Diagrammatic theory for twofold degenerate Anderson impurity model

0.15 MB·English

by  V. A. Moskalenko

Diagrammatic theory for twofold degenerate Anderson
impurity model
V. A. Moskalenko1,2, L. A. Dohotaru3, D. F. Digor1 and I. D. Cebotari1
January 21, 2013
1Institute of Applied Physics, Moldova Academy of Sciences, Chisinau 2028, Moldova,
2BLTP, Joint Institute for Nuclear Research, 141980 Dubna, Russia,
3
1 3 Technical University, Chisinau 2004, Moldova
0
2 Abstract

n
The twofold degenerate Anderson impurity model [1-4] is investigated and the strong
a
J electronic correlations of d-electrons of impurity ion are taken into account by elaborating

8 suitable diagram technique.
1
We discuss the properties of the Slater-Kanamori model [2-4] of d-impurity electrons.

] After finding the eigenfunctions and eigenvalues of all 16 local states, we determine the
l
e local one-particle propagator. Then we construct the perturbation theory around the
-
r atomic limit of the impurity ion and obtain the Dyson type equation for the renormalized
t
s one-particle propagator. Diagrammatic theory has been developed and correlation func-
.
t tion determined. Special diagrammatic approximation was discussed and summation of
a
m diagram has been considered.
-
d
PACS numbers: 71.27.+a, 71.10.Fd
n
o
c
[ 1 Introduction

1
v
The theory of strongly correlated electron systems plays a central role in contemporary con-
6
8 densed matter physics. The essence of the problem is the competition between the localization
3 tendency originated by the Coulomb repulsion of d electrons and itinerancy tendency arising
4
. as a result of hybridization of electron orbitals.
1
The orbital degeneracy can be completely eliminated in solid substances but in many of
0
3 them, for example, new superconductors based on Fe and AnC materials orbital degeneracy
60
1
is not completely eliminated and orbital effects are important. For instance, orbital degeneracy
:
v
plays essential role in the Mott metal-insulator transition. Here the effects of Hund’s rule
i
X
coupling in our orbitally degenerated model are studied with diagrammatic approach.
r
We study the influence of the intra-atomic Coulomb interactions of the two electrons with
a
opposite spins situated on the same or different orbitals and intra-atomic exchange is analyzed.
Our investigation is based on the diagram theory elaborated for strongly correlated electron
systems as in non-degenerated [5-9,11-14]and as in twofold degenerated ones [10].
The paper has the following structure. In Sec. 1 we describe the twofold degenerate An-
derson impurity model. The local properties of our model are considered in Sec. 2. The
perturbation theory around the atomic limit of impurity ion is formulated in Sec. 3. In this
section wediscuss theprocess ofdelocalizationandrenormalizationofthedynamical quantities.
In Sec. 4 the simplest irreducible Green’s function is calculated. Sec. 5 is devoted to discussion
of the Mott-Hubbard phase transition. Sec. 6 is devoted to the conclusions.
1
The Anderson Impurity model with twofold orbital degeneracy has the Hamiltonian com-
posedonepart ofconduction electrons -onepart ofinteracting localized andstronglycorrelated
electrons and of hybridization term between these two parts [1-4]:
H = H0 +H , (1)
int
H0 = H0 +HL, (2)
c d
H0 = ǫ (~k)C+ C , (3)
c l ~klσ ~klσ
X~klσ
HdL = ǫdd+lσdlσ +U nl nl +U′n1n2 +IH d+1σd+2σd1σ′d2σ′ (4)
l,σ l ↑ ↓ σσ′
X X X
+ I (d+d+d d +H.C.),
H′ 1 1 2 2
↑ ↓ ↓ ↑
1
H = (V d+C +V C+ d ), (5)
int √N ~kl lσ ~klσ ~k∗l ~klσ lσ
X~klσ
where the local Hamiltonian HL is standard Slater-Kanamori [2-4] form, C is conduction
d ~klσ
~
electronannihilationoperatorwithmomentumk, orbitalnumberl = 1,2andspinσ = 1( , ),
± ↑ ↓
d operator for localized d electron. Conduction electron of l th orbital state hybridizes only
lσ
−
with the local electron of the same orbital state. n = d+d , n = n , V is matrix element
lσ lσ lσ l lσ ~kl
σ
of hybridization. U is Coulomb repulsion between the d-electrons inPthe same orbital state and
U - between electrons in different orbital states. I is coefficient of the Hund’s rule coupling
′ H
and pair hopping terms, ǫ (~k) is the band dispersion and ǫd - is impurity ion energy evaluated
l
from the chemical potential µ. N is a number of lattice sites.
In the following we assume that the symmetry of the system is such that exist the relation:
U = U 2I ,I = I . (6)
′ − H H′ H
The Coulomb interactions are far too large to be treated as perturbation and they must be
included in H0 - zero order Hamiltonian. The hybridization term (5) is considered as the per-
turbation of the system. In the following the main ideas of the perturbation theory elaborated
fornon-degeneratestronglycorrelatedsystems areextended fordegenerated systems. Suchgen-
eralization has been discussed, for example, in the case of twofold degenerate Hubbard model
[10]. As is known the new elements of this perturbation theory of strongly correlated systems
are the irreducible correlation functions which contain all charge, spin and pairing quantum
fluctuations.
2 Local properties
In the main approximation of the Anderson model one has free conduction and strongly inter-
acting localized electrons described by the Hamiltonian H . The localized part of the Hamil-
0
tonian, HL, can be diagonalized by using Hubbard transfer operators χmn = m n where m
0 | ih | | i
is eigenvector of operator HL [5].
d
Becauseorbitalquantumnumber takestwovaluesl = 1,2thetotalnumber oflocalquantum
states is equal to 16.
There are the following eigenvectors of operator HL. The first quantum state 1 is the
d | i
vacuum state 0 with energy E = 0. There are 4 one particle states with spin S = 1 and
| i 1 2
S = 1:
z ±2
2 = d+ 0 , 3 = d+ 0 , 4 = d+ 0 and 5 = d+ 0 . The energies of all these states are
| i 1 | i | i 2 | i | i 1 | i | i 2 | i
E = E ↑= E = E =↑ǫ . ↓ ↓
2 3 4 5 d
2
Then there are six states with two particles. Three of them are singlet states with spin
S = 0 and others 3 triplet states with S = 1 and S = 1,0,1,
z
−
6 = 1 (d+d+ d+d+) 0 , 7 = 1 (d+d+ +d+d+) 0 ,
| i √2 1 1 − 2 2 | i | i √2 1 1 2 2 | i
↑ ↓ ↑ ↓ ↑ ↓ ↑ ↓
8 = 1 (d+d+ d+d+) 0 , 9 = d+d+ 0 ,
| i √2 1 2 − 1 2 | i | i 1 2 | i
↑ ↓ ↓ ↑ ↑ ↑
10 = 1 (d+d+ +d+d+) 0 , 11 = d+d+ 0 .
| i √2 1 2 1 2 | i | i 1 2 | i
The eigenval↑ues↓of th↓ese↑quantum state↓s a↓re
E = 2ǫ +U I , E = 2ǫ +U +I , E = 2ǫ +U +I ,
6 d − H′ 7 d H′ 8 d ′ H
E = E = E = 2ǫ +U I .
9 10 11 d ′ H
−
Then there are four states composed from three particles
12 = d+d+d+ 0 , 13 = d+d+d+ 0 ,
| i 1 1 2 | i | i 2 2 1 | i
14 = d+d+↑d+↓0 ↑, 15 = d+d+↑d+↓0 ↑,
| i 1 1 2 | i | i 2 2 1 | i
with ener↑gy↓val↓ue E = E ↑= E↓ ↓= E = 3ǫ +U +2U I .
12 13 14 15 d ′ H
−
The last local state is singlet
16 = d+d+d+d+ 0 with energy value E = 4ǫ +2U +4U 2I .
| i 1 1 2 2 | i 16 d ′ − H
When equ↑al↓itie↑s (6↓) take place we obtain more simple forms:
E = E = 2ǫ +U I , E = 2ǫ +U +I , E = 2ǫ +U 3I , E = 3ǫ +3U 5I ,
6 8 d H 7 d H 9 d H 12 d H
− − −
E = 4ǫ +6U 10I .
16 d H
−
The triplet states 9 , 10 and 11 are the lowest by energy.
| i | i | i
Quantum states enumerated above permit us to organize Hubbard transfer operators χmn
and establish the relation with fermion impurity operators [10]:
σ 1
d+ = χ2+l σ,1 + [( 1)l+1χ6,2+l+σ +χ7,2+l+σ]+ [σχ8,5 l+σ +( 1)l+1χ10,5 l+σ] (7)
lσ − √2 − √2 − − −
1 1
+ [ χ12+l σ,8 +σ( 1)l+1χ12+l σ,10]+ [ (1)lχ15 l σ,6 +χ15 l σ,7]+
− − −− −−
√2 − − √2 −
( 1)l+1X10 σ,5 l σ +( 1)l+1σχ12+l+σ,10+σ +σχ16,15 l+σ.
− −− −
− −
Equation (7) allows to calculate all the local dynamical quantities. For example quantum
electron number has the form:
1
n = χ2+l σ,2+l σ + [χ6,6 +( 1)l+1χ6,7 +( 1)l+1χ7,6 +X7,7] (8)
lσ − −
2 − −
1
+ [χ8,8 +σ( 1)l+1χ8,10 +σ( 1)l+1χ10,8 +χ10,10]
2 − −
χ10 σ,10 σ +χ12+l σ,12+l σ +χ12+l+σ,12+l+σ
− − − −
+χ15 l σ,15 l σ +χ16,16,
−− −−
and
n n = χ1+l,1+l χ3+l,3+l+( 1)l+1[χ8,10+χ10,8]+χ9,9 χ11,11+χ14 l,14 l χ16 l,16 l. (9)
l l − − − −
↑− ↓ − − − −
For τ dependent quantity A(τ) = eτH0Ae τH0 we have the equation:
−
n (τ) n (τ) = χ1+l,1+l χ3+l,3+l +( 1)l+1[χ8,10eτ(E8 E10) (10)
l l −
↑ − ↓ − −
+χ10,8eτ(E10 E8)]+χ9,9 χ11,11 +χ14 l,14 l χ16 l,16 l.
− − − − −
− −
The correlationbetween quantities with different orbitalnumbers is determined by theequation
(l = 1,2):
(nl (τ) nl (τ))(nl′ (0) nl′ (0)) = δll′[χ1+l,1+l +χ3+l,3+l +χ14−l,14−l + (11)
↑ χ16−l,16↓l]+( ↑1)l+l−′[χ8,8↓eτ(E8 E10) +χ10,10eτ(E10 E8)]+χ9,9 +χ11,11,
− − − −
−
3
4
(nl (τ) nl (τ))(nl′ (0) nl′ (0)) = (e−βE2 +e−βE12 +2e−βE9). (12)
ll′ ↑ − ↓ ↑ − ↓ Z0
X
In special case l = 1,l = 2 we have:
′
(n (τ) n (τ))(n (0) n (0)) = [χ8,8eτ(E8 E10) +χ10,10eτ(E10 E8)]+χ9,9 +χ11,11, (13)
1 1 2 2 − −
↑ − ↓ ↑ − ↓ −
which is the d electron susceptibility [2-4].
We now define the Matsubara one-particle Green’s function of localized d-electrons:
g0(lστ,l′σ′τ′) = gl0σ,l′σ′(τ τ′) = Tdlσ(τ)d¯l′σ′(τ′) 0, (14)
− −h i
where d (τ) = eτH0d e τH0, d¯ (τ) = eτH0d+e τH0.
lσ lσ − lσ lσ −
The Fourier components of this Green’s function are:
1
g(0)(τ) = e iωnτg(0)(iω ). (15)
− n
β
Xωn
Using (8) andthe properties of Hubbard operators we obtain the equation for local function:
g(0) (iω ) = δll′δσσ′ e−βE1 +e−βE2 + e−βE2 +e−βE6 + (16)
lσl′σ′ n Z {iω +E E iω +E E
0 n 1 2 n 2 6
− −
1 e βE2 +e βE7 3 e βE2 +e βE9 e βE6 +e βE12
− − − − − −
+ + +
2iω +E E 2iω +E E iω +E E
n 2 7 n 2 9 n 6 12
− − −
1 e βE7 +e βE12 3 e βE9 +e βE12 e βE12 +e βE16
− − − − − −
+ + ,
2iω +E E 2iω +E E iω +E E }
n 7 12 n 9 12 n 12 16
− − −
where Z is partition function in atomic limit
0
Z = e βE1 +4e βE2 +2e βE6 +e βE7 +3e βE9 +4e βE12 +e βE16. (17)
0 − − − − − − −
The spectral function of impurity d-electron in local approximation is equal to
A(0)(E) = 2Img0(E +iδ), (18)
−
where g0(E +iδ) with δ = +0 is analytical continuation of the Matsubara to retarded Green’s
function.
Using (14) we obtain
2π
A(0)(E) = (e βE1 +e βE2)δ(E +E E )+(e βE2 +e βE6)δ(E +E E )+
− − 1 2 − − 2 6
Z { − −
0
1 3
(e βE2 +e βE7)δ(E +E E )+ (e βE2 +e βE9)δ(E +E E )+ (19)
− − 2 7 − − 2 9
2 − 2 −
1
(e βE6 +e βE12)δ(E +E E )+ (e βE7 +e βE12)δ(E +E E )+
− − 6 12 − − 7 12
− 2 −
3
(e βE9 +e βE12)δ(E +E E )+(e βE12 +e βE16)δ(E +E E ),
− − 9 12 − − 12 16
2 − −
with property
∞A(0)(E)dE = 2π. (20)
Z∞
4
3 Delocalization processes
We use the perturbation theory elaborated previously for strongly correlated electron systems
both of non degenerate [5-9,11-14] and of degenerate forms [10]. We study the process of
renormalization of Green’s function resulting from intra- and inter-orbital flips of tunneling
electrons.
The full Matsubara Green’s function in the interaction representation for conduction and
impurity electrons are:
G(~klστ k~l σ τ ) = TC (τ)C¯ (τ )U(β) c, (21)
| ′ ′ ′ ′ −h ~klσ ~k′l′σ′ ′ i0
g(lστ|l′σ′τ′) = −hTdlσ(τ)d¯l′σ′(τ′)U(β)ic0.
The anomalous functions are defined as
F(~klστ ~k l σ τ ) = TC (τ)C (τ )U(β) c, (22)
| ′ ′ ′ ′ −h ~klσ ~k′l′σ′ ′ i0
F¯(~klστ ~k l σ τ ) = TC¯ (τ)C¯ (τ )U(β) c,
| ′ ′ ′ ′ −h ~klσ ~k′l′σ′ ′ i0
f(lστ|l′σ′τ′) = −hTdlσ(τ)dl′σ′(τ′)U(β)ic0,
f¯(lστ|l′σ′τ′) = −hTd¯lσ(τ)d¯l′σ′(τ′)U(β)ic0.
Here τ and τ stand for imaginary time with 0 τ β, β is inverse temperature, T is
′
≤ ≤
chronological ordering operator.
The evolution operator is
β
U(β) = T exp( H (τ)dτ). (23)
int
−
Z0
Thestatistical averagingiscarriedoutin(21)and(22)withrespect tothezero-orderdensity
matrix of the conduction and impurity electrons. Index c means connected diagrams.
In the zero order approximation we have
16 16
HL = E χnn, χnn = 1, (24)
0 n
n=1 n=1
X X
G(lσ0)l′σ′(~kk~′|τ −τ′) = δ~k~k′δll′δσσ′G(lσ0)(~k|τ −τ′),
1 (2n+1)π
G(0)(~k iω ) = ,ω = .
lσ | n iω ǫ(~k) n β
n
−
and g(0)(iω ) is determined by the equation (16).
n
Hybridization between the conduction and d impurity electrons results in renormalization
of their propagators. Because the number of conduction electrons N is much larger than the
single impurity state, the effect of the latter on the conduction band scales as 1.
N
The renormalized conduction electron propagator is
Glσl′σ′(~kk~′|iωn) = δ~k~k′δll′δσσ′G(lσ0)(~k|iωn)+ (25)
V V
~k∗lN~k′l′G(lσ0)(~k|iωn)glσ,l′σ′(iωn)G(l′0σ)′(~k′|iωn),
where glσl′σ′(iωn) is the full impurity electron propagator.
A similar equation holds for the anomalous function of conduction electrons in supercon-
ducting state:
V V
Flσl′σ′(~k,−k~′|iωn) = ~k∗lN~k′l′G(lσ0)(~k|iωn)flσl′σ′(iωn)G(l′0σ)′(−~k′|−iωn).
5
g VV11 G(0) VV22∗∗ gg
= Λ + ΛΛ
1 2
lστ lστ
′ ′ ′
V1∗ G(0) V2 f¯
Y
− 1 2
f¯ V1∗ G(0) V2 f¯
= Y¯ + ΛΛ
1 2
lστ lστ
′ ′ ′
V1 G(0) V2∗ g
+ Y¯
1 2
Figure 1: Dyson type equation for Green’s function of impurity electrons. Λ, Y, Y¯ are corre-
lation functions.
The equations for the full functions g and f of impurity electrons have the diagrammatical
form shown in Fig.1.
The structure representative of the diagrams in Fig. 1 is given by the following equation
V V
~k1l1 ~k∗2l2G0 (~k k~ iω ) = (26)
N l1σ1l2σ2 1 2| n
X~k1 X~k2
1
V 2G(0) (~k iω )δ δ = δ δ (0) (iω ),
N | ~k1l1| l1σ1 1| n l1l2 σ1σ2 l1l2 σ1σ2Gl1σ1 n
X~k1
where
1 1 V 2
(0)(iω ) = V 2G(0)(~k iω ) = | ~kl| . (27)
Glσ n N | ~kl| lσ | n N iω ǫ(~k)
X~k X~k n −
The renormalization quantity is
1
~~
Glσl′σ′(iωn) = N V~klV~k∗′l′Glσl′σ′(kk′|iω). (28)
X~k~k′
In the Fig. 1 the double dashed lines with arrows depict renormalized g and f propagators
of localized electrons and solid thin lines represent 0 function of conduction electrons. The
G
function V means V and summation by repeated indices is assumed.
1 ~k1l1
Λ and Y¯ are correlation functions. They contain a sum of strongly connected irreducible
diagrams. The simplest examples of such diagrams are shown on Fig. 2.
The analytical form of equations in Fig.1 is the following:
glσl′σ′(iωn) = Λlσl′σ′(iωn)+Λlσl1σ1(iωn)Gl(10σ)1(iωn)gl1σ1l′σ′(iωn)−
6
V V
1∗ 2
1 2
1 G(20)ir[lστ,l1σ1τ1 l2σ2τ2,l′σ′τ′]G(0)[~k2l2σ2τ2 ~k1l1σ1τ1]
− | |
lστ l σ τ
′ ′ ′
V V
1∗ 2∗
1 2
1 G(0)ir[l σ¯ τ ,l σ τ lσ¯τ,l σ τ ]F¯(0)[ ~k l σ¯ τ ~k l σ τ ]
−2 2 1 1 1 2 2 2| ′ ′ ′ − 1 1 1 1| 2 2 2 2
l σ τ lστ
′ ′ ′
Figure 2: The simplest examples of correlation functions Λ and Y¯.
Ylσl1σ1(iωn)Gl(10σ)1(−iωn)f¯l1σ1l′σ′(iωn), (29)
f¯lσl′σ′(iωn) = Y¯lσl′σ′(iωn)+Λl1σ1lσ(−iωn)Gl(10σ)1(−iωn)f¯l1σ1l′σ′(iωn)+
Y¯lσl1σ1(iωn)Gl(10σ)1(iωn)gl1σ1l′σ′(iωn).
This system of equations is rather general and admit different phases. We shall discuss one of
the most simple form with singlet superconductivity on the paramagnetic background.
For this special case we use the new notations (σ¯ = σ):
−
glσl′σ′(iωn) = δσσ′gσll′(iωn),f¯lσl′σ′(iωn) = δσσ¯′f¯σ¯llσ′(iωn),
Λlσl′σ′(iωn) = δσσ′Λlσl′(iωn),Y¯lσl′σ′(iωn) = δσσ¯′Y¯σ¯lσl′(iωn), (30)
g(0)(iω ) = g(0)l(iω ).
lσ n σ n
By using these definitions we obtain:
gll′(iω ) = Λll′(iω )+Λll1(iω ) l1(0)(iω )gl1l′(iω ) Yll1(iω ) l1(0)( iω )f¯l1l′(iω ), (31)
σ n σ n σ n Gσ n σ n − σσ¯ n Gσ¯ − n σ¯σ n
f¯ll′(iω ) = Y¯ll′(iω )+Λl1l( iω ) l1(0)( iω )f¯l1l′(iω )+Y¯ll1(iω ) l1(0)(iω )gl1l′(iω ).
σ¯σ n σ¯σ n σ¯ − n Gσ¯ − n σ¯σ n σ¯σ n Gσ n σ n
In the absence of orbital degeneracy this system of equation has the known solution [14]
Λ (iω ) (0)( iω )[Λ (iω )Λ ( iω )+Y (iω )Y¯ (iω )]
g (iω ) = σ n −Gσ¯ − n σ n σ¯ − n σσ¯ n σ¯σ n ,
σ n
d (iω )
σ n
Y¯ (iω ) Y (iω )
f¯ (iω ) = σ¯σ n , f (iω ) = σσ¯ n , (32)
σ¯σ n d (iω ) σσ¯ n d (iω )
σ n σ n
d (iω ) = (1 Λ (iω ) (0)(iω ))(1 Λ ( iω ) (0)( iω ))+
σ n σ n σ n σ¯ n σ¯ n
− G − − G −
(0)(iω ) (0)( iω )Y (iω )Y¯ (iω ).
σ n σ¯ n σσ¯ n σ¯σ n
G G −
Solutions of the equation (31) for the normal state of the degenerate system has the form:
Λ11(iω ) 2(0)(iω )[Λ11(iω )Λ22(iω ) Λ12(iω )Λ21(iω )]
g11(iω ) = σ n −Gσ n σ n σ n − σ n σ n ,
σ n d (iω )
σ n
Λ21(iω )
g21(iω ) = σ n , d (iω ) = (1 1(0)(iω ) (33)
σ n d (iω ) σ n −Gσ n ×
σ n
Λ11(iω ))(1 2(0)(iω )Λ22(iω )) 1(0)(iω ) 2(0)(iω )Λ12(iω )Λ21(iω ).
σ n −Gσ n σ n −Gσ n Gσ n σ n σ n
7
The other two functions are obtained by changing the indexes 1 2. These equations are of
↔
Dyson type. They determine Green’s functions through correlation functions Λ = g(0) +Z, Y
and Y¯ ones. The last three can only be given in a form of infinite diagram series, since the
exact solution does not exist.
An example of efficient summation of diagram and determination of the correlation function
Z, Y and Y¯ is presented on the Fig. 3.
V V
1∗ 2
l σ τ l σ τ
1 1 1 2 2 2
Zll′ (τ τ )= 1 G(0)ir[lστ,l σ τ l σ τ ,lστ ]V G[~k l σ τ ~k l σ τ ]V
σσ′ − ′ − 2 1 1 1| 2 2 2 ′ ′ ′ 1∗ 2 2 2 2| 1 1 1 1 2
lστ lστ
′ ′ ′
V V
1 2
l σ τ l σ τ
1 1 1 2 2 2
Yll′σσ¯′(τ −τ′) =−12 G(20)ir[lστ,l′σ¯′τ′|l1σ1τ1,l2σ¯2τ2]V1F[~k1l1σ1τ1|−~k2l2σ¯2τ2]V2
lσ¯ τ lστ
′ ′ ′
V V
1∗ 2∗
l σ τ l σ τ
1 1 1 2 2 2
Y¯ll′σσ¯′(τ −τ′) =−12 G(20)ir[l1σ¯1τ1,l2σ2τ2|lσ¯τ,l′σ′τ′]V1∗F¯[−~k1l1σ¯1τ1|~k2l2σ2τ2]V2∗
lστ lσ¯ τ
′ ′ ′
Figure 3: The main approximation for the correlation functions. The solid double lines with
arrows depict the full Green’s functions of conduction electrons. The rectangles depict the
irreducible Green’s functions of the impurity electrons.
The diagrams of Fig. 3 differ from the ones of Fig. 2 by the presence of the full conduction
electron Green’s function instead of the bare one of Fig. 2. This difference is the result of
ladder summation of main diagrams.
4 Correlation functions
The simplest correlation function is determined as
Girr[1,2 ¯3,¯4] = g(0)(1,2 ¯3,¯4) g(0)(1 ¯4)g(0)(2 ¯3)+g(0)(1 ¯3)g(0)(2 ¯4), (34)
2 | 2 | − 1 | 1 | 1 | 1 |
g(0)(1,2 3,4) = Td d d¯d¯ ,g(0)(1 ¯4) = Td d¯ ,1 = (l ,σ ,τ ),
2 | h 1 2 3 4i0 1 | −h 1 4i0 1 1 1
with two- and one-particle bare Green’s functions of localized electrons.
Because the presence of the Coulomb interactions in zero order Hamiltonian, equation (34)
is different of zero and contains charge, spin and pairing fluctuations.
The two-particle Green’s function g(0) is the sum of 4! terms of different time ordered
2
electron operators products. The statistical averages of these quantities are calculated by using
Hubbard transfer operators representation.
8
We need the Fourier representation of these functions
1
Girr[l σ τ ;l σ τ l σ τ ;l σ τ ] = Girr[l σ iω ;l σ iω l σ iω ;l σ iω ]
2 1 1 1 2 2 2| 3 3 3 4 4 4 β4 2 1 1 1 2 2 2| 3 3 3 4 4 4 ×
ω1ωX2ω3ω4
e iω1τ1 iω2τ2+iω3τ3+iω4τ4,
− −
1
g(0)(l σ τ l σ τ ) = g(0)(l σ ;l σ iω )e iω1(τ1 τ2) (35)
1 1 1 1| 2 2 2 β 1 1 1 2 2| 1 − −
Xω1
δ δ 1 1
g(0)(l σ ;l σ iω ) δ δ m(iω ) = l1l2 σ1σ2 + .
1 1 1 2 2| 1 ≈ l1l2 σ1σ2 1 2 iω +E E iω +E E
(cid:18) 1 2 − 9 1 9 − 12(cid:19)
Girr[l σ iω ;l σ iω l σ iω ;l σ iω ] = g(0)[l σ iω ;l σ iω l σ iω ;l σ iω ]
2 1 1 1 2 2 2| 3 3 3 4 4 4 2 1 1 1 2 2 2| 3 3 3 4 4 4 −
βδ(ω +ω ω ω )[βδ(ω ω )g(0)(l σ ;l σ iω )g(0)(l σ ;l σ iω ) (36)
1 2 − 3 − 4 1 − 4 1 1 1 4 4| 1 1 2 2 3 3| 2 −
βδ(ω ω )g(0)(l σ ;l σ iω )g(0)(l σ ;l σ iω )].
1 − 3 1 1 1 3 3| 1 1 2 2 4 4| 2
There exists the law of frequency conservation
Girr[l σ iω ;l σ iω l σ iω ;l σ iω ] = (37)
2 1 1 1 2 2 2| 3 3 3 4 4 4
βδ(ω +ω ω ω )Girr[l σ iω ;l σ iω l σ iω ;l σ iω ].
1 2 − 3 − 4 2 1 1 1 2 2 2| 3 3 3 4 4 4
The statistical averages of chronologeically ordered products of the electron operators of the
function g(0) have different weights of the form e−βEn, where E are the energies determined in
2 Z0 n
previous section. Because E is the lowest energy the weight e βE9 is the main of them and
9 −
only such terms are taken into account.
Just such considerations determined us to use instead initial exact equation (16) for zero
order Green’s function g(0) the approximate value (35). Zero order partition function Z (17)
lσl′σ′ 0
concomitant is approximated as 3e βE9.
−
Forexamplethecontributiontofunctiong(0)[l σ iω ;l σ iω l σ iω ;l σ iω ]withtimeorder
2 1 1 1 2 2 2| 3 3 3 4 4 4
β > τ > τ > τ > τ > 0 and with weight e βE9 is
1 3 2 4 −
1
δ δ ( δ δ +δ δ δ +δ δ δ δ )I(1)
− l1l3 l2l4 4 σ1σ3 σ2σ4 σ1,−σ3 σ2,−σ4 σ2σ3 σ1σ3 σ2σ4 σ1σ4 σ2σ3 1¯32¯4 −
1 1
(δ δ +( 1)l1+l4δ δ )( σ σ δ δ + δ δ δ )I(2) (38)
3−l1−l3,0 3−l2−l4,0 − l1l3 l2l4 4 1 4 σ1σ3 σ2σ4 2 σ1,−σ3 σ2,−σ4 σ1σ4 1¯32¯4 −
1 1
( 1)l1+l4δ δ ( σ σ δ δ + δ δ δ )I(3) ,
− 3−l1−l3,0 3−l2−l4,0 4 1 4 σ1σ3 σ2σ4 2 σ1,−σ3 σ2,−σ4 σ1σ4 1¯32¯4
where
e βE9 β τ1 τ3 τ2
I(1) = − dτ dτ dτ dτ e(E9 E12)(τ1+τ2 τ3 τ4)eiω1τ1+iω2τ2 iω3τ3 iω4τ4,
1¯32¯4 Z 1 3 2 4 − − − − −
0 Z Z Z Z
0 0 0 0
e βE9 β τ1 τ3 τ2
I(2) = − dτ dτ dτ dτ e(E9 E12)(τ1 τ4)+(E6 E12)(τ2 τ3)eiω1τ1+iω2τ2 iω3τ3 iω4τ4, (39)
1¯32¯4 Z 1 3 2 4 − − − − − −
0 Z Z Z Z
0 0 0 0
e βE9 β τ1 τ3 τ2
I(3) = − dτ dτ dτ dτ e(E9 E12)(τ1 τ4)+(E7 E12)(τ2 τ3)eiω1τ1+iω2τ2 iω3τ3 iω4τ4.
1¯32¯4 Z 1 3 2 4 − − − − − −
0 Z Z Z Z
0 0 0 0
These 4-fold multiple integrals by time variable τ can be transformed in contour integral
by using the method of Claude Bloch [15]. With this purpose it is necessary to introduce the
exponential form
e(β τ1)E¯0+(τ1 τ3)E¯1+(τ3 τ2)E¯2+(τ2 τ4)E¯3+(τ4 0)E¯4, (40)
− − − − −
9
which must be compared with exponential form of our integrals I(n) . Comparison with I(1)
1¯32¯4 1¯32¯4
give us the result
E¯ = E ,E¯ = E +iω iω ,E¯ = E +iΩ,E¯ = E +iω , (41)
0 9 2 9 1 3 4 9 1 12 1
− − − − −
E¯ = E +iω +iω iω ,Ω = ω +ω ω ω .
3 12 1 2 3 1 2 3 4
− − − −
Our integral I(1) is transformed in the contour integral
1¯32¯4
1 1 dze βz
I(1) = − , (42)
2πiZ (z +E¯ )(z +E¯ )(z +E¯ )(z +E¯ )(z +E¯ )
0 I 0 1 2 3 4
C+
where contour C+ surrounds the real axis in the positive direction. The integrals I(2) and
I(3) have the same form (42) but differ in the definition of energy E¯ . For I(2) the energy
2
E¯ = E +iω iω and for I(3), E¯ = E +iω iω . Other parameters coincide.
2 6 1 3 2 7 1 3
− − − −
The contour integral (42) is evaluated by the method of residues. The simple results are
obtained when the parameters E¯ are different. The existence of multiple poles is possible for
n
the special values of frequencies ω .
n
For example in the case when ω ω = 0 and Ω = 0 we have E¯ = E¯ = E¯ and the pole
1 3 0 2 4
z = E¯ is 3-fold multiple with the r−esidue
0
−
1 e−βZ ′′
. (43)
2 (z +E¯1)(z +E¯3)!z= E¯0
−
To find all possible multiple poles we consider different values of frequencies using the
identity 1 = δ(ω)+ψ(ω), where ψ(ω) = 1 δ(ω). For example we consider the possibility when
−
Ω can be equal to zero and ω = ω . We have the identity:
1 3
1 = (δ(Ω)+ψ(Ω))(δ(ω ω )+ψ(ω ω )) = (44)
1 3 1 3
− −
δ(Ω)δ(ω ω )+δ(Ω)ψ(ω ω ))+ψ(Ω)δ(ω ω )+ψ(Ω)ψ(ω ω )).
1 3 1 3 1 3 1 3
− − − −
The first term in the right-hand part of this equation admits the existence of triple pole, the
next two terms admit double poles and last term admit double and single poles.
We shall take into account these residues, statistical weights of which is e−βE9, and shall
Z0
omit the other ones. In such approximation we have
Z I(1) = 1δ(Ω)δ(ω ω ) e−βZ ′′ +
0 1¯32¯4 2 1 − 3 (z +E¯1)(z +E¯3)!z= E¯0
−
e−βZ ′ e−βZ
δ(Ω)ψ(ω ω ) + +
1 − 3 " (z +E¯1)(z +E¯2)(z +E¯3)!z= E¯0 (z +E¯0)2(z +E¯1)(z +E¯3)!z= E¯2#
− −
e−βZ ′ e−βZ
δ(ω ω )ψ(Ω) + +
1 − 3 " (z +E¯1)(z +E¯3)(z +E¯4)!z= E¯0 (z +E¯0)2(z +E¯1)(z +E¯3)!z= E¯4#
− −
ψ(Ω)ψ(ω ω )δ(ω ω ) (45)
1 3 2 4
− − ×
e−βZ ′ e−βZ
+ +
" (z +E¯0)(z +E¯1)(z +E¯3)!z= E¯2 (z +E¯2)2(z +E¯1)(z +E¯3)!z= E¯0#
− −
e βZ
−
ψ(Ω)ψ(ω ω )ψ(ω ω )[ +
1 − 3 2 − 4 (z +E¯1)(z +E¯2)(z +E¯3)(z +E¯4)!z= E¯0
−
e βZ e βZ
− −
+ ].
(z +E¯0)(z +E¯1)(z +E¯3)(z +E¯4)!z= E¯2 (z +E¯0)(z +E¯1)(z +E¯2)(z +E¯3)!z= E¯4
− −
10