134
L. Didukh et al.
If now we apply the mean-field approximation
X
↓
r + X
2
r =< X
↓
r + X
2
r >= n ↓ ,
X
↑
p + X
0
p =< X
↑
p + X
0
p >= 1 − n ↓ = n ↑ ,
then the Green functions in containing operator expressions X p and ˜
X p turn to
zero. We also neglect processes of simultaneous creation and annihilation of two
electrons on a site. As a result, we obtain from (18)
(E + μ − U )
X p X
↓2
r
X
↑0
s
= − n ↓ t(r p)
X
0↑
p
X
↑0
s
+
+
i
t(ri)
X p X
↓2
r
X
↑0
s
,
(19)
and from (17)
(E + μ)
˜
X p X
0↑
r
X
↑0
s
=− < X
↑
p + X
0
p > t(r p)
X
↓2
p
X
↑0
s
+
+
i
t(ri)
˜
X p X
0↑
r
X
↑0
s
.
(20)
In
k-space, we obtain the closed system of equations
E + μ − ε 1
k
G
k =
1
2π
+ ε 2
k
T
k −
1
N
p κ
T p
κ
k t( κ),
(21)
E + μ − U − ˜
ε 1
k
T
k = ˜
ε 2
k
G
k −
1
N
κ
t( κ) ˜
T p
κ
k ,
(22)
E + μ − U − t
k
T p
κ
k = n ↓
r
t(r p) exp
i
κr − i
k p
G
k ,
(23)
E + μ − t
k
˜
T p
κ
k = n ↑
r
t(r p) exp
i
κr − i
k p
T
k .
(24)
In the above equations
ε 1 =
n ↓
N
k
t
2
k
E + μ − U − t
k
,
(25)
ε 2 =
n ↑
N
k
t
2
k
E + μ − t
k
.
(26)