Strong coupling Hartree-Fock approximation in the dynamical mean-field theory
In the limit of infinite spatial dimensions, a thermodynamically consistent theory, which is valid for arbitrary value of the Coulombic interaction ( U < ∞ ), is built for the Hubbard model when the total auxiliary single-site problem exactly splits into four subspaces with different “vacuum s...
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irk-123456789-1197532017-06-09T03:03:02Z Strong coupling Hartree-Fock approximation in the dynamical mean-field theory Shvaika, A.M. In the limit of infinite spatial dimensions, a thermodynamically consistent theory, which is valid for arbitrary value of the Coulombic interaction ( U < ∞ ), is built for the Hubbard model when the total auxiliary single-site problem exactly splits into four subspaces with different “vacuum states”. Some analytical results are given for the Hartree-Fock approximation when the 4- pole structure for Green’s function is obtained: two poles describe contribution from the Fermi liquid component, which is ferromagnetic and dominant for small electron and hole concentrations (“overdoped case” of high- Tc ’s), whereas other two describe contribution from the non-Fermi liquid, which is antiferromagnetic and dominant close to half filling (“underdoped case”). Для моделі Хаббарда побудовано термодинамічно самоузгоджену теорію, яка застосовна для довільних значень кулонівської кореляції ( U < ∞ ) в границі безмежної розмірності простору, коли повна допоміжна задача точно розпадається на чотири підпростори з різними “вакуумними станами”. Наведено ряд аналітичних результатів для наближення Хартрі-Фока, коли отримується чотириполюсна структура для функцій Гріна: два полюси описують внески Фермі-рідинної компоненти, яка є феромагнітною і домінує при малих концентраціях електронів або дірок (“перелегований випадок” ВТНП), а інші два – внески від не-Фермі рідини, яка є антиферомагнітною і домінує поблизу половинного заповнення (“недолегований випадок”). 2001 Article Strong coupling Hartree-Fock approximation in the dynamical mean-field theory / A.M. Shvaika // Condensed Matter Physics. — 2001. — Т. 4, № 1(25). — С. 85-92. — Бібліогр.: 16 назв. — англ. 1607-324X PACS: 71.10.Fd, 71.15.Mb, 05.30.Fk, 71.27.+a DOI:10.5488/CMP.4.1.85 http://dspace.nbuv.gov.ua/handle/123456789/119753 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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English |
| description |
In the limit of infinite spatial dimensions, a thermodynamically consistent
theory, which is valid for arbitrary value of the Coulombic interaction ( U <
∞ ), is built for the Hubbard model when the total auxiliary single-site problem
exactly splits into four subspaces with different “vacuum states”. Some
analytical results are given for the Hartree-Fock approximation when the 4-
pole structure for Green’s function is obtained: two poles describe contribution
from the Fermi liquid component, which is ferromagnetic and dominant
for small electron and hole concentrations (“overdoped case” of high- Tc ’s),
whereas other two describe contribution from the non-Fermi liquid, which
is antiferromagnetic and dominant close to half filling (“underdoped case”). |
| format |
Article |
| author |
Shvaika, A.M. |
| spellingShingle |
Shvaika, A.M. Strong coupling Hartree-Fock approximation in the dynamical mean-field theory Condensed Matter Physics |
| author_facet |
Shvaika, A.M. |
| author_sort |
Shvaika, A.M. |
| title |
Strong coupling Hartree-Fock approximation in the dynamical mean-field theory |
| title_short |
Strong coupling Hartree-Fock approximation in the dynamical mean-field theory |
| title_full |
Strong coupling Hartree-Fock approximation in the dynamical mean-field theory |
| title_fullStr |
Strong coupling Hartree-Fock approximation in the dynamical mean-field theory |
| title_full_unstemmed |
Strong coupling Hartree-Fock approximation in the dynamical mean-field theory |
| title_sort |
strong coupling hartree-fock approximation in the dynamical mean-field theory |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2001 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119753 |
| citation_txt |
Strong coupling Hartree-Fock approximation in the dynamical mean-field theory / A.M. Shvaika // Condensed Matter Physics. — 2001. — Т. 4, № 1(25). — С. 85-92. — Бібліогр.: 16 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT shvaikaam strongcouplinghartreefockapproximationinthedynamicalmeanfieldtheory |
| first_indexed |
2025-07-08T16:32:21Z |
| last_indexed |
2025-07-08T16:32:21Z |
| _version_ |
1837097122615263232 |
| fulltext |
Condensed Matter Physics, 2001, Vol. 4, No. 1(25), pp. 85–92
Strong coupling Hartree-Fock
approximation in the dynamical
mean-field theory
A.M.Shvaika
Institute for Condensed Matter Physics
of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 79011 Lviv, Ukraine
Received August 23, 2000
In the limit of infinite spatial dimensions, a thermodynamically consistent
theory, which is valid for arbitrary value of the Coulombic interaction ( U <
∞ ), is built for the Hubbard model when the total auxiliary single-site prob-
lem exactly splits into four subspaces with different “vacuum states”. Some
analytical results are given for the Hartree-Fock approximation when the 4-
pole structure for Green’s function is obtained: two poles describe contribu-
tion from the Fermi liquid component, which is ferromagnetic and dominant
for small electron and hole concentrations (“overdoped case” of high- Tc ’s),
whereas other two describe contribution from the non-Fermi liquid, which
is antiferromagnetic and dominant close to half filling (“underdoped case”).
Key words: dynamical mean-field theory, Hartree-Fock approximation,
Hubbard model, (anti)ferromagnetism, strong coupling
PACS: 71.10.Fd, 71.15.Mb, 05.30.Fk, 71.27.+a
1. Introduction
In the last decade, essential achievements in the theory of strongly correlated
electron systems are connected with the development of the dynamical mean-field
theory (DMFT) proposed by Metzner and Vollhardt [1] for the Hubbard model (see
also [2,3] and references therein). There are no restrictions on the U value within this
theory and it turns out to be useful for intermediate coupling (U ∼ t) for which it
ensures a correct description of the metal-insulator phase transition and determines
the region of the Fermi-liquid behaviour of the electron subsystem. Moreover, some
classes of binary-alloy-type models (e.g., the Falicov-Kimball model) can be studied
almost analytically within DMFT [4]. But in the case of the Hubbard model, the
treatment of the effective single impurity Anderson model is very complicated and
mainly computer simulations are used, which calls for the development of analytical
approaches [5].
c© A.M.Shvaika 85
A.M.Shvaika
Such approaches can be built with a systematic perturbation expansion in terms
of the electron hopping [6] using a diagrammatic technique for Hubbard operators
[7,8]. One of them was proposed for the Hubbard (U = ∞ limit) and t − J models
[9]. The lack of such an approach is connected with the concept of a “hierarchy”
system for the Hubbard operators when the form of the diagrammatic series and
the final results strongly depend on the system of the pairing priority for Hubbard
operators. On the other hand, it is difficult to generalize it to the case of arbitrary U .
In our previous paper [10] we developed, for the Hubbard-type models, a rigorous
perturbation theory scheme in terms of electron hopping which is based on the
Wick theorem for Hubbard operators [7,8], which is valid for arbitrary values of U
(U < ∞) and which does not depend on the “hierarchy” system for X operators. In
the limit of infinite spatial dimensions, this analytical scheme permits us to build
a self-consistent Kadanoff-Baym type theory [11] for the Hubbard model and some
analytical results are given for simple approximations. Here we shall consider possible
magnetic orderings in the Hartree-Fock type approximation.
2. Perturbation theory in terms of electron hopping
We consider the lattice electronic system described by the statistical operator:
ρ̂ = e−βĤ0 σ̂(β), σ̂(β) = T exp
−
β∫
0
dτ
β∫
0
dτ ′
∑
ijσ
tσij(τ − τ ′)a†iσ(τ)ajσ(τ
′)
, (2.1)
where Ĥ0 =
∑
i Ĥi is a sum of the single-site contributions, and for the Hubbard
model we must put
Hi = Uni↑ni↓ − µ(ni↑ + ni↓)− h(ni↑ − ni↓), tσij(τ − τ ′) = tijδ(τ − τ ′). (2.2)
It is supposed that we know the eigenvalues and eigenstates of the zero-order
Hamiltonian: Hi|i, p〉 = λp|i, p〉, and one can introduce Hubbard operators X̂pq
i =
|i, p〉〈i, q| in terms of which the zero-order Hamiltonian is diagonal
H0 =
∑
i
∑
p
λpX̂
pp
i . (2.3)
For the Hubbard model we have four states |i, p〉 = |i, ni↑, ni↓〉: |i, 0〉 = |i, 0, 0〉
(empty site), |i, 2〉 = |i, 1, 1〉 (double occupied site), |i, ↑〉 = |i, 1, 0〉 and |i, ↓〉 =
|i, 0, 1〉 (sites with spin-up and spin-down electrons) with energies λ0 = 0, λ2 =
U − 2µ, λ↓ = h − µ, and λ↑ = −h − µ. The connection between the electron
operators and the Hubbard operators is the following:
niσ = X22
i +Xσσ
i ; aiσ = X0σ
i + σX σ̄2
i . (2.4)
The expression for 〈σ(β)〉0 is a series of terms that are products of the hopping
integrals and averages of the electron creation and annihilation operators or Hubbard
86
Strong coupling Hartree-Fock approximation in the dynamical mean-field theory
operators, that are calculated with the use of the corresponding Wick’s theorem [7,8],
and can be written as [10]:
〈σ̂(β)〉0 =
〈
exp
{
− − 1
2
− 1
3
− . . . (2.5)
− © − © − . . .− © − . . .
}〉
0
,
where arrows denote the zero-order Green’s functions
gpq(ωn) =
1
iωn − λpq
, (2.6)
wavy lines denote hopping integrals and �, . . . stand for complicated “n vertices”.
Each vertex (many-particle single-site Green’s function) is multiplied by a diagonal
Hubbard operator denoted by a circle.
3. Irreducible many-particle Green’s functions
For the Hubbard model, expressions for the two-vertex
=
∑
p
X̂pp
i gσ(p)(ωn), (3.1)
the four-vertex
© =
∑
p
X̂pp
i gσ(p)(ωn)gσ(p)(ωn+m)Ũσσ̄(p)(ωn, ωl|ωm)gσ̄(p)(ωl)gσ̄(p)(ωl+m), (3.2)
and for the vertices of higher order possess one significant feature [10]. They decom-
pose into four terms with different diagonal Hubbard operators X pp, which project
our single-site problem onto certain “vacuum” states (subspaces), and zero-order
Green’s functions
gσ(p)(ωn) =
{
gσ0(ωn) for p = 0, σ
g2σ̄(ωn) for p = σ̄, 2
, (3.3)
which describe all possible excitations and scattering processes around these “vac-
uum” states. Here
Ũσσ̄(p)(ωn, ωl|ωm) =
{
U ± U2g20(ωn+l+m) for p = 0, 2
U ± U2gσσ̄(ωn−l) for p = σ, σ̄
(3.4)
is a renormalized Coulombic interaction in the subspaces. In diagrammatic nota-
tions, expression (3.2) can be represented as
= ± , (3.5)
87
A.M.Shvaika
where dots denote the Coulombic correlation energy U and the dashed arrows denote
bosonic zero-order Green’s functions: doublon g20(ωm) or magnon gσσ̄(ωm). The
contributions to the six-vertex can be presented by the following diagrams:
(3.6)
where the first three diagrams contain the internal vertices of the same type as in
equation (3.5). So, we can introduce primitive vertices , , by which one
can construct all n vertices in the expansion (2.5).1
4. Dynamical mean-field theory
Within the framework of the considered perturbation theory in terms of electron
hopping a single-electron Green’s function can be presented in the form
Gσ(ωn,k) =
1
Ξ−1
σ (ωn,k)− tk
, (4.1)
where we introduce an irreducible part Ξσ(ωn,k) of Green’s function which, in gen-
eral, is not local. In the case of infinite dimensions d → ∞ one should scale the
hopping integral according to tij → tij/
√
d in order to obtain finite density-of-states
and it was shown by Metzner in his pioneer work [12] that in this limit the irreducible
part becomes local Ξijσ(τ − τ ′) = δijΞσ(τ − τ ′) or Ξσ(ωn,k) = Ξσ(ωn) and such a
site-diagonal function can be calculated by mapping the d → ∞ lattice problem
(2.1) onto the atomic model with the auxiliary Kadanoff-Baym field [4]
tσij(τ − τ ′) = δijJσ(τ − τ ′). (4.2)
The self-consistent set of equations for Ξσ(ωn) and Jσ(ωn) (e.g., see [3] and references
therein) is the following:
1
N
∑
k
1
Ξ−1
σ (ωn)− tk
=
1
Ξ−1
σ (ωn)− Jσ(ωn)
= G(a)
σ (ωn, {Jσ(ωn)}), (4.3)
where G
(a)
σ (ωn, {Jσ(ωn)}) is the Green’s function for the atomic limit (4.2).
The grand canonical potential for the lattice is connected with the one for the
atomic limit by the expression [4]
Ω
N
= Ωa −
1
β
∑
nσ
{
lnG(a)
σ (ωn)−
1
N
∑
k
lnGσ(ωn,k)
}
. (4.4)
On the other hand, we can write, for the grand canonical potential for the atomic
limit Ωa, the same expansion as in equation (2.5) but with diagonal X operators at
1For n vertices of higher order a new primitive vertices can appear but we do not check this
due to the rapid increase of the algebraic calculations with the increase of n.
88
Strong coupling Hartree-Fock approximation in the dynamical mean-field theory
the same site. We can reduce their product to a single X operator that can be taken
outside the exponent in (2.5), and its average is equal to 〈X pp〉0 = e−βλp/
∑
q e
−βλq .
Finally, for the grand canonical potential for the atomic limit we get [10]
Ωa = − 1
β
ln
∑
p
e−βΩ(p), (4.5)
where Ω(p) are the “grand canonical potentials” for the subspaces.
Now we can find the single-electron Green’s function for the atomic limit by
G(a)
σ (τ − τ ′) =
δΩa
δJσ(τ − τ ′)
=
∑
p
wpGσ(p)(τ − τ ′), (4.6)
where Gσ(p)(τ − τ ′) are single-electron Green functions for the subspaces character-
ized by the “statistical weights” wp = e−βΩ(p)/
∑
q e
−βΩ(q) and our single-site atomic
problem exactly splits into four subspaces p = 0, 2, ↓, ↑.
The fermionic Green’s function in subspaces can be written as
Gσ(p)(ωn) =
1
Ξ−1
σ(p)(ωn)− Jσ(ωn)
, Ξ−1
σ(p)(ωn) = iωn+µσ−Un
(0)
σ̄(p)−Σσ(p)(ωn), (4.7)
where n
(0)
σ(p) = −dλp/dµσ = 0 for p = 0, σ̄ and 1 for p = 2, σ is an occupation of the
state |p〉 by the electron with spin σ, and the self-energy Σσ(p)(ωn) depends on the
hopping integral Jσ′(ωn′) only through quantities
Ψσ′(p)(ωn′) = Gσ′(p)(ωn′)− Ξσ′(p)(ωn′). (4.8)
Now, one can reconstruct the expressions for the grand canonical potentials Ω(p)
in the subspaces from the known structure of Green’s functions:
Ω(p) = λp−
1
β
∑
nσ
ln
(
1− Jσ(ωn)Ξσ(p)(ωn)
)
− 1
β
∑
nσ
Σσ(p)(ωn)Ψσ(p)(ωn)+Φ(p), (4.9)
where Φ(p) is some functional, such that its functional derivative with respect to Ψ
produces the self-energy: δΦ(p)/δΨσ(p)(ωn) = Σσ(p)(ωn). So, if we can find or con-
struct the self-energy Σσ(p)(ωn), we can find Green’s functions and grand canonical
potentials for the subspaces and, according to equations (4.5) and (4.6), we can solve
atomic problems.
From the grand canonical potential (4.5) and (4.9) we get the following mean
values
nσ =
∑
p
wpnσ(p), nσ(p) = n
(0)
σ(p) +
1
β
∑
n
Ψσ(p)(ωn)−
∂Φ(p)
∂µσ
, (4.10)
where in the last term the partial derivative is taken over µσ not in the chains (4.8).
For the Falicov-Kimball model J↓(ωn) = 0 and, as a result, Σ↑(p)(ωn) ≡ 0 and
Ξ↑(p)(ωn) = g↑(p)(ωn) which immediately gives results of [4] (see also [13]).
For the Hubbard model, there is no exact expression for the self-energy but the
set of equations (4.7) and (4.9) permits one to construct different self-consistent
approximations.
89
A.M.Shvaika
5. Hartree-Fock approximation
One of the possible approximations is to construct the equation for the self-energy
in the following form:
Σσ(p)(ωn) =
1
β
∑
n′
UΨσ̄(p)(ωn′), (5.1)
which, together with the expression for mean values
nσ(p) = n
(0)
σ(p) +
1
β
∑
n
Ψσ(p)(ωn), (5.2)
gives, for the Green’s functions in the subspaces, an expression in the Hartree-Fock
approximation:
Gσ(p)(ωn) =
1
iωn + µσ − Unσ̄(p) − Jσ(ωn)
. (5.3)
Now, the grand canonical potentials in the subspaces are equal
Ω(p) = λp−
1
β
∑
nσ
ln
[
1− Jσ(ωn)Ξσ(p)(ωn)
]
−U
(
nσ(p) − n
(0)
σ(p)
)(
nσ̄(p) − n
(0)
σ̄(p)
)
(5.4)
and the Green’s function for the atomic problem (4.6) yield a four-pole structure,
that, in contrast to the alloy-analogy solution Σσ(p)(ωn) = 0, possesses the correct
Hartree-Fock limit for a small Coulombic interaction U ≪ t. On the other hand, in
the same way as an alloy-analogy solution, it describes the metal-insulator transition
with the change of U .
In [10] it was shown that the main contributions into the total spectral weight
function ρσ(ω) =
1
π
ℑG(a)
σ (ω−i0+) come from the subspaces p = 0 for the low electron
concentrations (n < 2
3
, µ < 0), p = 2 for the low hole concentrations (2 − n < 2
3
,
µ > U) and p = σ, σ̄ for the intermediate values. For the small electron or hole
concentrations, the Green’s function for the atomic problem (4.6) possess the correct
Hartree-Fock limits as well. It was supposed that the Hubbard model describes
strongly-correlated electronic systems that contain four components (subspaces).
Subspaces p = 0 and p = 2 describe the Fermi-liquid component (electron and hole,
respectively) which is dominant for the small electron and hole concentrations, when
the chemical potential is close to the bottom of the lower band or top of the upper
one (“overdoped regime” of high-Tc’s). On the other hand, subspaces p =↑ and ↓
describe the non-Fermi-liquid (strongly correlated, e.g., RVB) component, which is
dominant close to half-filling (“underdoped regime”).
Here we consider possible magnetic orderings. At low temperatures, the p =
0 and p = 2 components for the low electron or hole concentrations are in the
ferromagnetic state, while the non-Fermi-liquid one is antiferromagnetic (AF) close
to half-filling, see figure 1. For the intermediate concentration values, the picture
is very complicated, even frustrated. It is due to the fact that equations for the
mean values (5.2) have several solutions in this region, which, on the other hand,
are mutually connected with the dynamical mean field Jσ(ωn). It is difficult to
90
Strong coupling Hartree-Fock approximation in the dynamical mean-field theory
0.0 0.5 1.0 1.5 2.0
0.00
0.02
0.04
0.06
0.08
0.10
0.12
mF
n 0.0 0.5 1.0 1.5 2.0
0.0
2.0x10
-5
4.0x10
-5
6.0x10
-5
8.0x10
-5
1.0x10
-4
2.0x10
-3
3.0x10
-3
mAF
n
Figure 1. Ferromagnetic mF and antiferromagnetic mAF order parameters vs
electron concentration for U = 1.56, T = 0.14.
determine the ground state for this, possibly “pseudo-gap”, region, which is located
between the ferromagnetic and antiferromagnetic phases.
In figure 2 we presented the phase diagram (T, U) – the temperature of the
AF ordering vs correlation energy U , which is in a qualitative agreement with the
results of [14,15] and reproduces the results of the Hartree-Fock theory and mean
field approximation for U ≪ t and U ≫ t, respectively. Our results for the AF
critical temperature for small U are higher than the one of the Quantum Monte
Carlo simulations [14] by about a factor at three that describes the reduction of the
Hartree-Fock solution by the lowest order quantum fluctuations [16].
0 1 2 3 4 5
0.00
0.05
0.10
0.15
PM
AF
T
U
Figure 2. Phase diagram (T,U) at half-filling n = 1 (AF – antiferromagnetic
phase, PM – paramagnetic phase).
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5. Gebhard F. The Mott Metal-Insulator Transition: Models and Methods. Berlin,
Springer-Verlag, 1997.
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Phys. USSR, 1974, vol. 19, p. 616].
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tems. New York, Consultants Bureau, 1989.
9. Izyumov Yu.A., Letfulov B.M. // J. Phys.: Condens. Matter, 1990, vol. 2, p. 8905;
Izyumov Yu.A., Letfulov B.M., Shipitsyn E.V., Bartkowiak M., Chao K.A. // Phys.
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vol. 127, p. 1391.
12. Metzner W. // Phys. Rev. B, 1991, vol. 43, p. 8549.
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14. Pruschke T., Cox D.L., Jarrell M. // Phys. Rev. B, 1993, vol. 47, p. 3553.
15. Kakehashi Y., Hasegawa H. // Phys. Rev. B, 1988, vol. 37, p. 7777; Rozenberg M.J.,
Kotliar G., Zhang X.Y. // ibid., 1994, vol. 49, p. 10181.
16. van Dongen P.G.J. // Phys. Rev. Lett., 1991, vol. 67, p. 757.
Наближення сильного зв’язку типу Хартрі-Фока
в теорії динамічного середнього поля
А.М.Швайка
Інститут фізики конденсованих систем НАН України,
79011 Львів, вул. Свєнціцького, 1
Отримано 23 серпня 2000 р.
Для моделі Хаббарда побудовано термодинамічно самоузгоджену
теорію, яка застосовна для довільних значень кулонівської кореляції
( U < ∞ ) в границі безмежної розмірності простору, коли повна до-
поміжна задача точно розпадається на чотири підпростори з різни-
ми “вакуумними станами”. Наведено ряд аналітичних результатів для
наближення Хартрі-Фока, коли отримується чотириполюсна струк-
тура для функцій Гріна: два полюси описують внески Фермі-рідинної
компоненти, яка є феромагнітною і домінує при малих концентраці-
ях електронів або дірок (“перелегований випадок” ВТНП), а інші два
– внески від не-Фермі рідини, яка є антиферомагнітною і домінує по-
близу половинного заповнення (“недолегований випадок”).
Ключові слова: теорія динамічного середнього поля, наближення
Хартрі-Фока, модель Хаббарда, (анти)феромагнетизм, сильний
зв’язок
PACS: 71.10.Fd, 71.15.Mb, 05.30.Fk, 71.27.+a
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