A generating functional approach to the sd-model with strong correlations
A Kadanoff-Baym-type generating functional approach, earlier developed by the authors to strongly correlated systems, is applied to the sd-model with strong sd-coupling. Formalism of the Hubbard X -operators was used, and equation for electron Green’s function was derived with functional derivati...
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| Цитувати: | A generating functional approach to the sd-model with strong correlations / Yu.A. Izyumov, N.I. Chaschin, D.S. Alexeev // Condensed Matter Physics. — 2005. — Т. 8, № 4(44). — С. 801–812. — Бібліогр.: 11 назв. — англ. |
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irk-123456789-1210552017-06-14T03:04:49Z A generating functional approach to the sd-model with strong correlations Izyumov, Yu.A. Chaschin, N.I. Alexeev, D.S. A Kadanoff-Baym-type generating functional approach, earlier developed by the authors to strongly correlated systems, is applied to the sd-model with strong sd-coupling. Formalism of the Hubbard X -operators was used, and equation for electron Green’s function was derived with functional derivatives over external fluctuating fields. Iterations in this equation generate a perturbation theory near the atomic limit. Hartree-Fock type approximation is developed within the framework of this theory, and the problem of a metal–insulator phase transition in sd-model is discussed. Підхід генеруючого функціоналу типу Каданофа-Байма, розробленого раніше авторами для сильно скорельованих систем, застосовується до sd-моделі з сильною sd-взаємодією. Використовувався формалізм X -операторів Габбарда, і було отримано рівняння для електронних функцій Гріна з функціональними похідними по зовнішніх флуктуюючих полях. Ітерації в цьому рівнянні генерують теорію збурень біля атомної границі. В рамках цієї теорії розробляється наближення типу Хартрі-Фока, і обговорюється проблема фазового переходу метал-діелектрик в sd-моделі. 2005 Article A generating functional approach to the sd-model with strong correlations / Yu.A. Izyumov, N.I. Chaschin, D.S. Alexeev // Condensed Matter Physics. — 2005. — Т. 8, № 4(44). — С. 801–812. — Бібліогр.: 11 назв. — англ. 1607-324X PACS: 71.10.-w, 71.10.Fd, 71.27.+a DOI:10.5488/CMP.8.4.801 http://dspace.nbuv.gov.ua/handle/123456789/121055 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| description |
A Kadanoff-Baym-type generating functional approach, earlier developed
by the authors to strongly correlated systems, is applied to the sd-model
with strong sd-coupling. Formalism of the Hubbard X -operators was used,
and equation for electron Green’s function was derived with functional derivatives
over external fluctuating fields. Iterations in this equation generate
a perturbation theory near the atomic limit. Hartree-Fock type approximation
is developed within the framework of this theory, and the problem of a
metal–insulator phase transition in sd-model is discussed. |
| format |
Article |
| author |
Izyumov, Yu.A. Chaschin, N.I. Alexeev, D.S. |
| spellingShingle |
Izyumov, Yu.A. Chaschin, N.I. Alexeev, D.S. A generating functional approach to the sd-model with strong correlations Condensed Matter Physics |
| author_facet |
Izyumov, Yu.A. Chaschin, N.I. Alexeev, D.S. |
| author_sort |
Izyumov, Yu.A. |
| title |
A generating functional approach to the sd-model with strong correlations |
| title_short |
A generating functional approach to the sd-model with strong correlations |
| title_full |
A generating functional approach to the sd-model with strong correlations |
| title_fullStr |
A generating functional approach to the sd-model with strong correlations |
| title_full_unstemmed |
A generating functional approach to the sd-model with strong correlations |
| title_sort |
generating functional approach to the sd-model with strong correlations |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2005 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/121055 |
| citation_txt |
A generating functional approach to the sd-model with strong correlations / Yu.A. Izyumov, N.I. Chaschin, D.S. Alexeev // Condensed Matter Physics. — 2005. — Т. 8, № 4(44). — С. 801–812. — Бібліогр.: 11 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT izyumovyua ageneratingfunctionalapproachtothesdmodelwithstrongcorrelations AT chaschinni ageneratingfunctionalapproachtothesdmodelwithstrongcorrelations AT alexeevds ageneratingfunctionalapproachtothesdmodelwithstrongcorrelations AT izyumovyua generatingfunctionalapproachtothesdmodelwithstrongcorrelations AT chaschinni generatingfunctionalapproachtothesdmodelwithstrongcorrelations AT alexeevds generatingfunctionalapproachtothesdmodelwithstrongcorrelations |
| first_indexed |
2025-07-08T19:06:55Z |
| last_indexed |
2025-07-08T19:06:55Z |
| _version_ |
1837106848007716864 |
| fulltext |
Condensed Matter Physics, 2005, Vol. 8, No. 4(44), pp. 801–812
A generating functional approach to the
sd-model with strong correlations
Yu.A.Izyumov, N.I.Chaschin, D.S.Alexeev
Institute for Metal Physics of the RAS, Ural Division
620219 Ekaterinburg, Russia
Received July 18, 2005, in final form October 18, 2005
A Kadanoff-Baym-type generating functional approach, earlier developed
by the authors to strongly correlated systems, is applied to the sd-model
with strong sd-coupling. Formalism of the Hubbard X -operators was used,
and equation for electron Green’s function was derived with functional de-
rivatives over external fluctuating fields. Iterations in this equation generate
a perturbation theory near the atomic limit. Hartree-Fock type approxima-
tion is developed within the framework of this theory, and the problem of a
metal–insulator phase transition in sd-model is discussed.
Key words: theories and models of many-electron systems, lattice
fermion models, strongly correlated electron systems
PACS: 71.10.-w, 71.10.Fd, 71.27.+a
1. Introduction
In a series of papers [1–3] we suggested the generating functional approach (GFA)
for the basic models of strongly correlated systems: the Hubbard model, tJ-model,
periodic Anderson model. The GFA is actually a generalization of a well-known
Kadanoff and Baym [4] approach, suggested for the conventional fermi-systems, to
more complicated models with Hamiltonians written in terms of the spin- or the
Hubbard X-operators. The above mentioned models are just the models for which
GFA is to be effectively applied.
The idea of the method is based on introducing the generalization of the partition
function Z to the systems in external fields fluctuating in time and space. Z is
a functional of these fields. Different Green’s functions (GF) of a system can be
presented as functional derivatives over such fields. It is possible for each model to
derive equations for basic GFs in terms of the functional derivatives. It turns out
that for different models of strongly correlated systems these equations have a similar
structure, which indicates the tight relations between them. It is remarkable that
iterations in these equations generate a perturbation theory near the atomic limit.
The GFA and its applications to different models were discussed in detail in a
c© Yu.A.Izyumov, N.I.Chaschin, D.S.Alexeev 801
Yu.A.Izyumov, N.I.Chaschin, D.S.Alexeev
review [5] and a monograph [6], and here we briefly reproduce the main steps of the
method. The generating functional is determined by relation
Z[V ] = Tr
(
e−βHT e−V
)
≡ eΦ[V ], (1.1)
where H is Hamiltonian of the system, V is an operator of interaction with external
fields, T is a symbol of the ordering on the thermodynamic times −β < τ < β =
1/kT ; trace is taken over all variables of the system. V -operator is a linear com-
bination of the spin- or the X-operators and the coefficients in these combinations
are just the fluctuating fields. It is clear that functional derivatives over these fields
generate statistical averages of T -product of spin- or X-operators, which are just
different GFs of the system.
In this paper we develop GFA for sd-exchange model described by Hamiltonian
H =
∑
ijσ
tijc
+
iσciσ − J
2
∑
i
(Siσi) . (1.2)
Here the first term presents electron hopping on the lattice and the second one
describes interaction of a localized spin Si with electron spin σi/2, where σ is vector
with Pauli matrices. Under strong sd-coupling J & W , W is a width of the band, the
sd-exchange can be treated as Hamiltonian of zero approximation, and hopping as
a perturbation. Practical realization of the perturbation theory over the parameter
W/J is based on the fact that sd-exchange Hamiltonian is the one-site one.
Eigen functions of sd-exchange Hamiltonian −(Siσi) · J/2 are known; there are
four of them [7]:
|M 0〉 = |M〉|0〉 , (1.3)
|M 2〉 = |M〉|2〉 , (1.4)
|M+ +〉 = uM |M − 1
2
〉| ↑〉 + υM |M + 1
2
〉| ↓〉 , (1.5)
|M−−〉 = υM |M − 1
2
〉| ↑〉 − uM |M + 1
2
〉| ↓〉 . (1.6)
Here |0〉|, ↑〉, | ↓〉|, 2〉 describe the states without an electron, having a one electron
with spin σ =↑, ↓, and with two electrons on a site, respectively; |M〉 is a wave
function of an ion with spin projection M = −S,−S + 1, . . . , S; uM and υM are
Klebsh-Gordan coefficients:
u2
M =
S + M + 1
2
2S + 1
, υ2
M =
S − M + 1
2
2S + 1
, u2
M + υ2
M = 1 . (1.7)
The wave functions |M+〉 and |M−〉 describe a state of an ion with total spin
S + 1/2 and S − 1/2 and its projection may be equal to
−
(
S + 1
2
)
< · · · < M+ < · · · <
(
S + 1
2
)
j = S + 1
2
−
(
S − 1
2
)
< · · · < M− < · · · <
(
S − 1
2
)
j = S − 1
2
}
. (1.8)
802
A generating functional approach to the sd-model
In the basis of functions (1.3)–(1.6) sd-exchange Hamiltonian is diagonalized, and
two eigen-energies are equal to
E+ = −1
2
JS , j = S + 1
2
E− =
1
2
J(S + 1) , j = S − 1
2
. (1.9)
We will denote relations (1.5), (1.6) as
|Mα α〉 =
∑
σ
Θσα(Mα)|Mα − σ
2
〉 c+
σ |0〉 , (1.10)
where
Θσ+(M) =
√
S + σM + 1
2
2S + 1
, Θσ−(M) = σ
√
S − σM + 1
2
2S + 1
. (1.11)
2. Introducing X-operators
Arbitrary one-site operator Âi can be decomposed over the system of X-operators,
determined based on the wave functions |p〉. By definitions
Xpq = |p〉〈q|. (2.1)
This decomposition is
Âi =
∑
pq
〈p|Â|q〉Xpq
i . (2.2)
In the basis of functions (1.3)–(1.6) we have the following representation for electron
operator [8]
ciσ =
∑
Mα
[
Θσα(M + σ
2
) XM0 ; (M+σ
2
)α + σΘσ̄α(M − σ
2
) X(M−
σ
2
)α ; M2
]
. (2.3)
We have a similar representation for operator of total spin Stot = S + 1
2
σ on a
site [8]:
Sη
tot(i) =
∑
M
νη
S(M)
[
X
(M+η)0 ; M0
i + X
(M+η)2 ; M2
i
]
+
∑
αMα
νη
S+α
2
(Mα)X
(Mα+η)α ; Mαα
i ,
(2.4)
Sz
tot(i) =
∑
M
M
[
XM0 ; M0
i + XM2 ; M2
i
]
+
∑
αMα
MαXMαα ; Mαα
i . (2.5)
Here instead of two operators S+ and S− we introduce one operator Sη = 1/
√
2 ·
(Sx + iηSy), η = ±1. We also use a notation
νη
S(M) =
1√
2
√
(S − ηM)(S + ηM + 1) . (2.6)
803
Yu.A.Izyumov, N.I.Chaschin, D.S.Alexeev
One can see that the electron operator ciσ is presented by X-operators changing
the number of electrons on a site by 1. They are Fermi-like X-operators, obeying
anticommutation permutative relations. The both spin operators Sη
i and Sz
i are
presented by X-operators, changing electron number by 0 or 2. They are considered
to be Bose-like X-operators. The completeness condition should be fulfilled
∑
M
(XM0 ; M0
i + XM2 ; M2
i ) +
∑
αMα
XMαα ; Mαα
i = 1 . (2.7)
One-site part H1 of the Hamiltonian (with magnetic field) can be written as
H1 =
∑
i
[∑
M
(
E0
MXM0 ; M0
i + E2
MXM2 ; M2
i
)
+
∑
αMα
Eα
MαXMαα ; Mαα
i
]
, (2.8)
where eigen-energies of site states are equal [7]
E0
M = −hM , E2
M = −hM − 2µ , (2.9)
Eα
Mα = −hMα − µ + Eα , (2.10)
(Eα is determined in (1.9), h – magnetic field). In the expression for H1 we added
a term −µN with chemical potential.
To present a two-site part of the Hamiltonian H2 =
∑
ijσ
tijc
+
iσcjσ it is worth intro-
ducing two-component spinors composed of Fermi-like X-operators:
Ψ+
i (σ α M) =
(
X
(M+σ
2
)α ; M0
i , σX
M2 ; (M−
σ
2
)α
i
)
(2.11)
and column Ψi composed of conjugated X-operators. Then H2 is written as
H2 =
∑
ij
tij
∑
I1I2
Ψ+
i (I1)T(I1I2)Ψj(I2) , (2.12)
where
T(I1I2) = δσ1σ2
Θσ1α1(M1 + σ1
2
)Θσ2α2(M2 + σ2
2
) Θσ1α1(M1 + σ1
2
)Θσ̄2α2(M2− σ2
2
)
Θσ̄1α1(M1 − σ1
2
)Θσ2α2(M2 + σ2
2
) Θσ̄1α1(M1 − σ1
2
)Θσ̄2α2(M2− σ2
2
)
.
(2.13)
Here we introduced a combined index I = (σMαν), where ν = 1, 2 numerates the
components of the spinor. Relations (2.8) and (2.12) present Hamiltonian of the sd-
model in terms of X-operators. The motion between two different sites is described
by one-particle electron GF.
3. Equation of motion for electron Green’s function
Let us introduce one-particle electron GF
L12(I1 , I2) = −
〈
TΨ1(I1)Ψ
+
2 (I2)
〉
V
〈
TΨ1(I1)Ψ2(I2)
〉
V
〈
TΨ+
1 (I1)Ψ
+
2 (I2)
〉
V
〈
TΨ+
1 (I1)Ψ2(I2)
〉
V
, (3.1)
804
A generating functional approach to the sd-model
where 〈T . . . 〉V means a statistical average of the system in the fluctuating fields V ,
that is
〈T . . . 〉V =
1
Z[V ]
Tr
(
e−βHT . . . e−V
)
. (3.2)
Integer index is a combined one including a site number i and time τ , for example
1 = (i1, τ1).
We have to write the equation of motion for each matrix element in (3.1). For
matrix element “11” one can use the identity
∂
∂τ1
((
TΨ1(I1)Ψ
+
2 (I2)
))
= δ(τ1 − τ2)
((
T
[
Ψ1(I1), Ψ
+
2 (I2)
]
+
))
+
((
T Ψ̇1(I1)Ψ
+
2 (I2)
))
−
((
T
[
Ψ1(I1), V
]
−
Ψ+
2 (I2)
))
.(3.3)
Here we use a short notation:
((T . . . )) = Tr(e−βHT . . . e−V ) . (3.4)
Now we must determine the operator V , describing the interaction with fluctu-
ating fields. According to general conception of GFA, now we take V in the form:
V = v
M ′
1
0 ; M ′
1
0
1′ X
M ′
1
0 ; M ′
1
0
1′ + v
M ′
1
2 ; M ′
1
2
1′ X
M ′
1
2 ; M ′
1
2
1′
+ v
M ′
1
0 ; M ′
1
2
1′ X
M ′
1
2 ; M ′
1
0
1′ + v
M ′
1
2 ; M ′
1
0
1′ X
M ′
1
0 ; M ′
1
2
1′
+ v
(M ′
1
+
σ
′
2
2
)α′
2
; (M ′
1
+
σ
′
1
2
)α′
1
1′ X
(M ′
1
+
σ
′
1
2
)α′
1
; (M ′
1
+
σ
′
2
2
)α′
2
1′ . (3.5)
Here summation over all repeated primed indexes is implied.
As it is seen, we have to calculate first anticommutators of Ψ -operators. We have
δ(τ1 − τ2)
[
Ψ1(I1), Ψ
+
2 (I2)
]
+
= δ12F(I1 , I2)
δ(τ1 − τ2)
[
Ψ+
1 (I1), Ψ2(I2)
]
+
= δ12F+(I1 , I2)
δ(τ1 − τ2)
[
Ψ1(I1), Ψ2(I2)
]
+
= δ12Q(I1 , I2)
δ(τ1 − τ2)
[
Ψ+
1 (I1), Ψ
+
2 (I2)
]
+
= δ12Q+(I1 , I2)
. (3.6)
Here F and Q are 2 × 2 matrices:
F1(I1 , I2) =
δα1α2
δM1+
σ1
2
, M2+
σ2
2
XM10 ; M20
1
+δM1 , M2
X
(M2+
σ2
2
)α2 ; (M1+
σ1
2
)α1
1
0
0
σ1σ2δα1α2
δM2−
σ2
2
, M1−
σ1
2
XM22 ; M12
1
+σ1σ2δM1 , M2
X
(M1−
σ1
2
)α1 ; (M2−
σ2
2
)α2
1
,
(3.7)
Q1(I1 , I2) =
0 σ2δM1+
σ1
2
, M2−
σ2
2
XM10 ; M22
1
σ1δM2+
σ2
2
, M1−
σ1
2
XM20 ; M12
1 0
. (3.8)
805
Yu.A.Izyumov, N.I.Chaschin, D.S.Alexeev
Matricies F+ and Q+ are Hermitian conjugated to matrices F and Q.
We also need equations of motion for operators Ψ and Ψ+.
Ψ̇1(I) = −E(I , I′)Ψ1(I′) −F1(I , I′)T 12′(I′ , I′′)Ψ2′(I′′) + Q1(I , I′)T 12′(I′ , I′′)Ψ+
2′ (I′′) , (3.9)
Ψ̇+
1 (I) = E(I , I′)Ψ+
1 (I′)−Q+
1 (I , I′)T 12′(I′ , I′′)Ψ2′(I′′) + F+
1 (I , I′)T 12′(I′ , I′′)Ψ+
2′ (I′′) . (3.10)
In this equations we use the notation
T 12(I1 , I2) = T(I1 , I2)ti1i2δ(τ1 − τ2) , (3.11)
E(I1 , I2) =
Eα − µ 0
0 −Eα − µ
. (3.12)
Now we substitute the results (3.9), (3.11), (3.6) in equation (3.3) and in the similar
equation for three other matrix elements of (3.1). As a result we come to the following
equation for the matrix GF L:
[(
L−1
0V
)
11′
(I , I′) − (ÂΦY )11′(I , I′) − (ÂY )11′(I , I′)
]
L1′2(I′ , I′′) = (Â12Φ)(I , I′′). (3.13)
Here  is a matrix
Â12(I , I′) = δ12
F̂1(I , I′) Q̂1(I , I′)
Q̂+
1 (I , I′) F̂+
1 (I , I′)
, (3.14)
where quantities F̂1 and Q̂1 (and their conjugated ones) are given by matrixes (3.7)
and (3.8), in which X-operators should be replaced by the corresponding functional
derivatives according to a receipt
Xpq
1 → − δ
δvqp
1
. (3.15)
Thus matrix  is composed of the expressions, including functional derivatives. This
is marked by cups over the letters.
Quantity Y is the following 2 × 2 matrix
Y12(I , I′) =
T 12(I , I′) 0
0 −T 12(I , I′)
. (3.16)
Finally, the quantity L−1
0V is also 2 × 2 matrix
(
L−1
0V
)
11′
(I , I′) =
(
G−1
0V
)
11′
(I , I′) δ11′W
02
1 (I , I′)
−δ11′W
20
1 (I , I′)
(
G̃−1
0V
)
11′
(I , I′)
. (3.17)
806
A generating functional approach to the sd-model
Here
(
L−1
0V
)
11′
(I , I′) = −
(
∂
∂τ1
δII′ + E1δII′
)
δ11′ − δ11′W1(I , I′) , (3.18)
where
W1(I1 , I′) = δM1,M2
−δσ1σ2
δα1α2
vM10 ; M10
1 +
+v(M1+
σ2
2
)α2 ; (M1+
σ1
2
)α1
0
0
δσ1σ2
δα1α2
vM12 ; M12
1 −
−v
(M1−
σ1
2
)α1 ; (M1−
σ2
2
)α2
1
, (3.19)
W 02
1 (I1 , I′) = δM1,M2
δσ̄1σ2
δα1α2
vM10 ; M12
1
(
0 1
−1 0
)
. (3.20)
So, all quantities in equation (3.13) have been determined. Remind that in this
equation the summation over all repeated primed indexes is implied. Separate terms
in the equation have the following physical meaning. The first term is a reversed GF
of zero approximation (respectively hopping), but including the fluctuating fields.
The second term determines Hartree-Fock correction due to hopping, and the third
term includes the functional derivatives acting on the GF L. In the right hand side
there is ÂΦ quantity, which includes some averages of X-operators.
The form of equation (3.13) coincides with the form of equation for GF in the
Hubbard model [9]. The only difference is in the form of matrices Â, Y and L−1
0V as
well as in the structure of the combined index I. The Hubbard model I includes spin
σ and spinor index ν, while the sd-model I = (σMαν) includes two more indexes M
and α.
We present an explicit form of matrix F̂ and Q̂, determining the operator matrix
Â. For simplicity we write them only in a particular but important case, when
σ1 = σ2 ≡ σ.
F̂(I1 , I2) = −δM1,M2
δα1α2
δ
δvM10 ; M10
1
+ δ
δv(M1+
σ
2
)α2 ; (M1+σ
2
)α1
0
0
δα1α2
δ
δvM12 ; M12
1
+ δ
δv(M1−
σ
2
)α1 ; (M1−
σ
2
)α2
, (3.21)
Q̂(I1 , I2) = −σ
0 δM1−M2+σ,0
δ
δv
M10 ; (M1+σ)2
1
δM1−M2−σ,0
δ
δv
(M1−σ)0 ; M12
1
0
. (3.22)
807
Yu.A.Izyumov, N.I.Chaschin, D.S.Alexeev
4. The Hubbard-I type approximation
Consider a simple approximation, when in the basic equation (3.13) the term with
functional derivatives of GF L is neglected. We rewrite the approximate equation
with short notation
[
L−1
0V (1 1 ′ ) − (ÂΦY )(1 1 ′ )
]
L(1 ′2 ) = (ÂΦ)(1 2 ) , (4.1)
where underlined indexes mean: 1 = (1I1), etc. In a normal phase off-diagonal ele-
ments of matrix (3.1) vanish. Let us denote a diagonal element L11 = G, then from
the matrix equation (4.1) one can write an equation for GF G:
G(1 2 ) = G(1 1′ )Λ(1′ 2 ) , (4.2)
where G(1 2 ) is a propagator part of G, obeying the Dyson equation
G−1(1 2 ) = G−1
0V (1 2 ) − Σ(1 2 ) , (4.3)
and the terminal part
Λ(1 2 ) = (ÂΦ)(1 2 ) . (4.4)
The self-energy part in our approximation is equal to
Σ(1 2 ) = (ÂΦY )11(1 2 ) = t12(F̂1TΦ)(I1 , I2 ) . (4.5)
Quantities F̂ and T are determined by equation(3.21) and (2.13), respectively. The
matrix, standing in equation(4.5) is factorized, which means that it can be written
in the form
(F̂1TΦ)(I1 , I2 ) = Λ0(I1)θ(I1 )Tθ(I2 ) . (4.6)
Here T is a 2 × 2 matrix with all matrix elements equal to 1.
Λ0(I ) =
〈0〉σα(M) 0
0 〈2〉σ̄α(M)
, (4.7)
θ(I ) =
θσα(M + σ
2
) 0
0 θσ̄α(M − σ
2
)
. (4.8)
Here we introduce the notation
〈0〉σα(M) = 〈XM0 ; M0〉 + 〈X(M+σ
2
)α ; (M+σ
2
)α〉
〈2〉σα(M) = 〈XM2 ; M2〉 + 〈X(M−
σ
2
)α ; (M−
σ
2
)α〉
}
. (4.9)
The factorization (4.6) allows one to resolve the matrix equation (4.3). After Fourier
transformation over the variable (1 − 2) we obtain an equation for Gk(I1 , I2) and
Gk(I1 , I2) = Gk(I1 , I2)Λ(I2 ).
808
A generating functional approach to the sd-model
We present a solution of the obtained equation in the form:
Gk(I1 , I2) = G0k(I1)δI1 , I2 + G0k(I1)θ(I1 )D(k)Tθ(I2 )G0k(I2). (4.10)
Here
G0k(I) =
〈θ〉σα(M)
iωn − Eα + µ
0
0
〈2〉σ̄α(M)
iωn + Eα + µ
, (4.11)
D(k) =
[
1 − εk
∑
I
Tθ(I )G0k(I )θ(I )
]−1
. (4.12)
Notice that all quantities involved in equation (4.10) and equation (4.12) are 2 × 2
matrices.
For further analysis it is useful to determine a quantity
Ḡ12 =
∑
I1I2
θ(I1 )G12(I1 , I2)θ(I2 ) , (4.13)
being a GF determined on electronic operators ci =
∑
σ
ciσ, which are devided into
two values: ci = gi+hi, where gi includes only the first term in equation (2.3), but hi
– the second one, is connected with on-site transitions from double occupied states.
It is easy to derive the following expression for it
Ḡk =
∑
I
θ(I )G0k(I)θ(I )D(k) . (4.14)
After multiplying the matrices in the expressions (4.14) and (4.12) we obtain
∑
I
θ(I )G0k(I)θ(I ) =
∑
I
(
θ2〈0〉
)
(I )
iωn − Eα + µ
0
0
(
θ̄2〈2〉
)
(I )
iωn + Eα + µ
, (4.15)
D(k) =
1
d(k)
1 − εk
∑
I
(
θ̄2〈2〉
)
(I )
iωn + Eα + µ
εk
∑
I
(
θ2〈0〉
)
(I )
iωn − Eα + µ
εk
∑
I
(
θ̄2〈2〉
)
(I )
iωn + Eα + µ
1 − εk
∑
I
(
θ2〈0〉
)
(I )
iωn − Eα + µ
, (4.16)
where
d(k) = 1 − εk
∑
I
[ (
θ2〈0〉
)
(I )
iωn − Eα + µ
+
(
θ̄2〈2〉
)
(I )
iωn + Eα + µ
]
. (4.17)
Expressions (4.14)–(4.17) determine the matrix electron GF in the Hubbard-I ap-
proximation. All expressions include averaged values of diagonal X-operators deter-
mined by relations (4.9). In order to calculate them one has to know the electron
GF, constructed on X-operators for each expression (4.10).
809
Yu.A.Izyumov, N.I.Chaschin, D.S.Alexeev
The poles of electron GF Gk(I1 , I2) and Ḡk are determined by zeroes of quantity
d(k). Since α takes two values, it is clear that dispersion relation d(k) = 0 has
four solutions unlike the Hubbard model, where two solutions exist, corresponding
to lower and upper Hubbard subbands. The doubling of the solution numbers is
connected with the fact that in sd-model, on each site there are two allowed states
characterized by the value of total spin j = S + 1/2, j = S − 1/2.
5. Limit of classical spin
In the limit S → ∞ |E+| = |E−|, and the dispersion relation of the fourth
order reduces to a square equation, determining two subbands E1k and E2k. In
paramagnetic phase the solution of equation d(k) = 0 in the limit S → ∞ gives two
roots:
E1,2 k = εk ∓ Qk , Qk =
√(
SJ
2
)2
+ ε2
k
. (5.1)
Calculation using the equations (4.14)–(4.17) leads to the result for matrix elements
of GF Ḡ(k); (Ω = iωn + µ):
Ḡµν(k) = ±K(Ω) +
Pµν
1k
Ω − E1k
+
Pµν
2k
Ω − E2k
, (5.2)
where
K(Ω) = 〈0〉〈2〉
1
Ω − SJ
2
+
1
Ω + SJ
2
. (5.3)
Sign “+” in (5.2) stands for diagonal matrix elements, and “−” stands for off diag-
onal ones, and
P11
1,2 k
= ∓〈0〉E1,2 k
Qk
±
(
SJ
2
)2 〈0〉〈2〉
QkE1,2 k
± 2〈0〉〈2〉 εk
Qk
P22
1,2 k
= ∓〈2〉E1,2 k
Qk
±
(
SJ
2
)2 〈0〉〈2〉
QkE1,2 k
± 2〈0〉〈2〉 εk
Qk
P12
1,2 k
= P21
1,2 k
= ∓
(
SJ
2
)2 〈0〉〈2〉
QkE1,2 k
∓ 2〈0〉〈2〉 εk
Qk
. (5.4)
We have to know the electron GF, constructed on Fermi-operators
G(k) = −〈Tc1c
+
2 〉k =
∑
µν
Ḡµν(k) . (5.5)
Substituting here the relations (5.4) and taking the equality 〈0〉 + 〈2〉 = 1 (due to
the completeness condition (2.7)) we find
G(k) =
(
1 − εk
Qk
)
1
Ω − E1k
+
(
1 +
εk
Qk
)
1
Ω − E2k
. (5.6)
810
A generating functional approach to the sd-model
It is remarkable that in the Hubbard-I type approximation statistical weights
of the quasiparticle states do not depend on the electron concentration n; it enters
only in the chemical potential µ. However, at finite S it should not be so, and it is
necessary to solve the equation d(k) = 0 of fourth order to define the quasiparticle
energies.
An expression of the type (5.6) was obtained in [11] by decoupling the double-
time GFs in the first step. Obviously, the result (5.6) cannot give a metal-insulator
phase transition because two subbands of quasiparticles with energies (5.1) are sep-
arated and should not be overlapped at any values of parameters. In the next step
the authors [11] took into account the static contribution of spin fluctuations to the
self-energy and show that it can lead to such a phase transition at some reasonable
relations JS ∼ W between the parameters. However, there is some violation of sum
rules for GFs, that demands a more precise approximation.
A final aim of our work is to study phase-transitions in the sd-model when
dynamical fluctuations in the system are included. For this purpose the first order
corrections over W/SJ in the terminal part Λ1 and the second order in the self-energy
Σ2 over hopping will be calculated elsewhere. Similarly to what was done by us for
the Hubbard model [9] we extract from Σ2 a static contribution with some adjustive
parameters, which are determined from fundamental conditions for electron GF [10].
Preliminary analysis shows that quasiparticle subbands can be overlapped which
leads to a metal-insulator phase transition. Details of such a transition should be
determined by interaction of quasiparticles with dynamical fluctuations. All these
discussions will be a subject of next publication.
Authors thank Russian Foundation of Support of Science Schools, grant NS–
747.2003.2 and Division of Physical Sciences of the RAS, grant N 10104–71/OFN–
03/032–348/140705–126/01.06.2005.
811
Yu.A.Izyumov, N.I.Chaschin, D.S.Alexeev
References
1. Izyumov Yu.A., Chaschin N.I., Yushankhai V.Yu., Phys. Rev. B, 2002, 65, 214425.
2. Izyumov Yu.A., Chaschin N.I., Phys. Met. Metallogr., 2002, 94, No. 6, 527; 2002, 94,
No. 6, 539; 2004, 97, No. 3, 225.
3. Izyumov Yu.A., Alexeev D.S., Phys. Met. Metallogr., 2004, 97, No. 1, 5.
4. Kadanoff L.P., Baym G. Quantum Statistical Mechanics: Green’s Function Methods
in Equilibrium and Noneqvilibrium Problems. Benjamin, New York, 1962.
5. Izyumov Yu.A. – In: Lectures on the Physics of Highly Correlated Electron Systems
VII, Seventh Training Course in the Physics of Correlated Electron Systems and High-
Tc Superconductors, ed. Avella A., Mancini F., AIP Conference Proceedings 678, 2003.
6. Izyumov Yu.A., Skryabin Yu.N. Basic Models in Quantum Theory of Magnetism. Ural
Otd. Ros. Akad. Nauk, Ekaterinburg, 2002 (in Russian).
7. Erukhimov M.S., Ovchinnikov S.G., Phys. Stat. Sol. B, 1984, 123, 105.
8. Valkov V.V., Ovchinnikov S.G. Quasiparticles in Strongly Correlated Systems. Sib.
Otd. Ros. Akad. Nauk, Novosibirsk, 2001 (in Russian).
9. Izyumov Yu.A., Chaschin N.I., Alexeev D.S., Manchini F., Eur. Phys. J. B, 2005, 45,
69.
10. Mancini F., Avella A., Adv. Phys., 2004, 53, No. 5–6, 537.
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Підхід на основі генеруючого функціоналу до
sd-моделі з сильними кореляціями
Ю.A.Ізюмов, Н.I.Чащін, Д.С.Алєксєєв
Інститут фізики металів РАН, Уральське відділення
620219 Єкатеринбург, Росія
Отримано 18 липня, 2005, в остаточному варіанті – 18 жовтня,
2005
Підхід генеруючого функціоналу типу Каданофа-Байма, розробле-
ного раніше авторами для сильно скорельованих систем, застосо-
вується до sd-моделі з сильною sd-взаємодією. Використовувався
формалізм X -операторів Габбарда, і було отримано рівняння для
електронних функцій Гріна з функціональними похідними по зовніш-
ніх флуктуюючих полях. Ітерації в цьому рівнянні генерують теорію
збурень біля атомної границі. В рамках цієї теорії розробляється
наближення типу Хартрі-Фока, і обговорюється проблема фазового
переходу метал-діелектрик в sd-моделі.
Ключові слова: теорії і моделі багатоелектронних систем, моделі
граткового ферміону, сильно скорельовані електронні системи
PACS: 71.10.-w, 71.10.Fd, 71.27.+a
812
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