tJ -model in terms of equations with variational derivatives
For a tJ -model in the X -operators representation a generating functional of the field describing fluctuations of matrix elements of electron hopping on a lattice is presented. The first order functional derivative with respect to this field determines the electron Green function, while the seco...
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Інститут фізики конденсованих систем НАН України
1998
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| Назва видання: | Condensed Matter Physics |
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| Цитувати: | tJ -model in terms of equations with variational derivatives / Yu.A. Izyumov, N.I. Chashchin // Condensed Matter Physics. — 1998. — Т. 1, № 1(13). — С. 41-56. — Бібліогр.: 7 назв. — англ. |
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irk-123456789-1186312017-05-31T03:04:29Z tJ -model in terms of equations with variational derivatives Izyumov, Yu.A. Chashchin, N.I. For a tJ -model in the X -operators representation a generating functional of the field describing fluctuations of matrix elements of electron hopping on a lattice is presented. The first order functional derivative with respect to this field determines the electron Green function, while the second order derivatives determine the boson Green functions of collective excitations in the system. Thus, the Kadanoff-Baym approach in the theory of fermi system with a weak Coulomb interaction is generalized on the opposite limit of systems with strong correlations. A chain of equations for different order variational derivatives were obtained, and a method was suggested based on iterations over the parameters of a tJ -model: the hopping matrix element and the exchange integral. This approach corresponds to a self-consistent Born approximation, not for the effective but for the original Hamiltonian. A scheme of calculation of the dynamical spin susceptibility is analyzed with self-consistent corrections of the first and second order. Connection of this approach with the diagram technique for X -operators is discussed. Для tJ моделі в представленні X -операторів введено узагальнений функціонал поля, що описує флуктуації матричних елементів електронних перескоків на гратці. Функціональна похідна першого порядку по цьому полю визначає електронну функцію Гріна, тоді як похідні другого порядку визначають бозонні функції Гріна колективних збуджень в системі. Таким чином узагальнено підхід Каданова- Бейма в теорії фермі-систем з слабою кулонівською взаємодією на протилежний випадок систем з сильними кореляціями. Отримано ланцюжок рівнянь для варіаційних похідних різного порядку і запропоновано метод, що базується на ітераціях за параметрами tJ моделі: матричним елементом перескоку і обмінним інтегралом. Цей підхід відповідає самоузгодженому наближенню Борна, але не для ефективного, а для вихідного гамільтоніану. Аналізується схема розрахунку динамічної спінової сприйнятливості з самоузгодженими поправками першого і другого порядку. Обговорюється зв’язок цього підходу з діаграмною технікою для X -операторів. 1998 Article tJ -model in terms of equations with variational derivatives / Yu.A. Izyumov, N.I. Chashchin // Condensed Matter Physics. — 1998. — Т. 1, № 1(13). — С. 41-56. — Бібліогр.: 7 назв. — англ. 1607-324X PACS: 71.27.+a, 74.90.+n DOI:10.5488/CMP.1.1.41 http://dspace.nbuv.gov.ua/handle/123456789/118631 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 |
For a tJ -model in the X -operators representation a generating functional
of the field describing fluctuations of matrix elements of electron hopping
on a lattice is presented. The first order functional derivative with respect
to this field determines the electron Green function, while the second order
derivatives determine the boson Green functions of collective excitations
in the system. Thus, the Kadanoff-Baym approach in the theory of fermi
system with a weak Coulomb interaction is generalized on the opposite
limit of systems with strong correlations. A chain of equations for different
order variational derivatives were obtained, and a method was suggested
based on iterations over the parameters of a tJ -model: the hopping matrix
element and the exchange integral. This approach corresponds to a
self-consistent Born approximation, not for the effective but for the original
Hamiltonian. A scheme of calculation of the dynamical spin susceptibility
is analyzed with self-consistent corrections of the first and second order.
Connection of this approach with the diagram technique for X -operators
is discussed. |
| format |
Article |
| author |
Izyumov, Yu.A. Chashchin, N.I. |
| spellingShingle |
Izyumov, Yu.A. Chashchin, N.I. tJ -model in terms of equations with variational derivatives Condensed Matter Physics |
| author_facet |
Izyumov, Yu.A. Chashchin, N.I. |
| author_sort |
Izyumov, Yu.A. |
| title |
tJ -model in terms of equations with variational derivatives |
| title_short |
tJ -model in terms of equations with variational derivatives |
| title_full |
tJ -model in terms of equations with variational derivatives |
| title_fullStr |
tJ -model in terms of equations with variational derivatives |
| title_full_unstemmed |
tJ -model in terms of equations with variational derivatives |
| title_sort |
tj -model in terms of equations with variational derivatives |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
1998 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/118631 |
| citation_txt |
tJ -model in terms of equations with variational derivatives / Yu.A. Izyumov, N.I. Chashchin // Condensed Matter Physics. — 1998. — Т. 1, № 1(13). — С. 41-56. — Бібліогр.: 7 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT izyumovyua tjmodelintermsofequationswithvariationalderivatives AT chashchinni tjmodelintermsofequationswithvariationalderivatives |
| first_indexed |
2025-07-08T14:21:13Z |
| last_indexed |
2025-07-08T14:21:13Z |
| _version_ |
1837088872880668672 |
| fulltext |
Condensed Matter Physics, 1998, Vol. 1, No 1(13), p. 41–56
tJ -model in terms of equations with
variational derivatives
Yu.A.Izyumov, N.I.Chashchin
Institute for Metal Physics of the Russian Academy of Sciences,
Ural Division, 18 S.Kovalevskaya St., 620219 Ekaterinburg, Russia
Received June 17, 1998
For a tJ -model in the X -operators representation a generating functional
of the field describing fluctuations of matrix elements of electron hopping
on a lattice is presented. The first order functional derivative with respect
to this field determines the electron Green function, while the second order
derivatives determine the boson Green functions of collective excitations
in the system. Thus, the Kadanoff-Baym approach in the theory of fermi
system with a weak Coulomb interaction is generalized on the opposite
limit of systems with strong correlations. A chain of equations for different
order variational derivatives were obtained, and a method was suggested
based on iterations over the parameters of a tJ -model: the hopping ma-
trix element and the exchange integral. This approach corresponds to a
self-consistent Born approximation, not for the effective but for the original
Hamiltonian. A scheme of calculation of the dynamical spin susceptibility
is analyzed with self-consistent corrections of the first and second order.
Connection of this approach with the diagram technique for X -operators
is discussed.
Key words: tJ -model, Hubbard operators, Green‘s functions,
antiferromagnetism
PACS: 71.27.+a, 74.90.+n
1. Introduction
A tJ-model is the basic working model in the theory of strongly correlated
electron systems. It is convenient for the study of an interaction of charge and
spin degrees of freedom, because the model describes the correlated motion of
electrons on the lattice. The model is given by the Hamiltonian (see [1]):
H =
∑
ijσ
tij(1− niσ̄)C
†
iσCjσ(1− njσ̄) +
∑
ij
Jij
(
SiSj −
1
4
ninj
)
. (1.1)
Here Ciσ(C
†
iσ) is an operator of annihilation (creation) of an electron on site i with
spin σ, Si — operator of spin, and ni — operator of the electron number on a
c© Yu.A.Izyumov, N.I.Chashchin 41
Yu.A.Izyumov, N.I.Chashchin
site. Hopping matrix elements tij and exchange integrals Jij are usually taken in
the nearest neighbours approximations, so the model contains only two energy
parameters: t and J .
At half-filling, when the number of electrons on a site is n = 1, the model (1.1)
reduces to the Heisenberg model and has a dielectric antiferromagnetic ground
state. When a deviation from half-filling takes place then some concentration of
holes δ = 1 − n appears. They strongly interact with the magnetic order and lo-
cally deform it. As a result, a compound quasiparticle (magnetic polaron) appears,
being a carrier of an electron charge in the system. A mathematical description
of this situation is achieved by the transformation of Hamiltonian (1.1) into an
effective Hamiltonian of a hole-magnon interaction. The self-consistent Born ap-
proximation (SCBA) is an approximation leading to a magnetic polaron picture
inside the antiferromagnetic phase. The last one exists when δ < δc , where crit-
ical concentration δc ≪ 1 [1]. It is well known that in high-Tc-compounds the
superconducting state appears outside the antiferromagnetic phase when δ > δc ,
therefore, the question about applicability of the magnetic polaron picture for this
concentration region is left open. For this reason a lot of new attempts are made
to study the properties of the model (1.1) at δ > δc [1].
We find rather perspective here an approach on the basis of equations with vari-
ational derivatives in the spirit of the Kadanoff-Baym scheme [2]. At the beginning
it was applied to the description of usual Fermi systems with a weak Coulomb in-
teraction. Our aim is to develop a similar approach for the opposite case of systems
with a strong Coulomb interaction, for example, for Hamiltonian (1.1) describing
the correlated motion of electrons. The most convenient way to do that is to use
the Hamiltonian of the tJ-model in terms of the Hubbard X-operators. Then, by
the analogy with usual Fermi operators, one can introduce fluctuating fields, cor-
responding to the hopping term in the Hamiltonian, in contrast to Kadanoff-Baym
who introduced fluctuating fields of the potential interaction. As the result of such
an approach, equations for generating functional Z and variational derivatives of
Z over these fields are derived. These equations are convenient for iterations with
respect to parameters t and J , in contrast to Kadanoff-Baym equations convenient
for iterations with respect to Coulomb interaction U .
The obtained equations allow one to develop a scheme of the self-consistent
Born approximation (SCBA), not for the effective Hamiltonian, adopted for the
description of the antiferromagnetic phase, but for the original Hamiltonian (1.1)
and for the paramagnetic state as a ground state of the system.
The first attempt of such an approach was given by us in [3].
2. Generating potential in terms of X -operators
Because of the projection factors (1− niσ) in the hopping term of the Hamil-
tonian, it is convenient to rewrite it in terms of X-operators:
Xσ0
i = (1− niσ̄)C
†
iσ , Xσσ
i = C†
iσCiσ , Xσσ̄
i = C†
iσCiσ̄ . (2.1)
42
tJ -model in terms of equations with variational derivatives
Here the first X-operator is Fermi-like (f-type), and the others are Bose-like (b-
type). One can get then [4]
H =
∑
iσ
εσX
σσ
i +
∑
ijσ
tσijX
σ0
i X0σ
j +
1
2
∑
ijσ
Jij(X
σ̄σ
i Xσσ̄
j −Xσσ
i X σ̄σ̄
j ) . (2.2)
We included external magnetic field h, that is why term εσ = −σh/2 appeared.
We also introduced in a formal way hopping matrix element tij by spin index σ.
We shall denote by figures complex indices including site i and imaginary time
τ , so that 1 = (iτ), etc. Equations of motion for f- and b-operators are written in
the form:
−
dX0σ
1
dτ1
= [X0σ
1 , H − µN ] = (εσ − µ)X0σ
1 + (1− niσ̄)t
σ
11′X
0σ
1′ +
+X σ̄σ
1 tσ̄11′X
0σ̄
1′ + J11′(X
0σ̄
1 X σ̄σ
1′ −Xσσ
1 X σ̄σ̄
1′ ) , (2.3)
−
dXσσ̄
1
dτ1
= [Xσσ̄
1 , H − µN ] = (εσ̄ − εσ)X
σσ̄
1 − (Xσ0tσ)(1)X0σ̄
1 +
+Xσ0
1 (tσ̄X0σ̄)(1) + (Xσσ̄J)(1)σm1 −Xσσ̄
1 (Jσm)(1) . (2.4)
As usual, the summation (integrating) is implied over repeated indices because
we introduced the quantities
t11′ = δ(τ − τ ′)tii′ , J11′ = δ(τ − τ ′)Jii′ .
Here and further we use a notation for matrix products, for example,
(tX0σ)(1) ≡ t11′X
0σ
1′ , . . .
Finally, by niσ and mi we denote the number of electrons on a site and local
magnetization
n1σ = Xσσ
1 , σm1 = n1σ − n1σ̄ .
Instead of quantities n1σ it is convenient to use F σ
1 = 1 − n1σ̄, being an anti-
commutator of two f-operators
[X0σ
1 , Xσ0
1 ]+ = F σ
1 . (2.5)
Now we introduce the generating potential
Z[tσ] = Sp
{
e−β(H−µN)T̂e−F [tσ ]
}
, (2.6)
F [tσ] =
∑
σ
Xσ0
1′ t
σ
1′2′X
0σ
2′ , (2.7)
where ησ1′2′ is an auxiliary external field depending on thermodynamic time and
spin. Here β = 1/T and T̂ is a time-ordering operator. Since F [ησ] coincides in
43
Yu.A.Izyumov, N.I.Chashchin
the form with the hopping term in the Hamiltonian, the quantity tσ1′2′ is to be
considered as a fluctuating field of the electron hopping on the lattice.
The quantity Z[tσ] is useful for determining the Green functions of the system,
as its variational derivatives over tσ12
Gσ(12) = −
〈
T̂X0σ
1 Xσ0
2
〉
= −
1
Z
δZ
δtσ12
, (2.8)
Dσ
1 (12) =
〈
T̂Xσσ̄
1 X σ̄σ
2
〉
= −
1
Z
δ2Z
δtσ̄12δt
σ
21
, (2.9)
Dσσ′
0 (12) =
〈
T̂n1σn2σ′
〉
=
1
Z
δ2Z
δtσ11δt
σ′
22
. (2.10)
It is clear from these definitions that Gσ is the Green function of itinerant
electrons, and two others are the Green functions of collective Bose-like excitations:
magnons and plasmons. The Bose-like Green functions are actually two-particle
functions because b-operator may be presented as a product of f-operators: Xσσ̄
1 =
Xσ0
1 X0σ̄
1 , n1σ = Xσ0
1 X0σ
1 .
For the generating functional one can derive the equation of motion with vari-
ational derivatives:
Kσ(11
′)
δZ
δtσ1′2
= −δ12 〈F
σ
1 〉Z + γσ(1; 1
′3′)
δ2Z
δtσ1′2δt
σ̄
3′3′
+ γσ̄(1; 1
′3′)
δ2Z
δtσ3′2δt
σ̄
1′3′
(2.11)
or in the conjugated form
δZ
δtσ12′
Kσ(2
′2) = −δ12 〈F
σ
1 〉Z +
δ2Z
δtσ12′δt
σ̄
3′3′
γ̃σ(3
′2′; 2) +
δ2Z
δtσ13′δt
σ̄
3′2′
γ̃σ̄(3
′2′; 2). (2.12)
Here Kσ is a differential operator, γσ and γ̃σ are matrix elements of the Hamilto-
nian.
Kσ(12) = −
(
∂
∂τ1
+ εσ
)
δ(1− 2)− tσ12 , (2.13)
γσ(1; 23) = tσ12δ13 + δ12J13
γ̃σ(32; 1) = δ31t
σ
21 + J31δ21
. (2.14)
One must add the obvious relation to equations (2.11) and (2.12):
〈F σ
1 〉 = 1 +
1
Z
δZ
δtσ̄11
. (2.15)
Equation (2.11) is the basic equation corresponding to the Kadanoff-Baym
method applied to a strongly correlated system. In contrast to the theory of a
usual Fermi-system here variational derivatives are taken over the fluctuating field
of the hopping term but not over the potential one. For this reason equation (2.11)
is convenient for iterations with respect to parameters t and J , while in the usual
44
tJ -model in terms of equations with variational derivatives
Fermi-system variational derivatives are taken over the parameters of an electron
interaction.
The basic equation (2.11) connects the first derivative of Z with the second one.
To find an equation for the second derivative it is necessary to take a variational
derivative of equation (2.11) over tσ̄34. Then one obtains an equation connecting
the second derivative with the third one:
Kσ(11
′)
δ2Z
δtσ1′2δt
σ̄
34
= −δ12
(
δZ
δtσ̄34
+
δ2Z
δtσ̄11δt
σ̄
34
)
+ δ14
δ2Z
δtσ12δt
σ̄
31
+ (2.16)
+γσ(1; 1
′3′)
δ3Z
δtσ1′2δt
σ̄
3′3′δt
σ̄
34
+ γσ̄(1; 1
′3′)
δ3Z
δtσ3′2δt
σ̄
1′3′δt
σ̄
34
.
We write the following equation of the infinite chain obtained by the differen-
tiation of the previous equation with respect to tσ̄56:
Kσ(11
′)
δ3Z
δtσ1′2δt
σ̄
34δt
σ̄
56
= −δ12
(
δ2Z
δtσ̄34δt
σ̄
56
+
δ3Z
δtσ̄11δt
σ̄
34δt
σ̄
56
)
+
+δ14
δ3Z
δtσ12δt
σ̄
34δt
σ̄
56
+ δ16
δ3Z
δtσ12δt
σ̄
34δt
σ̄
56
+
+γσ(1; 1
′3′)
δ4Z
δtσ1′2δt
σ̄
3′3′δt
σ̄
34δt
σ̄
56
+ γσ̄(1; 1
′3′)
δ4Z
δtσ3′2δt
σ̄
1′3′δt
σ̄
34δt
σ̄
56
. (2.17)
After differentiation in these equations one has to put variable tσ12 equal to
matrix element t12 standing in the hopping term of the Hamiltonian.
In figures 1 and 2 a graph representation of the basic equation is given. A four-
Figure 1. Graphic elements.
tail diagram denoting a second order variational derivative is not a vertex part
but a two-particle Green function. We shall see later that the zero approximation
over γσ contains a nonconnected part, generating in the equation the first order
graphs, corresponding in the usual Fermi system theory to Hartree-Fock terms.
The connected part of the zero approximation generates new type graphs absent
in the Fermi system theory. Corrections of the first and next orders generate vertex
parts, more complicated in comparison with the usual Fermi systems [5,6].
45
Yu.A.Izyumov, N.I.Chashchin
Figure 2. Graphic representation of the basic equation (2.11).
3. Calculation of variational derivatives
In accordance with the graph representation for the second derivative shown
in figure 1, we shall look for the form:
1
Z
δ2Z
δtσ12δt
σ̄
34
=
δΦ
δtσ11′
δΦ
δtσ̄33′
Γ̂2(σ1
′2′; σ̄3′4′)
δΦ
δtσ2′2
δΦ
δtσ̄4′4
. (3.1)
The quantity δΦ/δt is an electron Green function (Z = eiφ), while Γ̂2 is to
be considered a vertex part. In the theory of the usual Fermi system Γ̂2 should
be a real vertex part of an electron-electron interaction. In the case of strongly
correlated electrons the situation is different and in representation (3.1) Γ̂2 should
be an operator. It acts on the Green function standing by the left- and the right-
hand side and can transform them into other Green functions.
The representation (3.1) is, however, convenient because it opens a way to
calculate the second and higher derivatives. For this purpose equation (3.1) has to
be multiplied by the left- and the right-hand side operator quantities (2.13). Then
it is written:
Kσ(11
′)Kσ̄(33
′)
1
Z
δ2Z
δtσ1′2′δt
σ̄
3′4′
Kσ(2
′2)Kσ̄(4
′4) =
(
Kσ
δΦ
δtσ
)
(11′)
(
Kσ̄
δΦ
δtσ
)
(33′)Γ̂2(σ1
′2′; σ̄3′4′)
(
δΦ
δtσ
Kσ
)
(2′2)
(
δΦ
δtσ̄
Kσ̄
)
(4′4). (3.2)
In the right-hand side of this relation we may use equations (2.11) and (2.12),
and in the left-hand side – equation (2.16) (and a conjugated one). It allows one
to find Γ̂2 as a series in powers of γσ and γσ̄. After rather cumbersome calculations
we find quantities Γ̂2 for two second variation derivatives with equal and opposite
spin indices. In the zero approximation
Γ̂ 0
2 (σ12; σ34) =
K0
σ(12)K
0
σ(34)−K0
σ(14)K
0
σ(32)
〈F σ
1 〉 〈F
σ
2 〉 〈F
σ
3 〉 〈F
σ
4 〉
[dσ̄σ̄0 (13) + 〈F σ
1 〉 〈F
σ
3 〉] ,
(3.3)
46
tJ -model in terms of equations with variational derivatives
Γ̂ 0
2 (σ12; σ̄34) = δ13δ34
K0
σ(12)
〈F σ
1 〉 〈F
σ
2 〉 〈F
σ̄
1 〉
+ δ12δ23
K0
σ̄(34)
〈F σ
4 〉 〈F
σ̄
3 〉 〈F
σ̄
4 〉
+
+
K0
σ̄(14)K
0
σ(32)D
σ
1 (31)
〈F σ
1 〉 〈F
σ
2 〉 〈F
σ̄
3 〉 〈F
σ̄
4 〉
+
K0
σ(12)K
0
σ̄(34) [d
σ̄σ
0 (13) + 〈F σ
1 〉 〈F
σ̄
3 〉]
〈F σ
1 〉 〈F
σ
2 〉 〈F
σ̄
3 〉 〈F
σ̄
4 〉
. (3.4)
Both equations contain Bose-like Green functions for magnons and plasmons,
however, now instead of quantities (2.10) we have
dσσ
′
0 (12) = Dσσ′
0 (12)− 〈F σ
1 〉
〈
F σ′
2
〉
=
〈
T̂[n1σ − 〈n1σ〉][n2σ′ − 〈n2σ′〉]
〉
.
We see that expressions (3.3) and (3.4) contain operators K0
σ which act in
relation (3.1) on electron Green functions. In this way the expression for the second
derivatives of Z splits into a number of terms containing not four electron Green
functions but a smaller number of them, three or two. We have used the fact that
at the zero approximation, according to equation (2.11), K0
σ(δΦ/δt
σ) is just 〈F σ〉.
As a result, the second derivative of Z can be written in a graph form (figures 3
and 4). When writing analytical expressions, one has to keep in mind the following
Figure 3. Graphic representation of
1
Z
δ2Z
δtσ12δt
σ
34
in zero approximation. Dasked
lines denot plasmon Green functions (2.10).
rule: a complex vertex creates a numerical factor corresponding to each outgoing
and ingoing electron line. For example, to the first connected graphs in figure
3 there corresponds factor 〈F σ
1 〉
−1 〈F σ
4 〉
−1. Coefficients Aσ and Bσ for the triple
graphs in figure 4 are equal to
Aσ
1 = 〈F σ
1 〉+ 〈F σ̄
1 〉 −
1
2
σ〈m1〉 , Bσ
2 = 〈F σ̄
2 〉 − 〈F σ
2 〉+
1
2
σ〈m1〉 .
47
Yu.A.Izyumov, N.I.Chashchin
Besides, according to the mentioned rule, to these two graphs additional factors
〈F σ̄〉−2 〈F σ〉−1 or 〈F σ〉−1 〈F σ〉−2 should be prescribed.
Figure 4. Graphic representation of
1
Z
δ2Z
δtσ12δt
σ̄
34
in zero approximation. Dasked
lines with an arrow denotes the magnon Green function (2.9).
The third order variational derivative is calculated in the same way. It is neces-
sary to start from the relation of type (3.1) not with four but six factors δΦ/δt. The
complete expression is too complicated to write it down. It contains nonconnected
diagrams corresponding to the product of the second derivative and the first one.
Among the connected diagrams there are graphs constructed only from electron
lines (five lines) and graphs with one boson line and four electron lines converging
to one point (as in the pure electron graphs in figure 4). Finally, there are peculiar
graphs containing a structure corresponding to a six-tailed one with pairs of joint
electron lines (a flower with three clovers). Such type of graphs is shown in figure
5. One can check that a complete set of graphs for the third derivative provides
the necessary symmetry arising from the possibility to change the order of differ-
entiation. Besides, the symmetry is achieved according to the permutation of pair
indices (3,5) and (4,6).
In principle, in the same way it would be possible to obtain a graphical repre-
sentation for the fourth derivative (eight-tailed) of the zero approximation.
48
tJ -model in terms of equations with variational derivatives
Figure 5. The third order variational derivative in zero approximation.
4. The magnon Green function
The magnon Green function can be found if the second derivative δ2Z/δtσ̄12δt
σ
34
is calculated in some approximation. According to definition (2.9), it is enough to
equate indices 4 = 1 and 3 = 2. We calculate first Dσ
1 (12) at the zero approxima-
tion. In the left- and the right-hand sides of the graph relation in figure 4 we make
these indices equal to each other. Then, an equation for Dσ
1 (12) appears, which we
write in the analytical form:
[
1−
(
1
2
∑
σ1
〈nσ1
1 〉
〈F σ1
1 〉
)(
1
2
∑
σ2
〈nσ2
2 〉
〈F σ2
2 〉
)]
Dσ
1 (12) =
−Gσ̄(12)Gσ(21)
[
1 +
1
2
(
Aσ̄
1 〈n
σ
1 〉
〈F σ̄
1 〉 〈F
σ
1 〉
2 +
Aσ
2 〈n
σ̄
2 〉
〈F σ
2 〉 〈F
σ̄
2 〉
2 +
Bσ
1 〈n
σ̄
1 〉
〈F σ
1 〉 〈F
σ̄
1 〉
2 +
Bσ̄
2 〈n
σ
2 〉
〈F σ̄
2 〉 〈F
σ
2 〉
2
)]
−
1
4
∑
σ′
1〈
F σ′
1
〉 〈
F σ̄′
2
〉Gσ̄(12)d
σ̄σ
0 (21)Gσ(21). (4.1)
One can check here the rules of writing analytical expressions.
For the paramagnetic phase we have from here:
Dσ
1 (12) = −
1 + n2/4
1− n
G(12)G(21) . (4.2)
We neglect the last term in expression (4.2) which changes only a numerical
coefficient. If in this expression sites 1 and 2 are put to be the nearest neighbours
49
Yu.A.Izyumov, N.I.Chashchin
and also time τ1 and τ2 to be equal, we obtain a relation between the spin and
electron correlators coinciding with the one obtained in [4].
To include corrections of the first order in the magnon Green function we
must calculate the second derivative with the accuracy up to the first order with
respect to t and J . Then, making the indices equal in the same way as in the zero
approximation, we come to the equation, presented in figure 6. Its solution gives
Figure 6. Equation for magnon Green function.
the magnon Green function of the form:
Dσ
1 (q) =
−Π(q)
1− n−
n
2
[Q(q) + Λ(q)]− J(q)Π(q)
, (4.3)
where
{Π(q), Q(q), Λ(q)} =
∑
k
{1, ε(k), ε(k− q)}G(k− q)G(k) . (4.4)
corresponds to an electron loop and the loops with an inserted wavy line. Such
expressions appeared earlier in the diagram technique with X-operators [7].
Result (4.3) is consistent with the expression obtained by us [3], with a differ-
ence in numerical coefficients. At q = 0 the last term in the denominator can be ne-
glected, thus, the ferromagnetic instability of the system is determined by the hop-
ping term in the Hamiltonian. In contrast, at q = Q = (π, π, . . .)/a Q(q)+Λ(q) =
0 the antiferromagnetic instability is determined by the exchange term. In [3] we
showed that in the Hubbard-1 approximation for electrons Π(Q, 0) ∼ (1− n) and
antiferromagnetic instability occurs only when J ∼ t, which is not consistent with
the known idea [1] that critical value J/t for the appearance of antiferromagnetic
order should be small, as half-filling (1 − n) is approached. It means that near
n = 1 for the analysis of antiferromagnetic instability it is probably necessary to
50
tJ -model in terms of equations with variational derivatives
go beyond the mean field approximation for electrons. The other possibility is to
take into account the second order corrections for t and J . In the next chapter we
will give a preliminary analysis of these corrections without explicit calculations.
5. On the second order corrections in the magnon Green func-
tion
It would be difficult to look for these corrections by the method of [3] because
one should know self-energy of electrons up to the third order. In the present
approach it is sufficient to have corrections up to only the second order for the
second derivative. We need for this the forth order derivative (eight-tailed) in the
zero approximation. As we saw earlier, the number of terms increases very fast
with the order of the derivative, and a selection of actual diagrams is necessary. In
our case (nearly half-filling) this selection might be done on the basis of parameter
1− n≪ 1.
Let us look for the magnon Green function (4.2) of the zero approximation.
Factor 1 − n in the denominator does not yet mean a singularity because it may
be cancelled by the numerator. However, it is necessary to pay attention to all the
terms containing this factor in the denominator. Notice that this factor appeared
due to the coefficient in the left-hand side of equation (4.1). We try to search for
such factors in expressions for the third and fourth derivatives, however, we shall
use now another method based on the Wick theorem for X-operators [5,7].
We start from the second order derivative written in the form:
1
Z
δ2Z
δtσ12δt
σ̄
34
=
〈
T̂ψσ(1)ψ
†
σ(2)ψσ̄(3)ψ
†
σ̄(4)
〉
, (5.1)
where ψσ(1) ≡ X0σ
1 is a Fermi-like operator. When averaging in (4.4) with the
Hamiltonian of the zero approximation, the average of T-product would be reduced
to the pair-averages by the procedure based on the Wick theorem. We should take
all the possible systems of pairing. First, consider the systems with the pairing of
only ψ and ψ† operators, for example:
〈T̂ψσ(1)ψ
†
σ(2)ψσ̄(3)ψ
†
σ̄(4)
✛
✲
〉 = −Gσ̄(32)Gσ(14)〈T̂X
σσ̄
2 X σ̄σ
4 〉 . (5.2)
Here Gσ(12) is the fermion Green function of the zero approximation. Our
approximation now is the one, in which we take only ψ-operators pairing and in
expressions of type (5.2) the Green functions are replaced by the exact ones. They
are related to the electron Green functions by the identity
Gσ(12) = Gσ(12) 〈F
σ
2 〉 . (5.3)
Taking into account in (5.1) only two systems of pairing, we can write
1
Z
δ2Z
δtσ12δt
σ̄
34
≈
[
Gσ(12)Gσ̄(34)
〈
T̂F σ
2 F
σ̄
4
〉
− Gσ̄(32)Gσ(14)
〈
T̂Xσσ̄
2 X σ̄σ
4
〉] 1
〈F σ
2 〉 〈F
σ̄
4 〉
.
(5.4)
51
Yu.A.Izyumov, N.I.Chashchin
In this approximate relation we put indices 4=1 and 3=2, then we have an equation
for the magnon Green function:
[
1−
〈nσ
1 〉
〈F σ̄
1 〉
〈nσ̄
2 〉
〈F σ
2 〉
]
Dσ
1 (12) = −Gσ̄(12)Gσ(21)
1 + dσ̄σ0 (21)
〈F σ̄
1 〉 〈F
σ
2 〉
. (5.5)
Here we use the value for the electron Green function with coincided arguments:
Gσ(11) = 〈Xσσ
1 〉 = 〈nσ
1 〉 .
For the paramagnetic phase the coefficient in the left-hand side is proportional
to 1 − n, and we come to the result consistent with (4.2). Thus, the decoupling
of the Wick theorem type leads for the second derivative to the same result as on
the basis of representation (3.1).
Consider now the third order derivative
1
Z
δ3Z
δtσ12δt
σ̄
34δt
σ̄
56
=
〈
T̂ψσ(1)ψ
†
σ(2)ψσ̄(3)ψ
†
σ̄(4)ψσ̄(5)ψ
†
σ̄(6)
〉
. (5.6)
Let us consider a system of pairing, when only ψ and ψ† operators are paired,
for example,
〈T̂ψσ(1)ψ
†
σ(2)ψσ̄(3)ψ
†
σ̄(4)
✛
✲
ψσ̄(5)ψ
†
σ̄(6)
✲
〉 = Gσ(14)Gσ̄(32)Gσ̄(56)〈T̂X
σσ̄
2 X σ̄σ
4 F σ̄
6 〉
= −Gσ(14)Gσ̄(32)Gσ̄(56)
[
1
Z
δ3Z
δtσ̄24δt
σ
42δt
σ
66
−Dσ
1 (24)
]
. (5.7)
Taking only this system of pairing in (5.6) we have approximately
1
Z
δ3Z
δtσ12δt
σ̄
34δt
σ̄
56
≈ −Gσ(14)Gσ̄(32)Gσ̄(56)
[
1
Z
δ3Z
δtσ̄24δt
σ
42δt
σ
66
−Dσ
1 (24)
]
+Symm. (5.8)
Putting arguments 4=1, 3=2, 5=6 we get an equation for a six-tailed diagram
with equal arguments. For the paramagnetic phase this equation is as follows:
1 +
(
n/2
1− n/2
)3
1
Z
δ3Z
δtσ̄24δt
σ
42δt
σ
66
≈
(
n/2
1− n/2
)3
Dσ
1 (24) . (5.9)
This third order derivative with equal arguments corresponds to a six-tailed dia-
gram with joint lines. Comparing relation (5.8) with the graphs in figure 5 shows
that decoupling by the Wick theorem corresponds to the two graphs explicitly
shown in figure 5. One of them is a flower with three clovers. As we see from
equation (5.9), the flower does not have a singular factor 1− n.
A different situation occurs if the forth order derivative (eigt-tailed) is studied
by the same method. When calculating T-product of eight ψ-operators, we take
into account only two systems of pairing:
〈
T̂ψσ(1)ψ
†
σ(2)ψσ̄(3)ψ
†
σ̄(4)ψσ(5)ψ
†
σ(6)ψσ̄(7)ψ
†
σ̄(8)
〉
=
= Gσ(14)Gσ̄(32)Gσ(58)Gσ̄(76)D
σ
1 (24)D
σ
1 (68) 〈F
σ
4 〉 〈F
σ
8 〉+
+Gσ(14)Gσ̄(32)Gσ(58)Gσ̄(76)
〈
T̂Xσσ̄
2 X σ̄σ
4 Xσσ̄
6 X σ̄σ
8
〉
+ . . . (5.10)
52
tJ -model in terms of equations with variational derivatives
Figure 7. The forth order variational derivative in zero approximation.
Graphic expressions of these terms are shown in figure 7. The average of T-product
of four b-operators corresponds to the fourth order derivative with coinciding argu-
ments in pairs. Putting in (5.10) the corresponding arguments equal, one obtains
an equation:
1−
(
n/2
1− n/2
)4
1
Z
δ4Z
δtσ42δt
σ̄
24δt
σ
86δt
σ̄
68
=
(
n/2
1− n/2
)4
Dσ
1 (24)D
σ
1 (68) 〈F
σ
4 〉 〈F
σ
8 〉 .
(5.11)
We see from here that the derivative has a singular factor 1−n in the denominator.
The difference in the third and fourth derivatives is caused by different signs in
diagrams of the flower type in six-tailed and eight-tailed diagrams. It is easy to
Figure 8. A second order correction to
magnon Green function.
see that the singular factor should al-
ways appear in the flower type dia-
grams with an even number of clovers.
If in expression (5.10) for the fourth
derivative we take only the last term,
we obtain a second order correction in
the magnon Green function presented
in figure 8. For this graph an analyti-
cal contribution into the denominator
of expression (4.3) is
1
8(1− n)
∑
k
J(k− q)Gσ(k− q)Gσ̄(k)
(
∑
q1
J(q1)D
σ
1 (q1)Gσ(k− q1)
)
. (5.12)
Of course, a symmetrical graph in figure 8 should be added which gives expression
(5.12) with the change q → −q.
Due to singular factor 1 − n, the second order corrections are as important,
when n→ 1, as the first order ones. Zero in the denominator of the magnon Green
function (4.3) at ω = 0 and q=Q determines a boundary of the paramagnetic phase
stability with respect to the appearance of an antiferromagnetic order. As one can
53
Yu.A.Izyumov, N.I.Chashchin
see from (4.3) and (5,12), this boundary goes along the line of type 1− n ∼ J/t,
as it should be. Detailed analysis of the magnetic phase diagram will be given
elsewhere.
6. Conclusions
We have generalized the Kadanoff-Baym approach to systems with strong elec-
tron correlations and applied it to a tJ-model. The X-operator representation al-
lows one to derive equations for variational derivatives of the generating functional
in a form convenient for iteration with respect to t and J . Within the framework
of the general approach we suggested an approximation of the SCBA type, where
electron and boson Green functions are considered as exact, and vertex parts are
expanded over t and J . In such an approach the problem of calculation of the
Green functions reduces to the calculation of the variational derivatives in the
”zero” approximation with exact Green functions. It opens a possibility to obtain
self-consistent equations for Green functions.
We have suggested a method for the calculation of variational derivatives in the
zero approximation, which gives a correct symmetry arising from the possibility to
change the order of differentiation. The results for such calculations are too cum-
bersome and we presented them only for the second derivatives. It is remarkable
that they are consistent with the diagram technique for X-operators based on the
generalized Wick theorem [7].
We used this formalism for the calculation of the magnon Green function in the
paramagnetic phase. To avoid cumbersome calculations of higher order derivatives
based on the representation of the (3.1)-type we calculated the third order deriva-
tive by using the Wick theorem, because in this case the exact symmetry is not
important. We have found graphs giving a growing contribution when the system
tends to half-filling. In n → 1 limit only graphs of the second order containing
singular factor (1 − n)−1 should be taken into account. They produce instability
of the paramagnetic phase with respect to antiferromagnetic ordering. Singular
contribution (5.12) in the denominator of the magnon Green function comes from
the graph in figure 8. One can see that it involves a magnon in the intermediate
state. The corresponding physical process is the following: a magnon creates an
electron-hole pair with the same spins and still it exists itself. Then, one of parti-
cles of the pair absorbs this magnon. As a final result, an electron-hole pair with
the opposite spins appears. Notice that the possibility of such a virtual process
is connected with the existence of two types of vertices for the electron-magnon
interaction in the diagram technique for the tJ-model: elastic and inelastic ones.
Both vertices are involved in the second order graph.
We consider the results of the calculation of the magnon Green function as
preliminary ones. It is necessary to look for singular terms among high order
diagrams. It is also necessary to calculate the plasmon Green functions (2.10),
particularly for the longitudinal spin deviations, and also to calculate the second
order correction to the electron Green function. In this way a self-consistent system
54
tJ -model in terms of equations with variational derivatives
of equations for boson and electron Green functions can be obtained.
This work was supported by the State Science and Technology Committee, the
program Superconductivity, grant No 95-056.
References
1. Izyumov Yu.A. The tJ-model for strongly correlated electrons and high-Tc-supercon-
ductors. // Uspekhi Fiz. Nauk, 1997, vol. 167, No 5, p. 465–497 (in Russian).
2. Kadanoff L., Baym G. Quantum statistical mechanics: Green’s function methods in
equilibrium and nonequilibrium problems. New–York: Benjamin, 1962.
3. Izyumov Yu.A., Chashchin N.I. Selfconsistent interaction of quasiparticles with fluctu-
ations of magnetic order in the tJ-model. // Fiz. Met. Metallography, 1998, vol. 85,
No 3, p. 5–19 (in Russian).
4. Plakida N.M., Yushankhai V.Yu., Stasyuk I.V. On the role of kinematic and exchange
interactions in superconducting pairing of electrons in the Hubbard model. // Physica
C, 1989, vol. 160, p. 80–88.
5. Slobodyan P.M., Stasyuk I.V. Diagram technique for Hubbard operators. // Teor. Mat.
Fiz., 1974, vol. 19, No 3, p. 423–431 (in Russian).
6. Izyumov Yu.A., Katsnelson M.I., Scryabin Yu.N. Magnetism of itinerant electrons.
Moscow, Fiziko-Matematicheskaya Literatura, 1994, 366 p.
7. Izyumov Yu.A., Letfulov B.M. A diagram technique for Hubbard operators: the mag-
netic phase diagram in the tJ-model. // J. Phys.: Cond. Matt., 1990, vol. 2, No 45,
p. 8905–8923.
tJ -модель в термінах рівнянь з варіаційними
похідними
Ю.О.Ізюмов, Н.І.Чащін
Уральське відділення Інституту металофізики РАН,
Росія, 620219 Єкатєрінбург, вул. С.Ковалевської, 18
Отримано 17 червня 1998 р.
Для tJ моделі в представленні X -операторів введено узагальнений
функціонал поля, що описує флуктуації матричних елементів елек-
тронних перескоків на гратці. Функціональна похідна першого по-
рядку по цьому полю визначає електронну функцію Гріна, тоді як
похідні другого порядку визначають бозонні функції Гріна колектив-
них збуджень в системі. Таким чином узагальнено підхід Каданова-
Бейма в теорії фермі-систем з слабою кулонівською взаємодією на
протилежний випадок систем з сильними кореляціями. Отримано
ланцюжок рівнянь для варіаційних похідних різного порядку і запро-
поновано метод, що базується на ітераціях за параметрами tJ мо-
делі: матричним елементом перескоку і обмінним інтегралом. Цей
підхід відповідає самоузгодженому наближенню Борна, але не для
55
Yu.A.Izyumov, N.I.Chashchin
ефективного, а для вихідного гамільтоніану. Аналізується схема роз-
рахунку динамічної спінової сприйнятливості з самоузгодженими по-
правками першого і другого порядку. Обговорюється зв’язок цього
підходу з діаграмною технікою для X -операторів.
Ключові слова: tJ -модель, оператори Хаббарда, функції Гріна,
антиферомагнетизм
PACS: 71.27.+a, 74.90.+n
56
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