To the theory of spatially nonuniform Bose systems with broken symmetry
The model of the self-consistent field for spatially nonuniform many-particle Bose-systems with broken symmetry is constructed. The self-consistent coupled equations for wave functions of the quasiparticles and of the particles of a condensate, and also system of the equations for normal and anomalo...
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irk-123456789-800512015-04-10T03:02:18Z To the theory of spatially nonuniform Bose systems with broken symmetry Poluektov, Yu.M. Quantum fluids The model of the self-consistent field for spatially nonuniform many-particle Bose-systems with broken symmetry is constructed. The self-consistent coupled equations for wave functions of the quasiparticles and of the particles of a condensate, and also system of the equations for normal and anomalous one-particle of density matrixes are deduced. The many-particle wave function is found. The thermodynamic of many-particle Bose-system on the basis of microscopic consideration in the self-consistent field model is constructed. We emphasize on the essential distinction of states with a Bose-condensate in model of ideal gas and in system of interating Bose-particles caused by obligatory presence alongside with one-particle, also of pair condensate, even at anyhow weak interaction. 2001 Article To the theory of spatially nonuniform Bose systems with broken symmetry / Yu.M. Poluektov // Вопросы атомной науки и техники. — 2001. — № 6. — С. 360-364. — Бібліогр.: 17 назв. — англ. 1562-6016 PACS: 67.40.Db, 05.30.Jp http://dspace.nbuv.gov.ua/handle/123456789/80051 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Quantum fluids Quantum fluids Poluektov, Yu.M. To the theory of spatially nonuniform Bose systems with broken symmetry Вопросы атомной науки и техники |
| description |
The model of the self-consistent field for spatially nonuniform many-particle Bose-systems with broken symmetry is constructed. The self-consistent coupled equations for wave functions of the quasiparticles and of the particles of a condensate, and also system of the equations for normal and anomalous one-particle of density matrixes are deduced. The many-particle wave function is found. The thermodynamic of many-particle Bose-system on the basis of microscopic consideration in the self-consistent field model is constructed. We emphasize on the essential distinction of states with a Bose-condensate in model of ideal gas and in system of interating Bose-particles caused by obligatory presence alongside with one-particle, also of pair condensate, even at anyhow weak interaction. |
| format |
Article |
| author |
Poluektov, Yu.M. |
| author_facet |
Poluektov, Yu.M. |
| author_sort |
Poluektov, Yu.M. |
| title |
To the theory of spatially nonuniform Bose systems with broken symmetry |
| title_short |
To the theory of spatially nonuniform Bose systems with broken symmetry |
| title_full |
To the theory of spatially nonuniform Bose systems with broken symmetry |
| title_fullStr |
To the theory of spatially nonuniform Bose systems with broken symmetry |
| title_full_unstemmed |
To the theory of spatially nonuniform Bose systems with broken symmetry |
| title_sort |
to the theory of spatially nonuniform bose systems with broken symmetry |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2001 |
| topic_facet |
Quantum fluids |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/80051 |
| citation_txt |
To the theory of spatially nonuniform Bose systems with broken symmetry / Yu.M. Poluektov // Вопросы атомной науки и техники. — 2001. — № 6. — С. 360-364. — Бібліогр.: 17 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT poluektovyum tothetheoryofspatiallynonuniformbosesystemswithbrokensymmetry |
| first_indexed |
2025-07-06T03:59:15Z |
| last_indexed |
2025-07-06T03:59:15Z |
| _version_ |
1836868547095035904 |
| fulltext |
TO THE THEORY OF SPATIALLY NONUNIFORM BOSE - SYSTEMS
WITH BROKEN SYMMETRY
Yu.M. Poluektov
National Science Centre "Kharkov Institute of Physics and Technology", Kharkov, Ukraine
The model of the self-consistent field for spatially nonuniform many-particle Bose-systems with broken sym-
metry is constructed. The self-consistent coupled equations for wave functions of the quasiparticles and of the
particles of a condensate, and also system of the equations for normal and anomalous one-particle of density
matrixes are deduced. The many-particle wave function is found. The thermodynamic of many-particle Bose-system
on the basis of microscopic consideration in the self-consistent field model is constructed. We emphasize on the
essential distinction of states with a Bose-condensate in model of ideal gas and in system of interating Bose-particles
caused by obligatory presence alongside with one-particle, also of pair condensate, even at anyhow weak
interaction.
PACS: 67.40.Db, 05.30.Jp
1. INTRODUCTION
The self-consistent field model frequently are used
for account of atomic shells [1], structures of atomic nu-
cleus [2], of properties of molecules and solid [3]. By
the important feature of self-consistent field model is an
opportunity of the description in his frameworks of sta-
teses with lower symmetry, than symmetry initial Ha-
miltonian. In particular, Bogolybov generalize Hartee-
Fock model on states with broken symmetry concerning
phase transformation [4], that has allowed describing
supercunducting state Fermi-systems with s-pairing.
The self-consistent field model was used basically for
theoretical research of properties Fermi systems. Only
later works have appeared in wich self-consistent field
model apply for study of Bose-systems [5]. The
generalization semifenomenological of the Fermi-liquid
approach on a case of a superfluid Bose-liquid is carried
out in work [6]. The self-consistent field model bring to
a conclusion about existence the one-particle excitations
with energy of activation at a zero momentum, on
necessity of that for many-particle Bose-systems is
inverted attention in the book N.N. Bogolybov and
N.N. Bogolybov (Jr.) [7]. The excitations with the
sound law dispersion, predicted Landau [8], can be
found from the non-stationary self-consistent equations.
As the many-particle Bose-systems at low temperatures
always, irrespective from the nature interparticle of
interaction, passes in states with the broken phase sym-
metry, it is natural to use self-consistent field model,
that with success used for the description of states with
spontaneously broken symmetries, as initial at
construction of the microscopic theory many-particle
Bose-systems. In this work in the general form the self-
consistent field model for Bose-systems by finite tem-
perature is constructed. That model allow theoretically
to investigate and the spatially nonuniform states. The
method of consideration is analogous to that what was
advanced for Fermi-systems in [9].
Is shown, that the states of system with the broken
phase invariancy, even is anyhow weak interacting of
Bose-particles, considerable different from ideal Bose-
gas with condensate. Is constructed termodynemic of
Bose-system with a condensate in self-consistent field
approach. The many-particle wave function is found. To
the advantages of the offered approach it is necessary to
attribute that in it all the particles of system are
considered on equally basis. The condensate of particles
with a zero momentum arises in spatially uniform states
as a consequence of the general theory.
2. THE SELF-CONSISTENT FIELD
EQUATIONS
Let's consider system of Bose-particles with spin a
zero interacting by means of pair potential ( )',rrU
Hamiltonian, which is
( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ,,'2
1
,
xxxxUxxdxdx
xxxHxxddxH
Ψ′Ψ∫ ′′ΨΨ+
∫ +′Ψ′Ψ′=
++
+
(1)
where
( ) ( ) −−+−∆−= )'()'(2', 0
2
xxxUxxmxxH δδ
( ) ,'xx −− δµ (2)
{ }r=x , ( )xU 0 − potential of an external field, µ - cemi-
cal potential. The field operators we define by the form-
ula ( ) ( )∑=Ψ
j jj axx .ϕ . The Bоse-operators jj aa ,+ are
creation and destruction operators of particles in a state
j . The wave functions ( )xjϕ satisfy the one-particle
Schroedinger equation. For transition to self-consistent
field approximation initial Hamiltonian (1) represent in
form of the sum two composed
,0 CHHH += (3)
where first composed - self-consistent Hamiltonian, in-
cluding the terms not above square-law on the field ope-
rators:
( ) ( ) ( )[ ]{ ( )
( ) ( ) ( ) ( ) ( ) ( ) +
Ψ∆Ψ++Ψ∆+Ψ+
+Ψ++Ψ∫=
xxxxxxxx
xxxWxxHxdxdxH
',*'2
1'',2
1
'',','0
( ) ( ) ( ) ( ) ,0* ExxFxxFdx ′+∫
Ψ++Ψ+ (4)
104 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 360-364.
and second – the correlation Hamiltonian that take into
account of the correlations of the particles which have
been not included in self-consistent field approximation.
Hamiltonian (4), as against a case Fermi-system [9],
contains also linear the members +ΨΨ , . The self-
consistent fields ( ) ( ) ( )',,',, xxxxWxF ∆ and a nonoperator
part 0E′ in 0H found from a condition best approximati-
on of Hamiltonian 0H to initial Hamiltonian H . So, in
self-consistent field approach many-particle systems are
characterized 0H and the influence the correlation
Hamiltonian can be taken into account under the pertru-
bation theory. In this work we shall be limited to the
consideration of Bose-system in the framework of self-
consistent field model, neglecting the effects, caused the
correlation Hamiltonian.
Hamiltonian (4) is resulted in the diagonal form, if it
to write down in the terms of the "displace" Bоsе-opera-
tors ( ) ( )xx +ΦΦ , , so
( ) ( ) ( )xxx Φ+=Ψ χ . (5)
The function ( )xχ is selected so that in 0H have drop-
ped out linear on the field operators the terms. In result
we receive a condition:
( ) ( ) ( ) ( )[ ] ( ) ,0'','',' * =+∆+Ω∫ xFxxxxxxxd χχ (6)
where ( ) ( ) ( ).',',', xxWxxHxx +=Ω . Take account to the
last condition self-consistent Hamiltonian by means of
Bogolybov transformation
( ) ( ) ( )[ ]∑ +=Φ +
i
iiii xvxux ,* γγ (7)
we transformated in a diagonal form:
,00 ∑+= +
i iiiEH γγε (8)
where i - complete set of quantum numbers describing
of a quasiparticle state. As we see, the self-consistent
field approximation naturally leads to appear of the
quasiparticles in Bоsе-systems. The conditions of
transformation from Hamiltonian (4) to (8) are the
equations on coefficients of the Bogolybov
transformation, which have sense of the components of
the quasiparticle wave function:
( ) ( ) ( ) ( )[ ] ( ) ,'','',' xiuixivxxxiuxxdx ε=∫ ∆+Ω (9)
( ) ( ) ( ) ( )[ ] ( ).'','',' ** xivixiuxxxivxxdx ε−=∫ ∆+Ω (10)
The self-consistent fields can be found from a con-
dition minimum of difference self-consistent Hamiltoni-
an 0H from H , that gives
( ) ( ) ( )
( ) ( ) ( ),,~,'
',~',',
xxxxUxdxx
xxxxUxxW
′′′′∫ ′′′′−+
+=
ρδ
ρ
(11)
( ) ( ) ( ) , ',',', xxxxUxx τ=∆ (12)
( ) ( ) ( ) ( ) ,'','2 2xxxUxdxxF χχ ∫−= (13)
where complete the one-particle density-matrixes are
defined by ratio
( ) ( ) ( ) ( ) ( ) ( ),'','',~ *0 xxxxxxxx χχρρ +=〉ΨΨ〈= + (14)
( ) ( ) ( ) ( ) ( ) ( ) .'','',~
0 xxxxxxxx χχττ +=〉ΨΨ〈= (15)
In (14), (15) averaging are made with the statistical
operator
( ) ,/1,exp 000 TH =−Ω= ββρ (16)
T − temperature. The normalization constant
[ ] ,)exp(ln 00 HSpT β−−=Ω (17)
is determined by a condition 10 =ρSp , it is meaningful
the thermodynamic potential of system in a self-
consistent field approximation. The overcondensate
density matrixes is:
( ) ( ) ( ) ( ) ( ) ( )[ ] ,1''', **∑ ++=
i
iiiiii fxvxvfxuxuxxρ (18)
( ) ( ) ( ) ( ) ( )( )[ ] ,1''', **∑ ++=
i
iiiiii fxuxvfxvxuxxτ (19)
where
( ) ( )[ ] 1
0 1exp −+ −==〉〈= iiiii ff εβεγγ (20)
is the Bоsе-quasiparticle distribution function. From eq.
(8) lead, that ( ) ( ) 000 =〉Φ〈=〉Φ〈 + xx and consequently
( ) ( ) ( ) ( ) ., 0*0 〉Ψ〈=〉Ψ〈= + xxxx χχ (21)
Thus, it is possible to treat ( )xχ as the function which
determining the density of number particles in an one-
partial Bоsе-condensate in the self-consistent field
model.
By the account of eq. (11), (12), the equations of the
self-coordination (9), (10) accept a form:
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )[ ] ( ) ,'',~'',~','
','~','2 0
2
xuxvxxxuxxxxUxd
xuxxxxUxdxUm
iiii
i
ετρ
ρµ
=+∫+
+
∫+−+∆−
(22)
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )[ ] ( )xvxuxxxvxxxxUxd
xvxxxxUxdxUm
iiii
i
ετρ
ρµ
−=+∫+
∫+−+∆−
'',~'',~','
','~','2
**
0
2
(23)
Besides the equations (22), (23) is necessary to
receive some one equation, as uncertain Bоsе-
condensate function. From (6) and (13) we find
( ) ( ) ( ) ( ) ( )xxxxxxUdxxU
m
χχρµ
∫ −+−+∆− 2
0
2
'2','~','
2
+
( ) ( ) ( ) ( ) ( )[ ] .0'',~'',~',' * =++ ∫ xxxxxxxxUdx χτχρ (24)
The equation (24), together with (22), (23) and (20)
completely describes system of many Bоsе-particles in
the self-consistent approximation. This system of the
equations has three type of the solutions:
I) ( ) ( ) ( ) ;0,0 ≠== xiuxivxχ
II) ( ) ( ) ( ) ;0,0,0 ≠≠= xiuxivxχ
III) ( ) ( ) ( ) .0,0,0 ≠≠≠ xiuxivxχ
The first type of the solutions (I) is described stateses
with not broken symmetry to phase transformation
( ) ( ) phase)arbitrary -( ξξiexx Ψ→Ψ . (25)
The system does not contain in this "normal" state
neither one-particle or pair condensates and has no pro-
perty of superfluidity. The second type of the solutions
(II) describes stateses with the broken concerning trans-
formation (25) symmetry in the consequence
appearance of the pair condencate, which is analogous
the pair condensate in superfluid Fermi-systems [9]. In
this case Bоsе-system has property of the superfluidity.
The superfluidity of Bоsе-system, cause by pair
correlations was investigated in works [10,11]. The
solutions such as III describe the superfluid stateses
with the broken phase symmetry containing as one-
105
partial and pair Bоsе-condensates. Let's pay attention
that there are absent the solutions, in which
( ) ( ) ( ) .0,0,0 ≠=≠ xiuxivxχ (26)
Such solution would respond a case of the ideal Bоsе-
gas below than point of the Bоsе-transition, in which
there is a Bоsе-condensate and of the overcondensate
particles. Thus, the system of noninteracting particles
with a Bоsе-condensate and system of interacting (even
ahyhow is weak) Bоsе-particles with the broken phase
symmetry are two essential various systems. The
application of the ideal gas model with a condensate as
base, lead to the difficulties at construction of the
consecutive theory many-particle of Bоsе-systems with
broken symmetries. It is connected with that wtat it is
impossible to describe in the ideal gas model the pair
correlations that always existing in the superfluid
systems of the interacting particles and playing a not
less essential role, than one-particle Bоsе-condensate.
3. THE TERMODYNAMIC PROPERTIES
The complete energy of a system of the particles in
self-consistent field approximation can be submitted as
the sum of three contributions ,321 EEEE ++= where
1E −energy is determined by overcondensating particles,
2E −energy of the condensating particles, 3E − energy
of the"interaction" condensating and overcondensating
of the particles. The first contribution can be written
down as the sum
,)1()1()1()1()1(
1 CEXDE UUUUTE ++++=
where: ( ) ( )',''
2
2
)1( xxxxdxdx
m
T ρδ ∆∫ −−= − kinetic ener-
gy, ( ) ( )∫= xnxUxdU QE 0
)1( − energy in external a field,
( ) ( ) ( )'','2
1)1( xnxnxxUdxdxU QQD ∫= − energy direct of the
interaction, ( ) ( ) 2)1( ',','2
1 ∫= xxxxUdxdxUEX ρ − energy of
the exchange interaction, ( ) ( ) 2',','2
1 ∫= xxxxUdxdxUC τ −
energy of the pair condensate . Here ( ) ( )xxxnQ ,ρ= − the
density of the number overcondensate particles. A
condensate part of energy can be written as the sum
)2()2()2(
2 DE UUTE ++= ,
where ( ) ( ) ( ) ( )[ ]xxxxxdmT **
2)2(
4 χχχχ ∆+∆∫−= − kinetic
energy of a condensate, ( ) ( ) 2
0
)2( xxUxdUE χ∫= − energy
of the one-particle condensate in the external field,
( ) ( ) ( ) 22)2( '','2
1 xxxxUxdxdUD χχ∫= − energy of the
interaction of the condensate particles. The third
contribution to the complete energy is determined by
interaction overcondensate of the particles and the
condensate:
( ) ( ) ( ) ( ) ( ) ( )[∫ +χ+χχρ= 2*
3 ''',',' xxnxxxxxxUdxdxE Q
( ) ( ) ( ) ( ) ( ) ( )
χχτ+χχτ+ '',2
1'',2
1 *** xxxxxxxx (27)
The thermodynamic potential of the system of Bose-
particles in self-consistent field approximation shall
present as:
( ) ( ) −∑ ∫−+++−=Ω
i
xivdxiUUUU DCEXD
2)2()1()1()1(
0 ε
( ) ( ) ( ) ( )[ ( ) ( ) ++− ∫ 2* ''',',' xxnxxxxxxUdxdx Q χχχρ
( ) ( ) ( ) ( ) ( ) ( ) +
++ '',2
1'',2
1 *** xxxxxxxx χχτχχτ
.1ln∑
−+ −
i
eT iεβ (28)
It is possible to show, that the variation of the thermo-
dynamic potential which determined by the formula
(17), is equal average on the self-consistent state from a
variation of the Hamiltonian:
.000 Hδδ =Ω (29)
Having expressed self-consistent Hamiltonian through
( ) ( ) ( )',,',, xxxxx τρχ and varying it with the account (29),
we receive:
( ) ( ) ( ) ( ) ==Ω==Ω
0
00
0
*
0
*
0
',', xx
H
xxx
H
x ρδ
δ
ρδ
δ
χδ
δ
χδ
δ
( ) ( ) . 0
',', 0*
0
*
0 ==
Ω
=
xx
H
xx τδ
δ
τδ
δ
(30)
Thus, the connections of fields ( ) ( ) ( )',,',, xxxxWxF ∆ with
the condensate wave function ( )xχ and the one-particle
density matrixes ( ) ( )',,', xxxx τρ (11), (12), (13),
established with help the variational principe leads to
extremeness of thermodynamic potential concerning its
variation on δ τδ ρδ χ ,, .
From the equations (18), (19), (22), (23) follows
system of the equations for a finding one-particle of the
density matrixes:
( ) ( ) ( ) ( )[ ] ( )
( ) ( )[ ] ( ) ( ) ( ) ( )[ +′′′′+′′′′∫ ′′−′′′′
+−+∆ ′−∆−
xxxxxxxxxxUxxUxd
xxxUxUxx
m
,~',~',~,~,',
',~'',~
2 00
2
ρρρρ
ρρ
( ) ( ) ( ) ( ) ( ) ,0'2',~,~ 2** =
′′−′′′′+ xxxxxxx χχχττ (31)
( ) ( ) ( ) ( ) ( )[ ] ( )
( ) ( )[ ] ( ) ( ) ( ) ( )[∫ ′′′′+′′′′′′+′′′′+
−+++∆ ′+∆−
',~,~',~,~,',
',~2','',~
2 00
2
xxxxxxxxxxUxxUxd
xxxxUxUxUxx
m
τρτρ
τµτ
( ) ( ) ( ) ( ) ( ) .0'2,~,'~ 2 =′′−′′′′+
xxxxxxx χχχτρ (32)
To Eq. (31), (32) it is necessary to attach and the
equation (24). The correlation Hamiltonian
0HHHC −= is expressed through of the overconden-
sate density matrixes and can be written compactly
through normal products of the operators. The
perturbation theory can be advanced for Bose-systems,
is similar, how it is realized for Fermi-systems [9].
4. SPATIALLY UNIFORM STATE
Let's take advantage of the received equations for
the analysis of spatially uniform system. It is consider
this important special case. In the spatially uniform
systems of the states of the particles is characterized by
their momentum { }k≡= ki and the wave functions look
like plane waves. Let's consider the short-range
106
interparticle potential: ( ) ( )'', 0 xxUxxU −= δ . In normal
Bosе-system (the values with a stroke) the quasiparticle
wave functions, the excitation spectra and the
distribution function look like:
( ) ,~
2,1 22
µε ′−=′−=′ m
k
k
kxe
V
xku
,1
1−
−′=′ kekf εβ (33)
where '2~
0 nU−′=′ µµ − the effective chemical potential,
V – volume occupied by a system. The connection
chemical potential with the density of number particles
is determined by the formula conterminous to the
formula for the ideal Bose-gas higher of a point
condensation, if in last to replace chemical potential µ ′
on effective chemical potential µ ′~ . The condition of
Bosе-condensation is .2 00 nU=′=′ µµ Temperature of
the Bosе-condensation is defined the same formula, as
well as in a case of ideal Bosе-gas. Below point of the
Bosе - condensation the self-consistent equations (26),
(27), (29) suppose the solutions of a kind of the plane
waves:
( ) ( ) .,, conste
V
kv
xkve
V
ku
xku ikxikx === −− χ (34)
Factors kk vu , , agrees (9), (10), satisfy of the algebraic
equations system
,022 0
22
* =
++−+∆ kvknUm
k
ku εµ (35)
,022 0
22
=∆+
−+− kvkuknUm
k εµ (36)
where ( )xxU ,~
0τ=∆ , and ( )xxn ,~ρ= − complete density
of number particles. In view of a condition normaliza-
tion 122 =− kvku is received:
,
2
,1
2
1,1
2
1 *22
k
kk
k
k
k
k
k
k vuvu
εε
ξ
ε
ξ ∆−=
−=
+= (37)
where ( ) nUmkk 0
22 22 +−= µξ . From (35), (36) the
energy of quasiparticles follows
,22 ∆−= kk ξε (38)
and from (24) establishing connection of the chemical
potential with the condensate wave function :
[ ] .02 *0 =∆++− χχµ QnU (39)
The complete density of number particles can be presen-
td as ,2
pq nnn ++= χ where first composed is the
density of number particles in a Bоsе-condensate,
second ∑= −
k
kq fVn 1 − density of number particles that
forms the quasiparticle excitations, and third
( ) ( ) ( )k
k
kkp fVn 2112 1 +∑ −= − εξ − the density of number
particles, that Cooper-pair is correlated. Thus, density of
number of particles which are not included in an one-
particle condensate is .pqQ nnn += The equation,
determining ∆ , looks like:
,2
0
20
1
χ
χ
Θ=
+
=∆
JU
U
(40)
where
.1,21
2
1
0
0
JU
Uf
VJ
k k
k
+=Θ∑
+= ε (41)
In the states with a Bоsе-condensate the chemical
potential, agrees (39), is determined by the formula
.2 2
0 χµ Θ+= QnU (42)
Substituting (40) and (42) in (38), we come to a final
ratio determining the quasiparticle energy
( ) .2
2
2
2
2
0
222
0
22
+
Θ−+= χχε U
m
kU
m
k
k
(43)
As see the quasiparticle energy at 0=k does not turn
into zero, and accepts finite value
.
21
0
0
0
2
∑ 〉−〈=
≠k
kaka
V
U χ
ε (44)
i.e. the spectrum has a energy gap at 0=k . It is obvi-
ous, that the energy spectrum is stable at pushing away
between particles 00 >U (positive scatterind length).
The energy (44) has clear physical sense, namely, is that
minimal energy, which is necessary for spending to pull
out a particle from a condensate and by that to create
new quasiparticle. It is quite natural, that in case of a
Bose-condensate of the interacting particles this energy
has final value. It is possible to establish a ratio
,1
1
0
02
0 JU
JUaaV k
kk +=∑ 〉〈
≠
− χ (45)
which determines the connection between the densities
of an one-particle condensate and pairing condensate.
So, the gap in a quasiparticle spectrum is defined the
constant of the interparticle interaction, the density of
number particles in condensate and the anomalous
pairing averages, that describing pair correlations in the
interacting system of Bose-particles with the broken
phase symmetry. The discussion of the solutions having
a gap in a quasipartial spectrum, also contains in work
[6]. The distribution function of quasiparticles, both are
higher and lower the temperatures of Bose-transition,
has in a point 0=k finite magnitude, except for
temperature of condensation 0T , where the distribution
function diverge at 0→k as 2−→ kkf .
For square-law self-consistent Hamiltonian (8), the
own vectors of stateses can be found. For this purpose it
is convenient pass to new Hamiltonian, connected with
initial one unitary transformation. New Hamiltonian
,
~
00 UHUH += has same own of the value as 0H , but
these values correspond new own vectors IUI +=
~
.
By means of two successive unitary transformations 1U
and 2U we shall pass from 0H to
∑ ++== ++
k kakakEUUHUUH .~
0210120 ε (46)
The unitary transformation
−+
= 0*0
1
aa
eU
χχ (47)
107
eliminates the linear on the particle operators 00 , aa +
terms and the unitary transformation
∑
ψ−ψ −
+
−
+
= k
aaaa kkkkkk
eU
2
1
2
, ψk=ψ-k (48)
eliminates the square-law terms, which are not invariant
concerning the phase transformations. The parameters
kψχ , in (47), (48) are determined by equations
.,
,2
*
2
0
k
k
k
kkk shvchu
UVF
ψ
ψ
ψ
=ψ=
χχ−=
(49)
To each own vector of the Schreodinger equation with
Hamiltonian (46) corresponds an own vector of the
Schreodinger equation with Hamiltonian 0H . The
vacuum vector of the self-consistent field Hamiltonian
is received in result the action of the operator 21UU on a
vacuum vector of particles:
.
0
0
0
2
00
0
*022
02
12*0
+
−
+ΛΓ−+Λ
×
×
+Λ−+Γ−Λ
≠−≠
Π
=
kakakka
aVVV
eee
ee
kk
q
χχχχ
(50)
In eq. (50) the designations are used:
.ln,;
,ln,
2 000
0
0
0
kkk
k
k
k chth
chth
ψ=Γψ
ψ
ψ=Λ
ψ=Γψ
ψ
ψ
=Λ
5. CONCLUSION
The statical self-consistent field model describes the
contribution in the thermodynamic of the multiparticle
system of the one-particle overcondensate excitftions
and of the condensate stateses. For Fermi-systems at
low temperature this contribution is determining.
Therefore theory of Fermi-systems based on the one-
particle description, is suitable in this case for research
of the real systems. For Bose-systems it not so, as here
with downturn of temperature the contribution of one-
particle exitations in the thermodynamic falls, and the
contribution collective exitations are grows. The
realistic theory of many-partile Bose-systems should
take into account alongside with one-particie exitations,
as well collective excitations. Though the static self-
consistent field model not satisfy to this requirement,
his theoretical study is important for several reasons.
First, one allows better to understand structure of the
states of Bose-system with broken by the phase
invariancy, in particular demonstrate essential
difference of such states from a state of ideal Bose-gas
with a condensate. Secondly, allows finding the
contribution of one-particle degrees of freedom to the
observable characteristics of a system. Thirdly, the
offered model serves natural initial approximation for
construction the quantum field perturbation theory and
the diagram techniques for Bose-systems with
spontaneously broken simmetries, similar by that is
advanced for Fermi-systems in work [9].
ACKNOWLEDGMENT
The author thanks I.V. Bogoyavlenskii and
L.V. Karnatsevich for discussions experiments on
research of a Bose-condensate and spectrum of the
exitations in liquid Не4 by a method of non-elastic
scattering of neutrons, A.S. Bakai and S.V. Peletminskii
for useful discussion of work.
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ACKNOWLEDGMENT
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