Sd-model with strong exchange coupling and a metal-insulator phase transition
Sd-exchange model (Kondo lattice model) is formulated for strong sd-coupling within the framework of the Xoperators technique and the generating functional approach. The X-operators are constructed based on the exact eigen functions of a single-site sd-exchange Hamiltonian. Such representation ena...
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Інститут фізики конденсованих систем НАН України
2006
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| Цитувати: | Sd-model with strong exchange coupling and a metal-insulator phase transition / Yu.A. Izyumov, N.I. Chaschin, D.S. Alexeev // Condensed Matter Physics. — 2006. — Т. 9, № 3(47). — С. 545–555. — Бібліогр.: 9 назв. — англ. |
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irk-123456789-1213582017-06-15T03:04:30Z Sd-model with strong exchange coupling and a metal-insulator phase transition Izyumov, Yu.A. Chaschin, N.I. Alexeev, D.S. Sd-exchange model (Kondo lattice model) is formulated for strong sd-coupling within the framework of the Xoperators technique and the generating functional approach. The X-operators are constructed based on the exact eigen functions of a single-site sd-exchange Hamiltonian. Such representation enables us to develop a perturbation theory near the atomic level. A locator-type representation was derived for the electron Green’s function. The electron self-energy includes interaction of electrons and spin fluctuations. An integral equation for the self-energy was obtained in the limit of infinite localized spins. A solution of this equation in the static approximation for spin fluctuations leads to a structure of electron Green’s function showing a metal-insulator phase transition. This transition is similar to that in the Hubbard model at half filling. Sd-обмiнна модель (модель Кондо гратки) сформульована для сильної sd-взаємодiї в рамках технiки X-операторiв та пiдходу твiрного функцiоналу. X-оператори побудованi на базисi точних власних функцiй одновузлового sd-обмiнного гамiльтонiану. Таке представлення дозволяє розвинути теорiю збурень поблизу атомного рiвня. Представлення локаторного типу було отримане для електронної функцiї Грiна. Власна енергiя електронiв включає взаємодiю електронiв та спiнових флуктуацiй. Отримано iнтегральне рiвняння для власної енергiї в границi безмежних локалiзованих спiнiв. Розв’язок цього рiвняння в статичному наближеннi для спiнових флуктуацiй веде до структури електронної функцiї Грiна, що виявляє фазовий перехiд метал-дiелектрик. Цей перехiд є схожим до переходу в моделi Хаббарда при половинному заповненнi. 2006 Article Sd-model with strong exchange coupling and a metal-insulator phase transition / Yu.A. Izyumov, N.I. Chaschin, D.S. Alexeev // Condensed Matter Physics. — 2006. — Т. 9, № 3(47). — С. 545–555. — Бібліогр.: 9 назв. — англ. 1607-324X PACS: 71.10.-w, 71.10.Fd, 71.27.+a DOI:10.5488/CMP.9.3.545 http://dspace.nbuv.gov.ua/handle/123456789/121358 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
Sd-exchange model (Kondo lattice model) is formulated for strong sd-coupling within the framework of the Xoperators
technique and the generating functional approach. The X-operators are constructed based on the
exact eigen functions of a single-site sd-exchange Hamiltonian. Such representation enables us to develop a
perturbation theory near the atomic level. A locator-type representation was derived for the electron Green’s
function. The electron self-energy includes interaction of electrons and spin fluctuations. An integral equation
for the self-energy was obtained in the limit of infinite localized spins. A solution of this equation in the static
approximation for spin fluctuations leads to a structure of electron Green’s function showing a metal-insulator
phase transition. This transition is similar to that in the Hubbard model at half filling. |
| format |
Article |
| author |
Izyumov, Yu.A. Chaschin, N.I. Alexeev, D.S. |
| spellingShingle |
Izyumov, Yu.A. Chaschin, N.I. Alexeev, D.S. Sd-model with strong exchange coupling and a metal-insulator phase transition Condensed Matter Physics |
| author_facet |
Izyumov, Yu.A. Chaschin, N.I. Alexeev, D.S. |
| author_sort |
Izyumov, Yu.A. |
| title |
Sd-model with strong exchange coupling and a metal-insulator phase transition |
| title_short |
Sd-model with strong exchange coupling and a metal-insulator phase transition |
| title_full |
Sd-model with strong exchange coupling and a metal-insulator phase transition |
| title_fullStr |
Sd-model with strong exchange coupling and a metal-insulator phase transition |
| title_full_unstemmed |
Sd-model with strong exchange coupling and a metal-insulator phase transition |
| title_sort |
sd-model with strong exchange coupling and a metal-insulator phase transition |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2006 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/121358 |
| citation_txt |
Sd-model with strong exchange coupling and a metal-insulator phase transition / Yu.A. Izyumov, N.I. Chaschin, D.S. Alexeev // Condensed Matter Physics. — 2006. — Т. 9, № 3(47). — С. 545–555. — Бібліогр.: 9 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT izyumovyua sdmodelwithstrongexchangecouplingandametalinsulatorphasetransition AT chaschinni sdmodelwithstrongexchangecouplingandametalinsulatorphasetransition AT alexeevds sdmodelwithstrongexchangecouplingandametalinsulatorphasetransition |
| first_indexed |
2025-07-08T19:42:47Z |
| last_indexed |
2025-07-08T19:42:47Z |
| _version_ |
1837109104396468224 |
| fulltext |
Condensed Matter Physics 2006, Vol. 9, No 3(47), pp. 545–555
Sd-model with strong exchange coupling and a
metal-insulator phase transition
Yu.A.Izyumov, N.I.Chaschin, D.S.Alexeev
Institute for Metal Physics, S.Kovalevskaya str., 18, Ekaterinburg, Russia
Received May 18, 2006, in final form June 14, 2006
Sd-exchange model (Kondo lattice model) is formulated for strong sd-coupling within the framework of the X-
operators technique and the generating functional approach. The X-operators are constructed based on the
exact eigen functions of a single-site sd-exchange Hamiltonian. Such representation enables us to develop a
perturbation theory near the atomic level. A locator-type representation was derived for the electron Green’s
function. The electron self-energy includes interaction of electrons and spin fluctuations. An integral equation
for the self-energy was obtained in the limit of infinite localized spins. A solution of this equation in the static
approximation for spin fluctuations leads to a structure of electron Green’s function showing a metal-insulator
phase transition. This transition is similar to that in the Hubbard model at half filling.
Key words: Kondo lattice model, strongly correlated electron systems, metal-insulator phase transitions
PACS: 71.10.-w, 71.10.Fd, 71.27.+a
1. Introduction
The sd-exchange model is one of the fundamental models in the theory of strongly correlated
systems. This model is frequently referred to as the Kondo lattice model in the West. Its Hamil-
tonian takes into account the hopping of conduction electrons over a lattice and the exchange
interaction of them with the localized atomic moments (value spin S) placed at the lattice sites.
The Hamiltonian is written as
H =
∑
ijσ
tijc
†
iσciσ − J
2
∑
i
(
Siσi
)
. (1.1)
Here ciσ (c†iσ) is a Fermi-operator of annihilation (creation) of an electron on a site i with spin
projection σ.
For the case of strong sd-exchange coupling S|J | � W (W is width of the electron band)
the sd-model was applied for description of a metal-insulator phase transition [1,2] and physics
of magnetic semiconductors [3]. Exact solution of eigenvalue problem for a single site exchange
Hamiltonian was applied [1–3]. Two energy levels E+ and E− for the states with the total spin on
a site j+ = S +1/2 and j+ = S − 1/2 are spread into two bands by hopping term, that determines
the physics of the model in the strong coupling limit.
In recent works [4,5] we applied the generating functional approach (GFA) to this problem
which, in a most general form, permits to construct a perturbation theory “near the atomic limit”.
Equations for electron Green’s functions (GF) were derived in terms of functional derivatives over
fluctuating fields introduced in GFA, and the simplest approximations for their solution were
tested. In the present paper we return to the problem on a most general basis. Particularly, we
introduced four-component electron operator instead of two-component one used in works [4,5].
Such generalization of mathematical structure makes it possible to construct more correctly the
perturbation theory near the atomic limit and to consider the states with more complicated order
parameters such as magnetic ordering or superconductivity.
c© Yu.A.Izyumov, N.I.Chaschin, D.S.Alexeev 545
Yu.A.Izyumov, N.I.Chaschin, D.S.Alexeev
In the case of strong sd-exchange coupling one has to use the exact solution of the problem on
one site. Wave functions of the Hamiltonian J/2
(
Sσ
)
are known [1,2]:
|M0〉 = |M〉|0〉 , (1.2)
|M2〉 = |M〉|2〉 , (1.3)
|Mαα〉 =
∑
σ
Θσα(Mα)|Mα − σ
2 〉c†σ|0〉 , α = + , − , (1.4)
where
Θσ+ =
√
S + σM + 1
2
2S + 1
, Θσ− = σ
√
S + σ̄M + 1
2
2S + 1
. (1.5)
A wave function |M0〉 describes the state of a site without an electron when the atom is in |M〉
state with projection of spin M; |M2〉 is a state with two electrons on a site, and |Mαα〉 with only
one electron with projection of total spin jα and
−(S + α
2 ) < · · · < Mα < · · · < (S + α
2 ) , jα = S + α
2 . (1.6)
In |Mαα〉 state the wave function is a superposition with both projections of an electron spin with
mixing coefficients being Klebs-Gordan coefficients. Selfenergy of one site exchange Hamiltonian is
as follows:
E+ = −1
2
JS , j = S + 1
2
E− =
1
2
J(S + 1) , j = S − 1
2
. (1.7)
Because the self-energy is known it is possible to develop a perturbation theory over small
parameter W/JS. The best way to do it is GFA combined with the formalism of X-operators.
2. Formalism of X-operators
Any one-site operator Âi can be expanded over the system of X-operators
Xpq
i = |p〉〈q| , (2.1)
determined on the base of functions |p〉. This explanation is written as
Âi =
∑
pq
〈
p |Â | q
〉
Xpq
i . (2.2)
The calculation of a matrix element of an annihilation operator ciσ leads to its representation in
terms of X-operators [2,4–7]:
ciσ =
∑
Mα
[
Θσα
(
M + σ
2
)
X
M0 ; (M+ σ
2 )α
i + σΘσ̄α
(
M − σ
2
)
X
(M−σ
2 )α ; M2
i
]
. (2.3)
Let us introduce four-component construction of Fermi-like X-operators
Ψ1σ(I) =
X
M0 ; (M+ σ
2 )+
1
σX
(M−σ
2 )+ ; M2
1
X
M0 ; (M+ σ
2 )−
1
σX
(M−σ
2 )− ; M2
1
, (2.4)
as well as a line Ψ †
iσ(I) including the conjugated X-operators. Operator Ψ1σ(I) describes the creation
of an electron on site i at time τ (1 = iτ) with spin σ when the atom has an angular momentum
546
Sd-model with strong exchange coupling and a metal-insulator phase transition
projection M, with two possible combinations of these momenta labeled by indexes α = +,−.
Components of spinor are numbered by index ν = 1, 2, 3, 4. Its combined index I = (Mν).
In such representation the Hamiltonian of sd-model is written as
H = Hsd − µN = H1 + H2
H1 = −2
∑
iM
µXM2 ; M2
i +
∑
iαMα
(Eα − µ)XMαα ; Mαα
i , (2.5)
H2 =
∑
ij
tij
∑
σ1I1σ2I2
Ψ †
1σ1
(I1)T(σ1I1 ; σ2I2)Ψjσ2
(I2) . (2.6)
where
T(σ1I1 ; σ2I2) = δσ1σ2
ϑσ1(I1)Tϑσ2(I2) , (2.7)
ϑσ(I) = diag
{
Θσ+(M + σ
2 ) , Θσ̄+(M − σ
2 ) , Θσ−(M + σ
2 ) , Θσ̄−(M − σ
2 )
}
, (2.8)
and T is a 4 × 4 matrix with all elements being equal to 1.
Let us introduce a supermatrix GF
Lσ1σ2
12 (I1 , I2) = −
(〈
TΨ1σ1
(I1)Ψ
†
2σ2
(I2)
〉
V
〈
TΨ1σ1
(I1)Ψ2σ2
(I2)
〉
V〈
TΨ †
1σ1
(I1)Ψ
†
2σ2
(I2)
〉
V
〈
TΨ †
1σ1
(I1)Ψ2σ2
(I2)
〉
V
)
, (2.9)
each of its elements is a matrix 4 × 4.
Here 〈T · · · 〉V is a statistical average of some chronological product of operators of the system
in fluctuating fields depending on both sites and times
〈T · · · 〉V =
1
Z[V ]
Tr
(
e−βHT · · · e−V
)
, (2.10)
where
Z[V ] = Tr
(
e−βHT e−V
)
(2.11)
is the partition sum which is a functional of the fluctuating fields. V is the operator of interaction
of the fields with our system. According to general concept GFA operator V is a linear combination
of bose-like X-operators and the mixing coefficients are just the fluctuating fields. In our case V
is taken in a form:
V = v
M′
10 ; M′
10
1′ X
M′
10 ; M′
10
1′ + v
M′
12 ; M′
12
1′ X
M′
12 ; M′
12
1′ + v
M′
10 ; M′
12
1′ X
M′
12 ; M′
10
1′ + v
M′
12 ; M′
10
1′ X
M′
10 ; M′
12
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′ . (2.12)
Herein below a sum over the repeated primed indexes is implied. The rest of the definition corre-
sponds to the standard technique of the temperature GFs [8].
The introduction of the fluctuating fields makes it possible to derive an equation of motion for
electron GF in terms of functional derivatives with respect to these fields. This is a principle of
GFA.
We have to write an equation of motion for each matrix element of GF (2.9). For example for
an element “11” a general form of the equation is:
∂
∂τ1
((
TΨ1σ2
(I1)Ψ
†
2σ2
(I2)
))
V
= δ(τ1 − τ2)
((
T
[
Ψ1σ2
(I1) , Ψ †
2σ2
(I2)
]
+
))
V
+
((
T Ψ̇1σ2
(I1)Ψ
†
2σ2
(I2)
))
V
−
((
T
[
Ψ1σ2
(I1) , V
]
−
Ψ †
2σ2
(I2)
))
V
, (2.13)
where we introduced a short notation
((
T . . .
))
V
= Tr
(
e−βHT . . . e−V
)
. (2.14)
547
Yu.A.Izyumov, N.I.Chaschin, D.S.Alexeev
Having matrix elements for GF constructed on X-operators we are able to write down the
electron GF determined on Fermi operators. To this end, keeping in mind formula (2.3) we must
sum up over indexes M1 and M2 giving the states of atoms. It is reasonable to do this summation
directly in equation (3.1). Then one can obtain the equation for “averaged” GF if we define it by
relation:
Lσ1σ2
12 =
∑
M1M2
θσ1(M1)Lσ1σ2
12 (M1M2)θ
σ2(M2) , (3.7)
where
θσ(M) =
(
ϑσ(M) 0
0 ϑσ(M)
)
(3.8)
with diagonal matrix ϑσ(M) (2.8). Multiplicative character of matrix (2.7), for the electron hopping
on the lattice makes it possible to derive an equation for averaged GF L. We can write this equation
as follows: [(
L
−1
0V
)σ1σ′
1
11′
−
(
Â
σ1σ′
1
1 Φ
)
T 11′ − Â
σ1σ′
1
1 T 11′
]
Lσ′
1σ2
1′2 = δ12
(
Â
σ1σ2
1 Φ
)
. (3.9)
Here all overlined quantities are determined by the relations of type (3.7). For example operator
Â1 and effective hopping T 12 are given by matrices:
Â
σ1σ2
1 =
F̂
σ1σ2
1 Q̂
σ1σ2
1
Q̂
†σ1σ2
1 F̂
†σ1σ2
1
, (3.10)
T 12 = t12
(
T 0
0 −T
)
, (3.11)
(matrix T was defined following the equation (2.8)). In equation (3.10) quantity F̂ is defined by
relation:
F̂
σ1σ2
1 =
∑
M1M2
ϑσ1(M1)F̂σ1σ2
1 (M1M2)ϑ
σ2(M2) , (3.12)
and so on.
An explicit form of matrix elements F̂ , Q̂ and their conjugated ones can be easily written if
necessary. Notice only that for 4 × 4 matrix F̂
F̂
12
= F̂
14
= F̂
21
= F̂
23
= F̂
32
= F̂
34
= F̂
41
= F̂
43
= 0 , (3.13)
and in matrix Q̂ only elements
Q̂
12
, Q̂
21
, Q̂
34
, Q̂
23
, (3.14)
do not vanish (spin indexes σ1 and σ2 are dropped).
Zero order GF L0V in the fluctuating fields is given by a relation
(
L
−1
0V
)σ1σ2
12
= δ12
(
G−1
0 δσ1σ2
− W
σ1σ2
S
−1 (W
02
)σ1σ2S
−1
−(W
20
)σ1σ2S
−1 G̃−1
0 δσ1σ2
− W̃
σ1σ2
S
−1
)
,
S = diag{S + 1, S + 1, S, S}.
4. Solution of equation for the electron Green’s function
Equation (3.9) has a standard form of equation for basic models of strongly correlated systems.
As usual we look for its solution in the multiplicative form:
L = LΠ , (4.1)
550
Sd-model with strong exchange coupling and a metal-insulator phase transition
where a propagator L obeys the Dyson equation
L
−1
= L
−1
0V − Σ , Σ = Σ
′
+ Π Y . (4.2)
Here in the expression for the self-energy part Σ we extracted a term Π Y , which can be cut over
“the interaction line” Y . From the basic equation (3.9) one obtains a coupled equation for the
terminal part Π and uncutable self-energy part Σ
′
of the GF L. We write them down in a form,
where
Π(12) =
(
ÂΦ
)
(12) +
(
Y L
)
(4′3′)Â(14′)Π(3′2) , (4.3)
Σ
′
(12) = −
(
Y L
)
(4′3′)Â(14′)
[
L
−1
0V (3′2) − Σ
′
(3′2)
]
. (4.4)
If we are interested only in the normal GF (first matrix element in (2.9)) which we denote by G,
corresponding to G propagator part of GF is G, and its zero order approximation is G0V , terminal
part is Λ and self-energy part is Σ, then from general equation (4.1)–(4.4) it follows
Gσ1σ2
12 = G
σ1σ′
1
11′ Λ
σ′
1σ2
1′2 ,
(
G
−1)σ1σ2
12
=
(
G
−1
0V
)σ1σ2
12
− Σ
σ1σ2
12 , Σ
σ1σ2
12 = Σ
′σ1σ2
12 −
(
ΛT
)′σ1σ2
12
. (4.5)
Here all quantities G, G, G0V , Λ, Σ, Σ
′
and T are 4 × 4-matrices.
First iteration in equation (4.3) leads to an expression for self-energy in the first and second
order, over hopping respectively. Let us calculate the self-energy part
Σ
σσ
12 =
(
ΛT
)σσ
12
. (4.6)
Taking the iteration expression of the terminal part Π one can write an expression for Σ:
Σ
σσ
12 =
(
Λ
(0)
T
)σσ
t12 +
[
F̂
σσ′
1 T
(
tG
)σ′σ′
12′
F̂
σ′σ
2′ t2′2 − Q̂
σσ′
1 T
(
tG̃
)σ′σ′
12′
Q̂
†σ′σ
2′ t2′2
]
Φ . (4.7)
Each matrix here is a 4 × 4-matrix. After substituting expressions for F and Q one can write a
result for Σ
σσ
12 in a form:
Σ =
λ1 λ1 λ1 λ1
λ2 λ2 λ2 λ2
λ3 λ3 λ3 λ3
λ4 λ4 λ4 λ4
, (4.8)
where
(
λ1
)σσ
12
=
〈
F11
1 + F13
1
〉σσ
t12 +
〈
T
(
F11
1 + F13
1
)σσ′
Aσ′σ
12′
〉
t2′2 −
〈
T
(
Q12
1
)σσ′
Bσ′σ
12′
〉
t2′2 ,
(
λ2
)σσ
12
=
〈
F22
1 + F24
1
〉σσ
t12 +
〈
T
(
F22
1 + F24
1
)σσ′
Aσ′σ
12′
〉
t2′2 −
〈
T
(
Q21
1
)σσ′
Bσ′σ
12′
〉
t2′2 ,
(
λ3
)σσ
12
=
〈
F33
1 + F31
1
〉σσ
t12 +
〈
T
(
F33
1 + F31
1
)σσ′
Aσ′σ
12′
〉
t2′2 −
〈
T
(
Q34
1
)σσ′
Bσ′σ
12′
〉
t2′2 ,
(
λ4
)σσ
12
=
〈
F44
1 + F42
1
〉σσ
t12 +
〈
T
(
F44
1 + F42
1
)σσ′
Aσ′σ
12′
〉
t2′2 −
〈
T
(
Q43
1
)σσ′
Bσ′σ
12′
〉
t2′2 .
Here we introduce a notation:
Aσ′σ
12′ =
(
tG(1)
)σ′σ′
12′
(
F11
2′ + F13
2′
)σ′σ
+
(
tG(2)
)σ′σ′
12′
(
F22
2′ + F24
2′
)σ′σ
+
(
tG(3)
)σ′σ′
12′
(
F33
2′ + F31
2′
)σ′σ
+
(
tG(4)
)σ′σ′
12′
(
F44
2′ + F42
2′
)σ′σ
, (4.9)
Bσ′σ
12′ =
(
tG̃(1)
)σ′σ′
12′
(
Q†12
2′
)σ′σ
+
(
tG̃(2)
)σ′σ′
12′
(
Q†21
2′
)σ′σ
+
(
tG̃(3)
)σ′σ′
12′
(
Q†34
2′
)σ′σ
+
(
tG̃(4)
)σ′σ′
12′
(
Q†43
2′
)σ′σ
. (4.10)
551
Yu.A.Izyumov, N.I.Chaschin, D.S.Alexeev
We see that the first order terms in λi (i = 1, 2, 3, 4) are expressed by averages of X-operators and
the second order terms by T -products of two X-operators, being bose-type GFs.
In equations (4.9) and (4.10) some combinations of electron GFs were introduced, namely
(
G(j)
)σσ
12
=
∑
i
(
G(ij)
)σσ
12
. (4.11)
We shall look for a solution of Dyson equation for propagator G
σ
(k), when the self-energy Σ
is determined by equation 4.7) and equation (4.8). Due to a specific matrix (4.8), the solution can
be written as
G
σ
(k) =
[
1 +
1
ds(k)
Gσ
0 (k)Λ
σ
Tεk
]
Gσ
0 (k) , (4.12)
where
ds(k) = 1 −
(
λ1(k)
zn − E+
+
λ2(k)
zn + E+
+
λ3(k)
zn − E−
+
λ4(k)
zn + E−
)
, (4.13)
zn = iωn +µ. Explicit expressions for matrices G and G are complicated and we do not write them
down.
One can see that in expressions for λi determining the electron GF, some linear combinations of
X-operators may appear which determine the components of the total spin on a site Stot
i = Si+
1
2σ
[1,2]:
Sz
tot =
S∑
M=−S
M
(
XM0 ; M0 + XM2 ; M2
)
+
∑
α
S+ α
2∑
Mα=−(S+ α
2 )
MαXMαα ; Mαα , (4.14)
Sσ
tot =
S∑
M=−S
νσ
S(M)
(
X(M+σ)0 ; M0 + X(M+σ)2 ; M2
)
+
∑
α
S+ α
2∑
Mα=−(S+ α
2 )
νσ
S+ α
2
(Mα)X(Mα+σ)α; Mαα,(4.15)
where
νσ
s (M) =
1√
2
√
(S − σM)(S + σM + 1) , (4.16)
as well as the components for the pseudospin [5]
ρz =
S∑
M=−S
(
XM2 ; M2 − XM0 ; M0
)
, (4.17)
ρ+ =
S∑
M=−S
XM2 ; M0 ; ρ− = (ρ+)† . (4.18)
Along with these combinations in expressions for λi there are terms containing operators of the
type
XMαα ; Mᾱᾱ , (4.19)
which describe the transfers on a site with change of the total spin S + α
2 on S− α
2 . At large values
of sd-exchange integral such transfers give a small contribution to statistical averages. We shall
neglect such terms. Therefore, all off diagonal matrix elements F ij
can be omitted. In the rest of
the expressions for λi, the following combinations of matrix elements appear:
(
F11
1 + F44
1
)σ1σ2
= δ12
1
2
(
1 +
2σ1
2S + 1
Sz
1 tot −
1
2S + 1
ρz
1
)
+ δσ̄1σ2
√
2
2S + 1
Sσ̄1
1 tot , (4.20)
(
F22
1 + F33
1
)σ1σ2
= δ12
1
2
(
1 − 2σ1
2S + 1
Sz
1 tot −
1
2S + 1
ρz
1
)
− δσ̄1σ2
√
2
2S + 1
Sσ̄1
1 tot , (4.21)
552
Sd-model with strong exchange coupling and a metal-insulator phase transition
(
Q12
1 + Q43
1
)σ1σ2
= −
(
Q21
1 + Q34
1
)σ1σ2
= −δσ̄1σ2
1
2S + 1
ρ−1 , (4.22)
(
Q†12
1 + Q†43
1
)σ1σ2
= −
(
Q†21
1 + Q†34
1
)σ1σ2
= δσ̄1σ2
σ1
2S + 1
ρ+
1 . (4.23)
We see that in expressions for λi, in fact, there appear components of total spin and pseudospin.
However, some combinations of X-operators of the type (4.19) still remain. Their role, however,
decreases with parameter J increasing. Further proceedings with an analytical study can be done
in a limit of large atomic spin S.
5. Limit S = ∞
In this limit both eigenvalues of the single site sd-exchange Hamiltonian coincide, so that
|E−| = |E+| = SJ
2 and a condition ds(k) = 0 for poles of the electron GF reduce to an equation of
second order but not fourth order as in general case. In this situation, matrix (4.13) and electron
GF constructed on Fermi operators can be easily written explicitly. Using the identity
gσ(k) = −
〈
Tc1σc†2σ
〉
=
∑
ij
[
Gσ
(k)
]ij
, (5.1)
we can write the result of the calculation in the following form
gσ(k) =
1
F σ(k) − εk
, (5.2)
where
F σ(k) =
z2
n − (I/2)2
zn(λσ
1 + λσ
2 + λσ
3 + λσ
4 ) − I/2(λσ
1 + λσ
4 − λσ
2 − λσ
3 )
, (5.3)
and all quantities λσ
i ≡ λσ
i (k) depend on spin and 4-momentum. We introduced notation I = JS,
which should be finite in the limit S = ∞.
Linear combination of λi in formula (5.3) in the site representation is equal to:
(λσ
1 + λσ
2 + λσ
3 + λσ
4 )σσ
12 = δ12 , (5.4)
(λσ
1 + λσ
4 − λσ
3 − λσ
4 )σσ
12 = σ〈Sz
1 〉t12 +
1
2
[
tG
(1) − tG
(2)]σσ
12′
〈TSz
2′Sz
1 〉 t2′2
+
[
tG
(1) − tG
(2)
]σ̄σ̄
12′
〈
TSσ
2′Sσ̄
1
〉
t2′2 , (5.5)
where we introduced normalized spin operators Sα
1 = 1
S
Sα
1 tot.
In momentum representation, the expressions (5.4) and (5.5) are given
λ1(k) + λ2(k) + λ3(k) + λ4(k) = 1 , (5.6)
λ1(k) + λ4(k) − λ2(k) − λ3(k) = εk
∑
q
εk+qG(k + q)D(q) ≡ ξ(k)εk , (5.7)
where D(q) is a Fourier-component of GF for spin fluctuations
D12 = −〈TS1S2〉 . (5.8)
In paramagnetic phase, electron GF g(k) and a function G(k), involved in (5.7) can be written in
a form
g(k) =
zn − ξ(k)I/2
z2
n − (I/2)2 − [zn − ξ(k)I/2]εk
. (5.9)
G(k) =
I
z2
n − (I/2)2 − [zn − ξ(k)I/2]εk
, (5.10)
553
Yu.A.Izyumov, N.I.Chaschin, D.S.Alexeev
where k = {k, iωn}. G(k) denotes a Fourier transform of the quantity
[
tG
(1) − tG
(2)
]
involved in
expression (5.5).
Using the spectral representation for Fermi-like GFs g(k) and G(k) and Bose-like GF D(k) we
present the expression (5.7) for ξ(k) in a form
ξ(k) = − 1
N
∑
k′
εk′
∫
ω′
π
∫
Ω′
π
ImG(k′, ω′)ImD(k′ − k,Ω′)
iωn − ω′ + Ω′
(f [ω′] + n[Ω′]) , (5.11)
where f [ω′] and n[Ω′] are functions of Fermi and Bose, respectively. After analytical continuation
iωn → ω + iδ we obtain an equation for ξk(ω), which is an integral one over frequency and
momentum. It should be solved together with the equation for chemical potential
n = 2
1
N
∑
k′
εk′
∫
ω′
[
− 1
π
Img(k′, ω′)
]
f [ω′]. (5.12)
Now we do an estimation of quantity ξ(k) under the following approximation: consider limit of
static fluctuations
ImD(q,Ω) ≈ π aqδ(Ω) (5.13)
and neglect q-dependence of the spectral density aq ≈ a. Then for temperature T = 0 from (5.11)
it follows
ξ(ω) = −aI
W
ε∫
−W
2
dε
ε
ε2 + I2
1
ω + E1(ε) − E1(ε)
, (5.14)
where we use a rectangular form of the bare band and mean field approximation for electron GF
g(k), equation (5.9) with ξ(k) = 0.
g(q, ω) =
1
2
(
1 − ε
k
Qk
)
1
ω + µ − E1(k)
+
1
2
(
1 +
ε
k
Qk
)
1
ω + µ − E2(k)
, (5.15)
with Qk = E2(k) − E1(k) , and
E1,2(k) =
1
2
(
ε
k
∓
√
ε2
k
+ I2
)
, (5.16)
determine two branches of quasiparticle spectrum.
When electron density n 6 1 chemical potential µ lies in the lower band, and ε means energy
εq corresponding to fixed µ; ε obeys equaion E1(ε) = µ. One can see that ξ(ω) has logarithmic
singularity at ω = 0 for account of electrons near Fermi-level.
In conclusion, we notice that, contrary to paper [4], here we developed an approximation for
the sd-model with strong sd-exchange coupling beyond the mean-field theory to take into account
fluctuations in the system. Electron GF (5.9) contains contribution of magnetic fluctuations via
ξ(k)-quantity. Further analysis of the problem reduces now to the calculation ξ(k) as a function of
momentum and frequency.
Preliminary analysis of equations (5.9)–(5.12) shows that at half filling a gap between two
branches of quasiparticle spectrum vanishes at large enough exchange parameter I, and an insulator-
metal phase transition occurs. A detailed numerical analysis of equations (5.9)–(5.12) at half filling
and beyond it will be done elsewhere.
This work was supported by Russian Foundation for Support of Scientific School, grant NS–
4640.2006.2.
References
1. Erukhimov M.S., Ovchinnikov S.G., Phys. Stat. Sol. (b), 1984, 123, 105.
2. Erukhimov M.S., Ovchinnikov S.G., Physica C, 1989, 157, 209.
554
Sd-model with strong exchange coupling and a metal-insulator phase transition
3. Nagaev E.L. Physics of Magnetic Semiconductors. Nauka, Moscow, 1979.
4. Izyumov Yu.A., Chaschin N.I., Alexeev D.S., Condensed Matter Physics, 2005, 123, No. 4, 801.
5. Izyumov Yu.A., Chaschin N.I., Alexeev D.S. Theory of Strongly Correlated Systems. The generating
Functional Approach, Regular and Chaotic Dynamics. Moscow-Izhevsk, 2006.
6. Anokhin A.O., Irkhin V.Yu., Phys. Stat. Sol. (b), 1991, 165, 129.
7. Anokhin A.O., Irkhin V.Yu., Katsnelson M.I., J. Phys.: Condens. Matter, 1991, 3, 1475.
8. Abrikosov A.A., Gorkov L.P., Dzyaloshinskii I.E. Methods of Quantum Field Theory in Statistical
Physics. Dover, New York, 1975.
9. Izyumov Yu.A., Chaschin N.I., Alexeev D.S., Mancini F., Eur. Phys. J. B, 2005, 45, 69.
Sd-модель з сильною обмiнною взаємодiєю та фазовим
переходом метал-дiелектрик
Ю.О.Iзюмов, Н.I.Чащiн, Д.С.Алексеєв
Iнститут фiзики металiв, вул. С. Ковалевської, 18, Єкатеринбург, Росiя
Отримано 18 травня 2006 р.б в остаточному виглядi – 14 червня 2006 р.
Sd-обмiнна модель (модель Кондо гратки) сформульована для сильної sd-взаємодiї в рамках технi-
ки X-операторiв та пiдходу твiрного функцiоналу. X-оператори побудованi на базисi точних власних
функцiй одновузлового sd-обмiнного гамiльтонiану. Таке представлення дозволяє розвинути теорiю
збурень поблизу атомного рiвня. Представлення локаторного типу було отримане для електрон-
ної функцiї Грiна. Власна енергiя електронiв включає взаємодiю електронiв та спiнових флуктуа-
цiй. Отримано iнтегральне рiвняння для власної енергiї в границi безмежних локалiзованих спiнiв.
Розв’язок цього рiвняння в статичному наближеннi для спiнових флуктуацiй веде до структури еле-
ктронної функцiї Грiна, що виявляє фазовий перехiд метал-дiелектрик. Цей перехiд є схожим до
переходу в моделi Хаббарда при половинному заповненнi.
Ключовi слова: модель Кондо гратки, сильноскорельованi електроннi системи, фазовi переходи
метал-дiелектрик
PACS: 71.10.-w, 71.10.Fd, 71.27.+a
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