Temperature-induced metal-insulator transition in a non-symmetric Hubbard model at half-filling
In the present paper metal-to-insulator transition with the increase of temperature is studied in a narrow-band model with non-equivalent Hubbard subbands at half-filling. It is shown that the results obtained in the considered model are essentially distinct from those obtained in the Hubbard model...
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1999
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irk-123456789-1205232017-06-13T03:05:18Z Temperature-induced metal-insulator transition in a non-symmetric Hubbard model at half-filling Didukh, L. Hankevych, V. In the present paper metal-to-insulator transition with the increase of temperature is studied in a narrow-band model with non-equivalent Hubbard subbands at half-filling. It is shown that the results obtained in the considered model are essentially distinct from those obtained in the Hubbard model. The results are applied to the interpretation of some experimental data. У цій роботі вивчається температурно-індукований перехід метал-діелектрик у вузькозонній моделі з нееквівалентними габбардівськими підзонами при половинному заповненні. Показано, що результати, отримані у розглядуваній моделі, суттєво відрізняються від результатів моделі Габбарда. 1999 Article Temperature-induced metal-insulator transition in a non-symmetric Hubbard model at half-filling / L. Didukh, V. Hankevych // Condensed Matter Physics. — 1999. — Т. 2, № 3(19). — С. 447-452. — Бібліогр.: 24 назв. — англ. 1607-324X DOI:10.5488/CMP.2.3.447 PACS: 71.30.+h, 71.28.+d, 71.10.Fd, 71.27.+a http://dspace.nbuv.gov.ua/handle/123456789/120523 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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In the present paper metal-to-insulator transition with the increase of temperature is studied in a narrow-band model with non-equivalent Hubbard
subbands at half-filling. It is shown that the results obtained in the considered model are essentially distinct from those obtained in the Hubbard
model. The results are applied to the interpretation of some experimental
data. |
| format |
Article |
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Didukh, L. Hankevych, V. |
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Didukh, L. Hankevych, V. Temperature-induced metal-insulator transition in a non-symmetric Hubbard model at half-filling Condensed Matter Physics |
| author_facet |
Didukh, L. Hankevych, V. |
| author_sort |
Didukh, L. |
| title |
Temperature-induced metal-insulator transition in a non-symmetric Hubbard model at half-filling |
| title_short |
Temperature-induced metal-insulator transition in a non-symmetric Hubbard model at half-filling |
| title_full |
Temperature-induced metal-insulator transition in a non-symmetric Hubbard model at half-filling |
| title_fullStr |
Temperature-induced metal-insulator transition in a non-symmetric Hubbard model at half-filling |
| title_full_unstemmed |
Temperature-induced metal-insulator transition in a non-symmetric Hubbard model at half-filling |
| title_sort |
temperature-induced metal-insulator transition in a non-symmetric hubbard model at half-filling |
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Інститут фізики конденсованих систем НАН України |
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1999 |
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http://dspace.nbuv.gov.ua/handle/123456789/120523 |
| citation_txt |
Temperature-induced metal-insulator transition in a non-symmetric Hubbard model at half-filling / L. Didukh, V. Hankevych // Condensed Matter Physics. — 1999. — Т. 2, № 3(19). — С. 447-452. — Бібліогр.: 24 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT didukhl temperatureinducedmetalinsulatortransitioninanonsymmetrichubbardmodelathalffilling AT hankevychv temperatureinducedmetalinsulatortransitioninanonsymmetrichubbardmodelathalffilling |
| first_indexed |
2025-07-08T18:01:28Z |
| last_indexed |
2025-07-08T18:01:28Z |
| _version_ |
1837102728840478720 |
| fulltext |
Condensed Matter Physics, 1999, Vol. 2, No. 3(19), pp. 447–452
Temperature-induced metal-insulator
transition in a non-symmetric Hubbard
model at half-filling
L.Didukh, V.Hankevych
Ternopil State Technical University, Department of Physics,
56 Rus’ka Str., 282001 Ternopil, Ukraine
Received June 8, 1998, in final form June 22, 1998
In the present paper metal-to-insulator transition with the increase of tem-
perature is studied in a narrow-band model with non-equivalent Hubbard
subbands at half-filling. It is shown that the results obtained in the con-
sidered model are essentially distinct from those obtained in the Hubbard
model. The results are applied to the interpretation of some experimental
data.
Key words: metal-insulator transition, narrow energy bands,
non-equivalent Hubbard subbands
PACS: 71.30.+h, 71.28.+d, 71.10.Fd, 71.27.+a
1. Introduction
It is known that the electron-hole symmetry is peculiar to the Hubbard model [1].
One of the manifestations of this symmetry is the equivalence of the lower and
upper Hubbard bands. It is the result of the equality of hopping integrals describing
both “translational” hopping of holes and doubly occupied sites (doublons) and
the processes of their paired creation and destruction. Equality of noted hopping
integrals is caused by neglecting the matrix elements of electron-electron interaction
J(ikjk) =
∫ ∫
φ∗(r−Ri)φ(r−Rj)
e2
|r− r′|
|φ(r′ −Rk)|
2
drdr
′, (1.1)
in the Hamiltonian of model (the matrix elements (1.1) describe hopping of electrons
between i and j lattice sites; φ-function is the Wannier function).
However, the theoretical analysis, on the one hand, and the available experimen-
tal data, on the other hand, indicate the fact that the Hubbard model generalization
is of a principle necessity taking into account the correlated hopping (1.1) [2–6]. In
such a model, hopping integrals describing “translational” hopping of holes and
doublons are different. These hopping integrals also differ from the hopping integral
c© L.Didukh, V.Hankevych 447
L.Didukh, V.Hankevych
which is connected with the processes of paired creation and destruction of holes
and doublons. Consequently, the lower and upper Hubbard bands are non-equivalent
(non-symmetric). In recent years, similar models have been studied intensively [7–
14].
An important puzzle arising in studying these models is a metal-insulator tran-
sition problem which is one of the most essential in narrow-band physics [15–17].
In this connection special interest is challenged by the observable metal-to-insulator
transitions in some narrow-band materials with the increase of temperature (see, for
example [17–22]).
Based on the approach proposed in the papers [5,23] we have studied metal-
insulator transition in a model of narrow-band material with non-equivalent Hub-
bard subbands (the so-called “non-symmetric Hubbard model”) at half-filling and
zero temperature in the paper [24]. The present paper is devoted to a further study
of metal-insulator transition in a model with non-equivalent Hubbard subbands, in
particular, to the study of temperature-induced metal-to-insulator transition.
2. Results
We start from the following natural generalization of the Hubbard model [1] at
half-filling including correlated hopping (1.1) [4,5]
H = −µ
∑
iσ
a+iσaiσ + (t0 + T1)
∑
ijσ
′
a+iσajσ + T2
∑
ijσ
′ (
a+iσajσniσ̄ + h.c.
)
+ U
∑
i
ni↑ni↓, (2.1)
with i, j nearest-neighbours sites, µ is the chemical potential, a+
iσ, (aiσ) is the creation
(destruction) operator of an electron of spin σ (σ =↑, ↓) on i-site (σ̄ denotes spin
projection which is opposite to σ), niσ = a+iσaiσ is the number operator of electrons
of spin σ on i-site, U is the intra-atomic Coulomb repulsion, t0, T1, T2 are the
integrals of electron hopping between the nearest neighbours,
T1 =
∑
k 6=i,j
J(ikjk), T2 = J(iiij);
the primes at the sums in Hamiltonian (2.1) signify that i 6= j.
Using a generalized mean-field approximation [5,23] in Green function method
we obtain for a paramagnetic state the single-particle energy spectrum as
E1,2(k) = −µ+
(1− 2d)(tk + t̃k) + U
2
∓
1
2
Fk, (2.2)
Fk =
√
[
B(tk − t̃k)− U
]2
+ (4dt′k)
2, B = 1− 2d+ 4d2, (2.3)
where d is the concentration of polar states (holes or doublons), tk, t̃k, t′k are
the Fourier transforms of the respective nearest-neighbour hopping integrals t =
448
Temperature-induced metal-insulator transition in a non-symmetric Hubbard model. . .
t0 + T1, t̃ = t + 2T2, t′ = t + T2; t and t̃ are terms describing hopping of quasipar-
ticles within the lower and upper Hubbard bands (hopping of holes and doublons)
respectively, t′ describes quasiparticle hopping between hole and doublon bands (the
processes of paired creation and destruction of holes and doublons).
The peculiarity of single-particle energy spectrum (2.2) is the dependence on
temperature (d is a function of temperature). Thus, the proposed approach makes
it possible to study temperature-induced metal-insulator transition.
The energy gap width (difference of energies between bottom of the upper and
top of the lower Hubbard bands) is
∆E = −(1− 2d)(w + w̃) +
1
2
(Q1 +Q2), (2.4)
Q1 =
√
[B(w − w̃)− U ]2 + (4dzt′)2, (2.5)
Q2 =
√
[B(w − w̃) + U ]2 + (4dzt′)2, (2.6)
where w = z|t|, w̃ = z|t̃| (z is the number of the nearest neighbours to the site).
The concentration of polar states (obtained using the Green function
〈〈aiσniσ̄|a
+
jσ〉〉) is [24]
d =
1
4w
w
∫
−w
[
Cε
exp E1(ε)
kBT
+ 1
+
Dε
exp E2(ε)
kBT
+ 1
]
dε, (2.7)
Cε =
1
2
−
U
2Fε
−
Bε
2Fε
(
t̃
t
− 1
)
, Dε =
1
2
+
U
2Fε
+
Bε
2Fε
(
t̃
t
− 1
)
,
and chemical potential of narrow-band model with non-equivalent Hubbard sub-
bands is given by the equation
w
∫
−w
[
1
exp −E2(ε)
kBT
+ 1
−
1
exp E1(ε)
kBT
+ 1
]
dε = 0, (2.8)
E1,2(ε), Fε are obtained from respective formulae (2.2), (2.3) for E1,2(k), Fk by
substitution of tk → ε, t̃k → (t̃/t)ε, t′k → (t′/t)ε. Here we have used the rectangular
density of states.
3. Discussions and conclusions
The temperature dependence of chemical potential of a narrow-band model with
non-equivalent Hubbard subbands obtained from equation (2.8) is plotted in fig-
ure 1. One can see that in the considered model chemical potential is essentially
dependent not only on the parameters w and w̃ but also on temperature (in con-
trast to the Hubbard model where µ = U/2), and moreover, the chemical potential
rapidly increases with the decrease of temperature depending on the parameters of
449
L.Didukh, V.Hankevych
Figure 1. The temperature depen-
dence (θ = kBT ) of chemical poten-
tial µ for U/2w = 1: the upper curve
corresponds to τ1 = τ2 = 0.3; the
lower curve – τ1 = τ2 = 0.1; the
straight line corresponds to values of
chemical potential in the Hubbard
model (τ1 = τ2 = 0).
Figure 2. The dependence of energy
gap on temperature at U/w = 0.9.
The upper curve corresponds to τ1 =
τ2 = 0.2, the middle curve – τ1 =
τ2 = 0.1, the lower curve – τ1 = τ2 =
0 (the Hubbard model).
non-equivalence of Hubbard subbands τ1, τ2. In a high temperature region in the
proposed model, the chemical potential tends to U/2 with the increase of tempera-
ture; really, at T → ∞ the probabilities of an electron finding within the lower and
upper Hubbard bands (independent of their bandwidth ratio) are equal.
At a given U, w, w̃, t′ (constant exterior pressure), the concentration of polar
states (2.7) increases with the increase of temperature. It leads to the fact that
the system can undergo transition from the state with ∆E 6 0 to the state with
∆E > 0, i.e. metal-to-insulator transition can occur. In this case the results obtained
in the Hubbard model and those obtained in non-symmetric Hubbard model can be
essentially different (figure 2 illustrates it). Let us take for example U/w = 0.9.
One can see that at T = 0 K the energy gap width in both models is ∆E < 0 (a
metallic state). With the increase of temperature, metal-to-insulator transition does
not occur in the Hubbard model, in a non-symmetric model the values of parameters
τ1, τ2 exist at which metal-to-insulator transition occurs.
In case the metal-to-insulator transition occurs in both models from figure 2
one can see that at given values of U/w in a model with non-equivalent Hubbard
subbands, metal-to-insulator transition occurs at a smaller temperature than in the
Hubbard model. So, for example, for w0 = z|t0| ≈1.05 eV (such bandwidth of NiS2
was estimated in the paper [19]) in the considered model in a paramagnetic state
metal-to-insulator transition occurs at T ≈ 280 K for U/w0 = 1.94 and τ1 = τ2 =
0.01 (observable the transition temperature of NiS2 is T ∼280 K at p ∼3 MPa [20]).
450
Temperature-induced metal-insulator transition in a non-symmetric Hubbard model. . .
For the same value of U/w0 metal-to-insulator transition occurs at T ≈940 K when
τ1 = τ2 = 0 (neglecting the correlated hopping, the Hubbard model) and at T =0 K
when τ1 = τ2 = 0.015. If U/w0 = 1.98 then transition from a metallic state to
an insulating state is realized at T ≈ 290 K for τ1 = τ2 = 0; at T = 0 K when
τ1 = τ2 = 0.005. Note that at U ∼ 2w the temperatures of metal-to-insulator
transition found in both models are essentially different; with a deviation from this
ratio the difference decreases.
The obtained temperature dependence of energy gap can explain observable tran-
sition from the state of a paramagnetic metal to the paramagnetic Mott-Hubbard in-
sulator state in the (V1−xCrx)2O3 compound [17,18] in NiS2 [20] and in the NiS2−xSex
system [20–22] with the increase of temperature.
To conclude, metal-to-insulator transition observable in the materials with nar-
row energy bands can be explained based on the proposed approach at realistic
values of the parameters characterizing non-equivalence of the Hubbard subbands.
This way, the results obtained are essentially distinct from those obtained with-
out taking into account this non-equivalence (the Hubbard model). In particular,
the temperature of metal-to-insulator transition is essentially smaller than in the
Hubbard model (this fact agrees to the observable transition temperature), and the
chemical potential distinguishes from the value µ = U/2 (obtained in the Hubbard
model) and it is temperature dependent.
The authors are grateful to Prof. I. Stasyuk for valuable discussions as well as to
the Organizing Committee of the INTAS-Ukraine Workshop on Condensed Matter
Physics for grant.
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Температурно-індукований перехід
метал-діелектрик у вузькозонній моделі з
нееквівалентними габбардівськими підзонами при
половинному заповненні
Л.Дідух, В.Ганкевич
Тернопільський державний технічний університет ім. І.Пулюя,
кафедра фізики, 282001 Тернопіль, вул. Руська, 56
Отримано 8 червня 1998 р., в остаточному вигляді – 22 червня
1998 р.
У цій роботі вивчається температурно-індукований перехід метал-
діелектрик у вузькозонній моделі з нееквівалентними габбардівсь-
кими підзонами при половинному заповненні. Показано, що резуль-
тати, отримані у розглядуваній моделі, суттєво відрізняються від ре-
зультатів моделі Габбарда.
Ключові слова: вузькі зони провідності, перехід метал-діелектрик,
нееквівалентні габбардівські підзони
PACS: 71.30.+h, 71.28.+d, 71.10.Fd, 71.27.+a
452
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