Three-particle correlation function in the electron-plasmon model
The method of calculating the n-particle electron correlation functions for the electron-plasmon model is demonstrated. We have proposed this model earlier for the description of the strongly non-ideal electron liquid. The three-particle dynamic correlation function is calculated and presented in...
Збережено в:
| Дата: | 2005 |
|---|---|
| Автори: | , , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут фізики конденсованих систем НАН України
2005
|
| Назва видання: | Condensed Matter Physics |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/121046 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Three-particle correlation function in the electron-plasmon model / M.V. Vavrukh, S.B. Slobodyan, N.L. Tyshko // Condensed Matter Physics. — 2005. — Т. 8, № 4(44). — С. 711–722. — Бібліогр.: 16 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-121046 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1210462017-06-14T03:04:01Z Three-particle correlation function in the electron-plasmon model Vavrukh, M.V. Slobodyan, S.B. Tyshko, N.L. The method of calculating the n-particle electron correlation functions for the electron-plasmon model is demonstrated. We have proposed this model earlier for the description of the strongly non-ideal electron liquid. The three-particle dynamic correlation function is calculated and presented in the elementary functions. The differences from the similar correlation function of the ordinary reference system approach in the electron liquid theory are investigated. Наведено спосіб розрахунку n-частинкових електронних кореляційних функцій електрон-плазмонної моделі, яка була запропонована авторами раніше для опису сильно неідеальної електронної рідини. Розраховано і представлено в елементарних функціях тричастинкову динамічну кореляційну функцію. Досліджено її відмінності від аналогічної кореляційної функції звичайного базисного підходу у теорії електронної рідини. 2005 Article Three-particle correlation function in the electron-plasmon model / M.V. Vavrukh, S.B. Slobodyan, N.L. Tyshko // Condensed Matter Physics. — 2005. — Т. 8, № 4(44). — С. 711–722. — Бібліогр.: 16 назв. — англ. 1607-324X PACS: 05.30.Fk DOI:10.5488/CMP.8.4.711 http://dspace.nbuv.gov.ua/handle/123456789/121046 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
The method of calculating the n-particle electron correlation functions for
the electron-plasmon model is demonstrated. We have proposed this model
earlier for the description of the strongly non-ideal electron liquid. The
three-particle dynamic correlation function is calculated and presented in
the elementary functions. The differences from the similar correlation function
of the ordinary reference system approach in the electron liquid theory
are investigated. |
| format |
Article |
| author |
Vavrukh, M.V. Slobodyan, S.B. Tyshko, N.L. |
| spellingShingle |
Vavrukh, M.V. Slobodyan, S.B. Tyshko, N.L. Three-particle correlation function in the electron-plasmon model Condensed Matter Physics |
| author_facet |
Vavrukh, M.V. Slobodyan, S.B. Tyshko, N.L. |
| author_sort |
Vavrukh, M.V. |
| title |
Three-particle correlation function in the electron-plasmon model |
| title_short |
Three-particle correlation function in the electron-plasmon model |
| title_full |
Three-particle correlation function in the electron-plasmon model |
| title_fullStr |
Three-particle correlation function in the electron-plasmon model |
| title_full_unstemmed |
Three-particle correlation function in the electron-plasmon model |
| title_sort |
three-particle correlation function in the electron-plasmon model |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2005 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/121046 |
| citation_txt |
Three-particle correlation function in the electron-plasmon model / M.V. Vavrukh, S.B. Slobodyan, N.L. Tyshko // Condensed Matter Physics. — 2005. — Т. 8, № 4(44). — С. 711–722. — Бібліогр.: 16 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT vavrukhmv threeparticlecorrelationfunctionintheelectronplasmonmodel AT slobodyansb threeparticlecorrelationfunctionintheelectronplasmonmodel AT tyshkonl threeparticlecorrelationfunctionintheelectronplasmonmodel |
| first_indexed |
2025-07-08T19:05:33Z |
| last_indexed |
2025-07-08T19:05:33Z |
| _version_ |
1837106769141170176 |
| fulltext |
Condensed Matter Physics, 2005, Vol. 8, No. 4(44), pp. 711–722
Three-particle correlation function in
the electron-plasmon model
M.V.Vavrukh, S.B.Slobodyan, N.L.Tyshko
The Ivan Franko National University of Lviv,
Departament for Astrophysics,
8 Kyrylo and Methodii Str., 79005 Lviv, Ukraine
Received August 3, 2005
The method of calculating the n-particle electron correlation functions for
the electron-plasmon model is demonstrated. We have proposed this mod-
el earlier for the description of the strongly non-ideal electron liquid. The
three-particle dynamic correlation function is calculated and presented in
the elementary functions. The differences from the similar correlation func-
tion of the ordinary reference system approach in the electron liquid theory
are investigated.
Key words: electron liquid, plasmon oscillation, n-particle dynamic
correlation functions, transition operator, collective variables
PACS: 05.30.Fk
1. Introduction
The practical calculations of the characteristics of degenerate electron systems
are based on the perturbation theory over the power of the Coulomb potential which
is connected with the local field conception. Since there is no rigorous microscopical
theory of the local field correlation function, the development of alternative methods
for description of the strongly non-ideal electrons systems remains one of the urgent
tasks in statistical physics. It is well known that a very promissing direction in this
field is a collective description of the interelectron interactions which presents the
real situation, namely the existence of the collective motions. One of the earlier
variants of such methods is described in the papers by Bohm and Pines [1–6]. This
method uses a series of canonical transformations for the transition to the expanded
space of the variables of electrons and plasmons. Consequently this approach bears
an approximate character. Furthermore, at that time the problem of describing the
strongly non-ideal systems was not considered to be urgent. Another variant of
collective description was developed in the papers by Yukhnovskii et al. This is the
method of displacements and collective variables (see [7–10]). In this variant of the
collective description, as opposed to the approach by Bohm and Pines, the transition
c© M.V.Vavrukh, S.B.Slobodyan, N.L.Tyshko 711
M.V.Vavrukh, S.B.Slobodyan, N.L.Tyshko
to the expanded space is made rigorously by means of a transition function. The
absence of the divergent diagrams is characteristic of both approaches as opposed
to the standard methods of perturbation theory.
The collective description has a deep physical basis as well as possesses some ad-
vantages over the other methods, especially in the strong non-ideality region. Based
on the example of the electron liquid model in paper [11] a new variant of collective
description is proposed, which differs from the variants by Bohm-Pines and from the
method of the displacement and collective variables. We start with the secondary
quantization representation. Transition to the expanded space is made using the
transition operator which was introduced in paper [12]. These collective variables
are an intermediate element in our approach. They serve for the introduction of
the operators of the creation and destruction of plasmons. Partition function of the
model in the electron and plasmon terms does not have any approximations. Per-
turbation theory relatively to the electron-plasmon interaction is built in terms of
the n-particle dynamic correlation functions. Short-range interelectron interactions
are taken into account in the local-field approximation [11].
2. Correlation functions of the electron-plasmon model
Due to the absence of the divergent diagrams the calculation of thermodynamic
functions within the framework of the electron-plasmon model in the intermediate
and strong non-ideality region is reduced to the calculation of only low order dia-
grams of the perturbation theory over the power of the operator of electron-plasmon
interaction [11]. In these diagrams, the dynamic electron correlation functions, the
so-called connected averages, are found, as in the following formulae:
ηn(x1, . . . , xn) = (
~2
2m
)nε−n
F
β−1
〈
T
{
f̂x1
f̂x2
· · · f̂xn
}〉c
0
, (1)
where εF = ~
2k2
F/2m is Fermi energy,
f̂x =
∑
k,s
∑
ν∗
(kq)a+
k+q,s(ν
∗ + ν)ak,s(ν
∗), (2)
ν∗
n = (2n + 1)πβ−1, νn = 2πnβ−1 are Matsubara frequencies, β = (kBT )−1, x ≡
(q, ν), s is spin variable. Here, ak,s(ν
∗) is superposition of the secondary quantization
operators on the plane wave base in the interaction representation [11,13]
ak,s(ν
∗) = β−
1
2
β
∫
0
ak,s(β
′
) exp(iν∗β
′
)dβ
′
. (3)
Similar functions also appear in other approaches with renormalization of interac-
tions (see [14]). The calculations of the functions of this type are not known in
the literature. We shall show that functions ηn(x1, . . . , xn) can eventually lead to
712
Correlation function in the electron-plasmon model
correlation functions which are constructed on the operator density of the particles
ρ̂x =
∑
k,s
∑
ν∗
a+
k+q,s(ν
∗ + ν)ak,s(ν
∗), namely
µ0
n(x1, . . . , xn) = β−1 〈T {ρ̂x1
ρ̂x2
. . . ρ̂xn
}〉c
0
, (4)
which at n > 3 was originally calculated in papers [14,15].
As it is shown in papers [13,14]
−
〈
T
{
ak1,s1
(ν∗
1), a
+
k2,s2
(ν∗
2)
}〉
0
= Ge
k1,s1
(ν∗
1)δk1,k2
δs1,s2
δν∗
1
,ν∗
2
, (5)
where Ge
k1,s1
(ν∗
1) = {iν∗ − εk + µ}−1 is spectral representation of the one-particle
Green’s function of the reference system. Relationship (5) makes it possible to
present functions ηn(x1, . . . , xn) in the form of such convolutions:
η2(x1, x2) = β−1δx1+x2,0(
~2
2mεF
)2Re
∑
k,s
∑
ν∗
Ge
k,s(ν
∗)Ge
k+q1,s(ν
∗ + ν1)
× (k,q1)(k + q1,q1); (6)
η3(x1, x2, x3) = −2β−1δx1+x2+x3,0(
~2
2mεF
)3Re
∑
k,s
∑
ν∗
Ge
k,s(ν
∗)
× Ge
k+q1,s(ν
∗ + ν1)G
e
k−q2,s(ν
∗ − ν2)
× (k,q1)(k + q1,q1 + q2)(k − q2,q2); . . . ,
where symbol Re is applied to Bose-Matsubara frequencies (ν1, ν2, . . .). Functions
ηn(x1, . . . , xn) are real functions of their arguments (q1, . . . ,qn; ν1, . . . , νn). Let us
do the sums over the frequency ν∗ using the rule [13]
β−1
∑
ν∗
Ge
k,s(ν
∗) = nk,s = {1 + exp [β(εk − µ∗)]}−1 (7)
represented by ηn(x1, . . . , xn) in the form of the sum over the wave vector
η2(x,−x) = 2(
~2
2mεF
)2Re
∑
k,s
nk,sy(x)(k,q)(k + q,q);
η3(x1, x2, x3) = −2δx1+x2+x3,0(
~2
2mεF
)3Re
∑
k,s
nk,s
× {y(x1)y(−x2)(k,q1)(k − q2,q2)(k + q1,q1 + q2)
+ y(x2)y(−x3)(k,q2)(k − q3,q3)(k + q2,q2 + q3)
+ y(x3)y(−x1)(k,q3)(k − q1,q1)(k + q3,q3 + q1) } ; . . . , (8)
where y(x) ≡ {iν + εk − εk+q}−1. It should be noticed, that the expression in the
brackets (8) is symmetrical. The second and the third terms are obtained from the
713
M.V.Vavrukh, S.B.Slobodyan, N.L.Tyshko
first one by means of cyclic transposition. Function η2(x,−x) is easily calculated.
Transiting from the sum over the vector k to the integral and using spherical coor-
dinate system (axis 0z is parallel to vector q), at the absolute zero temperature we
can obtain
η2(x,−x) = − N
2εF
q2 +
(
u2 +
1
4
q2
)
µ0
2(x,−x)
= − N
2εF
q2
{
1 − 3
[
u2 +
1
4
q2
]
I2,0(q, u)
}
, (9)
where q ≡ |q|k−1
F
; u ≡ ν(2εFq)−1, I2,0(q, u) is the dimensionless function of these
variables
I2,0(q, u) =
1
2
{
1 +
1
2q
(
1 + u2 − q2
4
)
∑
σ=±1
σ ln
[
(
1 + σ
q
2
)2
+ u2
]
− u
∑
σ=±1
arctan
[
1
u
(
1 + σ
q
2
)
]
}
. (10)
0 1 2 3 4 5 6 7 8
q/kF
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
µ2(q,u)
η2(q,u)
u = 0
0.5
1
u = 0
1
0.5
Figure 1. Two-particle correlation functions η2(q, u) = 2εF(3N)−1η2(x,−x) and
µ2(q, u) = 2εF(3N)−1µ0
2(x,−x) at different values of the dimensionless frequency
(u = 0; 0.5; 1).
Functions ηn(x1, . . . , xn) differ from µ0
n(x1, . . . , xn) due to the presence of the product
of scalar factors (k,qi). But this “trifle” strongly complicates the calculation of the
functions ηn(x1, . . . , xn) (at the n > 3) and forms the fundamental difference in the
dependence of the function (1) and (4) from wave vectors q1, . . . ,qn, namely their
asymptotes
|µ0
n(x1, . . . , xn)| →
{
NεF(εq1
. . . εqn
)−1 at qi � kF;
Nε1−n
F
at qi � kF; νi = 0;
|ηn(x1, . . . , xn)| →
{
Nε1−n
F
at qi � kF;
Nε1−n
F
q1q2 . . . qnk
−n
F
at qi � kF; νi = 0.
(11)
This asymptote is confirmed in figure 1 where dimensionless factors of the functions
η2(x,−x) and µ2(x,−x) as functions of wave vector q at the given frequencies are
714
Correlation function in the electron-plasmon model
shown. Function η2(x,−x) is also an oscillating function in the region of the low and
medium vectors, as opposed to µ0
2(x,−x).
The calculation method of the correlation functions at the n > 3 is illustrated
based on the example of three-particle function. At first, in each of the three terms
of the formula (8) transformation is made to decrease the number of the scalar
products by applying the identity type
~2
2m
(k,q1) = −1
2
{εk + εq1
− εk+q1
} ,
~2
2m
(k − q2,q2) = −1
2
{εk−q2
− εk + εq2
} . (12)
Due to the transformations from each of the terms in formula (8) there arise compo-
nents without energy denominators and components with one energy denominator
type:
µ2(x) = −2
∑
k,s
nk,s[iν + εk − εk+q]
−1,
ζ2(x1|q2) = −4
∑
k,s
nk,s
~2
2m
(k,q2)[iν1 + εk − εk+q1
]−1 (13)
and components with two energy denominators but without the scalar products in
the numerator of the fraction type
Γ3(x1;−x2) =
∑
k,s
nk,s[iν1 + εk − εk+q1
]−1[−iν2 + εk − εk−q2
]−1. (14)
Calculation of the components of the type (13) is easily done by integrating over
the vector k in the spherical coordinate system. In the dimensionless variables
µ2(x) =
3N
2εF
R2,0(q, u),
ζ2(x|q1) =
3N
q2
(q,q1)C(q, u), (15)
where dimensionless complex functions R2,0(q, u) and C(q, u) are shown in the Ap-
pendix. Introducing the notation ε(q, u) = q2 + 2iuq we present η3(x1, x2, x3) in the
dimensionless variables as follows:
η3 (x1, x2, x3) =
3N
(2εF)2
δq1+q2+q3,0δν1+ν2+ν3,0
{
−1
3
(
q2
1 + q2
2 + q2
3
)
+
1
2
I2,0 (q1, u1) Re [ε∗ (q1, u1) (ε∗ (q2, u2) + ε∗ (q3, u3))]
+
1
2
I2,0 (q2, u2) Re [ε∗ (q2, u2) (ε∗ (q3, u3) + ε∗ (q1, u1))]
+
1
2
I2,0 (q3, u3) Re [ε∗ (q3, u3) (ε∗ (q1, u1) + ε∗ (q2, u2))]
715
M.V.Vavrukh, S.B.Slobodyan, N.L.Tyshko
− Re
4
[R2,0 (q1, u1) (ε (q1, u1) ε (q3, u3) + ε∗ (q1, u1) ε (q2, u2))]
− Re
4
[R2,0 (q2, u2) (ε (q2, u2) ε (q1, u1) + ε∗ (q2, u2) ε (q3, u3))]
− Re
4
[R2,0 (q3, u3) (ε (q3, u3) ε (q2, u2) + ε∗ (q3, u3) ε (q1, u1))]
− (q1,q2)
2
Re
[
ε∗ (q1, u1) C (q1, u1) q−2
1 + ε∗ (q2, u2) C∗ (q2, u2) q−2
2
]
− (q2,q3)
2
Re
[
ε∗ (q2, u2) c (q2, u2) q−2
2 + ε∗ (q3, u3) c∗ (q3, u3) q−2
3
]
− (q1,q3)
2
Re
[
ε∗ (q3, u3) C (q3, u3) q−2
3 + ε∗ (q1, u1) C∗ (q1, u1) q−2
1
]
}
− 1
4
δq1+q2+q3,0δν1+ν2+ν3,0Re
{
ε∗ (q1, u1) ε∗ (q2, u2) ε∗ (q3, u3)
× [Γ3 (x1;−x2) + Γ3 (x2;−x3) + Γ3 (x3;−x1)]
}
. (16)
The last term in formula (16) is connected with the function µ0
3(x1, x2, x3), since
µ0
3(x1, x2, x3) = −2δq1+q2+q3,0δν1+ν2+ν3,0Re {Γ3(x1;−x2)
+Γ3(x2;−x3) + Γ3(x3;−x1)} . (17)
The difficulty in calculating Γ3(xi;−xj) as well as the functions of higher order
(together with a greater number of energy denominators) is caused by the necessity
to integrate over the vector k at the given configuration of the vectors q1, . . . ,qn.
However, the integration over the angled variables of the vector k is easily done if
one uses Feynman identity [16],
n
∏
j=1
A−1
j = (n − 1)!
1
∫
0
· · ·
1
∫
0
dα1 . . . dαn
{
n
∑
j=1
αjAj
}−n
δ
(
n
∑
j=1
αj − 1
)
. (18)
Let us consider the calculation function Γ3(x1; x2) for the case of frequencies
ν1 and ν2 of the same sign, since real part
∑
j
αjAj can be equal to zero at some
values α. Then, there appears a condition of positive distinctness of the imaginary
part
∑
j
αjAj. The formulae which permit to obtain the function Γ3(x1; x2) for
frequencies of different sign are given in the Appendix. In formula (14) we transit
to dimensionless variables qi = |qi|k−1
F
, ui = νi(2εFqi)
−1 and use the identity (18) at
n = 2. Thus, we obtain the following representation
Γ3(x1; x2) =
3N
4πq1q2(2εF)2
∫
dk nk,s
1
∫
0
dαF−2
α ; (19)
Fα = α[(k, e1) + ξ1] + (1 − α)[(k, e2) + ξ2],
716
Correlation function in the electron-plasmon model
where ei = qi|qi|−1, vector k measured pin the units of kF, ξj =
1
2
qj − iuj; j = 1, 2.
Let us introduce vector
ρα = αe1 + (1 − α)e2 , (20)
and mark
Ωα = α(ξ2 − ξ1) − ξ2 = Ωc
α − iΩs
α ;
Ωc
α =
1
2
[α(q1 − q2) + q2], Ωs
α = α(u1 − u2) + u2 , (21)
so that Ωc
α, Ωs
α > 0. In these notations
Fα = (k,ρα) + Ωα, (22)
therefore integration over the angled variables of the vector k is done in the spherical
coordinate system (axis 0z parallel to vector ρα). After integrating over the module
of the vector k we obtain Γ3(x1; x2) in the form of single integral over the parameter α
Γ3(x1; x2) = − 3N
q1q2(2εF)2
1
∫
0
dα
ρ2
α
{
1 − Ωα
2ρα
ln
∣
∣
∣
ρα + Ωα
ρα − Ωα
∣
∣
∣
}
,
ρα ≡ |ρα| =
{
1 − 2α(1 − t) + 2α2(1 − t)
}
1
2 , (23)
t ≡ t12 ≡ (e1, e2) is cosine of the angle between vectors q1 and q2.
The integral over the variable α is divided into two ones. The integral in which we
have logarithm is integrated by parts. Then we unite it with the integral in which the
subintegral function is equal to ρ−2
α . Thus, the integral (23) is considerably simplified
Γ3(x1; x2) =
3N
(2εF)2
[
2q1q2(1 − t2)
]−1
×
{
[ξ2 − tξ1] ln
[
1 − ξ1
1 + ξ1
]
+ [ξ1 − tξ2] ln
[
1 − ξ2
1 + ξ2
]}
− 3N
(2εF)2
[
2q1q2
(
1 − t2
)]−1
δ(ξ, t)
1
∫
0
dα
ρ2
α − Ω2
α
, (24)
where
δ(ξ, t) = 1 − t2 − ξ2
1 − ξ2
2 + 2tξ1ξ2;
ρ2
α − Ω2
α = α2
{
2(1 − t) − (ξ1 − ξ2)
2
}
+ 2α {ξ2(ξ2 − ξ1) − (1 − t)} + 1 − ξ2
2 . (25)
Let α1 = αc
1 + iαs
1, α2 = αc
2 + iαs
2 are roots of the equation ρ2
α − Ω2
α = 0:
αc
1 =
{
p2
c + p2
s
}−1 {pc(bc − η) + ps(bs + ζ)} ;
αs
1 =
{
p2
c + p2
s
}−1 {pc(bs + ζ) − ps(bc − η)} ;
αc
2 =
{
p2
c + p2
s
}−1 {pc(bc + η) + ps(bs − ζ)} ;
αs
2 =
{
p2
c + p2
s
}−1 {pc(bs − ζ) − ps(bc + η)} . (26)
717
M.V.Vavrukh, S.B.Slobodyan, N.L.Tyshko
Here the following notations are used
pc = 2(1 − t) − 1
4
(q1 − q2)
2 + (u1 − u2)
2;
ps = (q1 − q2)(u1 − u2);
bc = 1 − t − u2(u1 − u2) +
q2
4
(q1 − q2);
bs =
u2
2
(q2 − q1) −
q2
2
(u1 − u2); [δ(ξ, t)]
1
2 = ζ + iη;
ζ =
1√
2
{
δc +
[
δ2
c + δ2
s
]
1
2
}
1
2
; η =
δs√
2
{
δc +
[
δ2
c + δ2
s
]
1
2
}−
1
2
;
δc = 1 − t2 − 1
4
(q2
1 + q2
2 − 2tq1q2) + (u2
1 + u2
2 − 2tu1u2);
δs = u1(q1 − tq2) + u2(q2 − tq1). (27)
Let us divide the subintegral function into simple factors and present the integral
over the variable α in the following form:
δ (ξ, t)
1
∫
0
dα
ρ2
α − Ω2
α
= γc
3 + iγs
3 ,
γc
3 =
ζ
2
(
Ã1 − Ã2
)
+
η
4
(
L̃1 − L̃2
)
; γs
3 =
η
2
(
Ã1 − Ã2
)
− ζ
4
(
L̃1 − L̃2
)
;
Ãi = arctan
1 − αc
i
αs
i
+ arctan
αc
i
αs
i
; L̃i ≡ ln
(1 − αc
i)
2 + (αs
i )
2
(αc
i)
2 + (αs
i )
2
; (i = 1, 2) . (28)
Thus, the real and the imaginary components of the function Γ3(x1; x2) are deter-
mined in the following expressions:
Γc
3(x1; x2) =
3N
(2εF)2
[
2q1q2
(
1 − t2
)]−1
{
−1
4
(q2 − tq1)L(q1, u1) − (u2 − tu1)
×A(q1, u1) −
1
4
(q1 − tq2)L(q2, u2) − (u1 − tu2)A(q2, u2) − γc
3
}
;
Γs
3(x1; x2) = − 3N
(2εF)2
[
2q1q2
(
1 − t2
)]−1 1
2
{(q2 − tq1) A(q1, u1) − (u2 − tu1)
× L(q1, u1) + (q1 − tq2) A(q2, u2) − (u1 − tu2) L(q2, u2) + γs
3} , (29)
where functions A(q, u) and L(q, u) are shown in the Appendix.
Expressions (15), (16), (26)–(29) together with the formulae from the Appendix
determine the function η3(x1, x2, x3).
3. Conclusions
The proposed method for calculating the three-particle dynamic correlation func-
tion permits to calculate the degenerate function of the fourth order η4(x1,−x1, x2,−x2)
718
Correlation function in the electron-plasmon model
in the analytical form. Function η4(x1, x2, x3,−x1 − x2 − x3) and function
η5(x,−x, x1, x2,−x1 − x2) can be presented in the form of single integral over the
parameter α from elementary functions. Due to a good fit of the series of the per-
turbation theory there is no need in calculating the functions ηn(x1, . . . , xn) of the
higher order.
0.5 1 1.5 2 2.5 3 3.5 4
q10
0.5
1
1.5
2
2.5
3
3.5
4
q2
-1
-0.5
0
0.5
1
0.5 1 1.5 2 2.5 3 3.5 4
q10
0.5
1
1.5
2
2.5
3
3.5
4
q2
-1.5
-1
-0.5
0
0.5
1
1.5
(a) (b)
Figure 2. Functions η3(x1, x2, x3) (figure 2a) and µ0
3(x1, x2, x3) (figure 2b) at the
ν̄1 = ν̄2 = 0, 1; t12 = −1 (ν̄ = ν(2εF)−1).
0 0.5 1 1.5 2 2.5 3 3.5 4 q1
0
0.5
1
1.5
2
2.5
3
3.5
4
q2
-2.5
-2
-1.5
-1
-0.5
0
0.5
0 0.5 1 1.5 2 2.5 3 3.5 4 q1
0
0.5
1
1.5
2
2.5
3
3.5
4
q2
0
0.5
1
1.5
2
(a) (b)
Figure 3. Functions η3(x1, x2, x3) (figure 3a) and µ0
3(x1, x2, x3) (figure 3b) at the
ν̄1 = ν̄2 = 0, 1; t12 = 0.
Figure 1 and figures 2–6 clearly show that the function η3(x1, x2, x3) differs from
the correlation function of the ordinary reference system approach µ0
3(x1, x2, x3). In
the region of the small wave numbers the η3(x1, x2, x3) have much smaller values
(in absolute value) than µ0
3(x1, x2, x3). As a rule, η3(x1, x2, x3) is the oscillating
function (even in those cases when µ0
3(x1, x2, x3) have constant signs (see figures 3–
5)). Similarly to µ0
3(x1, x2, x3), the functions η3(x1, x2, x3) have a strong frequency
dependence. At the frequencies ν1 = ν2, the functions µ0
3(x1, x2, x3) are the surfaces
that are symmetrical relatively to the trussed q1 = q2 at any values of cos(q1,q2)
(figures 2b, 3b, 4b). At ν1 6= ν2 the functions µ0
3(x1, x2, x3) are asymmetrical.
719
M.V.Vavrukh, S.B.Slobodyan, N.L.Tyshko
0
0.5
1
1.5
2
2.5
3
3.5
4
q1
0 0.5 1 1.5 2 2.5 3 3.5 4 q2
-1
-0.5
0
0.5
1
0
0.5
1
1.5
2
2.5
3
3.5
4
q1
0 0.5 1 1.5 2 2.5 3 3.5 4 q2
0
0.5
1
1.5
(a) (b)
Figure 4. Functions η3(x1, x2, x3) (figure 4a) and µ0
3(x1, x2, x3) (figure 4b) at the
ν̄1 = ν̄2 = 0, 1; t12 = 1.
-2.5
-2
-1.5
-1
-0.5
0
0.5
0 1 2 3 4 5
q1
q2=0.1
q2=0.5
q2=1.0
q2=2.0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 1 2 3 4 5
q1
q2=0.1
q2=0.5
q2=1.0
q2=2.0
(a) (b)
Figure 5. Functions η3(x1, x2, x3) (figure 5a) and µ0
3(x1, x2, x3) (figure 5b) at the
constant waves numbers q2 for the case where ν̄1 = ν̄2 = 0, 1; t12 = 0 (q2 =
0, 1; 0, 5; 1; 2).
0
0.5
1
1.5
2
2.5
3
3.5
q1
0
0.5
1
1.5
2
2.5
3
3.5
q2
-2
-1.5
-1
-0.5
0
0.5
0 0.5 1 1.5 2 2.5 3 3.5 4
q1
0
0.5
1
1.5
2
2.5
3
3.5
4
q2
-1
-0.5
0
0.5
1
1.5
(a) (b)
Figure 6. Function η3(x1, x2, x3) for the case of unequal frequencies (ν̄1 =
0, 1; ν̄2 = 0, 3) at the t12 = 0.
720
Correlation function in the electron-plasmon model
The functions η3(x1, x2, x3) have a strong dependence on cos(q1,q2) and are asym-
metrical even at the ν1 = ν2. This is illustrated in figures 2a, 3a, 4a. From figures 2–7,
it arises that the asymptotes of the functions µ0
3(x1, x2, x3) and η3(x1, x2, x3) corre-
spond to the formula (11).
Appendix
Here we introduce the explicit expressions for the functions (13) in the dimensi-
onless variables q = |q|k−1
F
, u = ν(2εFq)−1:
µ2(x) =
3N
2εF
R2,0(q, u),
R2,0(q, u) = I2,0(q, u) + iJ2,0(q, u),
where I2,0(q, u) is determined by formula (10),
J2,0 (q, u) = − 1
2q
{
u− 1
2
(
1+u2− q2
2
)
A (q, u)
}
; ζ2 (x|q1) =
3N
q2
(q,q1) C (q, u) ,
C (q, u) = Cc (q, u) + iCs (q, u) ,
Cc (q, u) =
2
3
+
(
u2− q2
4
)
− q
8
(
1+3u2 − q2
4
)
L (q, u)−u
2
(
1 + u2 − 3
4
q2
)
A (q, u) ;
Cs (q, u) = qu +
u
4
(
1 + u2 − 3
4
q2
)
L (q, u) − 1
4
q
(
1 + 3u2 − q2
4
)
A (q, u) .
Above we use the following marks:
L (q, u) = ln
u2 +
(
1 +
q
2
)2
u2 +
(
1 − q
2
)2
; A (q, u) = arctan
1 +
q
2
u
+ arctan
1 − q
2
u
.
Proceeding from the definition (14) and the explicit expression Γ3(x1; x2) ≡
Γ3(q1, ν1;q2, ν2) at the ν1, ν2 > 0 or ν1, ν2 < 0 by way of elementary transforma-
tions we can obtain such relationships which permit to write Γ3(q1, ν1;q2, ν2) for
positive and negative wave vectors and frequencies. Let Γ3(x1; x2) for the case of the
frequencies of the same sign ν1, ν2 be written in the dimensionless form:
Γ3 (x1; x2) ≡
3N
(2εF)2
γ3
(
q1, q2, t
∣
∣
∣
∣
ν̄1
q1
,
ν̄2
q2
)
where t ≡ t12 = cos(q̂1,q2), ν̄i ≡ ν(2εF)−1.
Then
Γ3 (q1, ν1;−q2, ν2) =
3N
(2εF)2
γ3
(
q1, q2,−t
∣
∣
∣
∣
ν̄1
q1
,
ν̄2
q2
)
, (ν1, ν2 > 0) ,
Γ3 (q1, ν1;−q2,−ν2) =
3N
(2εF)2
γ3
(
q1,−q2, t
∣
∣
∣
∣
ν̄1
q1
,
ν̄2
q2
)
, (ν1, ν2 > 0) ,
Γ3 (q1, ν1;q2,−ν2) =
3N
(2εF)2
γ3
(
q1,−q2,−t
∣
∣
∣
∣
ν̄1
q1
,
ν̄2
q2
)
, (ν1, ν2 > 0) .
721
M.V.Vavrukh, S.B.Slobodyan, N.L.Tyshko
References
1. Bohm D., Pines D., Phys. Rev., 1951, 82, No. 4, 625–634.
2. Bohm D., Pines D., Phys. Rev., 1952, 85, No. 2, 338–353.
3. Bohm D., Pines D., Phys. Rev., 1953, 92, No. 3, 609–625.
4. Pines D., Phys. Rev., 1953, 92, No. 3, 626–636.
5. Bohm D., Huang K., Pines D., Phys. Rev., 1957, 107, No. 1, 71–80.
6. Nozieres P., Pines D., Phys. Rev., 1958, 111, No. 2, 442–454.
7. Yukhnovskii I.R., Ukr. Fiz. Zhurn., 1964, 9, No. 7, 702–714 (in Russian).
8. Yukhnovskii I.R., Ukr. Fiz. Zhurn., 1964, 9, No. 8, 827–838 (in Russian).
9. Yukhnovskii I.R., Tsiganenko V.V., Vavrukh M.V, Ukr. Fiz. Zhurn., 1965, 10, No. 2,
135–146 (in Russian).
10. Yukhnovskii I.R., Petrashko R.N., Teor. Mat. Fiz., 1973, 17, No. 1, 118–130
(in Russian).
11. Vavrukh M.V., Slobodyan S.B., Conden. Matt. Phys., 2005, 8, No. 3, 453.
12. Vavrukh M.V., Visnik Lviv. Univer. seriya fizichna, 1968, No. 3, 26–33 (in Ukrainian).
13. Vavrukh M.V., Slobodyan S.B., Journ. Phys. Stud., 2003, 7, No. 3, 275–282
(in Ukrainian).
14. Vavrukh M., Tyshko N., Koval’ S., Kushtay Ya., Journ. Phys. Stud., 2003, 7, No. 4,
375–387 (in Ukrainian).
15. Vavrukh M., Krokhmalskii T., Phys. Stat. Sol. (b), 1991, 168, 519–532.
16. Hwa R.C., Teplitz V.L. Homology and Feynman integrals. W.A. Benjamin, New–
York/Amsterdam, 1966.
Тричастинкова кореляційна функція в
електрон-плазмонній моделі
М.В.Ваврух, С.Б.Слободян, Н.Л.Тишко
Львівський національний університет імені Івана Франка,
кафедра астрофізики, 79005 Львів, вул. Кирила і Мефодія, 8
Отримано 3 серпня 2005 р.
Наведено спосіб розрахунку n-частинкових електронних кореляцій-
них функцій електрон-плазмонної моделі, яка була запропонована
авторами раніше для опису сильно неідеальної електронної рідини.
Розраховано і представлено в елементарних функціях тричастинко-
ву динамічну кореляційну функцію. Досліджено її відмінності від ана-
логічної кореляційної функції звичайного базисного підходу у теорії
електронної рідини.
Ключові слова: електронна рідина, плазмові коливання,
n-частинкові динамічні кореляційні функції, оператор переходу,
колективні змінні
PACS: 05.30.Fk
722
|