Green’s functions of infinite-U asymmetric Hubbard model: Falicov-Kimball limit
The asymmetric Hubbard model is used in investigating the lattice gas of the moving particles of two types. The model is considered within the dynamical mean-field method. The effective single-site problem is formulated in terms of the auxiliary Fermi-field. To solve the problem an approximate a...
Збережено в:
| Дата: | 2003 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут фізики конденсованих систем НАН України
2003
|
| Назва видання: | Condensed Matter Physics |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/120689 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Green’s functions of infinite-U asymmetric Hubbard model: Falicov-Kimball limit / I.V. Stasyuk, O.B. Hera // Condensed Matter Physics. — 2003. — Т. 6, № 1(33). — С. 127-143. — Бібліогр.: 24 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-120689 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1206892017-06-13T03:06:10Z Green’s functions of infinite-U asymmetric Hubbard model: Falicov-Kimball limit Stasyuk, I.V. Hera, O.B. The asymmetric Hubbard model is used in investigating the lattice gas of the moving particles of two types. The model is considered within the dynamical mean-field method. The effective single-site problem is formulated in terms of the auxiliary Fermi-field. To solve the problem an approximate analytical method based on the irreducible Green’s function technique is used. This approach is tested on the Falicov-Kimball limit (when the mobility of ions of either type is infinitesimally small) of the infinite-U case of the model considered. The dependence of chemical potentials on concentration is calculated using the one-particle Green’s functions, and different approximations are compared with the exact results obtained thermodynamically. The densities of states of localized particles are obtained for different temperatures and particle concentrations. The phase transitions are investigated for the case of the Falicov-Kimball limit in different thermodynamic regimes. Асиметрична модель Хаббарда використовується для дослідження двохсортного граткового газу рухомих частинок. Модель розглядається в рамках методу динамічного середнього поля. Ефективна одновузлова задача формулюється в термінах допоміжного фермі поля. Для її розв’язку використано наближений аналітичний метод, який базується на техніці незвідних функцій Гріна. Цей підхід апробується на моделі з безмежним U у границі Фалікова-Кімбала (коли рухливість іонів одного з сортів є безмежно малою). Концентраційні залежності хімічних потенціалів розраховуються з допомогою одно-частинкових функцій Гріна; різні наближення порівнюються з точними результатами, отриманими термодинамічним шляхом. Отримано густини станів для локалізованих частинок при різних температурах і концентраціях. Досліджено фазові переходи у границі Фалікова-Кімбала для різних термодинамічних режимів. 2003 Article Green’s functions of infinite-U asymmetric Hubbard model: Falicov-Kimball limit / I.V. Stasyuk, O.B. Hera // Condensed Matter Physics. — 2003. — Т. 6, № 1(33). — С. 127-143. — Бібліогр.: 24 назв. — англ. 1607-324X PACS: 71.10.Fd, 05.30.Fk, 05.70.Fh DOI:10.5488/CMP.6.1.127 http://dspace.nbuv.gov.ua/handle/123456789/120689 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 |
The asymmetric Hubbard model is used in investigating the lattice gas of
the moving particles of two types. The model is considered within the dynamical
mean-field method. The effective single-site problem is formulated
in terms of the auxiliary Fermi-field. To solve the problem an approximate
analytical method based on the irreducible Green’s function technique is
used. This approach is tested on the Falicov-Kimball limit (when the mobility
of ions of either type is infinitesimally small) of the infinite-U case of
the model considered. The dependence of chemical potentials on concentration
is calculated using the one-particle Green’s functions, and different
approximations are compared with the exact results obtained thermodynamically.
The densities of states of localized particles are obtained for
different temperatures and particle concentrations. The phase transitions
are investigated for the case of the Falicov-Kimball limit in different thermodynamic
regimes. |
| format |
Article |
| author |
Stasyuk, I.V. Hera, O.B. |
| spellingShingle |
Stasyuk, I.V. Hera, O.B. Green’s functions of infinite-U asymmetric Hubbard model: Falicov-Kimball limit Condensed Matter Physics |
| author_facet |
Stasyuk, I.V. Hera, O.B. |
| author_sort |
Stasyuk, I.V. |
| title |
Green’s functions of infinite-U asymmetric Hubbard model: Falicov-Kimball limit |
| title_short |
Green’s functions of infinite-U asymmetric Hubbard model: Falicov-Kimball limit |
| title_full |
Green’s functions of infinite-U asymmetric Hubbard model: Falicov-Kimball limit |
| title_fullStr |
Green’s functions of infinite-U asymmetric Hubbard model: Falicov-Kimball limit |
| title_full_unstemmed |
Green’s functions of infinite-U asymmetric Hubbard model: Falicov-Kimball limit |
| title_sort |
green’s functions of infinite-u asymmetric hubbard model: falicov-kimball limit |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2003 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120689 |
| citation_txt |
Green’s functions of infinite-U
asymmetric Hubbard model:
Falicov-Kimball limit / I.V. Stasyuk, O.B. Hera // Condensed Matter Physics. — 2003. — Т. 6, № 1(33). — С. 127-143. — Бібліогр.: 24 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT stasyukiv greensfunctionsofinfiniteuasymmetrichubbardmodelfalicovkimballlimit AT heraob greensfunctionsofinfiniteuasymmetrichubbardmodelfalicovkimballlimit |
| first_indexed |
2025-07-08T18:24:35Z |
| last_indexed |
2025-07-08T18:24:35Z |
| _version_ |
1837104186339098624 |
| fulltext |
Condensed Matter Physics, 2003, Vol. 6, No. 1(33), pp. 127–143
Green’s functions of infinite-U
asymmetric Hubbard model:
Falicov-Kimball limit
I.V.Stasyuk, O.B.Hera
Institute for Condensed Matter Physics
of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 79011 Lviv, Ukraine
Received February 24, 2003
The asymmetric Hubbard model is used in investigating the lattice gas of
the moving particles of two types. The model is considered within the dy-
namical mean-field method. The effective single-site problem is formulated
in terms of the auxiliary Fermi-field. To solve the problem an approximate
analytical method based on the irreducible Green’s function technique is
used. This approach is tested on the Falicov-Kimball limit (when the mo-
bility of ions of either type is infinitesimally small) of the infinite-U case of
the model considered. The dependence of chemical potentials on concen-
tration is calculated using the one-particle Green’s functions, and different
approximations are compared with the exact results obtained thermody-
namically. The densities of states of localized particles are obtained for
different temperatures and particle concentrations. The phase transitions
are investigated for the case of the Falicov-Kimball limit in different thermo-
dynamic regimes.
Key words: asymmetric Hubbard model, Falicov-Kimball model,
dynamical mean-field theory, Green’s functions, phase transitions
PACS: 71.10.Fd, 05.30.Fk, 05.70.Fh
1. Introduction
Strongly correlated electron systems have been a subject of growing interest in
recent years. These systems became especially interesting after the discovery of high-
Tc superconductivity. In general, the methods used for their description are based on
the Hubbard model and its generalizations taking into account strong short-range
correlations of particles. Similar models are considered in investigating the ionic
conductivity in crystalline materials. Fermi lattice gas models can be mentioned
among them. The asymmetric Hubbard model [1] arises by extending the models
to the case of the systems with moving ions of two types. The transfer parameters
and chemical potentials are different for the ions of different nature. As some spe-
cial cases, the asymmetric Hubbard model includes the Falicov-Kimball model and
c© I.V.Stasyuk, O.B.Hera 127
I.V.Stasyuk, O.B.Hera
the standard Hubbard model. Thereby, it should be mentioned that the asymmet-
ric Hubbard model has been recently obtained as a generalization of the electron
Falicov-Kimball model which describes the interactions between itinerant d-electrons
and localized f -electrons by including f–f hopping [2]. The model was used in de-
scribing the effects related to orbital ordering, as well as to Bose-condensation of
electron-hole pairs (which includes a spontaneous ferroelectric polarization) in oxide
compounds.
Despite the relative simplicity of these models, the theory of electron spectrum
and thermodynamic properties of such systems is far from its final completion. In
recent years the essential achievements of the theory of strongly correlated electron
systems have been connected with the development of the dynamical mean-field
theory (DMFT) [3]. This method is exact in the limit of infinite space dimension
(d = ∞) and is based on the fact that the irreducible (according to Larkin) part of
the one-particle Green’s function is single-sited (at appropriate scaling of transfer
parameter).
The central point in this method is the formulation and the solution of the
auxiliary single-site problem. In this case a separated lattice site is considered as
placed in some effective environment. Since the processes of the particle hopping
from the site and returning into the site are taken into account, the mean field
acting on the particle states of the site is of a dynamic nature. The field is described
by the coherent potential Jσ(ω) that should be determined in a self-consistent way
from the condition that the same total irreducible (according to Larkin) part defines
Green’s functions both in the atomic limit and for the lattice [4–6]:
G(a)
σ (ωn) =
1
Ξ−1
σ (ωn) − Jσ(ωn)
, (1)
Gσ(ωn,k) =
1
Ξ−1
σ (ωn) − tσ
k
, (2)
G(a)
σ (ωn) = Gσ
ii(ωn) =
1
N
∑
k
Gσ(ωn,k). (3)
Here σ is the type index (or a spin index for electron systems); there is a possibility
of the transfer parameter tσ
k
can depend on a particle type. Summation over the
wave vector can be changed by integration with the density of states. There are
cases that are usually investigated in the d = ∞ limit: the hypercubic lattice which
is the generalization of the cubic d = 3 lattice, and the Bethe lattice which is the
thermodynamic limit of the Cayley tree when the number of the nearest neighbours
tends to infinity. The density of states is Gaussian for a hypercubic lattice [3], while
on the Bethe lattice the density of states is semielliptic [4].
The single-site problem can be solved analytically only in some simple cases.
In general, the application of numerical methods turns out to be necessary. In the
case of the Falicov-Kimball model it is possible to investigate the thermodynamic
properties of the system by analytical calculation of the grand canonical potential
[7,8]. Also, the analytical expressions for the itinerant and for the localized electron
128
Green’s functions of infinite-U Hubbard model
Green’s functions have been obtained [9,10]. However, the topology of the localized
electron band has not been completely investigated yet.
Recently, an approximate analytical approach to the solving of a single-site prob-
lem has been developed [11,12]. This approach is based on the irreducible Green’s
function technique with projecting on the Hubbard basis of Fermi-operators. A sys-
tem of DMFT equations was obtained in the approximation which is a generalization
of the Hubbard-III approximation combined with a self-consistent renormalization
of the local electron levels (here this approach will be called GH3). It has been shown
that the proposed approach includes, as special cases, a number of known approxi-
mations (Hubbard-III, alloy-analogy (AA) and modified alloy-analogy (MAA)).
In the present work this approach is formulated for the asymmetric Hubbard
model. Its applicability within different approximations is tested on the infinite-
U spinless Falicov-Kimball model. The densities of states of the moving and the
localized particles are obtained for the Bethe lattice at different particle concen-
trations and temperatures. Dependence of the chemical potentials on concentration
is calculated using the Green’s functions. The results of different approximations
are compared with the exact results obtained thermodynamically. Some thermody-
namic properties (such as phase transitions and phase separations) of the system in
different thermodynamic regimes are investigated as well.
2. Model
We consider the case when the lattice gas model is used in describing the systems
with moving particles (ions or electrons) of two types. A single site can be in three
possible states: (i) no particles, (ii) one particle of the first (A) type, (iii) one particle
of the second (B) type. Motion of the particles can be described by the creation and
by the annihilation operators and by the transfer parameters dependent on the
particle type. Since the site can be occupied by only one particle of either type, the
operators are not of the Fermi type. It is possible to investigate the system using
the Fermi operators by formal adding the fourth state of a site occupied by two (A
and B) particles simultaneously. In this case, the asymmetric Hubbard model arises.
The Hamiltonian is
H =
∑
i
Hi +
∑
〈ij〉
tAija
A+
i aA
j +
∑
〈ij〉
tBija
B+
i aB
j , (4)
Hi = −µAnA
i − µBnB
i + UnA
i nB
i , (5)
with chemical potentials dependent on the particle type. It can be easily seen that
Hamiltonian (4) corresponds to the Hubbard model in an external magnetic field
with the spin-dependent electron transfer. The U term is the single-site interaction
energy between the particles and it should tend to infinity when we wish to return
to the initial lattice gas model.
In the limit of large dimensions, the problem is investigated using the DMFT
approach. The single-site problem is formulated in terms of the auxiliary Fermi-field
129
I.V.Stasyuk, O.B.Hera
[11]. Let us write the effective single-site Hamiltonian in the Hubbard operators
representation
Heff = −
∑
σ
[
µσ(Xσσ + X22)
]
+ UX22
+ V
∑
σ
[
(Xσ0 + ζX2σ̄)ξσ + ξ+
σ (X0σ + ζX σ̄2)
]
+ Hξ , (6)
where the notations are used: σ̄ = B, ζ = + for σ = A; σ̄ = A, ζ = − for σ = B.
The basis of single-site states |nA, nB〉 is
|0〉 = |0, 0〉, |A〉 = |1, 0〉,
|2〉 = |1, 1〉, |B〉 = |0, 1〉. (7)
The Green’s function G
(a)
σ (ω) ≡ 〈〈aσ|a+
σ 〉〉ω can be written in the following form:
G(a)
σ = 〈〈X0σ|Xσ0〉〉ω + ζ〈〈X0σ|X2σ̄〉〉ω + ζ〈〈X σ̄2|Xσ0〉〉ω + 〈〈X σ̄2|X2σ̄〉〉ω . (8)
The auxiliary Fermi-field describes the environment of the selected site and is for-
mally characterized by the Hamiltonian Hξ. The explicit form of this Hamiltonian
is unknown. However, to calculate the Green’s function G
(a)
σ (ω), the averaging over
the ξ, ξ+ operators is done with the help of the function
Gσ(ω) = 〈〈ξσ|ξ+
σ 〉〉
(Hξ)
ω . (9)
The relation
2πV 2Gσ(ω) = Jσ(ω) (10)
takes place in this case. Unlike the standard Hubbard model, the coherent potential
Jσ is dependent on the type of the particles (JA 6= JB for tAij 6= tBij).
3. Green’s functions for the effective single-site problem
Equations for functions (8) are written using the equations of motion for X-
operators. According to the method developed in [13,14], let us separate (in the
Green’s functions of a higher order) the irreducible parts expressing the derivatives
idX0σ(σ̄2)
s
dt = [X0σ(σ̄2), Heff ] as the sums of regular (projected on the subspace formed
by operators X0σ and X σ̄2) and irregular parts
[X0σ, Heff ] = −µσX0σ + α0σ
1 X0σ + α0σ
2 X σ̄2 + Z0σ,
[X σ̄2, Heff ] = (U − µσ)X
σ̄2 + ασ̄2
1 X0σ + ασ̄2
2 X σ̄2 + Z σ̄2. (11)
Operators Z0σ and Z σ̄2 are defined as orthogonal to the operators from the basic
subspace and we come to the expressions
Z0σ = V (X00 + Xσσ)ξσ + V X σ̄σξσ̄ + ζV X02ξ+
σ̄ ,
Z σ̄2 = V (X22 + X σ̄σ̄)ξσ − ζV X σ̄σξσ̄ − V X02ξ+
σ̄ , (12)
130
Green’s functions of infinite-U Hubbard model
where
(X00 + Xσσ)ξσ = (X00 + Xσσ)ξσ,
(X22 + X σ̄σ̄)ξσ = (X22 + X σ̄σ̄)ξσ,
X σ̄σξσ̄ = X σ̄σξσ̄ − 1
A0σ
〈ξσ̄X
σ̄0〉X0σ − 1
A2σ̄
〈X2σξσ̄〉X σ̄2,
X02ξ+
σ̄ = X02ξ+
σ̄ − 1
A0σ
〈Xσ2ξ+
σ̄ 〉X0σ − 1
A2σ̄
〈ξ+
σ̄ X0σ̄〉X σ̄2, (13)
and Apq = 〈Xpp + Xqq〉; A0σ = 1 − nσ̄, A2σ̄ = nσ̄.
In this case
α0σ
1 = −ζασ̄2
1 =
V
A0σ
ϕσ,
α0σ
2 = −ζασ̄2
2 = − V
A2σ̄
ζϕσ,
ϕσ = 〈ξσ̄X
σ̄0〉 + ζ〈Xσ2ξ+
σ̄ 〉. (14)
Using this procedure by differentiating both with respect to the left and to the
right time arguments, we come to the relations between the components of the
Green’s function G
(a)
σ and scattering matrix P̂σ. In a matrix representation, we have
Ĝσ = Ĝσ
0 + Ĝσ
0 P̂σĜ
σ
0 , (15)
where
Ĝσ = 2π
(
〈〈X0σ|Xσ0〉〉 〈〈X0σ|X2σ̄〉〉
〈〈X σ̄2|Xσ0〉〉 〈〈X σ̄2|X2σ̄〉〉
)
, (16)
and nonperturbed Green’s function Ĝσ
0 is
Ĝσ
0 =
1
Dσ
(
ω − bσ −ζ V
A2σ̄
ϕσ
−ζ V
A0σ
ϕσ ω − aσ
) (
A0σ 0
0 A2σ̄
)
, (17)
where
Dσ = (ω − aσ)(ω − bσ) − V 2
A0σA2σ̄
ϕ2
σ, (18)
aσ = −µσ +
V
A0σ
ϕσ, bσ = U − µσ +
V
A2σ̄
ϕσ. (19)
The scattering matrix
P̂σ = 2π
(
A−1
0σ 0
0 A−1
2σ̄
) (
〈〈Z0σ|Zσ0〉〉 〈〈Z0σ|Z2σ̄〉〉
〈〈Z σ̄2|Zσ0〉〉 〈〈Z σ̄2|Z2σ̄〉〉
) (
A−1
0σ 0
0 A−1
2σ̄
)
(20)
being expressed in terms of irreducible Green’s functions contains the scattering
corrections of the second and the higher orders in powers of V . The separation
of the irreducible parts in P̂σ enables us to obtain the mass operator M̂σ and the
single-site Green’s function expressed as a solution of the Dyson equation
Ĝσ = (1 − Ĝσ
0M̂σ)−1Ĝσ
0 . (21)
131
I.V.Stasyuk, O.B.Hera
We will restrict ourselves hereafter to the simple approximation in calculating
the mass operator P̂σ, taking into account the scattering processes of the second
order in V . In this case
M̂σ = P̂ (0)
σ , (22)
where the irreducible Green’s functions are calculated without allowance for corre-
lation between electron transition on the given site and environment. It corresponds
to the procedure of different-time decoupling [15], which means in our case an in-
dependent averaging of the products of X and ξ operators. Let us illustrate this
approximation by some examples.
1. The Green’s function 〈〈(X00 + Xσσ)ξσ |ξ+
σ (X00 + Xσσ)〉〉ω ≡ I1(ω).
Using the spectral theorem and performing the different-time decoupling:
〈ξ+
σ (t)(X00 + Xσσ)t(X
00 + Xσσ)ξσ〉ir ≈ 〈(X00 + Xσσ)t(X
00 + Xσσ)〉〈ξ+
σ (t)ξσ〉,
(23)
we obtain
I1(ω) = A0σ〈〈ξσ |ξ+
σ 〉〉ω =
A0σ
2πV 2
Jσ(ω). (24)
Here, X-correlators are in a zero approximation
〈(X00 + Xσσ)t(X
00 + Xσσ)〉 ≈ 〈(X00 + Xσσ)2〉 = A0σ . (25)
2. The Green’s function 〈〈X σ̄σξσ̄ |ξ+
σ̄ Xσσ̄〉〉ω ≡ I2(ω).
The time correlation function 〈ξ+
σ̄ (t)Xσσ̄(t)X σ̄σξσ̄ 〉ir is decoupled as
〈ξ+
σ̄ (t)Xσσ̄(t)X σ̄σξσ̄ 〉ir ≈ 〈Xσσ̄(t)X σ̄σ〉〈ξ+
σ̄ (t)ξσ̄ 〉. (26)
In the zero approximation
〈Xσσ̄(t)X σ̄σ〉 = exp [i(µσ̄ − µσ)t]〈Xσσ〉. (27)
Using these expressions we obtain
2πV 2I2(ω) =
〈Xσσ〉 + 〈X σ̄σ̄〉
2
Jσ̄(ω + µσ − µσ̄) −
〈Xσσ〉 − 〈X σ̄σ̄〉
4π
×
∫ +∞
−∞
dω′
ω − ω′ − µσ̄ + µσ
tanh
βω′
2
{ − 2 ImJσ̄(ω′ + iε)}. (28)
Let us notice that in the case with nA = nB, µA = µB, which corresponds to
the simple Hubbard model in the absence of an external magnetic field,
I2(ω) =
〈Xσσ〉
2πV 2
Jσ̄(ω). (29)
132
Green’s functions of infinite-U Hubbard model
3. The Green’s function 〈〈X02ξ+
σ̄ |ξσ̄ X20〉〉ω ≡ I3(ω).
In this case
I3(ω) =
〈X00〉 + 〈X22〉
2
〈〈ξ+
σ̄ |ξσ̄ 〉〉ω+µσ+µσ̄−U +
〈X00〉 − 〈X22〉
4π
×
∫ +∞
−∞
dω′
ω − ω′ + µσ̄ + µσ − U
tanh
βω′
2
{ − 2 Im 〈〈ξ+
σ̄ |ξσ̄ 〉〉ω′+iε}. (30)
Using the above results, the mass operator can be obtained in an explicit form.
According to (22), it is given by the expression (20) where
〈〈Z0σ|Zσ0〉〉ω = A0σJσ(ω) − Rσ(ω),
〈〈Z σ̄2|Z2σ̄〉〉ω = A2σ̄Jσ(ω) − Rσ(ω),
〈〈Z0σ|Z2σ̄〉〉ω = 〈〈Z σ̄2|Zσ0〉〉ω = Rσ(ω), (31)
and
Rσ(ω) = −〈Xσσ〉 + 〈X σ̄σ̄〉
2
Jσ̄(ω + µσ − µσ̄)
+
〈Xσσ〉 − 〈X σ̄σ̄〉
4π
∫ +∞
−∞
dω′
ω − ω′ − µσ̄ + µσ
{−2 ImJσ̄(ω′ + iε)} tanh
βω′
2
+
〈X00〉 + 〈X22〉
2
Jσ̄(U − µσ − µσ̄ − ω)
+
〈X00〉 − 〈X22〉
2π
∫ +∞
−∞
dω′
ω − ω′ + µσ̄ + µσ − U
{− Im Jσ̄(−ω′ − iε)} tanh
βω′
2
.
(32)
4. Set of DMFT equations in the infinite-U limit
The following consideration will be performed in the case of U → +∞ which
excludes simultaneous occupation by two particles of A and B types of the same
site, when the model is used in describing the lattice gas of the particles of two
types. Then functions Rσ for σ = A or B are as follows
RA(ω) = −nA + nB
2
JB(ω + µA − µB)
+
nA − nB
4π
∫ +∞
−∞
dω′
ω − ω′ − µB + µA
tanh
βω′
2
{ − 2 Im JB(ω′ + iε)}, (33)
RB(ω) = −nA + nB
2
JA(ω + µB − µA)
+
nB − nA
4π
∫ +∞
−∞
dω′
ω − ω′ − µA + µB
tanh
βω′
2
{ − 2 ImJA(ω′ + iε)}, (34)
133
I.V.Stasyuk, O.B.Hera
The single-site Green’s functions (8) are obtained using relations (21), (22) and (31)
in the U → +∞ limit, and respectively they equal to:
G
(a)
A (ω) =
1 − nB
ω + µA − V
1−nB
ϕA − JA(ω) + RA(ω)
1−nB
, (35)
G
(a)
B (ω) =
1 − nA
ω + µB − V
1−nA
ϕB − JB(ω) + RB(ω)
1−nA
. (36)
Parameter ϕσ, which is expressed by the average values of the products of X
and ξ operators, is a functional of the potential Jσ(ω). According to the spectral
theorem
V 〈Xσ0ξσ〉 = i
∫ +∞
−∞
dω
eβω + 1
[V 〈〈ξσ|Xσ0〉〉ω+iε − V 〈〈ξσ|Xσ0〉〉ω−iε]. (37)
The Green’s functions are found using linearized equations of motion and neglecting
the irreducible parts.
In this case
V ϕA = − 1
2π
∫ +∞
−∞
dω
eβω + 1
{
− 2 Im
(1 − nA)JB(ω)
ω + µB − V ϕB
1−nA
}
ω+iε
, (38)
V ϕB = − 1
2π
∫ +∞
−∞
dω
eβω + 1
{
− 2 Im
(1 − nB)JA(ω)
ω + µA − V ϕA
1−nB
}
ω+iε
. (39)
The coherent potential Jσ(ω) is self-consistently determined from the equations
(1)–(3) by eliminating the total irreducible part. Integration with the semielliptic
ρσ(ε) =
2
πW 2
σ
√
W 2
σ − ε2 (40)
density of states is done. In this case, we have simple expressions
G
(a)
A (ω) =
4JA(ω)
W 2
A
, (41)
G
(a)
B (ω) =
4JB(ω)
W 2
B
. (42)
The set of equations becomes a closed one by adding expressions for average particle
concentrations obtained with the help of the imaginary parts of the Green’s functions
(i.e. interacting densities of states):
nA =
1
2π
∫ +∞
−∞
dω
−2 Im G
(a)
A (ω + iε)
eβω + 1
, (43)
nB =
1
2π
∫ +∞
−∞
dω
−2 Im G
(a)
B (ω + iε)
eβω + 1
. (44)
134
Green’s functions of infinite-U Hubbard model
It has been shown in the case of the standard Hubbard model [11] that such a
set of equations corresponds to GH3 approximation, and includes, as some special
cases, a number of known approximations. The case with Rσ = 0, ϕσ = 0 corresponds
to the alloy-analogy (AA) approximation [16]. The modified alloy-analogy (MAA)
approximation is obtained by taking into account the renormalization of the local
electron levels ϕσ [16,17]. For Rσ 6= 0, ϕσ = 0 the system of equations corresponds
to the extension of the Hubbard-III approximation by the inclusion of the integral
terms responsible for scattering.
5. Lattice gas thermodynamics in the Falicov-Kimball limit
5.1. Exact solution of DMFT problem
In the limiting case of U → +∞, WB → 0 (tBij → 0), when mobility of the B type
ions is very small (this case corresponds to the spinless Falicov-Kimball model [18]
with the infinite on-site repulsion), the effective single-site problem can be solved ex-
actly. Thermodynamics of the model, equilibrium states and phase transitions were
investigated in a series of papers ([7,8,19], see [20] as well). There were considered
thermodynamic regimes with the constant n = nA +nB, ε = µB − µA values [8] and
with the fixed relative filling nB = const, nA = const · (1 − nB) [7]. However, other
thermodynamic regimes should be also considered: (i) constant chemical potentials
µA, µB; (ii) the mixed regimes µA = const, nB = const or µB = const, nA = const.
Let us note that in the case of the pseudospin-electron model (PEM) without tun-
nelling, the dynamics of pseudospins being analogous to the Falicov-Kimball model,
the regime of µB = const corresponds to the fixation of longitudinal asymmetry
field h acting on pseudospins. The PEM thermodynamics has been a subject of in-
vestigations within DMFT [21] as well as in the framework of approximate methods
(generalized random phase approximation (GRPA), [22–24]).
An exact solution of the single-site problem is given by the simple expression:
G
(a)
A (ω) =
1 − nB
ω + µA − JA(ω)
. (45)
It should be mentioned that this formula can be obtained by the above described
approach and corresponds to the AA approximation. The final expression for the
Green’s function is obtained by solving the quadratic equation formed by (41), (45):
G
(a)
A (ω ± iε) =
=
2
W 2
A
(ω + µA) + 2
W 2
A
√
(ω + µA)2 − W 2
A(1 − nB), ω + µA < −WA
√
1 − nB;
2
W 2
A
(ω + µA) ∓ i 2
W 2
A
√
W 2
A(1 − nB) − (ω + µA)2, |ω + µA| 6 WA
√
1 − nB;
2
W 2
A
(ω + µA) − 2
W 2
A
√
(ω + µA)2 − W 2
A(1 − nB), ω + µA > WA
√
1 − nB.
(46)
Here, the phase of the square root is chosen so that an imaginary part of the Green’s
function has the correct sign as well as by using the properties which follow from
the spectral theorem.
135
I.V.Stasyuk, O.B.Hera
-0,8 -0,4 0,0 0,4 0,8
0,0
0,2
0,4
0,6
0,8
T=0.04
T=0.01
T=0.06
µ
B
µ
A
-0,9 -0,8 -0,7 -0,6
-0,01
0,00
0,01
0,02
T=0.02T=0.01
T=0.002
µ
B
µ
A
Figure 1. (µA, µB)-phase diagrams.
The parameter values WA = 1, WB = 0,
the dashed line – T = 0.01, the solid line
– T = 0.04, the bold line – T = 0.06.
Figure 2. The part of (µA, µB)-phase
diagrams at different temperatures; the
appearance of the phase transitions for
µB < 0 (WA = 1, WB = 0) is shown.
Using this Green’s function the first equation for the particle concentrations is
obtained from (43)
nA =
2(1 − nB)
π
∫ +1
−1
dx
√
1 − x2
eβ(WA
√
1−nBx−µA) + 1
. (47)
Here, in the standard scheme, the second equation is obtained thermodynamically
by differentiating the grand canonical potential [12]
nB =
eβµB+Q
1 + eβµA + eβµB+Q
, (48)
where in our case
Q = −β
WA
√
1 − nB
π
∫ 1
−1
dx
e−β(xWA
√
1−nB−µA)) + 1
arctan
√
1 − x2
x
. (49)
Let us notice that parameter Q does not depend on the sign of chemical potential
of the moving particles, i.e., it does not depend on the interchange µA ↔ −µA.
Separating the solutions of a set of equations (47)–(49) that correspond to abso-
lute minima of the grand canonical potential, we will investigate equilibrium states in
the above-mentioned thermodynamic regimes. There are phase transitions between
homogeneous phases with different particle concentrations in the regime of constant
chemical potentials (µA = const, µB = const). Phase separation phenomena take
place in the regimes where either of the particle concentrations is constant.
1. µA = const, µB = const. In figure 1 phase diagrams are shown in (µA, µB)
coordinates. All energy quantities are given in the units of a half-bandwidth
(WA = 1). The phase transition curves terminate at critical points. They
136
Green’s functions of infinite-U Hubbard model
0,0 0,2 0,4 0,6 0,8 1,0
0,00
0,02
0,04
0,06
µ
A
=0.8
µ
A
=0.4
µ
A
=0
T
n
B
0,00 0,01 0,02 0,03 0,04 0,05
0,000
0,005
0,010
0,015
0,020
T
n
A
0,0 0,2 0,4 0,6 0,8 1,0
0,00
0,01
0,02
0,03
0,04
T
n
A
(a) (b) (c)
Figure 3. The phase separation diagrams for WA = 1, WB = 0. (a) – (nB, T )-
phase diagrams for the different chemical potentials of moving particles. (b), (c)
– (nA, T )-phase diagrams for µB = −0.006 and µB = 0.6 respectively.
become shorter (symmetrically with respect to the value µA = 0) with the
temperature growth and vanish at critical temperature. Numerical calculations
give the following values of critical parameters
T c ≈ 0.0601WA, µc
A = 0, µc
B ≈ 0.229WA. (50)
At the zero temperature, phase transitions are within the −WA < µA < WA,
0 < µB < WA ranges of chemical potentials. However, figure 2 shows that there
is a possibility of phase transitions with µB < 0 when temperature increases.
Let us note that diagrams analogous to those in figure 1 can be obtained in
DMFT for PEM (where µA corresponds to chemical potential µ of electrons,
and µB corresponds to the asymmetry field h). The phase coexistence curve
on (µ, h) diagram obtained for finite values of the coupling constant in [21]
consists of two parts corresponding to the chemical potential being in the
vicinity or inside of either electron subband (the band splitting in a spectrum
is caused by interaction with pseudospins). There is a direct correspondence
between the lower part of the phase diagram for PEM and the diagrams in
figure 2.
2. µA = const, nB = const (figure 3(a)). In this case, a phase separation takes
place. A topology of the phase diagrams (nB, T ) depends only on the absolute
value of the chemical potential of the moving particles. The highest value of
critical temperature is reached at µA = 0 and is given by the expression (50).
The critical temperature decreases when chemical potential of the moving
particles approaches either of the band edges. At the zero temperature for
−WA < µA < WA, the system is separated into two phases: n
(1)
B = 0 and
n
(2)
B = 1. The regime of relative half-filling (nB = const, nA = (1− nB)/2) has
been considered in [7]. This case corresponds to the nB = const regime with
µA = 0, so the diagram which is equivalent to figure 3(a) for µA = 0 has been
previously obtained.
137
I.V.Stasyuk, O.B.Hera
0,0 0,2 0,4 0,6 0,8 1,0
-0,8
-0,4
0,0
0,4
54
3
2
1
1
2
3
4
5
T=0.02
µ
A
=-0.4
µ
B
n
B
0,0 0,2 0,4 0,6 0,8 1,0
-0,4
0,0
0,4
0,8
1
2
3
4
5
5
4
3
2
1
T=0.02
µ
A
=0
µ
B
n
B
0,0 0,2 0,4 0,6 0,8 1,0
-0,4
0,0
0,4
0,8
5
43
2
1
1
2
3
4
5
T=0.02
µ
A
=0.4
µ
B
n
B
0,0 0,2 0,4 0,6 0,8 1,0
-1,0
-0,5
0,0
0,5
1,0
5
4
3
2
1
1
2
3
4
5
T=0.2
µ
A
=-0.4
µ
B
n
B
0,0 0,2 0,4 0,6 0,8 1,0
-0,3
0,0
0,3
0,6
5
4
3
2
1
1
2
3
4
5
T=0.2
µ
A
=0
µ
B
n
B
0,0 0,2 0,4 0,6 0,8 1,0
-0,6
0,0
0,6
1,2
5
4
3
2
1
1
2
3
4
5
T=0.2
µ
A
=0.4
µ
B
n
B
Figure 4. The nB dependence of µB in different approximations is compared with
the exact result obtained thermodynamically. The parameter values: WA = 1,
WB = 0. 1 – exact result; 2 – AA; 3 – MAA; 4 – H3; 5 – GH3.
3. nA = const, µB = const (figure 3(b,c)). There is a homogeneous state at
zero temperature, but a segregated state appears at higher temperatures for
−0.009WA < µB < 0 (figure 3(b)). The phase separation region decreases
continuously and there is a point at temperature T ≈ 0.01 WA when chemical
potential of localized particles approaches some critical value µB ≈ −0.009 WA.
The possibility of such a situation is illustrated by the behaviour of the left
end of the phase transition curve in the vicinity of µB = 0, µA = −WA
values (figure 2). The thermodynamic regime with ε0 = µA − µB = const,
nA + nB = const has been investigated in [8], and phase diagrams similar to
that in figure 3(b) have been obtained. It can be seen that both ends of the
phase transition curve (µA, µB) have the similar temperature behaviour (with
respect to µB = 0 and to µB − µA = 0 lines).
At µB > 0, the topological structure of phase diagram (nA, T ) is similar to the
one in figure 3(c). A phase separated state always exists for low temperatures
with non-integer values of nA in both phases at T = 0. Diagrams of this type
have been obtained for PEM in the strong coupling case (that corresponds to
U � WA) using a GRPA approach [23].
5.2. Approximate analytical approach
Let us consider the results obtained at the Falicov-Kimball limit of the considered
model with U → +∞ in the case when the above described approximate approach
is used instead of the exact one (48). In the Falicov-Kimball limit, JB = 0 and it
138
Green’s functions of infinite-U Hubbard model
leads to ϕA = 0, RA = 0. According to equation (44) the argument in the expression
for RB(ω) (34) should be replaced by ω+iε. Substituting the Green’s function given
by (46) into this expression we obtain
RB(ω + iε) = −nA + nB
2
JA(ω + iε + µB − µA)
+ i
nB − nA
2
Im JA(ω + iε + µB − µA) tanh
β(ω − µA + µB)
2
+
nB − nA
4π
W 2
A(1 − nB) Vp
∫ 1
−1
dx
√
1 − x2 tanh β
2
(xWA
√
1 − nB − µA)
ω + µB − xWA
√
1 − nB
, (51)
where the principle value of the integral is taken. The similar procedure for equation
(39) leads to
V ϕB = −WA(1 − nB)3/2
2π
Vp
∫ 1
−1
dx
√
1 − x2
x
[
1 + eβ(xWA
√
1−nB−µA)
]−1
. (52)
The RB(ω) function provides in this approximation a frequency dispersion of the
Green’s function of localized particles and leads to the broadening of the correspond-
ing density of states into a band, (which is shifted with respect to the initial level
on the − V
1−nA
ϕB value).
The dependence of the chemical potential on the concentration of localized par-
ticles is calculated using expressions (44), (47), (51), (52). In figure 4 the approxi-
mations such as AA, MAA, GH3 and H3 (in the last case the integral terms that
describe the processes of scattering by boson excitations are neglected in the ex-
pressions for Rσ) are compared with the exact results obtained thermodynamically.
The simple AA approximation cannot describe phase transitions because it gives a
monotonically increasing dependence µB = µB(nB). MAA and AA approximations
can describe the thermodynamics of the system at high temperatures or at a neg-
ative value of µA (small concentration of moving particles). The H3 approximation
gives better results for high temperatures but not for the low ones. The best results
are given by the GH3 approximation for high temperatures, as well as for low tem-
peratures in the cases of a nearly full or empty band of the moving particles (and
high values of nB).
In figure 5 the case of a nearly full band of the moving particles is shown. The
µB = µB(nB) dependence is illustrated together with the corresponding curves for
density of states of localized particles in H3 and GH3 approximations:
ρB(ω) = − 1
π
Im G
(a)
B (ω + iε). (53)
One can see that this band can have a complicated structure or even be split in
the H3 approximation, but it does not correspond to the satisfactory µB(UB) de-
pendence. In general, there is a considerable difference between densities of states
ρB(ω) given by the H3 and GH3 approximations, but the band edges are determined
properly even in the H3 approximation.
139
I.V.Stasyuk, O.B.Hera
0,0 0,2 0,4 0,6 0,8 1,0
0,0
0,6
1,2
1,8
2,4
5
4
32
1
1
2
3
4
5
T=0.02
µ
A
=1.2
µ
B
n
B
-1,2 -0,6 0,0 0,6 1,2
0,00
0,03
0,06
0,09
22
T=0.02
µ
A
=1.2
n
B
=0.1
2
1
ρ
B
(ω)
ω+µ
B
1
2
-1,0 -0,5 0,0 0,5 1,0
0
1
2
3
4
5
T=0.02
µ
A
=1.2
n
B
=0.9
2
1
ρ
B
(ω)
ω+µ
B
1
2
0,0 0,2 0,4 0,6 0,8 1,0
0,0
0,6
1,2
1,8
2,4
5
4
3
2
1
1
2
3
4
5
T=0.2
µ
A
=1.2
µ
B
n
B
-1,2 -0,6 0,0 0,6 1,2
0,00
0,03
0,06
0,09
0,12
0,15
22
T=0.2
µ
A
=1.2
n
B
=0.1
2
1
ρ
B
(ω)
ω+µ
B
1
2
-1,0 -0,5 0,0 0,5 1,0
0
1
2
3
4
5
T=0.2
µ
A
=1.2
n
B
=0.9
2
1
ρ
B
(ω)
ω+µ
B
1
2
Figure 5. µB as a function of nB in different approximations (1 – exact result; 2
– AA; 3 – MAA; 4 – H3; 5 – GH3) and corresponding densities of states ρB of
localized particles (1 – GH3; 2 – H3).
In figure 6 the densities of states of localized particles are presented for different
temperatures and particle concentrations. In the cases corresponding to the above-
mentioned criteria ((i) high T ; (ii) low T and chemical potential µA close to the
edges of the band of the moving particles), the obtained densities of states ρB can
be considered as close to the real ones due to a satisfactory description of the µB(nB)
dependences.
Calculation of densities of states of localized particles in the Falicov-Kimball
model had been done earlier within DMFT in [9,10]. The algorithm which leads to
exact results is rather complicated. The plots obtained in [9,10] correspond to finite
values of U and to the symmetrical case of half-filling (the hypercubic lattice with
Gaussian density of states was considered). The band of localized particles is split
and consists of two subbands for U larger than some critical value (U & 5 WA);
for U → +∞, the only lower subband remains. Our results correspond by the
general shape of the curves to the above-mentioned results in the cases illustrated
in figure 5 for T = 0.02 0.2; µA = 1.2; nB = 0.9 and in figure 6 for T = 0.2;
µA = 0; nB = 0.9 within the GH3 approximation (in these cases, there is the best
agreement with the exact results for µB(nB) dependences). Quantitative comparison
of densities of states of localized particles will be possible after doing calculations
with nonperturbed density of state like in [9,10] and for finite values of U .
6. Conclusions
Application of the approximate analytical method of solving the effective single-
site problem within DMFT gives a basic set of equations for evaluating single-particle
140
Green’s functions of infinite-U Hubbard model
-1,0 -0,5 0,0 0,5 1,0
0
3
6
9 T=0.02
µ
A
=0
n
B
=0.9
2
1
ρ
B
(ω)
ω+µ
B
1
2
-1,0 -0,5 0,0 0,5 1,0
0
2
4
6
T=0.2
µ
A
=0
n
B
=0.9
2
1
ρ
B
(ω)
ω+µ
B
1
2
-1,0 -0,5 0,0 0,5 1,0
0
1
2
3
4
5
T=0.02
µ
A
=-0.4
n
B
=0.7
2
1
ρ
B
(ω)
ω+µ
B
1
2
-2 -1 0 1 2
0,0
0,1
0,2
0,3
12
T=0.02
µ
A
=0
n
B
=0.07
2
1
ρ
B
(ω)
ω+µ
B
1
2
-2 -1 0 1 2
0,0
0,3
0,6
0,9 T=0.2
µ
A
=0
n
B
=0.07
2
1
ρ
B
(ω)
ω+µ
B
1
2
-1,0 -0,5 0,0 0,5 1,0
0,0
0,3
0,6
0,9
1,2
1,5
T=0.02
µ
A
=0.4
n
B
=0.7
2
1
ρ
B
(ω)
ω+µ
B
1
2
Figure 6. The approximate densities of states of localized particles (1 – GH3;
2 – H3).
Green’s functions and for investigating an energy spectrum of the asymmetric Hub-
bard model describing the system of particles of two types with different transfer
parameters (tA 6= tB) on a lattice.
The limiting case U = ∞ (exclusion of a double occupation of a site) and tB → 0
(the case of the infinitesimally small mobility of particles of one of the type, when
the model becomes the Falicov-Kimball model) is considered.
Phase transitions are investigated in thermodynamic regimes of the fixed values
of the following quantities: (i) µA, µB; (ii) µA, nB; or (iii) µB, nA. Phase diagrams
describing coexistence curves of homogeneous phases with different nA and nB val-
ues and phase separation regions are obtained. These results extending a space of
thermodynamic parameters complement the existing information about the thermo-
dynamics of the Falicov-Kimball model in the U → +∞ limit.
The set of self-consistent equations, which relates the concentrations (chemical
potentials) of particles of A and B types, coherent potentials JA,B(ω) and shift
constants of energy spectrum ϕA,B, is solved in two cases: (i) with an application
of the exact thermodynamic relations for the Falicov-Kimball model to derive an
equation for concentration nB of localized particles; (ii) by using the approximate
analytical scheme in DMFT. Calculation of µB(nB) dependences being done by the
two mentioned methods allows us to investigate an applicability of this approximate
approach. The results closest to the exact ones are obtained within the GH3 ap-
proximation for high temperatures, as well as for low temperatures in the cases of
nearly full or nearly empty band of the moving particles and for high values of nB.
Densities of states ρB(ω) of localized particles are calculated using the approxi-
mate DMFT scheme. The results obtained in the GH3 approximation can be con-
sidered as the closest to the real ones, when a satisfactory dependence of µB(nB) is
141
I.V.Stasyuk, O.B.Hera
achieved. Extending this approach to the cases of low temperatures and intermediate
band filling requires an improvement of the applied scheme, namely, a more precise
(probably self-consistent) way of taking into account the processes of scattering by
boson excitations for calculating the RB(ω) function which determines the energy
spectrum of localized particles.
Acknowledgements
This work was partially supported by the Fundamental Researches Fund of the
Ministry of Ukraine of Science and Education (Project No. 02.07/266).
References
1. Gruber Ch. Falicov-Kimball models: a partial review of the ground states problem.
Preprint of arXiv.org e-Print archive, cond-mat/9811299, 1998, 18 p.
2. Batista C.D. // Phys. Rev. Lett., 2002, vol. 89, p. 166403.
3. Metzner W., Vollhardt D. // Phys. Rev. Lett., 1989, vol. 62, p. 324.
4. Georges A., Kotliar G., Krauth W., Rosenberg M.J. // Rev. Mod. Phys., 1996, vol. 68,
p. 13.
5. Metzner W. // Phys. Rev. B, 1991, vol. 43, p. 8549.
6. Brandt U., Mielsch C. // Z. Phys. B, 1989, vol. 75, p. 365; 1990, vol. 79, p. 295; 1991,
vol. 82, p. 37.
7. Freericks J.K. // Phys. Rev. B, 1999, vol. 60, p. 1617.
8. Letfulov B.M. // Eur. Phys. J. B, 1999, vol. 11, p. 423–428.
9. Brandt U., Urbanek M.P. // Z. Phys. B, 1992, vol. 89, p. 297–303.
10. Zlatic V., Freericks J.K., Lemanski R., Czycholl G. // Philosophical Magazine B, 2001,
vol. 81, p. 1443–1467.
11. Stasyuk I.V. // Condens. Matter Phys., 2000, vol. 3, p. 437–456.
12. Stasyuk I.V., Shvaika A.M. // Ukrainian Journal of Physics, 2002, vol. 47, p. 975.
13. Tserkovnikov Yu.A. // Teor. Mat. Fiz., 1971, vol. 7, p. 260
14. Plakida N.M. // Phys. Lett. A, 1973, vol. 43, p. 471
15. Plakida N.M. Method of two-time Green’s functions in the theory of anharmonic
crystals. – In: Statistical Physics and Quantum Field Theory, Ed. by Bogolyubov N.N.,
Moscov, 1973, p. 205–240.
16. Potthoff M., Herrmann T., Wegner T., Nolting W. // Phys. Stat. Sol. (b), 1998,
vol. 210, p. 199.
17. Herrmann T., Nolting W. // Rhys. Rev. B, 1996, vol. 53, p. 10579.
18. Falicov L.M., Kimball J.C. // Phys. Rev. Lett., 1969, vol. 22, p. 997.
19. Letfulov B.M. // Eur. Phys. J. B, 1998, vol. 4, p. 447–457.
20. Freericks J.K., Zlatic V. Exact solution of the Falicov-Kimball model with dynamical
mean-field theory. Preprint of arXiv.org e-Print archive, cond-mat/0301188, 2003,
51 p.
21. Stasyuk I.V., Shvaika A.M. // J. Phys. Studies, 1999, vol. 3, p. 177–183.
22. Stasyuk I.V., Shvaika A.M. // Condens. Matter Phys., 1994, vol. 3, p. 133
23. Stasyuk I.V., Shvaika A.M., Tabunshchyk K.V. // Condens. Matter Phys., 1999, vol. 2,
p. 109
142
Green’s functions of infinite-U Hubbard model
24. Stasyuk I.V., Danyliv O.D. // Phys. Stat. Sol. (b), 1999, vol. 219, p. 299.
���������
�
��
� �����������������
��������������� !��"# %$&��'('���
) *�,+
'(��+����.-/�����
U 0�1
(�������(2,�3��"# ��4�
5.�(6879 ���'���"��
:�;=<!;?>#@BA�C)D�E4F%G�;=HI;=JLK�M�A
N O8PRQ�S�QUTVQ�WIX Y%SBZVS#ZU[�OV\^]BO8P8[�_R`BOaScb
PaSaPRQR]Bdfehg�e*i^ZVja`8klOaShm
nBoqp�rqr)s(tc_8X _qmV_uTVv^wVx�_cyqOazBX zBtcZU[�{|[^mar
}9~q�^�4�
�4�B�&�����.�
~8�������h�4�4���.�
g�PaSBd�]�Q�j�Sa��O�`�d�[8\^]�v�t
�^`%�B�q`Bj�\4`�_8SBZU[�j�SaPRQR[�_uT�ycQ�tVP��I\�vq�!\^[BPRv�X \���]BOaO��
\�_c[8b�P8[�j8Q�O8[�{|[�{�ja`�Q�ZU[�_c[�{|[�{�`%YVT!j8Tub�[�d�ScbI�V`aPRQ�SBO8[�Z�wB�![8\^]�v�t�jq[qY%{�vq�R\4`��
ycQ�tVP��9_(ja`Bd�Z�`cb(d�]�QR[8\�T�\�SBO�`Bd�X ��O8[�{|[�P8]Bjq]c\�OatV[�{|[��%[%vq��wR�aW�]BZ�Q�SB_8O�`
[8\.�
O8[�_uTcYVvB[�_R`#Yc`c\4`a�V`#W�[�j�d�TVv� (ycQ�tVP��I_�QR]Bj�d�X O�`cb(\^[��%[�d�X ��O8[�{|[IW�]Bj�d�Xu�%[h�
vq��w�¡�vq�¢k k�jq[qY%_q£ �8Y%Z�T¢_8SBZU[�j�SaPRQV`BO8[¤O�`%�%v�Sc��]BOaSB¥¦`BO�`�v�X Q�Sa��OaSB¥§d�]�QR[8\�m
�qZVSB¥��q`%YVT�ycQ�tVP��&O�`�QR]cb%OaX zBXaO8]%Y%_8X \�OaScb!W�T%OaZVzBX ¥�¨uj�X O�`hw�©)]B¥��qX \�b%X \ª`B�qjq[h�
�%T�ycQ�tVP��9O�`#d�[8\^]�v�X|Y(�B]%Y%d�]c��OaSBd
U
T�{�ja`BOaSBzBX�«�`�v�X ZU[�_R`��¬�X d��q`�vq`�®¯ZU[%v�S
j8TubVv�SB_8X PRQ�t�X [�OaX _�[8\�O8[�{|[IY
P8[�j8Q�X _�y��B]%Y%d�]c��O8[�d)`�vB[� 9°Lw8¬B[�Oaza]BORQ�ja`BzBX ¥BOaX
Yc`�vB]c��O8[BPRQ�Xb%X d�X ��OaScbI�%[%QR]BOazBX `�v�X _9jq[qY%ja`cb�[�_uT% �Q�tVP��
Y�\^[��%[�d�[�{|[� f[8\�O8[h�
�V`aPRQ�SBOaZU[�_8Scb9W�T%OaZVzBX ¥�¨uj�X O�`h±8j�X Y%OaXVO�`%�%v�Sc��]BOaO����%[�j�X _8Oa
�Q�tVP��9Y�QR[B��OaS��
d�S�jq]%YVTVv�tQV`�QV`Bd�Shma[%Q�j�SBd)`BOaSBd�S*QR]Bj�d�[8\�SBO�`Bd�X ��OaSBd¤²�vq�Rb�[�d�wB³�Q�j�SBd)`��
O8[�{´T�PRQ�SBOaSIPRQV`BOaX _�\�vq�(vB[�Z�`�v�X Y�[�_R`BOaScb��V`aPRQ�SBO8[�Z��qj�S#j�X Y%OaScb�QR]Bd��%]Bja`�QUTa�
ja`cb�X�ZU[�Oaza]BORQ�ja`BzBX �RbBw´¡
[BPRv�X \���]BO8[�W9`%Y�[�_8X��%]Bjq]cb�[8\�S)T�{�ja`BOaSBzBX´«�`�v�X ZU[�_R`��
¬�X d��q`�vq`�\�vq�#j�X Y%OaScb�QR]Bj�d�[8\�SBO�`Bd�X ��OaScb9jq]c��SBd�X _qw
µh¶�·�¸a¹^ºB»u¼a¶^¹^ºq½�¾8¿BÀ�ÁBÂ(ÃBÄLÅ4Á4Æ�Çq¿9Â(ÈuÉ�ÃRÊ�Ë)Ì^¿BÍBÍa¿�Å%É.¿qÎ�Â(ÈuÉ�ÃRÊ�Ë
Ï ¿cÊ�Ð ÑRÈaÒ8¿hÓLÔBÐ Â(Ía¿cÊ�¿qÎaÄÕÃqÈ�Å4Ð Ö)É.ÁBÇq¿aÂ�Ð Æ�Ç�Èa×�È�À%ÃcÅ�ÃUÉ.ÇqË8Èa×�È
Ø�ÈcÊBÖaÎaÙ�ÚRÇqÑuÛ�Ð Ü
Ý´Å4Ð Çq¿qÎ%Ù(¿aÞ8ÈaÒ8Ð�Ø�ÃcÅ�ÃVßaÈuÉ.Á
àBá�â�ã�¾qä�åÕæ�å�çqæéèaêBÎBçaëBæéì�çqæéè8í�Î�çaëBæ¯ä8çqæéè8î
143
144
|