Ground state ferromagnetism in a generalized Hubbard model with strong correlations
The present paper considers the ground state ferromagnetic ordering in a generalized Hubbard model with correlated hopping and interatomic exchange interaction in the case of strong correlations at non-integer band filling. The quasi-particle energy spectrum is obtained by means of a generalized...
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| Zitieren: | Ground state ferromagnetism in a generalized Hubbard model with strong correlations / L. Didukh, O. Kramar // Condensed Matter Physics. — 2003. — Т. 6, № 1(33). — С. 145-158. — Бібліогр.: 46 назв. — англ. |
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irk-123456789-1206992017-06-13T03:06:00Z Ground state ferromagnetism in a generalized Hubbard model with strong correlations Didukh, L. Kramar, O. The present paper considers the ground state ferromagnetic ordering in a generalized Hubbard model with correlated hopping and interatomic exchange interaction in the case of strong correlations at non-integer band filling. The quasi-particle energy spectrum is obtained by means of a generalized mean-field approximation. The ground state energy of the system and the condition of ferromagnetic state realization are calculated. The dependences of magnetization on energy parameters and band-filling are obtained. В роботі розглядається основний феромагнітний стан для узагальненої моделі Габбарда з корельованим переносом та міжатомною обмінною взаємодією у випадку сильних електронних кореляцій та нецілого заповнення зони. З використанням узагальненого наближення середнього поля отримано квазічастинковий енергетичний спектр. Розраховано енергію основного стану та умову реалізації феромагнітного впорядкування. Отримано залежності намагніченості системи від значень енергетичних параметрів моделі та заповнення зони. 2003 Article Ground state ferromagnetism in a generalized Hubbard model with strong correlations / L. Didukh, O. Kramar // Condensed Matter Physics. — 2003. — Т. 6, № 1(33). — С. 145-158. — Бібліогр.: 46 назв. — англ. 1607-324X PACS: 71.10.Fd, 75.10.-b, 75.20.En DOI:10.5488/CMP.6.1.145 http://dspace.nbuv.gov.ua/handle/123456789/120699 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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The present paper considers the ground state ferromagnetic ordering in
a generalized Hubbard model with correlated hopping and interatomic exchange
interaction in the case of strong correlations at non-integer band
filling. The quasi-particle energy spectrum is obtained by means of a generalized
mean-field approximation. The ground state energy of the system
and the condition of ferromagnetic state realization are calculated. The dependences
of magnetization on energy parameters and band-filling are obtained. |
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Article |
| author |
Didukh, L. Kramar, O. |
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Didukh, L. Kramar, O. Ground state ferromagnetism in a generalized Hubbard model with strong correlations Condensed Matter Physics |
| author_facet |
Didukh, L. Kramar, O. |
| author_sort |
Didukh, L. |
| title |
Ground state ferromagnetism in a generalized Hubbard model with strong correlations |
| title_short |
Ground state ferromagnetism in a generalized Hubbard model with strong correlations |
| title_full |
Ground state ferromagnetism in a generalized Hubbard model with strong correlations |
| title_fullStr |
Ground state ferromagnetism in a generalized Hubbard model with strong correlations |
| title_full_unstemmed |
Ground state ferromagnetism in a generalized Hubbard model with strong correlations |
| title_sort |
ground state ferromagnetism in a generalized hubbard model with strong correlations |
| publisher |
Інститут фізики конденсованих систем НАН України |
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2003 |
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http://dspace.nbuv.gov.ua/handle/123456789/120699 |
| citation_txt |
Ground state ferromagnetism in a
generalized Hubbard model with strong
correlations / L. Didukh, O. Kramar // Condensed Matter Physics. — 2003. — Т. 6, № 1(33). — С. 145-158. — Бібліогр.: 46 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT didukhl groundstateferromagnetisminageneralizedhubbardmodelwithstrongcorrelations AT kramaro groundstateferromagnetisminageneralizedhubbardmodelwithstrongcorrelations |
| first_indexed |
2025-07-08T18:25:45Z |
| last_indexed |
2025-07-08T18:25:45Z |
| _version_ |
1837104257578303488 |
| fulltext |
Condensed Matter Physics, 2003, Vol. 6, No. 1(33), pp. 145–158
Ground state ferromagnetism in a
generalized Hubbard model with strong
correlations
L.Didukh∗, O.Kramar
Ternopil State Technical University, Department of Physics,
56 Ruska Str., 46001 Ternopil, Ukraine
Received December 20, 2002
The present paper considers the ground state ferromagnetic ordering in
a generalized Hubbard model with correlated hopping and interatomic ex-
change interaction in the case of strong correlations at non-integer band
filling. The quasi-particle energy spectrum is obtained by means of a gen-
eralized mean-field approximation. The ground state energy of the system
and the condition of ferromagnetic state realization are calculated. The de-
pendences of magnetization on energy parameters and band-filling are ob-
tained.
Key words: narrow energy band, ground state energy, ferromagnetism
PACS: 71.10.Fd, 75.10.-b, 75.20.En
1. Introduction
In recent years, intensive studies have been carried out concerning the electrical
and the magnetic properties of the materials with a narrow energy band [1–3]. The
narrow band model proposed by Hubbard [4] and its generalizations still attract
great attention of the researchers (for review see references [5,6]). The Hubbard
model is very rich in its physical contents in spite of its relative simplicity. In the
framework of this model, an antiferromagnetic solution has been obtained, as well as
a metal to insulator transition has been described. The conception of the Hubbard
subbands is very useful in understanding the physical properties of narrow band
materials, in particular, of the materials with high-temperature superconductivity.
Nevertheless, there are a lot of problems in the physics of narrow energy bands,
which are convenient to study in the framework of the Hubbard model and its
generalizations but those problems have not yet been solved. One of them is the
problem of ferromagnetism in a narrow energy band.
∗E-mail: didukh@tu.edu.te.ua
c© L.Didukh, O.Kramar 145
L.Didukh, O.Kramar
The Hubbard model with a partially filled band and the intra-atomic Coulomb
repulsion energy U → ∞ is the most favorable situation for ferromagnetism. In this
case the antiferromagnetic exchange is suppressed and the effective Hamiltonian can
be rewritten in the form (for band filling n < 1)
Heff =
∑
ij
′
tijc
+
iσcjσ , (1)
where ciσ = (1 − niσ̄)aiσ, a+
iσ and aiσ are the creation and destruction operators
of electron on site i, σ =↑, ↓; σ̄ denotes the spin projection which is opposite to σ,
niσ = a+
iσaiσ is the operator of the number of electrons with spin σ on site i. In spite of
the large amount of papers where the Hamiltonian (1) has been studied, there is no
decisive evidence for the ferromagnetism existence in such a system, though within
some approximations (see below) the ferromagnetic solution has been obtained.
The Hamiltonian (1) yields the ferromagnetic solution already within Hubbard-I
approximation [4] if the density of states with peculiarities is used. It is caused by
the fact that the hopping of electrons in Hamiltonian (1) is described by quasi-fermi
operators and ferromagnetism in the Hubbard model at U → ∞ and n 6= 1 is the
kinematic effect due to the non-fermi character of the commutation relations for the
operators c+
iσ and ciσ (it should be noted in the latter context that the “kinematic
superconductivity” in the Hubbard model in some approximations [7] is realized).
For the model, described by Hamiltonian (1), the exact Nagaoka’s result [8] is
available; this result states that for the alternating lattices in the case of Coulomb
repulsion U → ∞ and exactly one electron above or below half-filling the ground
state is ferromagnetic with a maximum total spin. The authors of reference [9] have
generalized the Nagaoka’s theorem on ferromagnetism in the Hubbard model (with
one hole in a half-filled band) to the case where other nearest-neighbor Coulomb
interactions (the interatomic Coulomb interaction, some type of correlated hopping,
the exchange interaction and hopping of double occupancies) are included. The
obtained results show that for non-zero interatomic exchange interactions J the
saturated ferromagnetic ground state is stable at finite U > Uc (Uc is a critical value
of intra-atomic Coulomb repulsion dependent on the lattice type).
The study of ferromagnetism in the case of a finite number of holes in the frame-
work of the approximations which improve the Hubbard-I approximation (see, for
example [10–12]), leads to the conclusion that in the Hubbard model with a par-
tially filled band (within some concentration interval) and with the strong Coulomb
correlation, a ferromagnetic solution is possible. The important feature of the cit-
ed works (similar in the ideology to the work by Roth [13]) is the presence of a
spin-dependent shift of the band centers; this is only a distinction [14] of the noted
approximations from the Hubbard-I approximation. The developments of such ap-
proximations [15,16] in the framework of modified perturbation theory [14] make it
possible to obtain a ferromagnetic solution in the Hubbard model. The results of
these considerations agree with the results of numerical finite-temperature Monte-
Carlo simulations solving the dynamical mean-field theory (DMFT) equations [17].
The authors of references [18,19] have obtained the ferromagnetic solution at in-
146
Ground state ferromagnetism in a generalized Hubbard model
termediate U -values and in a broad range of electron concentrations using a non-
symmetrical density of states with the peak near the band edge (in the framework
of DMFT).
At the same time, in a series of papers the question concerning the existence
of ferromagnetic solution in the Hubbard model (in particular, for the case de-
scribed by Hamiltonian (1)) is disputed. It should be noted that the absence of
ferromagnetism was proved in the Hubbard model in one-dimensional [20] and in
two-dimensional [21] cases as well as in the case n = 1 and at arbitrary finite U [22];
in the latter context we also note the result of reference [23] where the absence of fer-
romagnetism has been stated in the Hubbard model at an arbitrary density of states.
The ferromagnetic ordering is not realized in the case of small electron concentration
n � 1 (the gas limit) [24] as well. The absence of ferromagnetism has been also stat-
ed in reference [25], where the calculation of magnetic susceptibility in the random
phase approximation has been done. The extension of Nagaoka’s theorem to the case
of a system with a finite number of holes is not trivial whereas the thermodynamic
limit for half-filling does not exist. Besides, in a series of papers [26–28] it has been
emphasized that the increase of hole concentration destabilizes the ferromagnetic
ordering. In particular, the results of reference [27] show that Nagaoka’s state is not
realized. It should be noted that approximations [4,13,29,30] (and the papers cited
above which follow the ideology of Roth) for the case of Mott-Hubbard insulator
(n = 1 and U → ∞) do not provide the Curie law for magnetic susceptibility [31].
It can be shown in the following way: if one generalizes Hamiltonian (1) by taking
into account the interatomic exchange interaction in the mean-field approximation
one obtains incorrect equations for magnetization and Curie temperature in the
noted approximations. In particular, for the Curie temperature one obtains (in this
connection see [32,33]) the expression
ΘC =
w
2 arcth (w/zJ)
. (2)
At n = 1 and U → ∞ the electrons are localized on the lattice sites and equation
(2) has to take the mean-field form
ΘC =
zJ
2
, (3)
which can be obtained from (2) only if the unrealistic condition w/zJ → 0 is satis-
fied.
Although there are arguments which corroborate the existence of ferromagnetism
in the Hubbard model, this problem requires further studies. In a series of papers [34–
37] the importance has been shown of generalization of the Hubbard model by tak-
ing into account other matrix elements of electron-electron interaction besides the
intra-atomic Coulomb repulsion U , namely the correlated hopping and interatomic
exchange interaction. In reference [19] it has been stated that taking into account
above mentioned interactions the ferromagnetic state realization is considerably ef-
fected. Indeed, in a partially filled narrow energy band the translational part of the
147
L.Didukh, O.Kramar
ground state energy ∼ nδw (here δ = 1−n, w is the half-width of unperturbed band)
and the exchange part of ground state energy ∼ zn2J . Close to half-filling (n → 1,
δ → 0) the exchange part is crucial. Thereby, the consideration of interatomic ex-
change interaction is very important regardless of the possibility of ferromagnetic
state occurrence due to the correlation shifts of the band centers. The importance
of exchange interaction for the ferromagnetism realization has been also emphasized
in references [9,38–41].
The distinctive feature of the Hubbard model, namely the electron-hole sym-
metry, is broken with taking into account the correlated hopping; this fact may
be important forasmuch as the magnetic properties of transition metals and their
compounds with less and with more than half-filled 3d-shell are asymmetrical. The
consideration of correlated hopping within the mean-field approximation in the case
of weak electron interactions shows [37,40] that there are band narrowing and spin-
dependent shifts of the band centers (due to correlated hopping) which favor the
ferromagnetism realization. In the case of strong interactions, taking into account
correlated hopping leads to renormalization of Hubbard bandwidth [34]. The ef-
fect of correlated hopping on the formation of ferromagnetic ordering has been also
studied in reference [37] by the exact diagonalization of one-dimensional chains. The
correlation effects (ferromagnetism and metal-insulator transition) connected with
correlated hopping effect in the framework of the Gutzwiller method have been con-
sidered in reference [42] (the properties of the model in the limiting case of infinite
space dimension d → ∞ are analyzed). It has been obtained that correlated hop-
ping favors the ferromagnetic ordering and lowers the critical value of intra-atomic
Coulomb repulsion required for the occurrence of ferromagnetic spin ordering. At
the same time it should be noted that the consideration of the correlated hopping
effect of ferromagnetism within DMFT is non-trivial [42]. Therefore, the methods
of linearization of Green function equations might be useful [5].
The present paper considers the model of narrow band material with electron-
hole asymmetry [43] for the study of ferromagnetism in narrow energy bands. In
the case of strong electron correlation, the effective Hamiltonian is obtained. The
treatment of the Hamiltonian is based on the use of a variant of second quantization
method in the procedure of linearization of Green function equations. The efficiency
of this approach has been approved earlier at the study of metal-insulator transi-
tion [44] at half-filling (in this case the obtained energy spectrum reproduces exact
atomic and band limits, provides the correct behavior of ground state energy, energy
gap and polar states concentration; at the basis of these results some peculiarities
of narrow band materials have been interpreted).
2. The model Hamiltonian
We write the Hamiltonian of generalized Hubbard model in the form [43]:
H = − µ
∑
iσ
a+
iσaiσ +
∑
ijσ
′
(tij + nT1(ij)) a+
iσajσ +
∑
ijσ
′
(
T2(ij)a
+
iσajσniσ̄ + h.c.
)
148
Ground state ferromagnetism in a generalized Hubbard model
+ U
∑
i
ni↑ni↓ +
J
2
∑
ijσσ
′
′
a+
iσa+
jσ′aiσ′ajσ , (4)
here µ is the chemical potential, n = 〈ni↑ + ni↓〉, tij is the matrix element which
describes the hopping of electrons from site j to site i, T1(ij) and T2(ij) are the in-
tegrals of correlated hopping of electrons, U is the integral of intra-atomic Coulomb
repulsion of electrons, J is the integral of interatomic exchange interaction of elec-
trons. The peculiarity of the model described by (4) is taking into account the effect
of the site occupation on the hopping process (the correlated hopping).
The effect of correlated hopping of the first type (T1(ij)) is determined by occu-
pancies of the neighboring sites which are not involved in the hopping process. This
fact may be taken into account in the following way: we consider the presence of
such occupancies by a peculiar mean-field which correlates the hopping of electrons.
Therefore, the presence of correlated hopping leads to the concentration dependence
of effective hopping integrals tij(n) = tij + nT1(ij). It should be noted that taking
into account the correlated hopping of the first type T1(ij) in the mean-field approx-
imation becomes exact in the homeopolar limit ni↑ + ni↓ = 1. The peculiar feature
of the correlated hopping of the second type T2(ij) is as follows: such a hopping
of electrons occurs with the involvement of doubly occupied lattice sites. This fact
leads to the electron-hole asymmetry in the model.
We write Hamiltonian (4) in the configurational representation using Hubbard
transition operators Xkl
i [29]; the relation between the electron and X-operators is
as follows:
a+
iσ = Xσ0
i − ησX2σ̄
i , aiσ = X0σ
i − ησX σ̄2
i , (5)
here η↑ = 1, η↓ = −1.
In the case of strong Coulomb correlation U � |tij|, using the perturbation
theory in the form proposed in reference [45] the effective Hamiltonian of generalized
Hubbard model can be written in the form:
H = H0 + Htr + Hex + H̃ex , (6)
where
H0 = −µ
∑
i
(
X↑
i + X↓
i + 2X2
i
)
+ U
∑
i
X2
i , (7)
Htr =
∑
ijσ
′
tij(n)Xσ0
i X0σ
j +
∑
ijσ
′
t̃ij(n)X2σ
i Xσ2
j , (8)
Hex = −
1
2
∑
ijσ
Jij
(
(Xσ
i + X2
i )(Xσ
j + X2
j ) + Xσσ̄
i X σ̄σ
j
)
, (9)
H̃ex = −
1
2
∑
ijσ
J̃ij
(
Xσ
i X σ̄
j − Xσσ̄
i X σ̄σ
j − X2
i X0
i
)
, (10)
here
J̃ij =
2(t′ij(n))2
U
(11)
149
L.Didukh, O.Kramar
is the integral of indirect (through the polar states) exchange,
t(n) = t(1 − τ1n), (12)
t̃(n) = t(1 − τ1n − 2τ2), (13)
t′(n) = t(1 − τ1n − τ2) (14)
are the hopping integrals in the lower Hubbard subband (hole subband), the upper
Hubbard subband (doublon subband) and between these bands, respectively. For
convenience the parameters of correlated hopping τ1 = T1/t and τ2 = T2/t are
introduced, here t is the band hopping integral for the nearest neighbors, T1 and
T2 are the integrals of correlated hopping of the first and the second types for the
nearest neighbors.
3. Single-particle energy spectrum of narrow band model
We use the retarded Green functions method [46] to obtain the single-particle
energy spectrum. We introduce the Green function Gpp′(E) = 〈〈X0σ
p |Xσ0
p′ 〉〉 and write
the equation of motion in the case n < 1 when the processes with doublons can be
neglected:
(E + µ + zJeffnσ)Gpp′(E) =
δpp′
2π
〈Xσ
p + X0
p〉 + 〈〈[X0σ
p , Htr]|X
σ0
p′ 〉〉, (15)
(in this case the exchange interaction is taken into account within the mean-field
approximation, z is the number of the nearest neighbor sites, Jeff = J− J̃). Similarly
to reference [43] to obtain a self-consist equation for the Green function Gpp′(E) in
accordance with the projection procedure we assume that
[X0σ
p , Htr] =
∑
j
εσ(pj)X0σ
j , (16)
where εσ(pj) is non-operator expression. After anticommutation of equation (16)
with Xσ0
k we obtain:
εσ(pk)(Xσ
k + X0
k) = t(n)(Xσ
p + X0
p)(Xσ
k + X0
k) + t(n)Xσσ̄
k X σ̄σ
p
− δpkt(n)
∑
j
X σ̄0
k X0σ̄
j . (17)
Then, equation (15) can be rewritten in the form:
(E + µ + zJeffnσ)Gpp′(E) =
δpp′
2π
〈Xσ
p + X0
p〉 +
∑
j 6=p
εσ(pj)〈〈X0σ
j |Xσ0
p′ 〉〉
+ εσ
p〈〈X
0σ
p |Xσ0
p′ 〉〉, (18)
The standard way of calculating the non-operator coefficients εσ(pj), εσ
p is to
average equation (17), that leads to the known approximations [4,13,30]. To obtain
150
Ground state ferromagnetism in a generalized Hubbard model
εσ(pj), εσ
p we use a variant of the second quantization method. The relation between
Hubbard operators and configurational operators has been found in reference [45]:
Xkl
i = α+
ikαil , (19)
where α+
ik, αil are the creation and destruction operators of |k〉-state and |l〉-state
on site i (Shubin-Vonsowsky operators), respectively. For non-integer band-filling
n and magnetization m, in accordance with quasi-classical approximation, one can
obtain:
〈α+
i↑αi↑〉 =
n − 2d + m
2
, 〈α+
i↓αi↓〉 =
n − 2d − m
2
,
〈α+
i0αi0〉 = c, 〈α+
i2αi2〉 = d. (20)
As in paper [43], substituting Hubbard operators by Shubin-Vonsowsky operators
in equation(17)
α+
i↑ = αi↑ =
(
n − 2d + m
2
)
1
2
, α+
i↓ = αi↓ =
(
n − 2d − m
2
)
1
2
,
α+
i0 = αi0 = (c)
1
2 , α+
i2 = αi2 = (d)
1
2 (21)
we postulate a non-operator character of the expressions εσ(pj), εσ
p . Here, we intro-
duce the notations
εσ(pj) = ασtpj(n), εσ
p = βσ(pi). (22)
After Fourier transformation of equation (18) we obtain the equation for Green
function
Gσ
k
=
1
2π
1 − nσ̄
E − Eσ
k
, (23)
where single-particle energy spectrum Eσ
k
has the form:
Eσ
k
= −µ + ασtk(n) + βσ − zJeffnσ . (24)
Here, the coefficient of correlation band-narrowing is
ασ = 1 − nσ̄ +
nσ̄nσ
1 − nσ̄
, (25)
and the correlation shift of the band center is
βσ =
w(n)(1 − n)nσ̄
1 − nσ̄
, (26)
w(n) = z|t(n)| is the half-width of the lower subband, nσ is the concentration of
electrons with spin σ and it has been used such that
tk(n) =
1
N
∑
ij
tij(n)eik(Ri−Rj) = t(n)γ(k), γ(k) =
∑
R
eikR (27)
151
L.Didukh, O.Kramar
(the sum goes over the nearest neighbors to a site). Analogously, for the case n > 1
we obtain
G̃σ
k
=
1
2π
nσ̄
E − Ẽσ
k
, (28)
Ẽσ
k
= −µ + α̃σ t̃k(n) + U + β̃σ − zJeffnσ . (29)
The expressions for α̃ and β̃ are obtained from (25)–(26) by substituting w(n) by
w̃(n) = z|t̃(n)| and n by 2 − n.
The important peculiarity of the obtained spectra (24) and (29) is the presence
of correlation narrowing of subbands, which has different widths due to correlated
hopping. Besides, there is a spin-dependent shift of each subband center which favors
the ferromagnetic state realization (see section 4).
4. The ground state energy of the system
Let us calculate the ground state energy of the system described by Hamiltoni-
an (6). In the case n < 1 averaging equation (6) we obtain
E0
N
=
1
N
∑
ijσ
tij(n)〈Xσ0
i X0σ
j 〉 − zJeffn2
↑ − zJeffn2
↓ , (30)
where the average in the last equation is determined by spectral intensity of Green
function (23).
After the transition to quasi-impulse representation we integrate equation (30)
using the density of states without peculiarities. In the case of rectangular density
of states
ρ(ε) =
1
N
∑
k
δ(ε − ε(k)) =
1
2w
θ(ε2 − w2) (31)
for the ground state energy (per lattice site) we obtain
E0
N
=
∑
σ
(
−
(1 − n)nσ
1 − nσ̄
w(n) − zJeffn2
σ
)
. (32)
The obtained expression can be rewritten in the form
E0
N
= −
(1 − n)(2n(2 − n) − 2m2)
(2 − n)2 − m2
w(n) −
zJeff
2
(n2 + m2), (33)
where m is magnetization of the system.
In the case of semi-elliptic density of states
ρ(ε) =
1
N
∑
k
δ(E − ε(k)) =
2
πw
√
1 −
(
E
w
)2
(34)
152
Ground state ferromagnetism in a generalized Hubbard model
one can obtain the following equations to calculate the ground state energy
E0
N
=
∑
σ
(
−
2(1 − nσ)
3πw(n)2
(w(n)2 − x2
σ)
3
2 − zJeffn2
σ
)
, (35)
nσ=
(1 − nσ)
πw(n)2
(
xσ
√
w(n)2 − x2
σ + w(n)2 arcsin (xσw(n)) + w(n)2 arcsin
(
w(n)2
)
)
,
here, xσ = (µ − βσ + zJeff)/ασ (the results of numerical calculations are shown in
figure 1).
Figure 1. The concentration depen-
dences of ground state energy of param-
agnetic state (solid curves) and saturat-
ed ferromagnetic state (dotted curves):
curves 1 and 3 correspond to the semi-
elliptic density of states, curves 2 and 4
correspond to the rectangular density of
states.
In figure 1 one can see that the
ground state of the Hubbard model at
U → ∞ is paramagnetic and this con-
clusion is valid for a rectangular density
of states (curves 2 and 4) as well as for
semi-elliptic density of states (curves 1
and 3). Figure 2 shows the dependences
of ground state energy on band-filling n
at different values of magnetization and
fixed parameters of effective exchange
and correlated hopping. The behavior of
the ground state energy (at the same
values of model parameters) indicates
that in the range of concentrations close
to half-filling, the ferromagnetic state
is indeed realized. The correlated hop-
ping substantially enlarges the range of
electron concentration where the spon-
taneous magnetic moment exists.
To calculate the magnetization (this
result and other ones are obtained for
the rectangular density of states) we
use the condition of ground state ener-
gy minimum dE0/dm = 0. The equation
for equillibrium value of magnetization has the form (for the case of n < 1)
8(1 − n)2(2 − n)
((2 − n)2 − m2)2
mw(n) − mzJeff = 0. (36)
The criterion of ferromagnetism realization is obtained from the expression
d2E0/dm2 < 0
zJeff >
8(1 − n)2
(2 − n)3
w(n). (37)
For the case n > 1 one can obtain corresponding equations from equations (33)–(37)
substituting w(n) by w̃(n) and n by 2 − n.
153
L.Didukh, O.Kramar
Figure 2. The concentration depen-
dences of ground state energy of param-
agnetic state (solid curve), ferromagnet-
ic state (m = 0.5 · n, dotted curve) and
saturated ferromagnetic state (dashed
curve) at zJeff/w = 0.1, τ1 = 0.1, τ2 =
0.4.
Using equation (36) one can get the
critical values of the effective exchange
integral which are required for the fer-
romagnetism realization. Figure 3 plots
the dependence of (zJeff/w) on band-
filling at fixed values of magnetization
(the case of generalized Hubbard mod-
el with exchange interaction). For the
case of strong Coulomb interaction we
obtain the correct behavior of critical
exchange integral as a function of elec-
tron concentration n (at distinct from
the results of Hubbard-I approximation
used in reference [41]). It should be not-
ed that the range of band fillings, which
is most favorable for ferromagnetism re-
alization, is located close to half-filling.
For the Hubbard model at half-filling
the antiferromagnetic ordering is a dis-
tinctive feature. The competition of dif-
ferent types of magnetic ordering can be
described by the parameter of effective
Figure 3. The critical values of an
effective exchange integral zJeff/w
as a function of band-filling n at
τ1 = τ2 = 0. Curves (beginning from
the lower one) correspond to the case
m = 0, m = 0.5 · n, m = 0.75 · n and
m = n, respectively.
Figure 4. The concentration depen-
dence of critical value of an effective
exchange integral zJeff/w. The upper
curve corresponds to τ1 = τ2 = 0;
the middle curve corresponds to τ1 =
τ2 = 0.1; the lower curve corresponds
to τ1 = τ2 = 0.15.
154
Ground state ferromagnetism in a generalized Hubbard model
Figure 5. The concentration depen-
dence of magnetization m at zJeff/w =
0.02. The solid curve corresponds to
τ1 = τ2 = 0; the dashed curve corre-
sponds to τ1 = τ2 = 0.2; the dotted
curve corresponds to τ1 = 0.1; τ2 = 0.4.
Figure 6. The dependence of magneti-
zation m on effective exchange interac-
tion at n = 0.9. The solid curve corre-
sponds to τ1 = τ2 = 0; the dashed curve
corresponds to τ1 = τ2 = 0.1; the dotted
curve corresponds to τ1 = τ2 = 0.2.
exchange interaction Jeff . The ferromagnetic ordering arises when Jeff > 0; at the
same time, if 2t2/U > J , then the indirect antiferromagnetic exchange dominates
(however, this problem in the present paper is not considered and will be studied
elsewhere).
The effect of correlated hopping on the critical value of the effective exchange
interaction is illustrated in figure 4. The correlated hopping favors the occurrence of
ferromagnetic ordering at non-integer band fillings n: with the increase of the values
of parameters τ1 and τ2 the critical value zJeff/w decreases. The case of electron
concentration n > 1 is more favorable for ferromagnetism which is in agreement with
Hirsch’s results [37]. In comparison with the case of weak Coulomb interactions [40],
where more a favorable situation for ferromagnetism is determined by domination
of some type of correlated hopping parameters, in the case of strong correlation,
the crucial factor is the correlated hopping of the second type τ2 which comes out
at n > 1 (in this connection, note that ferromagnetic metals have more than a
half-filled 3d-shell).
The magnetization of the system in the ground state obtained from equation (36)
is plotted in figure 5. The increase of electron concentrations (at n < 1) in the band
at fixed energy parameters of the model leads to the occurrence of a spontaneous
magnetic moment; when the band filling reaches some critical value (at n > 1)
the magnetization vanishes. The peculiar feature of the plotted dependences is the
enlargement of the concentration range where the ferromagnetic ordering is obtained
caused by the effect of correlated hopping as well as the asymmetry with respect to
half-filling which is characteristic of ferromagnetic materials. It should be also noted
155
L.Didukh, O.Kramar
that the increase of an effective exchange interaction (at a fixed band-filling) leads
to the increase of magnetization up to saturation values. Taking into account the
correlated hopping makes it possible to obtain realistic critical values of an exchange
integral (see figure 6).
5. Conclusions
The present paper studies the possibility of ferromagnetic state realization in the
model of the material with strong Coulomb correlations. Taking into account the
two types of the correlated hopping and direct exchange interaction is the peculiarity
of the considered model.
In the strong correlation regime U � |t| the single-particle energy spectrum is
calculated at non-integer band-fillings n. The peculiar feature of the obtained energy
spectrum is a narrowing of the band caused by electron correlation: the bandwidth
depends on the electron concentration and on the correlated hopping parameters.
The calculated ground state energy depends on energy parameters of the system,
on concentration of electrons n and on magnetization m. From the expressions for
the ground state energy, calculated using the density of states without peculiari-
ties (rectangular and semi-elliptic), one can see that the exchange interaction is the
crucial parameter for the stability of ferromagnetic ordering. The analysis of the
condition of ferromagnetic ordering obtained from the ground state energy shows
that both types of correlated hopping favor ferromagnetism and cause the asymme-
try of the properties of the system with respect to the half-filling point. The case
of more than half-filled band n > 1 is more favorable for ferromagnetism realization
than the opposite case n < 1; from this point of view it is attractive to interpret
the following fact: ferromagnetism of transitional metals and their alloys is realized
in the system with more than half-filled 3d-shell. The concentration dependence of
magnetization illustrates that the increase of the effective exchange interaction pa-
rameter (as well as correlated hopping parameters) causes the enlargement of the
range of electron concentration where ferromagnetic ordering is realized. At some
values of energy parameters of the system, the magnetization is saturated.
The results of this paper for the case of strong electron correlation supplements
the results obtained in references [39,40], where the problem of ferromagnetism was
studied in the case of weak and moderate electron interactions. The case of non-zero
temperature and an application of the obtained results to explain of the properties
of ferromagnetic materials will be considered in subsequent papers.
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