної суми системи по шарах фазового простору здійснюється в наближенні густини міри, яка містить в собі парні та непарні степені змінної до шостого включно (асиметрична модель ρ⁶). Одержано загальні рекурентні співвідношення між коефіцієнтами ефективних густин мір. Нові функції, що появляються в цих рекурентних співвідношеннях, подані у вигляді збіжних рядів. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics 3D Ising system in an external field. Recurrence relations for the asymmetric ρ⁶ model Tривимірна ізінгівська система в зовнішньому полі. Рекурентні співвідношення для асиметричної моделі ρ⁶ Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
3D Ising system in an external field. Recurrence relations for the asymmetric ρ⁶ model |
| spellingShingle |
3D Ising system in an external field. Recurrence relations for the asymmetric ρ⁶ model Pylyuk, I.V. Kozlovskii, M.P. |
| title_short |
3D Ising system in an external field. Recurrence relations for the asymmetric ρ⁶ model |
| title_full |
3D Ising system in an external field. Recurrence relations for the asymmetric ρ⁶ model |
| title_fullStr |
3D Ising system in an external field. Recurrence relations for the asymmetric ρ⁶ model |
| title_full_unstemmed |
3D Ising system in an external field. Recurrence relations for the asymmetric ρ⁶ model |
| title_sort |
3d ising system in an external field. recurrence relations for the asymmetric ρ⁶ model |
| author |
Pylyuk, I.V. Kozlovskii, M.P. |
| author_facet |
Pylyuk, I.V. Kozlovskii, M.P. |
| publishDate |
2001 |
| language |
English |
| container_title |
Condensed Matter Physics |
| publisher |
Інститут фізики конденсованих систем НАН України |
| format |
Article |
| title_alt |
Tривимірна ізінгівська система в зовнішньому полі. Рекурентні співвідношення для асиметричної моделі ρ⁶ |
| description |
The 3D one-component spin system in an external magnetic field is studied using the collective variables method. The integration of the partition function of the system over the phase space layers is performed in the approximation of the sextic measure density including the even and the odd powers of the variable (the asymmetric ρ⁶ model). The general recurrence relations between the coefficients of the effective measure densities are obtained. The new functions appearing in these recurrence relations are given in the form of a convergent series.
Тривимірна однокомпонентна спінова система в зовнішньому магнітному полі досліджується з допомогою методу колективних змінних. Інтегрування статистичної суми системи по шарах фазового простору здійснюється в наближенні густини міри, яка містить в собі парні та непарні степені змінної до шостого включно (асиметрична модель ρ⁶). Одержано загальні рекурентні співвідношення між коефіцієнтами ефективних густин мір. Нові функції, що появляються в цих
рекурентних співвідношеннях, подані у вигляді збіжних рядів.
|
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/119752 |
| citation_txt |
3D Ising system in an external field. Recurrence relations for the asymmetric ρ⁶ model / I.V. Pylyuk, M.P. Kozlovskii // Condensed Matter Physics. — 2001. — Т. 4, № 1(25). — С. 15-24. — Бібліогр.: 18 назв. — англ. |
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| first_indexed |
2025-11-25T21:20:34Z |
| last_indexed |
2025-11-25T21:20:34Z |
| _version_ |
1850556560411983872 |
| fulltext |
Condensed Matter Physics, 2001, Vol. 4, No. 1(25), pp. 15–24
3D Ising system in an external field.
Recurrence relations for the
asymmetric ρ
6 model
I.V.Pylyuk, M.P.Kozlovskii
Institute for Condensed Matter Physics
of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 79011 Lviv, Ukraine
Received July 26, 2000
The 3D one-component spin system in an external magnetic field is stud-
ied using the collective variables method. The integration of the partition
function of the system over the phase space layers is performed in the ap-
proximation of the sextic measure density including the even and the odd
powers of the variable (the asymmetric ρ
6 model). The general recurrence
relations between the coefficients of the effective measure densities are
obtained. The new functions appearing in these recurrence relations are
given in the form of a convergent series.
Key words: Ising model, external field, collective variables, recurrence
relations
PACS: 05.50.+q, 75.10.Hk
1. Introduction
The object of investigation is the Ising model on a simple cubic lattice with
an exponentially decreasing interaction potential. The Ising model, which is simple
and convenient for mathematical analysis, is widely used in the theory of phase
transitions for an analysis of properties of various magnetic and nonmagnetic systems
(binary mixtures, lattice model of liquids, ferromagnets, etc.).
The three-dimensional (3D) Ising system in an external magnetic field will be
studied here based on the collective variables (CV) method [1]. This original method
can be extended to the liquid-gas system [2], multicomponent fluids [3]. In this case,
the method makes it possible to use a convergent series by applying the solution
of the 3D Ising model in contrast to the asymptotic ǫ-expansions (see, for example,
[4,5]).
The CV method allows one to calculate the partition function of the Ising model
and to obtain not only the universal quantities but also analytic expressions for
c© I.V.Pylyuk, M.P.Kozlovskii 15
I.V.Pylyuk, M.P.Kozlovskii
nonuniversal characteristics as functions of the microscopic parameters of the sys-
tem. The statistical description of the spin system behaviour is performed in real
3D space on the microscopic level. The main results obtained in the CV method for
the 3D systems in the absence of an external field are presented in [1,6].
An important condition for describing the system properties by the CV method
is the use of non-Gaussian measure densities [1]. Such a measure density at a zero
external field is represented as an exponential function of the CV, the argument of
which contains, along with the quadratic term, higher even powers of the variable
with the corresponding coupling constants. The confinement to the 6th power of the
CV in the expression for a measure density corresponds to the ρ6 model. This model
provides a more adequate quantitative description of the 3D Ising system behaviour
in comparison with the ρ4 model (see [7–9]).
In the case of a nonzero external field, the non-Gaussian measure density includes
the terms proportional to odd powers of the CV in addition to the terms propor-
tional to even powers. The partition function of the 3D one-component system in an
external field is presented below in the approximation of the sextic measure density,
which involves the even and odd powers up to the 6th power of the variable (the
asymmetric ρ6 model). A partition function functional is similar to the Ginzburg-
Landau functional (see, for example, [10]). We calculate the partition function by
successively integrating its expression over the short-wave or the so-called unimpor-
tant variables. The corresponding renormalization group (RG) transformation can
be related to the Wilson type [11]. The general recurrence relations (RR) corre-
sponding to the asymmetric ρ6 model and the new special functions entering the
RR are considered.
In our earlier works (see, for example, [7–9]), an infinitely weak external field
was introduced in the course of calculation of the contribution from the long-wave
modes of the spin moment density oscillations to the 3D Ising system thermodynamic
characteristics. In this paper, we introduce an external field in the Hamiltonian from
the outset. Such an approach leads to the appearance of odd powers of the CV in
the expression for the partition function and makes it possible to describe a lot
of quantities (cumulants in the Jacobian of the transition to the CV set, initial
coefficients of the partition function, etc.) as functions of an external field.
2. Recurrence relations for the asymmetric ρ
6 model
We consider the Ising system on a simple cubic lattice with the period c. The
Hamiltonian of the system has the form
H = −1
2
∑
i,j
Φ(rij)σiσj − h
∑
i
σi. (2.1)
Here h is an external field, rij is the distance between particles at the sites i and j, σi
is the operator of the z-component of the spin at the ith site, having two eigenvalues
16
3D Ising system in an external field . . .
+1 and −1. The interaction potential is an exponentially decreasing function
Φ(rij) = A exp
(
−rij
b
)
, (2.2)
where A is a constant, b is the radius of effective interaction. For the Fourier trans-
form of the interaction potential, we use the following approximation:
Φ̃(k) =
{
Φ̃(0)(1− 2b2k2), k 6 B′,
0, B′ < k 6 B.
(2.3)
Here B = π/c is the boundary of the Brillouin half-zone, B ′ = (b
√
2)−1 is de-
termined from the condition for the application of the parabolic approximation
Φ̃(0)(1 − 2b2k2) = 0, Φ̃(0) = 8πA(b/c)3. At Φ̃(0) = 2dJ, b = bI = c/(2
√
d) (J is
the constant of the interaction between the nearest neighbours, d = 3 is the space
dimension) for small values of the wave vectors k, the parabolic approximation of
the Fourier transform of the exponentially decreasing interaction potential corre-
sponds to the analogous approximation of the Fourier transform for the interaction
potential between the nearest neighbours [12].
In the CV representation for the partition function of the 3D Ising model in an
external field, we have
Z =
∫
exp
[
1
2
∑
k
βΦ̃(k)ρkρ−k
]
J(ρ) (dρ)N , (2.4)
where the summation over the wave vectors k is carried out within the first Brillouin
zone, β = 1/(kBT ) is the inverse thermodynamic temperature, kB is the Boltzmann
constant. The CV ρk are introduced using the relations of the type of an analyt-
ic functional for operators of spin density oscillation modes ρ̂k = (
√
N)−1
∑
l σl ·
exp(−ikl) [1]. The Jacobian of the transition from the set of the N spin variables σ l
to the set of the CV ρk can be written as
J(ρ) = (2 cosh h′)N
∫
exp
[
2πi
∑
k
ωkρk
+
∑
n>1
(−2πi)nN−(n−2)/2Mn
n!
∑
k1,...,kn
ωk1
· · ·ωkn
δk1+···+kn
]
(dω)N. (2.5)
Here δk1+···+kn
is the Kronecker symbol, the variables ωk are conjugate to ρk. The
cumulants Mn are functions of a generalized field h′ = βh:
M1 = tanhh′ = x,
M2 = 1− tanh2 h′ = y,
M3 = −2xy,
M4 = −2y2 + 4x2y,
M5 = 16xy2 − 8x3y,
M6 = 16y3 − 88x2y2 + 16x4y. (2.6)
17
I.V.Pylyuk, M.P.Kozlovskii
We shall proceed from the expression for the partition function in the approxima-
tion of the asymmetric ρ6 model. Putting n = 6 in (2.5) and carrying out integration
in (2.4) with respect to the variables ρk and ωk with indices B ′ < k 6 B, followed
by the integration with respect to the N ′ variables ωk, we obtain
Z = (2 coshh′)N2(N
′−1)/2ea
′
0
N ′
∫
exp
[
−(N ′)1/2a′1ρ0 −
1
2
∑
k6B′
d′(k)ρkρ−k
−
6
∑
l=3
a′l
l!(N ′)l/2−1
∑
k1,...,kl6B′
ρk1
· · ·ρkl
δk1+···+kl
]
(dρ)N
′
, (2.7)
where N ′ = Ns−d
0 , s0 = B/B′ = π
√
2b/c, and
d′(k) = a′2 − βΦ̃(k). (2.8)
The coefficients
a′0 = lnQ(M), Q(M) = π−1
(
4!
M̄4
sd0
)1/4
I0c(η
′),
a′1 = −
(
4!
M̄4
sd0
)1/4
C1sc(η
′),
a′k =
(
4!
M̄4
sd0
)k/4
Cksc(η
′), k = 2, 3, 4, 5, 6 (2.9)
are functions of a field h′ and a ratio b/c. In this expressions, M̄4 = −M4. The set
η′ = {η′1, η′2, η′3, η′4, η′5, η′6} is the set of initial arguments
η′1 = M1
(
4!
M̄4
)1/4
s
3d/4
0 , η′2 =
M2
2
(
4!
M̄4
)1/2
s
d/2
0 ,
η′3 =
M3
3!
(
4!
M̄4
)3/4
s
d/4
0 , η′4 ≡ 1,
η′5 =
M5
5!
(
4!
M̄4
)5/4
s
−d/4
0 , η′6 =
M6
6!
(
4!
M̄4
)3/2
s
−d/2
0 . (2.10)
The special functions Clsc(η
′) in (2.9) are combinations of the functions
F(2l+1)sc(η
′) = I(2l+1)s(η
′)/I0c(η
′) and F2lcc(η
′) = I2lc(η
′)/I0c(η
′):
C1sc(η
′) = F1sc(η
′),
C2sc(η
′) = F2cc(η
′) + F 2
1sc(η
′),
C3sc(η
′) = F3sc(η
′)− 3F2cc(η
′)F1sc(η
′)− 2F 3
1sc(η
′),
C4sc(η
′) = −F4cc(η
′)− 4F3sc(η
′)F1sc(η
′) + 3F 2
2cc(η
′) + 12F2cc(η
′)F 2
1sc(η
′)
+6F 4
1sc(η
′),
C5sc(η
′) = −F5sc(η
′) + 5F4cc(η
′)F1sc(η
′) + 10F3sc(η
′)F2cc(η
′)
18
3D Ising system in an external field . . .
+20F3sc(η
′)F 2
1sc(η
′)− 30F 2
2cc(η
′)F1sc(η
′)− 60F2cc(η
′)F 3
1sc(η
′)
−24F 5
1sc(η
′),
C6sc(η
′) = F6cc(η
′) + 6F5sc(η
′)F1sc(η
′)− 15F4cc(η
′)F2cc(η
′)
−30F4cc(η
′)F 2
1sc(η
′) + 10F 2
3sc(η
′)− 120F3sc(η
′)F2cc(η
′)F1sc(η
′)
−120F3sc(η
′)F 3
1sc(η
′) + 30F 3
2cc(η
′) + 270F 2
2cc(η
′)F 2
1sc(η
′)
+360F2cc(η
′)F 4
1sc(η
′) + 120F 6
1sc(η
′). (2.11)
Here
I(2l+1)s(η
′) =
∞
∫
0
t2l+1fs(t) dt, I2lc(η
′) =
∞
∫
0
t2lfc(t) dt, l = 0, 1, 2, 3, (2.12)
and
fs(t) = sin(η′1t− η′3t
3 + η′5t
5) exp(−η′2t
2 − η′4t
4 − η′6t
6),
fc(t) = cos(η′1t− η′3t
3 + η′5t
5) exp(−η′2t
2 − η′4t
4 − η′6t
6). (2.13)
We use the method of layer-by-layer integration of (2.7) with respect to the
variables ρk proposed by I.R.Yukhnovskii [1]. The integration begins from the vari-
ables ρk with large values of k (of the order of the Brillouin half-zone boundary).
The phase space of the CV ρk is divided into layers with the division parameter
s. In each nth layer (corresponding to the region of wave vectors Bn+1 < k 6 Bn,
Bn+1 = Bn/s, s > 1), the Fourier transform of the potential Φ̃(k) is replaced by
its average value Φ̃(Bn+1, Bn). As a result of step-by-step calculation of the parti-
tion function, the number of integration variables in the expression for this quantity
decreases gradually. The partition function is then represented as a product of the
partial partition functions of individual layers and the integral of the “smoothed”
effective measure density. After the integration over the n+1 layers of the CV space,
we obtain
Z = (2 cosh h′)N2(Nn+1−1)/2Q0Q1 · · ·Qn[Q(Pn)]
Nn+1
∫
W(n+1)
6 (ρ) (dρ)Nn+1 , (2.14)
where Nn+1 = N ′s−d(n+1), and
Q0 =
[
ea
′
0Q(d)
]N ′
, Q1 = [Q(P )Q(d1)]
N1 , . . . , Qn = [Q(Pn−1)Q(dn)]
Nn ,
Q(dn) =
(
4!/a
(n)
4
)1/4
I0(h
(n)),
Q(Pn) = π−1
(
sda
(n)
4 /C4(h
(n))
)1/4
I0c(η
(n)). (2.15)
The components
h
(n)
1 = a
(n)
1
(
4!
a
(n)
4
)1/4
, h
(n)
2 =
dn(Bn+1, Bn)
2
(
4!
a
(n)
4
)1/2
,
19
I.V.Pylyuk, M.P.Kozlovskii
h
(n)
3 =
a
(n)
3
3!
(
4!
a
(n)
4
)3/4
, h
(n)
4 ≡ 1,
h
(n)
5 =
a
(n)
5
5!
(
4!
a
(n)
4
)5/4
, h
(n)
6 =
a
(n)
6
6!
(
4!
a
(n)
4
)3/2
(2.16)
of the set of basic arguments in the nth layer h(n) = {h(n)
1 , h
(n)
2 , h
(n)
3 , h
(n)
4 , h
(n)
5 , h
(n)
6 }
are determined by the average value of the coefficient dn(k), i.e., by dn(Bn+1, Bn) =
a
(n)
2 − βΦ̃(Bn+1, Bn) as well as the quantities a
(n)
1 , a
(n)
3 , a
(n)
4 , a
(n)
5 and a
(n)
6 . The com-
ponents of the set of intermediate arguments η(n) = {η(n)1 , η
(n)
2 , η
(n)
3 , η
(n)
4 , η
(n)
5 , η
(n)
6 }
are functions of the quantities (2.16). They are given by the formulas
η
(n)
1 = −(4!s3d)1/4C1(h
(n))C
−1/4
4 (h(n)), η
(n)
2 = (6sd)1/2C2(h
(n))C
−1/2
4 (h(n)),
η
(n)
3 =
(
32
3
sd
)1/4
C3(h
(n))C
−3/4
4 (h(n)), η
(n)
4 ≡ 1,
η
(n)
5 =
(4!s−d)1/4
5
C5(h
(n))C
−5/4
4 (h(n)), η
(n)
6 =
(6s−d)1/2
15
C6(h
(n))C
−3/2
4 (h(n)).
(2.17)
The special functions Cl(h
(n)) can be expressed in terms of Fl(h
(n)) =
Il(h
(n))/I0(h
(n)):
C1(h
(n)) = F1(h
(n)),
C2(h
(n)) = F2(h
(n))− F 2
1 (h
(n)),
C3(h
(n)) = F3(h
(n))− 3F2(h
(n))F1(h
(n)) + 2F 3
1 (h
(n)),
C4(h
(n)) = −F4(h
(n)) + 4F3(h
(n))F1(h
(n)) + 3F 2
2 (h
(n))− 12F2(h
(n))F 2
1 (h
(n))
+6F 4
1 (h
(n)),
C5(h
(n)) = −F5(h
(n)) + 5F4(h
(n))F1(h
(n)) + 10F3(h
(n))F2(h
(n))
−20F3(h
(n))F 2
1 (h
(n))− 30F 2
2 (h
(n))F1(h
(n)) + 60F2(h
(n))F 3
1 (h
(n))
−24F 5
1 (h
(n)),
C6(h
(n)) = F6(h
(n))− 6F5(h
(n))F1(h
(n))− 15F4(h
(n))F2(h
(n))
+30F4(h
(n))F 2
1 (h
(n))− 10F 2
3 (h
(n)) + 120F3(h
(n))F2(h
(n))F1(h
(n))
−120F3(h
(n))F 3
1 (h
(n)) + 30F 3
2 (h
(n))− 270F 2
2 (h
(n))F 2
1 (h
(n))
+360F2(h
(n))F 4
1 (h
(n))− 120F 6
1 (h
(n)). (2.18)
Here
Il(h
(n)) =
∞
∫
−∞
tlf (n)(t) dt, l = 0, 1, 2, 3, 4, 5, 6, (2.19)
and
f (n)(t) = exp(−h
(n)
1 t− h
(n)
2 t2 − h
(n)
3 t3 − h
(n)
4 t4 − h
(n)
5 t5 − h
(n)
6 t6). (2.20)
20
3D Ising system in an external field . . .
The effective sextic measure density of the (n + 1)th block structure W (n+1)
6 (ρ) is
written as
W(n+1)
6 (ρ) = exp
−N
1/2
n+1a
(n+1)
1 ρ0 −
1
2
∑
k≤Bn+1
dn+1(k)ρkρ−k
−
6
∑
l=3
a
(n+1)
l
l!N
l/2−1
n+1
∑
k1,...,kl6Bn+1
ρk1
· · · ρkl
δk1+···+kl
, (2.21)
where Bn+1 = B′s−(n+1), dn+1(k) = a
(n+1)
2 − βΦ̃(k), a
(n+1)
l are the renormalized
values of the coefficients a′l after the integration over the n + 1 layers of the phase
space of the CV.
The coefficients of the sextic measure densities of the (n + 1)th and nth block
structures are connected through the following general RR:
u
(n+1)
l = sl−
l−2
2
d
[
−qδl−2 + (u
(n)
4 )l/4Yl(h
(n))
]
, l = 1, 2, 3, 4, 5, 6. (2.22)
We have introduced new designations here:
u
(n)
1 = sna
(n)
1 ,
u
(n)
2 + q = s2ndn(Bn+1, Bn), u
(n)
2 = s2ndn(0),
u(n)
m = smna(n)m , m = 3, 4, 5, 6. (2.23)
The quantity q = q̄βΦ̃(0) determines the average value of the Fourier transform
of the potential in the nth layer βΦ̃(Bn+1, Bn) = βΦ̃(0) − q/s2n (q̄ corresponds to
the average value of k2 on the interval (1/s, 1]). The functions appearing in the RR
(2.22) can be defined by the expressions
Y1(h
(n)) = s−d/4C1sc(η
(n))C
−1/4
4 (h(n)),
Yk(h
(n)) = (−1)ks
3k−4
4
dCksc(η
(n))C
−k/4
4 (h(n)), k = 2, 3, 4, 5, 6. (2.24)
The functions Clsc(η
(n)) have the form similar to Clsc(η
′) (2.11) under condition that
the set η′ and the functions fs(t), fc(t) (2.13) should be replaced by η(n) and
Fs(t) = sin(η
(n)
1 t+ η
(n)
3 t3 + η
(n)
5 t5) exp(−η
(n)
2 t2 − η
(n)
4 t4 − η
(n)
6 t6),
Fc(t) = cos(η
(n)
1 t+ η
(n)
3 t3 + η
(n)
5 t5) exp(−η
(n)
2 t2 − η
(n)
4 t4 − η
(n)
6 t6), (2.25)
respectively. It should be noted that the formula for I2lc(η
′) (see (2.12)) allows us to
express the function I0c(η
(n)) entering (2.15) in the same way. The function C4(h
(n))
is given in (2.18).
The obtained expressions make it possible to investigate the properties of the 3D
one-component spin system in the vicinity of the critical point (T = T c, h = 0). In
the absence of an external field (h = 0), we have a
(n)
1 = a
(n)
3 = a
(n)
5 = 0 and relations
21
I.V.Pylyuk, M.P.Kozlovskii
(2.22) are indentical with the RR presented in [13]. The behaviour of the system
can also be studied at T = Tc and h → 0.
The components of the set of intermediate arguments η (n) and the functions
Clsc(η
(n)), Cl(h
(n)), Yl(h
(n)) are approximated by power series in deviations of the
basic arguments h
(n)
i from their values at a fixed point h∗
i . Taking into account the
linear deviations, we find the following forms for these series:
Al = Al0
[
1 + Al1(h
∗
1 − h
(n)
1 ) + Al2(h
∗
2 − h
(n)
2 ) + Al3(h
∗
3 − h
(n)
3 )
+Al5(h
∗
5 − h
(n)
5 ) + Al6(h
∗
6 − h
(n)
6 )
]
, l = 1, 2, 3, 4, 5, 6. (2.26)
Here Al are the above-mentioned intermediate arguments or functions. The coeffi-
cients Ali are ultimate functions of the basic arguments at a fixed point. It should be
emphasized that we necessarily arrive at asymptotic expansions when the Gaussian
density of measure is taken as the basis density. The use of the sextic non-Gaussian
measure density permits to obtain the rapidly converging series (2.26). Having (2.26),
we can write and study the approximate RR. On the basis of (2.14) and RR, the
free energy F = −kBT lnZ can be calculated.
3. Discussion and conclusions
The results obtained for 3D Ising model in an external field can be applied
to the description of liquid-gas critical points of both a one-component fluid [2,
14] and a binary fluid mixture (see, for example, [15]). The CV method with a
reference system (RS) is used for these systems. The expression for the partition
function contains the even and the odd powers of the variable with the corresponding
cumulants and is similar to (2.4), (2.5). Detailed investigation of the properties of
RS cumulants in [2] makes it possible to transform the grand partition function into
a functional form defined on the effective block lattice. The functional corresponds
to the partition function of the Ising model in an external field. A new point in the
problem of the liquid-gas critical point as compared to the case of the Ising model is
the dependence of the critical temperature and of all coefficients on the density and
chemical potential. The latter is equivalent to the insertion of a constant external
field into the Ising model. To describe the effects connected with the asymmetry for
the surface of coexistence of the phases, one should compute the partition function
using the sextic measure density [2,16].
In this paper, the 3D one-component spin system in a nonzero external field is
investigated using the CV method in the asymmetric ρ6 model approximation. An
initial expression for the partition function of the system is constructed in the form
of a functional with explicitly known coefficient functions (see (2.7)). The partition
function is integrated over the layers of the CV phase space. The main feature
is the integration of short-wave spin density oscillation modes, which is generally
done without using perturbation theory. The short-wave modes are characterized by
the presence of the RG symmetry and are described by the non-Gaussian measure
22
3D Ising system in an external field . . .
density. The general RR (2.22) for the coefficients of the even and the odd powers of
the variable in the two adjacent sextic measure densities are found. The new special
functions appearing in the construction of the theory using the asymmetric ρ6 model
are considered. These functions entering the RR are represented in the form of a
power series in small deviations of the basic arguments from their values at a fixed
point. Representing the RR in the form of a nonasymptotic series relates to rejecting
the traditional use of perturbation theory, which is based on the Gaussian measure
density.
The expressions under investigation allow us to correctly trace the dependence of
the results on a field within the framework of the CV method. As a consequence of
the presence of an external field h in the Hamiltonian (2.1), we obtain odd powers
of the variable in the expressions (2.5) for the Jacobian J(ρ) and (2.7) for the
partition function Z. The cumulants Mn in the Jacobian (2.5) are combinations
of the quantities x = tanh h′ and y = 1 − tanh2 h′, where h′ = βh is a generalized
field. The initial coefficients a ′
l (2.9) of the partition function are determined by these
cumulants and are depended by s0 on the microscopic parameters of the system (the
effective interaction radius b and the lattice constant c). The cumulants M1, M3,
M5 as well as the coefficients a′
1, a
′
3, a
′
5 are odd functions of an external field. The
quantitiesM2,M4,M6 and a′0, a
′
2, a
′
4, a
′
6 are even functions of a field. The RR (2.22)
make it possible to determine the elements of the block Hamiltonians from the initial
data. As follows from the obtained expressions, the cumulants M1, M3, M5 and
coefficients a′1, a
′
3, a
′
5, ... , a
(n)
1 , a
(n)
3 , a
(n)
5 vanish at h = 0. Then the RR (2.22) for the
asymmetric ρ6 model turn into the RR for the ρ6 model with even powers only [13].
In the case when h 6= 0, the measure density involves odd powers of the variable ρk in
addition to the even powers. Although the CV method as well as Wilson’s approach
exploit the RG ideas, it is based on the use of non-Gaussian measure densities. This
allows one to obtain a qualitatively new form of the RR between the coefficients of
the block Hamiltonians. In the limiting case (corresponding to the Gaussian basis
measure density; h = 0) these RR reduce to the Wilson RR [11]. As was shown
in [17], while this limiting case does not allow one to perform the calculation of
the expression for the free energy of the system, it provides reliable results for the
critical exponents of thermodynamic characteristics. The solutions of the RR under
consideration can be used for calculating the thermodynamic functions of the system
and for deriving the equation of state using the method proposed in [18].
References
1. Yukhnovskii I.R. Phase Transitions of the Second Order. Collective Variables Method.
Singapore, World Scientific, 1987.
2. Yukhnovskii I.R., Idzyk I.M., Kolomiets V.O. // J. Stat. Phys., 1995, vol. 80, No. 1/2,
p. 405–443.
3. Yukhnovskii I.R., Patsahan O.V. // J. Stat. Phys., 1995, vol. 81, No. 3/4, p. 647–672.
4. Vause C., Sak J. // Phys. Rev. A, 1980, vol. 21, No. 6, p. 2099–2114.
5. Nicoll J.F. // Phys. Rev. A, 1981, vol. 24, No. 4, p. 2203–2220.
23
I.V.Pylyuk, M.P.Kozlovskii
6. Pylyuk I.V. // Theor. Math. Phys., 1998, vol. 117, No. 3, p. 1459–1482.
7. Kozlovskii M.P., Pylyuk I.V., Dukhovii V.V. // Cond. Matt. Phys., 1997, No. 11,
p. 17–49.
8. Pylyuk I.V. // Low Temp. Phys., 1999, vol. 25, No. 11, p. 877–888.
9. Pylyuk I.V. // Low Temp. Phys., 1999, vol. 25, No. 12, p. 953–961.
10. Ma Shang-keng. Modern Theory of Critical Phenomena. Reading, Massachusetts, Ben-
jamin, 1976.
11. Wilson K.G., Kogut J. // Phys. Rep. C, 1974, vol. 12, No. 2, p. 75–199.
12. Kozlovskii M.P., Pylyuk I.V., Usatenko Z.E. // Phys. Stat. Sol. (b), 1996, vol. 197,
p. 465–477.
13. Kozlovskii M.P. // Theor. Math. Phys., 1989, vol. 78, No. 3, p. 300–308.
14. Yukhnovskii I.R. // Physica A, 1990, vol. 168, No. 3, p. 999–1020.
15. Kozlovskii M.P., Patsahan O.V. // Cond. Matt. Phys., 2000, vol. 3, No. 3(23), p. 607–
631.
16. Yukhnovskii I.R. // Proceedings of the Steklov Institute of Mathematics, 1992, Iss. 2,
p. 223–243.
17. Ilnytskyi Ya.M., Kozlovskii M.P., Yukhnovskii I.R. // Int. J. Mod. Phys. B, 1997,
vol. 11, No. 8, p. 1009–1022.
18. Kozlovskii M.P., Pylyuk I.V., Dukhovii V.V. // J. Magn. Magn. Mater., 1997, vol. 169,
p. 335–342.
Tривимірна ізінгівська система в зовнішньому полі.
Рекурентні співвідношення для асиметричної
моделі ρ
6
І.В.Пилюк, М.П.Козловський
Інститут фізики конденсованих систем НАН України,
79011 Львів, вул. Свєнціцького, 1
Отримано 26 липня 2000 р.
Тривимірна однокомпонентна спінова система в зовнішньому маг-
нітному полі досліджується з допомогою методу колективних змін-
них. Інтегрування статистичної суми системи по шарах фазового
простору здійснюється в наближенні густини міри, яка містить в со-
бі парні та непарні степені змінної до шостого включно (асиметрична
модель ρ
6). Одержано загальні рекурентні співвідношення між коефі-
цієнтами ефективних густин мір. Нові функції, що появляються в цих
рекурентних співвідношеннях, подані у вигляді збіжних рядів.
Ключові слова: модель Ізінга, зовнішнє поле, колективні змінні,
рекурентні співвідношення
PACS: 05.50.+q, 75.10.Hk
24
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