Kinetic equations in the theory of normal Fermi liquid
On the bases of the improved approximation for the spectral function of one-particle states the Landau– Silin kinetic equations for the normal Fermi liquids of neutral and electrically charged particles are shown to be valid at finite temperature above the temperature of superfluid transition.
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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irk-123456789-1168142017-05-17T03:02:44Z Kinetic equations in the theory of normal Fermi liquid Kondratyev, A.S. Siddique, I. Электpонные свойства металлов и сплавов On the bases of the improved approximation for the spectral function of one-particle states the Landau– Silin kinetic equations for the normal Fermi liquids of neutral and electrically charged particles are shown to be valid at finite temperature above the temperature of superfluid transition. 2008 Article Kinetic equations in the theory of normal Fermi liquid / A.S. Kondratyev, I. Siddique // Физика низких температур. — 2008. — Т. 34, № 2. — С. 179–184. — Бібліогр.: 22 назв. — англ. 0132-6414 PACS: 05.20.Dd;71.10.Ay;74.20.Mn http://dspace.nbuv.gov.ua/handle/123456789/116814 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Электpонные свойства металлов и сплавов Электpонные свойства металлов и сплавов |
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Электpонные свойства металлов и сплавов Электpонные свойства металлов и сплавов Kondratyev, A.S. Siddique, I. Kinetic equations in the theory of normal Fermi liquid Физика низких температур |
| description |
On the bases of the improved approximation for the spectral function of one-particle states the Landau–
Silin kinetic equations for the normal Fermi liquids of neutral and electrically charged particles are
shown to be valid at finite temperature above the temperature of superfluid transition. |
| format |
Article |
| author |
Kondratyev, A.S. Siddique, I. |
| author_facet |
Kondratyev, A.S. Siddique, I. |
| author_sort |
Kondratyev, A.S. |
| title |
Kinetic equations in the theory of normal Fermi liquid |
| title_short |
Kinetic equations in the theory of normal Fermi liquid |
| title_full |
Kinetic equations in the theory of normal Fermi liquid |
| title_fullStr |
Kinetic equations in the theory of normal Fermi liquid |
| title_full_unstemmed |
Kinetic equations in the theory of normal Fermi liquid |
| title_sort |
kinetic equations in the theory of normal fermi liquid |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2008 |
| topic_facet |
Электpонные свойства металлов и сплавов |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/116814 |
| citation_txt |
Kinetic equations in the theory of normal Fermi liquid / A.S. Kondratyev, I. Siddique // Физика низких температур. — 2008. — Т. 34, № 2. — С. 179–184. — Бібліогр.: 22 назв. — англ. |
| series |
Физика низких температур |
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AT kondratyevas kineticequationsinthetheoryofnormalfermiliquid AT siddiquei kineticequationsinthetheoryofnormalfermiliquid |
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2025-07-08T11:05:02Z |
| last_indexed |
2025-07-08T11:05:02Z |
| _version_ |
1837076529420435456 |
| fulltext |
Fizika Nizkikh Temperatur, 2008, v. 34, No. 2, p. 179–184
Kinetic equations in the theory of normal Fermi liquid
A.S. Kondratyev1,2 and I. Siddique1
1
School of Mathematical Sciences, GC University 68-B, New Muslim Town, Lahore, Pakistan
E-mail: imransmsrazi@gmail.com
2
Department of Physics, Herzen State Pedagogical University, 48 Moika River Embankment, St. Petersburg 191168, Russia
E-mail: kondrat98926@yahoo.com
Received October 3, 2007
On the bases of the improved approximation for the spectral function of one-particle states the Lan-
dau–Silin kinetic equations for the normal Fermi liquids of neutral and electrically charged particles are
shown to be valid at finite temperature above the temperature of superfluid transition.
PACS: 05.20.Dd Kinetic theory;
71.10.Ay Fermi-liquid theory and other phenomenological models;
74.20.Mn Nonconventional mechanisms.
Keywords: Fermi liquid, charged particles, spectral function, kinetic equations.
1. Introduction
The exclusive successfulness of the phenomenological
Landau normal Fermi liquid theory [1] in predicting and
describing a set of new phenomena, among them the zero
sound in liquid He 3 and spin waves in nonferromagnetic
metals, made this theory a subject of investigation on the
basis of strict microscopic theory. Most attention was de-
voted to the derivation of the kinetic equation for the
quasiparticle distribution function. The initial derivation
of this equation was produced by Kadanoff and Byam [2]
on the basis of the quasiparticle (QP) approximation for
the spectral function and was continued by some follow-
ers who used so called extended quasiparticle (EQP) ap-
proximation [3–7]. However, in all these cases the second
Poisson bracket in the left side of Kadanoff–Baym (KB)
generalized kinetic equation could not be eliminated in a
lawful mathematical way (see below). This fact made the
temperature range of validity of the kinetic equation for
quasiparticle distribution very narrow, strictly speaking
the theory was proved to be valid only in the vicinity of
absolute zero.
Experimental discovery of the superfluidity of He 3 at
the temperature lower than the temperature at which the
zero-sound in the normal Fermi liquid was discovered,
and theoretical works devoted to the description of the
superfluid state [8,9], left no room for the temperature
range of validity of the equation for normal Fermi liquid.
At the same time theoretical predictions of this theory
turned to be in a perfect numerical agreement with the ex-
perimental data. This fact stimulated a second wave of
attempts to deriving the kinetic equation for normal Fermi
liquid, but as it was mentioned above, the result was not
achieved to a satisfactory extent. Further development of
KB theory went in the direction of the constructing of a
nonlocal quasiparticle kinetic equations [10,11], a devel-
opment of a detailed selfconsistent microscopic treatment
of arbitrary initial correlations in the system [12], etc., but
the question of the temperature range of validity of the
Landau’s kinetic equation remained open.
2. Spectral function and kinetic equation
for normal Fermi liquid
The problem consisted in a mathematically lawful elim-
ination of the second generalized Poisson bracket (a «puz-
zling term» in the terminology accepted in [5]) in the gen-
eralized Kadanoff–Byam kinetic equation written in the
collisionless approximation in the case of slowly varying
in space and time disturbances when only the first deriva-
tives with respect to T and R are taken into account [2]:
[ ( ; ), ( ; )]� � �� ��e T g Tp R p R
� ��[Re ( ; ), ( ; )]g T Tp R p R� � � 0 , (1)
here [A,B] — the generalized Poisson bracket, defined
by the expression:
© A.S. Kondratyev and I. Siddique, 2008
[ , ]A B
A B
T
A
T
B
A B A B�
�
�
�
�
�
�
�
�
�
�� � � � �
� � p R R p , (2)
e T( ; )p R� — the energy of a particle, defined by the equality:
e T E T THF( ; ) ( ; ) Re ( ; )p R p R p R� � �� � , (3)
E HF is a one-particle energy in the Hartree–Fock approxi-
mation, Re� is a real part of the correlation energy of a par-
ticle related to the imaginary part through the Hilbert trans-
form,
Re ( ; )
' ( ' ; )
'
� �
�
�
� �
p R
p R
T P
d T
�
�
��
�
� 2
. (4)
Here, P refers to a principle value integration.
It was shown in [2] that the function g entering the Eq.
(1) can be taken in the form
g Z T Z E T Z THF( ; ) [ ( ; ) Re ( ; )] .p R p R p R� � � �� 1 (5)
Equations (4) and (5) lead to the following general form
for a spectral function of one-particle states in the system:
a T
T
e T T /
( ; )
( ; )
[ ( ; )] ( ; )
p R
p R
p R p R
�
�
� � �
�
� �
2 2 4
. (6)
The spectral function satisfies the exact sum rule:
d
a T
�
�
2
1
��
�
� �( ; )p R . (7)
Correlation function g � is related to a spectral function by
the equality [2]:
g T a T f T� �( ; ) ( ; ) ( ; )p R p R p R� � � . (8)
The complexity of the expression (6) makes to search for a
certain approximations for the spectral function which
could be successfully used in calculations.
Quasiparticle approximation for the spectral function
(6) corresponds to the case
�� , when the quasiparticles
are stable:
a Z T E TQP � �2
� �( ; ) [ ( ; )]p R p R , (9)
where E T( ; )p R is the solution of the equation:
E T E T THF
E T( ; ) ( ; ) Re ( ; )| ,( ; )p R p R p R p R� � �� � � (10)
and the renormalizing factor Z T( ; )p R is given by the ex-
pression:
Z T
T
E T
�
�� �
�
1 1( ; )
Re ( ; )
( ; )p R
p R
p R
� � �
� � . (11)
Using the approximation (9) one follows the KB way to de-
riving the Landau’s kinetic equation when the second Pois-
son bracket in Eq. (1) is dropped considering �� � f
to
be negligibly small what is valid only at � �� at zero
temperature.
The extended quasiparticle (EQP) [3–7] and improved
extended quasiparticle (iEQP) [4] approximations for
the spectral function are written in the form:
a Z E P
E
EQP � � �
�
2
2
� �
�
( )
( )
, (12)
a Z E ZP
E
iEQP � � �
�
2
2
� �
�
( )
( )
, (13)
where E and Z are determined by the Eqs. (10) and (11)
correspondingly.
It is easy to verify that the approximations (12) and
(13) do not satisfy the KB equation for a nonequilibrium
spectral function:
[ ( ; ), ( ; )]� � �� �e T a Tp R p R
� �[Re ( ; ), ( ; )]g T Tp R p R� �
0 , (14)
and do not lead to the elimination of the second Poisson
bracket in Eq. (1) in the case of finite
. Only such a
mathematically lawful elimination of this term for finite
values of
would testify the validity of the Landau’s
equation at a finite range of temperature.
The expressions (12) and (13) both were obtained on
the basis of the general expression (6) by means of the
Taylor expansion in powers of
in the frame of different
approximations [3–7]. As a result, some important fac-
tors were lost. In reality there does not exist a mathemat-
ically strict correct form for the expansion of (6) in
power series of
which starts with the delta function
when
� 0.
Another expression for the spectral function can be
offered on the basis of the following consideration. We
utilize a well known relation of the Fourier transform in
the case of a constant value of c:
e e�
��
�
� �
�
�| | ,t c itxdt
c
c x
c
2
0
2 2
. (15)
Expanding the first exponent in the left side of (15) in
Taylor series, we get:
2
1
22 2
2 2c
c x
c t
c t
dtitx
�
� � � �
��
�
� ( | |
!
... ) e . (16)
Now we use the formulas equivalent to those represented
in [13,14]:
t dt i xn itx n n( ) ( )( ) ( )2 2 22
��
�
� � �e
� , n � 0 1 2, , , ... , (17)
180 Fizika Nizkikh Temperatur, 2008, v. 34, No. 2
A.S. Kondratyev and I. Siddique
| | sin ( ) ( )!
| |
( )t dt n n
x
n itx2 1 2 2 1
2
2 1
1�
��
�
� � � ��
�
�
�
�
� �e
2 2n �
,
n � 0 1 2, , , ... . (18)
If the quantity
in Eq. (6) was not a function of �, then
the expressions (17), (18) would lead to a strict correct
expansion of the spectral function (6) in terms of the
power series of
. In the case of
depending on �, one can
rely only on the first two terms of the expansion: the delta
function independent of
and the term proportional to
.
Then, taking into account that
� � �� � ��e T Z T E T( ; ) ( ; )[ ( ; )]p R p R p R
1 , (19)
we come to the following approximation for the spectral
function (6)
a Z E Z P
E
� � �
�
2 2
2
� �
�
( )
( )
. (20)
It is not difficult to verify that the approximation (20)
satisfies the sum rule (7), with the same precision as ap-
proximations (12) and (13) do, but contrary to them the
approximation (20) satisfies the Eq. (14) for the spectral
function and eliminates the second Poisson bracket in the
left side of Eq. (1). Indeed, when we substitute Eq. (20)
into Eq. (14), the first term in the right of this expression
gives:
[ , ( )] [( ) , ( )]�
� �
� � �� � � � � ��e Z E Z E Z E2 2 01 , (21)
due to the property of the generalized Poisson bracket
[ , ( )]A f A � 0 .
The second term in the right side of Eq. (20) leads to the
expression:
�
� �
��
�
�
�
�
�
�
�
�
�
�
�
�e Z
E
Z
E
E,
( ) ( )
[ , ]2
2 2
1
, (22)
the second poisson bracket in Eq. (14) due to Eq. (5) gives
the expression:
[Re , ]
( )
,
( )
[ , ]g Z
E
Z
E
E
�
�
�
��
�
��
� �
�
�
1 1
2� �
� . (23)
Thus, the ansatz (20) satisfies the Eq. (14) exactly. Now
we consider the Eq. (1). Using Eq. (8), we get:
[ , ] [Re , ] .�� � �e af g f
0 (24)
The substitution of the first term in Eq. (20) into Eq.
(24) leads directly to the kinetic equation for the qua-
siparticle distribution function n T( ; )p R [2]:
�
�
� � � �� � �
n
T
E n E nP R R p 0 , (25)
n T f T E T( ; ) ( ; )| ( ; )p R p R p R� �� � .
The second term in Eq. (20), being substituted to Eq. (24),
gives the expression:
Z e
f
E
Z E f
E
2
2 2
1
�
�
�
�
�
�
�
�
�
�
�
�
�
�
� �
�
,
( )
[ , ]
( )
, (26)
which is eliminated by the second poisson bracket in
Eq (24):
[Re , ] , ,g
e
f Z
E
f�
� �
� �
�
�
��
�
��
�
�
�
��
�
��
�
1 1
� �
�
Z f E
E
[ , ]
( )
�
�
1
2
. (27)
Thus, the kinetic Eq. (25) is valid for finite values of
and
correspondingly, for nonzero temperatures. For qualitative
estimation of the precision, we can use the third term in the
expansion (16) which is proportional to
2. Substituting
this term into Eq. (24), it is not difficult to show that this
equation is valid up to the terms of order
2.
3. Kinetic equation for normal Fermi liquid
in a magnetic field
The kinetic equation for the normal Fermi liquid con-
sisting of charged particles in the presence of compen-
sating background was considered in [15] in the quasi-
particle approximation for the spectral function. It was
shown that the spin splitting of the energy levels being
neglected, the spectral function for a system in a non-
quantizing magnetic field can be written in the form:
a T
T
e T /c
m
u T THF
( ; )
( ; )
( ) )
( ) ( ; )
p R
p R
p A R
R p R
�
�
� �
�
�
�
� �
(
2
2
�
�
�
�
�
�
�
�
�
�Re ( ; )
( ; )
� �
�
p R
p R
T
T
2
2
4
, (28)
where p is the canonical momentum and u T T( ), ( )R A R are
scalar and vector potentials of the electromagnetic field,
A R( )T being chosen in the Coloumb gauge div A R( )T � 0.
The kinetic equation for quasiparticle distribution is writ-
ten in the form (25) [15] and acquires a usual form after the
transition to kinetic momentum P = p A R�( ) ( )e/c T [1].
Kinetic equations in the theory of normal Fermi liquid
Fizika Nizkikh Temperatur, 2008, v. 34, No. 2 181
We would like to stress that the kinetic equations for nor-
mal Fermi liquid were written in [16] right in the form, cor-
responding the expression (28) for the spectral function,
concerning the dependence of all the quantities on vector
potential A and canonical momentum p and then were
transformed into gauge-invariant form [17]. More details
about the development of transport theory of interacting
fermions in an electromagnetic field can be found in
[18–20].
The derivation of the phenomenological Landau–Silin
kinetic equations in the case when spin splitting of energy
levels is taken into account was produced in [21] also in
the quasiparticle approximation. In this case all quantities
become matrices in spin space, in particular the quantity
e T( ; )p R� has the form:
e T
e T /c
m
u T I( ; )
( ( ) )
( )p R
p A R
R� �
�
�
�
�
�
�
�
�
�
�
�
2
2
� �
e
mc
curl T T� ( ) ~( ; )S A R p R� � (29)
where I and �S are the unit matrix and the set of three spin
matrices [1], ~� is a Hermitian part of the self-energy matrix
�. The spectral function is still given by Eq. (28). The spec-
tral function a and the inverse life-time of a particle’s state
are Hermitian matrices in the case under consideration.
When
is considered to be finite (nonzero), the next
possibilities can occur. If the magnetic field is not strong,
so that
exceeds the spin splitting of the energy levels,
the last is not essential and e T( ; )p R� can be written in the
form:
e T
e T /c
m
( ; )
( ( ) )
p R
p A R
� �
�
�
2
2
� � �u T T THF( ) ( ; ) Re ( ; )R p R p R� � � . (30)
The approximation for the spectral function in this case
can be written in a way analogous to Eq. (20):
a T Z T E T Z P
E
( ; ) ( ; ) ( ( ; ))
( )
,p R p R p R�
� �
�
� � �
�
2 2
2
(31)
where E is a solution of Eq. (10) with e given by Eq. (30).
The situation turns to be more complicated when spin
splitting of the energy levels should be taken into ac-
count. More convenient form of the theory suitable for the
generalization of the quasiparticle approximation for the
spectral function, than that presented in [21], was devel-
oped in [22]. First of all , we should stress, that even in the
quasiparticle approximation for the spectral function the
kinetic equation of the phenomenological theory are valid
only with the precision to the squared ratio of spin split-
ting of the energy levels to the chemical potential of the
system [22]. Thus, producing the derivation of the kinetic
equation with the improved approximation for the spec-
tral function we should take into account only the terms
that do not exceed this precision.
We start with the expansion of the matrices g � and e
over the full set of matrices in spin space,
g g I� � �
1
2
0
�
�S g , (32)
e e I� � 1 2 �
�S e 2 . (33)
These expansions should be substituted into the gener-
alized KB kinetic equations which under consideration
are written in the form [22]:
( ) ,�
� �
�
�
I e g I e g� � � � ��
��
�
��
�� �
2 2 2
� ��
�
�
�
�
� � ��
��
�
��
�� ��
� �
�
�~ , ~g a g a
2 2 2
0 , (34)
g I e g I e� �� � � � ��
��
�
��
�( ) ,�
� �
�
�
2 2 2
� ��
�
�
�
�
� � ��
��
�
��
�� �~ ~ ,g a g a
�
�
� �
�
2 2 2
0 , (35)
where ~g is a Hermitian part of the matrix g Z T( ; )p R .
Separating in Eqs. (34) and (35) the terms corresponding
to the collisionless case, we get in the quasiparticle ap-
proximation
� 0) the equations:
[ , ] [ , ]�� � �e g e gi i1 0 2 0 , (36)
[ , ] [ , ]�� � � ! �e e g1 2 0 2 0g e g2 , (37)
( )�� � �e g1 0 0e g2 , (38)
( )�� � �e g e gi i1 2 0 0 . (39)
Here, the quantities with Latin indexes are the cartesian
projections of the corresponding vectors: Summation
over the repeated indexes is supposed to be done. The
Eqs. (36)–(39) lead to the kinetic equations of the phe-
nomenological Landau–Silin theory [22] in the limit
� 0: the Eqs. (38) and (39) give the expressions for the
spectral functions, while the Eqs. (36) and (37) give ki-
netic equations on the basis of the determined spectral
functions.
With the help of Eqs. (38) and (39) we get the follow-
ing expressions for the functions g0 and g in the qua-
siparticle approximations [22]:
g f e f e0 � � � �" " # #� � � �( ) ( ) , (40)
g f f f� � � � � �" " # #� � � � � �( ) ( ) ( )e e e1 , (41)
182 Fizika Nizkikh Temperatur, 2008, v. 34, No. 2
A.S. Kondratyev and I. Siddique
where
e e e e" #� � � �1 1| | ; | |e e2 2 . (42)
Vector f" is antiparallel to e f2, # is parallel to e 2, and f is
perpendicular to e 2.
The argument of all the delta functions in (40), (41) ex-
cept one of them, are not equal to �� e1 . Thus, the substi-
tution of expressions (40), (41) into the Eqs. (36) and (37)
will generate the terms with the derivatives of the �-func-
tions. As a result, after rather tedious transformations of
Eq. (36) we get the Eq. (19.41) from [22]. The analysis of
the distinction between different renormalizing factors
entering this equation, leads to the mentioned above con-
clusion about the precision of the validity of the
phenomenological kinetic equation in the case
� 0 , i.e.
at zero temperature. In the case of finite
, (but
being
less then the spin splitting) it is necessary to put down the
system of Eqs. (36)–(39) in the collisionless approxima-
tion, but saving the terms with
, which do not enter the
collision integrals. It can be done, tracing carefully what
terms with
in the equations in the absence of magnetic
field do not enter collision integrals. Thus, we come to the
system of equations:
( ) [ , ] ~ [ , ~]�
�
� �
�
�I e g I e g g g� � � � � �� � � �
2 2
0 , (43)
g I e g I e g g� � �� � � � � �( ) [ , ] ~ [~, ]�
�
� �
�
� �
2 2
0 , (44)
where the expansions of the type (32), (33) should be
done for the functions ~g and �� �
f :
~ ~
�
~g g I� �
1
2
0 Sg , (45)
� �� � �� � 1 22I �S � . (46)
Finally, taking into account the comment about the terms,
exceeding the precision of the equation’s validity, we get
the system of equations that will lead to the kinetic equa-
tions:
[ , ] [ , ] [~ , ]� �� � � ��e g e g gi i1 0 2 0 1 0 , (47)
[ , ] [ , ] ~ , ]�� � � ! ��e g g1 2 0 0 22 0g e e g2 + [ � , (48)
and the system of equations that will give the expressions
for spectral functions:
( ) ~ ,� ��� � � �e g g1 0 0 1 0e g2 (49)
( ) ~� �� � � ��e g e g gi i i1 2 0 0 2 0 . (50)
Now we consider the case e e" � �1 | |e 2 and choose the
approximations for g0 and g in the form:
g Z f E Z
E
0
2
2
� � �
�
" " "� �
�
�
$
�
( )
( )
, (51)
g f� � �
�
" " "Z E Z
E
� �
�
�
( )
( )
2 2
2
�
, (52)
where E" and E are the solutions of the equations:
E T e T E( ; ) ( ; )| ,p R p R� �1 � (53)
E T e T e T E" �� �( ; ) [ ( ; ) | ( ; )| | ,p R p R p R1 2� �] (54)
and renormalizing factors Z" and Z are defined by the ex-
pressions:
Z T
e T
E"
� "
�� �
�
� "
1 1( ; )
( ; )
p R
p R�
� � , (55)
Z T
e T
E
�
�� �
�
�
1 11( ; )
( ; )
p R
p R�
� � . (56)
If we substitute the expressions (51), (52) into Eqs. (49)
and (50), these equations would be satisfied with the pre-
cision to the terms of the order
2 2, | |e 2 and
| |e 2 . The
substitution of expressions (51), (52) into Eqs. (47) and
(48) lead to the Landau–Silin kinetic equations with the
precision to the terms of the indicated order. The same is
valid, if we consider the case e e# � �1 | |e 2 . Thus, the ki-
netic equations of the phenomenological theory, (the Eqs.
(7.21) and (7.23) in [22]) are valid in the case of finite
temperature up to the terms linear in
, provided
does
not exceed the spin splitting of the energy levels.
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