Pressure-temperature phase diagram of the generalized Hubbard model with correlated hopping at half-filling
In the present paper, the pressure-temperature phase diagram of a generalized Hubbard model with correlated hopping in a paramagnetic state at half-filling is determined by means of a generalized mean-field approximation in the Green function technique. The constructed phase diagram describes th...
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| Цитувати: | Pressure-temperature phase diagram of the generalized Hubbard model with correlated hopping at half-filling / L. Didukh, V. Hankevych // Condensed Matter Physics. — 2001. — Т. 4, № 1(25). — С. 93-100. — Бібліогр.: 33 назв. — англ. |
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irk-123456789-1197542017-06-09T03:04:52Z Pressure-temperature phase diagram of the generalized Hubbard model with correlated hopping at half-filling Didukh, L. Hankevych, V. In the present paper, the pressure-temperature phase diagram of a generalized Hubbard model with correlated hopping in a paramagnetic state at half-filling is determined by means of a generalized mean-field approximation in the Green function technique. The constructed phase diagram describes the metal-to-insulator transition with increasing temperature, and the insulator-to-metal transition under the action of external pressure. The phase diagram can explain the paramagnetic region of the phase diagrams of some transition metal compounds Робота присвячена побудові фазової діаграми тиск-температура узагальненої моделі Габбарда з корельованим переносом у парамагнітному стані при половинному заповненні зони з використанням узагальненого наближення Гартрі-Фока в методі функцій Ґріна. Побудована фазова діаграма описує перехід з металічного стану в діелектричний при збільшенні температури і перехід з діелектричного стану в металічний під дією зовнішнього тиску. Фазова діаграма може пояснити парамагнітні області фазових діаграм деяких сполук перехідних металів. 2001 Article Pressure-temperature phase diagram of the generalized Hubbard model with correlated hopping at half-filling / L. Didukh, V. Hankevych // Condensed Matter Physics. — 2001. — Т. 4, № 1(25). — С. 93-100. — Бібліогр.: 33 назв. — англ. 1607-324X PACS: 71.10.Fd, 71.30.+h, 71.27.+a DOI:10.5488/CMP.4.1.93 http://dspace.nbuv.gov.ua/handle/123456789/119754 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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English |
| description |
In the present paper, the pressure-temperature phase diagram of a generalized
Hubbard model with correlated hopping in a paramagnetic state
at half-filling is determined by means of a generalized mean-field approximation
in the Green function technique. The constructed phase diagram
describes the metal-to-insulator transition with increasing temperature, and
the insulator-to-metal transition under the action of external pressure. The
phase diagram can explain the paramagnetic region of the phase diagrams
of some transition metal compounds |
| format |
Article |
| author |
Didukh, L. Hankevych, V. |
| spellingShingle |
Didukh, L. Hankevych, V. Pressure-temperature phase diagram of the generalized Hubbard model with correlated hopping at half-filling Condensed Matter Physics |
| author_facet |
Didukh, L. Hankevych, V. |
| author_sort |
Didukh, L. |
| title |
Pressure-temperature phase diagram of the generalized Hubbard model with correlated hopping at half-filling |
| title_short |
Pressure-temperature phase diagram of the generalized Hubbard model with correlated hopping at half-filling |
| title_full |
Pressure-temperature phase diagram of the generalized Hubbard model with correlated hopping at half-filling |
| title_fullStr |
Pressure-temperature phase diagram of the generalized Hubbard model with correlated hopping at half-filling |
| title_full_unstemmed |
Pressure-temperature phase diagram of the generalized Hubbard model with correlated hopping at half-filling |
| title_sort |
pressure-temperature phase diagram of the generalized hubbard model with correlated hopping at half-filling |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2001 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119754 |
| citation_txt |
Pressure-temperature phase diagram of the generalized Hubbard model with correlated hopping at half-filling / L. Didukh, V. Hankevych // Condensed Matter Physics. — 2001. — Т. 4, № 1(25). — С. 93-100. — Бібліогр.: 33 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT didukhl pressuretemperaturephasediagramofthegeneralizedhubbardmodelwithcorrelatedhoppingathalffilling AT hankevychv pressuretemperaturephasediagramofthegeneralizedhubbardmodelwithcorrelatedhoppingathalffilling |
| first_indexed |
2025-07-08T16:32:26Z |
| last_indexed |
2025-07-08T16:32:26Z |
| _version_ |
1837097129093365760 |
| fulltext |
Condensed Matter Physics, 2001, Vol. 4, No. 1(25), pp. 93–100
Pressure-temperature phase diagram of
the generalized Hubbard model with
correlated hopping at half-filling
L.Didukh, V.Hankevych∗
Ternopil State Technical University, Department of Physics,
56 Rus’ka Str., 46001 Ternopil, Ukraine
Received August 14, 2000
In the present paper, the pressure-temperature phase diagram of a gen-
eralized Hubbard model with correlated hopping in a paramagnetic state
at half-filling is determined by means of a generalized mean-field approx-
imation in the Green function technique. The constructed phase diagram
describes the metal-to-insulator transition with increasing temperature, and
the insulator-to-metal transition under the action of external pressure. The
phase diagram can explain the paramagnetic region of the phase diagrams
of some transition metal compounds.
Key words: phase diagram, metal-insulator transition, correlated hopping
PACS: 71.10.Fd, 71.30.+h, 71.27.+a
1. Introduction
In recent years a generalized Hubbard model with correlated hopping has been
used widely to describe strongly correlated electron systems (see papers [1,2] and
references therein); the electron-hole asymmetry is a property of such a generalized
Hubbard model as a result of the dependence of the hopping integral on the occu-
pation of the sites involved in the hopping process. Recently, this model has been
extended to the case of a doubly orbitally degenerate band [3].
The generalized Hubbard model has much richer properties than the well-known
Hubbard model [4], and usage of the electron-hole asymmetry concept allows one
to interpret the peculiarities of physical properties of narrow-band materials which
are not explained by the Hubbard model. In particular, the experimentally observed
electron-hole asymmetries of metal oxides conductivity, of cohesive energy of tran-
sition 3d-metals and of superconducting properties of high-temperature supercon-
ductors have been explained within the generalized Hubbard model with correlated
hopping in papers [5–10] respectively.
∗E-mail: vaha@tu.edu.te.ua
c© L.Didukh, V.Hankevych 93
L.Didukh, V.Hankevych
Despite the fact that the phase diagram of the generalized Hubbard model has
been studied in works [11–17], researchers pay no attention to determining the mod-
el phase diagram in a paramagnetic state under the action of external effects, in
particular the pressure-temperature phase diagram. This task is related directly to
the problem of the metal-insulator transition description under the action of exter-
nal pressure and temperature, namely the constructed pressure-temperature phase
diagram of the model would allow us to describe the observed metal-insulator tran-
sitions in narrow-band materials with the change of pressure and temperature. An
interest to such transitions is caused by the theoretical point of view as well as by
the rich possibilities of its application (see, for example, monograph [18] and re-
view [19]). Consequently, the goal of the present paper, being a continuation of the
previous work [20] where the temperature-induced metal-insulator transition was
studied, is to determine the pressure-temperature phase diagram of the general-
ized Hubbard model with correlated hopping in a paramagnetic state at half-filling.
Based on this phase diagram, we describe the metal-insulator transitions under the
action of external pressure and temperature.
2. Pressure-temperature phase diagram of the model
Taking into account an external hydrostatic pressure p we write the model Hamil-
tonian in the following form [5] (in this connection also see [21]):
H = −µ
∑
iσ
a+iσaiσ + (1 + αu)t
∑
ijσ
′
a+iσajσ + T2
∑
ijσ
′
(
a+iσajσniσ̄ + h.c.
)
+ U
∑
i
ni↑ni↓ +
1
2
NV0κu
2, (2.1)
where i, j are the nearest-neighbours sites, µ is the chemical potential, a+
iσ, (aiσ) is
the creation (destruction) operator of an electron of spin σ (σ =↑, ↓) on site i (σ̄
denotes spin projection which is opposite to σ), niσ = a+iσaiσ is the number operator
of electrons of spin σ on site i, U is the intra-atomic Coulomb repulsion, t = t 0+T1,
with t0 being the matrix element of the electron-ion interaction, T1, T2 are the
correlated hopping integrals (matrix elements of electron-electron interaction), the
primes on the sums in Hamiltonian (2.1) signify that i 6= j.
The last term of the Hamiltonian has the meaning of an elastic energy of a
uniformly deformed crystal, where κ is the “initial” (purely lattice) bulk elasticity,
N is the number of lattice sites, u = ∆V0/V0 is the relative change of the volume
in uniform strain (V0 is the initial unit-cell volume). Formulating the Hamiltonian
we have used the result of paper [21]: the dependence of a bandwidth W on relative
change of the volume u in uniform strain can be written in the formW = 2w(1+αu),
where w = z|t| (z is the number of the nearest neighbours to a site), α = V0
2w
∂W
∂V
< 0.
We also assume that under the action of external pressure only the bandwidth
changes, and the matrix elements of the electron-electron interaction (the correlated
hopping integrals and intra-atomic Coulomb repulsion) do not depend on the relative
change of the volume.
94
Pressure-temperature phase diagram of the generalized Hubbard model. . .
As in papers [2,20], using the generalized mean-field approximation [5,22] (an
analog of the projection operation) in the Green function method, we obtain for a
paramagnetic state at half-filling the energy gap width as
∆E = −(1− 2d)(w + w̃)[1 + αu] +
1
2
(Q1 +Q2), (2.2)
Q1 =
√
[B(w − w̃)(1 + αu)− U ]2 + [4dzt′(1 + αu)]2, (2.3)
Q2 =
√
[B(w − w̃)(1 + αu) + U ]2 + [4dzt′(1 + αu)]2, (2.4)
where B = 1 − 2d+ 4d2, d is the concentration of polar states (holes or doublons)
which has been calculated in [2,20], w̃ = z|t̃|, t̃ = t + 2T2, t′ = t + T2; t and
t̃ are the terms describing hopping of quasiparticles within the lower and upper
Hubbard bands (hopping of holes and doublons) respectively, t′ describes a quasi-
particle hopping between hole and doublon bands (the processes of paired creation
and destruction of holes and doublons).
According to the method proposed for the s(d)-f model in paper [23], the equi-
librium value of relative change of the volume u is determined from the condition of
the minimum of the thermodynamic Gibbs’ potential
G = F +NpV0(1 + u), (2.5)
where F is the free energy. Using the known identity ∂F/∂u = 〈∂H/∂u〉, equa-
tion (2.5) for the parameter u can be represented as
〈
∂H
∂u
〉
+NpV0 = 0, (2.6)
with H being Hamiltonian (2.1). In the mean-field approximation passing to the
space of quasi-momenta we get the following equation for the relative change of the
volume u:
αu =
2α1V0
WN
∑
kσ
tk〈a
+
kσakσ〉+ τpV0, (2.7)
where α1 is the parameter which determines the quantity ∂W/∂V, 2α1V0/W ≈
0.1, τ ≈ 0.05 eV−1 [21].
Taking into consideration the fact that within the generalized mean-field approx-
imation the first term of the right-hand side of equation (2.7) is equal to zero [24]
at the point of the metal-insulator transition, we obtain the relation between the
relative change of the volume u and an external hydrostatic pressure p as
αu = τpV0. (2.8)
Note that within the generalized Hartree-Fock approximation this equation is valid
at the point of the metal-insulator transition as well as in an insulating phase.
To determine the pressure-temperature phase diagram of the model we use for-
mula (2.2) for the energy gap width and the expression for the concentration of
95
L.Didukh, V.Hankevych
polar states calculated in [20]. Let us consider, for instance, the Mott-Hubbard com-
pound NiS2. This has two electrons half filling an eg band, the half-width of the
initial (uncorrelated) band of this crystal is w0 = z|t0| ≈ 1.05 eV [26,27], and the
initial unit-cell volume is V0 ≈ 14.79 ·10−30 m3 [28]. It shows the transition from the
state of a paramagnetic insulator to the paramagnetic metal state at a hydrostatic
external pressure of 46 kbar and room temperatures. Thus the transition occurs for
a decrease in volume of about 0.4% with no change in crystal structure [29,30]. It
also becomes metallic on alloying with Ni 2Se, and the behaviour of this system is
discussed later in this section.
To calculate the model parameter U we fix one of the points (p = 22 kbar,
T = 100 K) of the experimental curve in the phase diagram (the dashed-line curve
of figure 1) and find the value of intra-atomic Coulomb repulsion U at which the
theoretical calculations within the present model reproduce this point. Thus, we
obtain: U/w=U/w0 = 2.0168 for the correlated hopping parameters τ1 = T1/|t0| =
0, τ2 = T2/|t0| = 0 (these values of τ1, τ2 correspond to the Hubbard model),
U/w = U/w0(1 − τ1) = 1.79107 for τ1 = τ2 = 0.1, and U/w = U/w0 = 1.81437 at
τ1 = 0, τ2 = 0.1. Using these values of the model parameters we find the values
of external hydrostatic pressure and temperature at which the energy gap width is
equal to zero (i.e. metal-insulator transition occurs).
Figure 1. The pressure-temperature
phase diagram of the metal-insulator
transition determined within the gener-
alized Hubbard model with correlated
hopping for NiS2 in comparison with ex-
periment (the dashed curve): τ1 = τ2 = 0
(the upper curve); τ1 = 0, τ2 = 0.1 (the
middle curve); τ1 = τ2 = 0.1 (the lower
curve). PI (PM) denotes paramagnetic
insulating (metallic) phase.
The calculated pressure-tempera-
ture phase diagram of the model (fig-
ure 1) describes metal-insulator tran-
sitions in a paramagnetic state un-
der the action of an external pres-
sure and temperature in NiS2, name-
ly the constructed phase diagram de-
scribes the metal-to-insulator transi-
tion with increasing temperature, and
the insulator-to-metal transition un-
der the action of external pressure.
Comparison of this theoretically deter-
mined phase diagram with the phase
diagram of the compound NiS2 shows
good agreement between the theory
and experiment. Besides, the theoret-
ical calculations within the model re-
produce the experimental data of pa-
per [29] which point out the presence
of an energy gap width ∆E > 0 in
the ground state of NiS2 and in the
absence of an external pressure. The
phase diagram shows that our taking
into account the correlated hopping
permits a much better description of
96
Pressure-temperature phase diagram of the generalized Hubbard model. . .
these experimental data than the Hubbard model; this also much better illustrates
the physics of the present model and the important role of correlated hopping.
Analogous phase diagrams can be constructed for other compounds:
(V1−xCrx)2O3 [18,25], NiS2−xSex [26,27] and Y1−xCaxTiO3 [31,32] exhibiting such
metal-insulator transitions. For example, the material (V 0.96Cr0.04)2O3 shows a me-
tal-insulator transition at a hydrostatic external pressure of 13 kbar and at room
temperatures; the transition occurs at a small decrease in volume of about 1% with
no change in crystal structure [25]. In (V1−xMx)2O3 (with M = Cr, Ti) the addition
of Ti3+ ions to V2O3 leads to an insulator-to-metal transition, whereas the addition
of Cr3+ ions results in a metal-to-insulator transition. The simplest explanation is
that the substitution of V3+ ion for Cr3+ ion leads to a band narrowing; the Cr3+ ion
is a localized impurity and it deletes a state from the 3d-bands. Deleting a state is
equivalent to a band narrowing or an external pressure decreasing [18] which drives
the system towards the insulating phase. Likewise the addition of Ti3+ impurities is
equivalent to an external pressure increase.
In the Mott-Hubbard compound NiS2−xSex electron hoppings between the sites
of Ni occur through the chalcogenide sites (this is caused by peculiarities of the
pyrite crystal structure [33]), the substitution of S2− ion for Se2− ion in NiS2 leads
to an increase of wave functions overlap, consequently the probability of an electron
hopping increases which is equivalent to a band broadening or to an external pressure
increasing. Therefore, the pressure-temperature phase diagram constructed for NiS2
can describe the experimental composition-temperature phase diagram [26,27] of the
compound NiS2−xSex.
Note that the doping may cause the disorder effects which can modify the elec-
tronic bands and contribute to disorder-induced localization. But for the compounds
listed above, apparently, it can be assumed that the doping effects a bandwidth only
or, equivalently, an effective pressure (especially for small values x). It should be also
pointed out that to construct the phase diagram of the system Y0.61Ca0.39TiO3 we
have to generalize the previous results obtained at half-filling to the case of a non
half-filled band because this compound is characterized by such a band [32].
In conclusion, in the present paper the pressure-temperature phase diagram of
the generalized Hubbard model with correlated hopping in a paramagnetic state at
half-filling has been determined. The constructed phase diagram describes a metal-
to-insulator transition with the increasing temperature, and an insulator-to-metal
transition under the action of external pressure. Comparison of this theoretically
determined phase diagram with the experimental data, in particular with the phase
diagram of the compound NiS2, shows good agreement between the theory and
experiment. We have found that our taking into account the correlated hopping
permits a much better description of these experimental data than just the Hubbard
model; this also better illustrates the physics of the present model and the important
role of correlated hopping.
The determined pressure-temperature phase diagram of the model can also ex-
plain the paramagnetic region of the phase diagrams of the transition metal com-
pounds: the systems NiS2−xSex and (V1−xCrx)2O3, calcium doped YTiO3.
97
L.Didukh, V.Hankevych
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99
L.Didukh, V.Hankevych
Фазова діаграма тиск-температура узагальненої
моделі Габбарда з корельованим переносом при
половинному заповненні
Л.Дідух, В.Ганкевич
Тернопільський державний технічний університет імені І.Пулюя,
кафедpа фізики, 46001 Тернопіль, вул. Руська, 56
Отримано 14 серпня 2000 р.
Робота присвячена побудові фазової діаграми тиск-температура
узагальненої моделі Габбарда з корельованим переносом у пара-
магнітному стані при половинному заповненні зони з використанням
узагальненого наближення Гартрі-Фока в методі функцій Ґріна. По-
будована фазова діаграма описує перехід з металічного стану в ді-
електричний при збільшенні температури і перехід з діелектричного
стану в металічний під дією зовнішнього тиску. Фазова діаграма мо-
же пояснити парамагнітні області фазових діаграм деяких сполук пе-
рехідних металів.
Ключові слова: фазова діаграма, перехід метал-діелектрик,
корельований перенос
PACS: 71.10.Fd, 71.30.+h, 71.27.+a
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