Effects of correlated hopping on electronic ferroelectricity in the extended Falicov-Kimball model in two dimensions
We use the Hartree-Fock (HF) approximation with the charge-density-wave (CDW) instability to study the influence of correlated hopping on the stability of electronic ferroelectricity in the extended Falicov-Kimball model (FKM) in two dimensions. Ми використовуємо наближення Гартрi-Фока (ГФ) iз нестi...
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| Cite this: | Effects of correlated hopping on electronic ferroelectricity in the extended Falicov-Kimball model in two dimensions / P. Farkašovský, J. Jurečková // Condensed Matter Physics. — 2015. — Т. 18, № 3. — С. 33704: 1–8. — Бібліогр.: 23 назв. — англ. |
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| citation_txt | Effects of correlated hopping on electronic ferroelectricity in the extended Falicov-Kimball model in two dimensions / P. Farkašovský, J. Jurečková // Condensed Matter Physics. — 2015. — Т. 18, № 3. — С. 33704: 1–8. — Бібліогр.: 23 назв. — англ. |
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| description | We use the Hartree-Fock (HF) approximation with the charge-density-wave (CDW) instability to study the influence of correlated hopping on the stability of electronic ferroelectricity in the extended Falicov-Kimball model (FKM) in two dimensions.
Ми використовуємо наближення Гартрi-Фока (ГФ) iз нестiйкiстю щодо хвиль зарядової густини до дослiдження впливу корельованого переносу на стiйкiсть електронної сегнетоелектрики в розширенiй моделi Фалiкова-Кiмбала (ФК) у двох вимiрах.
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Condensed Matter Physics, 2015, Vol. 18, No 3, 33704: 1–8
DOI: 10.5488/CMP.18.33704
http://www.icmp.lviv.ua/journal
Effects of correlated hopping on electronic
ferroelectricity in the extended Falicov-Kimball
model in two dimensions
P. Farkašovský∗, J. Jurečková
Institute of Experimental Physics, Slovak Academy of Sciences, Watsonova 47, 040 01 Košice, Slovakia
Received April 29, 2015
We use the Hartree-Fock (HF) approximation with the charge-density-wave (CDW) instability to study the effect
of correlated hopping on the stability of electronic ferroelectricity in the extended Falicov-Kimball model (FKM)
in two dimensions. It is shown that the effects of correlated hopping term are very strong, especially for nega-
tive values of the f -level energy E f , where they lead to: (i) stabilization of the ferroelectric ground state with a
spontaneous hybridization Pd f = 〈d+ f 〉 for positive values of correlated hopping parameter t ′
d
, (ii) stabiliza-
tion of the antiferroelectric phase for t ′
d
< t ′
d ,crit.
< 0 and (iii) suppression of the ferroelectric ground state for
t ′
d ,crit.
É t ′
d
< 0, The effects of correlated hopping on valence transitions are also discussed.
Key words: electronic ferroelectricity, charge density waves, valence transitions
PACS: 71.27.+a, 71.28.+d, 71.30.+h
1. Introduction
In the recent years the Falicov-Kimball model (FKM) [1] has been extensively studied in connection
with the exciting idea of electronic ferroelectricity [2–11] that is also directly related to the problem of an
excitonic insulator [12–18]. It is generally supposed that ferroelectricity in mixed-valent compounds is of
purely electronic origin, i.e., it results from an electronic phase transition, in contrast to the conventional
displacive ferroelectricity due to a lattice distortion. Since the FKM is probably the simplest model of
electronic phase transitions in rare-earth and transition-metal compounds it was natural to test the idea
of electronic ferroelectricity just on this model.
The FKM is based on the coexistence of two different types of electronic states in a given material:
localized, highly correlated ionic-like states and extended, uncorrelated, Bloch-like states. It is accepted
that insulator-metal transitions result from a change in the occupation numbers of these electronic states,
which remain themselves basically unchanged by their character. Taking into account only the intra-
atomic Coulomb interaction between the two types of states, the Hamiltonian of the spinless FKM can be
written as the sum of three terms:
H =
∑
i j
ti j d+
i d j +U
∑
i
f +
i fi d+
i di +E f
∑
i
f +
i fi , (1)
where f +
i
, fi are the creation and annihilation operators for an electron in the localized state at lattice
site i with binding energy E f , and d+
i
, di are the creation and annihilation operators of the itinerant
spinless electrons in the d -band Wannier state at site i .
∗E-mail: farky@saske.sk
© P. Farkašovský, J. Jurečková, 2015 33704-1
http://dx.doi.org/10.5488/CMP.18.33704
http://www.icmp.lviv.ua/journal
P. Farkašovský, J. Jurečková
The first term of (1) is the kinetic energy corresponding to quantum-mechanical hopping of the itin-
erant d electrons between sites i and j . These intersite hopping transitions are described by the matrix
elements ti j , which are −td if i and j are the nearest neighbors and zero otherwise (in what follows all
parameters are measured in units of td ). The second term represents the on-site Coulomb interaction
between the d -band electrons with density nd =
1
L
∑
i d+
i
di and the localized f electrons with density
n f = 1
L
∑
i f +
i
fi , where L is the number of lattice sites. The third term stands for the localized f electrons
whose sharp energy level is E f .
The first attempt to describe the electronic ferroelectricity within the FKM was made by Porten-
gen et al. [2, 3]. They studied the FKM with a k-dependent hybridization in Hartree-Fock (HF) approx-
imation and found, in particular, that the Coulomb interactionU between the itinerant d -electrons and
the localized f -electrons gives rise to a non-vanishing excitonic 〈d+ f 〉-expectation value even in the
limit of vanishing hybridization V → 0. As an applied (optical) electrical field provides for excitations
between d - and f -states and thus for an excitonic expectation value Pd f = 〈d+
i
fi 〉, the finding of a spon-
taneous Pd f (without hybridization or electric field) has been interpreted as the evidence for electronic
ferroelectricity. However, analytical calculations within well controlled approximation (forU small) per-
formed by Czycholl [4] in infinite dimensions do not confirm the existence of electronic ferroelectricity.
The same conclusion has been also obtained independently [5, 6] by small-cluster exact-diagonalization
and density-matrix renormalization group calculations in the one dimension for both small (U < 1) and
intermediate interactions (U ∼ 5).
Hybridization between the itinerant d and localized f states, however, is not the only way to de-
velop d– f coherence. Recent theoretical works by Batista et al. [8, 9] showed that the ground state
with a spontaneous electric polarization can also be induced by the nearest-neighbor f -electron hop-
ping (−t f
∑
〈i , j 〉 f +
i
f j ) for dimensions D > 1. Based on these results the authors postulated the following
conditions that favor the formation of the electronically driven ferroelectric state: (i) the system should
be in a mixed-valence regime, (ii) two bands involved should be of different parity, (iii) a local Coulomb
repulsionU between the f and d orbitals is required.
Later on this model was extensively used to describe different phases in the ground state and par-
ticularly the properties of the excitonic phase [12–19]. It was found that the ground state phase diagram
exhibits a very simple structure consisting of only four phases, and namely, the full d and f band in-
sulator (BI), the excitonic insulator (EI), the charge-density-wave (CDW) and the staggered orbital order
(SOO). The EI is characterized by a nonvanishing 〈d+ f 〉 average. The CDW is described by a periodicmod-
ulation in the total electron density of both f and d electrons, and the SOO is characterized by a periodic
modulation in the difference between the f and d electron densities.
In our recent paper [10], the HF approach with the CDW instability was used to study the ground-state
properties of the extended FKMwith f – f hopping in two and three dimensions. It was found that the HF
solutions with the CDW instability perfectly reproduce the two-dimensional intermediate coupling phase
diagram of the model calculated by constrained path Monte Carlo (CPMC) method [9], including phase
boundaries of ferroelectric state. On the other hand, it should be noted that the model discussed above
neglects all nonlocal interaction terms, and thus it is questionable whether the above mentioned results
also persist in more realistic situations when nonlocal interactions are turned on. An important nonlocal
interaction term obviously absent in the conventional FKM is the term of correlated hopping,
H =−t ′d
∑
〈i j 〉
d+
i d j ( f +
i fi + f +
j f j ), (2)
in which the d -electron hopping amplitudes between the neighboring lattice sites i and j explicitly de-
pend on the occupancy ( f +
i
fi ) of the f -electron orbitals. The importance of the correlated hopping term
has been already mentioned by Hubbard [20]. Later Hirsch [21] pointed out that this term may be rel-
evant in explaning the superconducting properties of strongly correlated electrons. Here, we examine
the effects of this term on the stability of the EI phase in the two-dimensional extended FKM. With the
exception of small deviations, the method used bellow is the same as the one described in our previous
paper [10] and, therefore, we summarize here only its main steps.
33704-2
Effects of correlated hopping on electronic ferroelectricity
2. The method
The Hamiltonian of the extended FKM with correlated hopping has the form
H =−td
∑
〈i j 〉
d+
i d j − t f
∑
〈i j 〉
f +
i f j +U
∑
i
f +
i fi d+
i di +E f
∑
i
f +
i fi − t ′d
∑
〈i j 〉
d+
i d j
(
f +
i fi + f +
j f j
)
. (3)
In accordance with our previous paper [10], we go here beyond the usual HF approach [22] in which
only homogeneous solutions are postulated and also consider inhomogeneous solutions modelled by a
periodic modulation of the order parameters:
〈n
f
i
〉 = n f +δ f cos(Q ·ri ), 〈nd
i 〉 = nd +δd cos(Q ·ri ), 〈 f +
i di 〉 =∆+∆P cos(Q ·ri ). (4)
Here, δd and δ f are the order parameters of the CDW state for the d - and f -electrons, ∆ is the excitonic
average and Q = (π,π) is the nesting vector for D = 2.
Using these expressions, the HF Hamiltonian of the extended FKM with correlated hopping can be
written as
H = (−td −2t ′d n f )
∑
〈i , j 〉
d+
i d j − t f
∑
〈i , j 〉
f +
i f j +E f
∑
i
n
f
i
+U
∑
i
[
n f +δ f cos(Q ·ri )
]
nd
i
+U
∑
i
[nd +δd cos(Q ·ri )]n
f
i
−U
∑
i
[∆+∆P cos(Q ·ri )]d+
i fi +h.c. , (5)
where the simple decoupling of the form
d+
i d j ( f +
i fi + f +
j f j ) → d+
i d j (〈 f +
i fi 〉+〈 f +
j f j 〉) = 2n f d+
i d j (6)
has been used for the correlated hopping term. Thus, one can see that within this decoupling scheme, the
effect of the correlated hopping is fully transformed to the renormalization of the static single electron
hopping amplitude according to td → t∗
d
= td+2t ′
d
n f . Next, we shall show that this relatively small change
has dramatical effects on the existence of the EI phase in the extended FKM.
This Hamiltonian can be diagonalized by the following canonical transformation [10, 23]
γm
k
= um
k
dk + vm
k
dk+Q +am
k
fk +bm
k
fk+Q , m = 1,2,3,4, (7)
where Ψm
k
= (am
k
,bm
k
,um
k
, vm
k
)T are solutions of the associated Bogoliubov-de Gennes (BdG) eigenequa-
tions:
HkΨ
m
k = E m
k Ψ
m
k , (8)
with
Hk =
ǫd
k
+Un f Uδ f −U∆ −U∆P
Uδ f ǫd
k+Q
+Un f −U∆P −U∆
−U∆ −U∆P ǫ
f
k
+Und +E f Uδd
−U∆P −U∆ Uδd ǫ
f
k+Q
+Und +E f
, (9)
and the corresponding energy dispersions ǫd
k
and ǫ
f
k
for d and f electrons and the HF parameters nd , δd ,
n f , δ f , ∆, ∆P are given by:
ǫ
f
k
=−2t f
[
cos(kx )+cos(ky )
]
, ǫd
k =−2t∗d
[
cos(kx )+cos(ky )
]
, (10)
nd =
1
N
∑
k
′
∑
m
{um
k um
k + vm
k vm
k } f (E m
k ), δd =
1
N
∑
k
′
∑
m
{vm
k um
k +um
k vm
k } f (E m
k ), (11)
n f =
1
N
∑
k
′
∑
m
{am
k am
k +bm
k bm
k } f (E m
k ), δ f =
1
N
∑
k
′
∑
m
{bm
k am
k +am
k bm
k } f (E m
k ), (12)
∆=
1
N
∑
k
′
∑
m
{am
k um
k +bm
k vm
k } f (E m
k ), ∆P =
1
N
∑
k
′
∑
m
{bm
k um
k +am
k vm
k } f (E m
k ). (13)
Here, the prime denotes summation over half the Brillouin zone and f (E ) = 1/{1+exp[β(E −µ)]} is the
Fermi distribution function.
33704-3
P. Farkašovský, J. Jurečková
3. Results and discussion
To examine the effects of correlated hopping on the ground-state phase diagram of the extended FKM
(in the E f –t f plane), and particularly on the stability of the electronic ferroelectricity, we use the zero
temperature variant of the method described above. The HF equations are solved self-consistently for
each pair of (E f , t f ) values for several selected values of t ′
d
. We use an exact diagonalization method to
solve the Bogoliubov-de Gennes equation. We start with an initial set of the order parameters. By solving
equation (8), the new order parameters are computed via equations (11) to (13) and are substituted back
into equation (8). The iteration is repeated until a desired accuracy is achieved. It should be noted that
the stability of different HF solutions has been also checked numerically by calculating the total energy
and it was found that all phases presented in the ground-state phase diagram represent the most stable
HF solutions.
First, we have examined the model in the intermediate coupling regime U = 2, since for this case,
there exists a comprehensive phase diagram of the extended FKM model (t f , 0, t ′
d
= 0) obtained by a
CPMC technique [9] as well as its HF version [10] and both accord very nicely (both qualitatively and
quantitatively). In figure 1, we have displayed typical examples of our HF solutions obtained for ∆, ∆P
and n f in the limit of small t f values (t f = −0.1) and t ′
d
< 0. One can see that going with t ′
d
from 0
to t ′
d ,crit. = −0.5, the effects of correlated hopping on the stability of excitonic phase (∆ , 0) increase
dramatically, especially in the limit of E f < 0. For negative E f , the stability region of excitonic phase
rapidly reduces with increasing |t ′
d
|, and at t ′
d
∼−0.4, it fully disappears. This is obviously a consequence
of the absence of amixed valence phase for n f > 0.5 and t ′
d
É−0.4 as it is demonstrated in Fig 1, where the
Figure 1. Dependence of the HF parameters ∆, ∆P and n f on the f -level energy E f calculated for
different values of t ′
d
(t ′
d
= 0, −0.1, −0.2, −0.3, −0.4, −0.5) andU = 2, t f =−0.1.
33704-4
Effects of correlated hopping on electronic ferroelectricity
average f -electron occupancy n f is plotted as a function of the f -level energy E f . In this case, the valence
transitions change discontinuously from n f = 1 to n f = 0.5 and the system exhibits a direct transition
from the BI phase to the CDW phase. At the same time, this result stresses the fact what a crucial role is
played by the term of correlated hopping in the mechanism of valence transitions, and that already very
small values of the correlated hopping parameter are capable of changing the type of valence transitions
from continuous to discontinuous. For this reason, the term of correlated hopping should be surely taken
into account in the correct description of valence transitions in rare-earth compounds.
In accordance with the case t ′
d
= 0, we have found two different excitonic phases for t ′
d
< 0. The first
one is homogeneous (∆> 0,∆P = 0) and the second one is inhomogeneous ( ∆> 0,∆P , 0). Analysing the
numerical data obtained for δd ,δ f and∆P (see figure 2), one can see that the appearance of an inhomoge-
neous phase is tightly connected with the formation of the CDW ordering in f and d electron subsystems
since in the corresponding regions where ∆P , 0, the order parameter δd (δ f ) changes continuously from
0 to its maximal (minimal) value. With increasing |t ′
d
|, the stability region of the inhomogeneous phase
strongly reduces and at t ′
d
∼−0.3 this phase practically disappears.
Figure 2. Dependence of the HF parameters δd , δ f and ∆P on the f -level energy E f calculated for differ-
ent values of t ′
d
(t ′
d
= 0, −0.1, −0.2, −0.3, −0.4, −0.5) andU = 2, t f =−0.1.
Going with t ′
d
to smaller values we have observed a completely different behavior of the model in
the limit of E f < 0 (see figure 3). For all the examined values of t ′
d
, we have found that ∆= 0 and ∆P > 0,
whichmeans that the ground state of themodel is antiferroelectric in this limit. Very interesting is also the
behavior of n f as a function of E f . As is shown in figure 4, the mixed-valent phase is again stabilized in
the region n f > 1/2 and the magnitude of the discontinuous valence transition from n f > 0.5 to n f = 1/2
gradually decreases with decreasing t ′
d
and a similar behaviour also exhibits the width of the n f = 1/2
phase. At a first glance, this result seems to be in contradiction with the rules favour the formation of the
electronically driven ferroelectric state postulated by Batista et al. [9]. The system is in a mixed-valence
33704-5
P. Farkašovský, J. Jurečková
Figure 3. Dependence of the HF parameters ∆ and ∆P on the f -level energy E f calculated for different
values of t ′
d
(t ′
d
=−0.6, −0.7, −0.8) andU = 2, t f =−0.1.
regime, the local Coulomb interaction between different orbitals is present, but the ferroelectric state is
absent. A more detailed analysis (see figure 4) of the behaviour of the renormalized d -electron hopping
parameter t∗
d
shows, however, that there in no contradiction with conditions postulated by Batista et al.,
since for t ′
d
<−0.5 and E f < 0, the d band changes its parity. In this region, the d and f bands are of the
same parity (t∗
d
< 0, t f < 0), which, in accordance with the above mentioned rules, does not support the
ferroelectric ground state.
The situation in the opposite limit t ′
d
> 0 is depicted in figure 5. Again one can recognize two different
regimes in the behaviour of n f ,∆ and ∆p . For positive E f , only weak effects of correlated hopping on
n f ,∆ and ∆p are observed, while in the opposite limit, these effects are quite dramatic and the stability
Figure 4. Dependence of the HF parameter n f and the renormalized d -electron hopping parameter t∗
d
on
the f -level energy E f calculated for different values of t ′
d
(t ′
d
=−0.6, −0.7, −0.8) andU = 2, t f =−0.1.
33704-6
Effects of correlated hopping on electronic ferroelectricity
Figure 5.Dependence of theHF parameters n f ,∆ and∆P on the f -level energy E f calculated for different
values of t ′
d
(t ′
d
= 0, 0.1, 0.2, 0.3) andU = 2, t f =−0.1.
regions of the mixed valence as well as excitonic phase are considerably enhanced by increasing t ′
d
. This
result independently confirms that the term of correlated hopping plays a very important role in the
mechanism of stabilizing the excitonic phase and, thus, it should be taken into account for the correct
description of this phenomenon. Finally, it should be noted that we have also found qualitatively the
same behaviour of the model for smaller values of t f and larger values ofU .
Thus, we can conclude that the effects of correlated hopping term on the stability of the excitonic
state in the extended FKM (within the HF approximation with the CDW instability) are very strong and
they lead to: (i) suppression of the excitonic phase for t ′
d ,crit. É t ′
d
< 0, E f < 0, (ii) stabilization of the
antiferroelectric phase for t ′
d
< t ′
d ,crit., E f < 0 and (iii) stabilization of excitonic phase for t ′
d
> 0, E f <
0. Similarly, the strong effects of correlated hopping are also observed on the mechanism of valence
transitions, where already very small values of the correlated hopping parameter are capable of changing
the type of valence transitions from continuous to discontinuous.
Acknowledgements
This work was supported by the Slovak Grant Agency VEGA under Grant No. 2/0077/13, Slovak Re-
search andDevelopment Agency (APVV) under Grant APVV–0097–12 and EU GrantsNo. ITMS26210120002
and ITMS26110230097.
33704-7
P. Farkašovský, J. Jurečková
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Вплив корельованого переносу на електронну
сегнетоелектрику в розширенiй моделi Фалiкова-Кiмбала
у двох вимiрах
П. Фаркашовскi, Ю. Юречкова
Iнститут експериментальної фiзики, Словацька академiя наук, 040 01 Кошицi, Словаччина
Ми використовуємо наближення Гартрi-Фока (ГФ) iз нестiйкiстю щодо хвиль зарядової густини до дослi-
дження впливу корельованого переносу на стiйкiсть електронної сегнетоелектрики в розширенiй моделi
Фалiкова-Кiмбала (ФК) у двох вимiрах. Показано, що внесок корельованого переносу є дуже сильний,
особливо для вiд’ємних значень енергiї f -рiвня E f , коли цей вплив приводить до: (1) стабiлiзацiї сегнето-
електричного основного стану зi спонтанною гiбридизацiєю Pd f = 〈d+ f 〉 для додатних значень пара-
метра корельованого переносу t ′
d
, (2) стабiлiзацiї антисегнетоелектричної фази при t ′
d
< t ′
d ,crit.
< 0 i (3)
приглушення сегнетоелектричного основного стану для t ′
d ,crit.
É t ′
d
< 0. Крiм того, обговорюються впли-
ви корельованого переносу на переходи зi змiною валентностi.
Ключовi слова: електронна сегнетоелектрика, хвилi зарядової густини, переходи зi змiною валентностi
33704-8
http://dx.doi.org/10.1103/PhysRevLett.22.997
http://dx.doi.org/10.1103/PhysRevLett.76.3384
http://dx.doi.org/10.1103/PhysRevB.54.17452
http://dx.doi.org/10.1103/PhysRevB.59.2642
http://dx.doi.org/10.1103/PhysRevB.59.9707
http://dx.doi.org/10.1103/PhysRevB.65.081102
http://dx.doi.org/10.1080/13642810110066470
http://dx.doi.org/10.1103/PhysRevLett.89.166403
http://dx.doi.org/10.1103/PhysRevLett.92.187601
http://dx.doi.org/10.1103/PhysRevB.77.155130
http://dx.doi.org/10.1140/epjb/e2008-00273-y
http://dx.doi.org/10.1103/PhysRevB.81.115122
http://dx.doi.org/10.1103/PhysRevB.81.205117
http://dx.doi.org/10.1103/PhysRevB.84.245106
http://dx.doi.org/10.1103/PhysRevB.85.121102
http://dx.doi.org/10.1103/PhysRevB.85.165135
http://dx.doi.org/10.1103/PhysRevB.88.035312
http://dx.doi.org/10.1007/s10909-014-1165-x
http://dx.doi.org/10.1103/PhysRevLett.112.026401
http://dx.doi.org/10.1098/rspa.1963.0204
http://dx.doi.org/10.1016/0921-4534(89)90225-6
http://dx.doi.org/10.1016/0038-1098(78)90399-X
http://dx.doi.org/10.1103/PhysRevB.72.125122
Introduction
The method
Results and discussion
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| id | nasplib_isofts_kiev_ua-123456789-155263 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1607-324X |
| language | English |
| last_indexed | 2025-12-07T16:36:08Z |
| publishDate | 2015 |
| publisher | Інститут фізики конденсованих систем НАН України |
| record_format | dspace |
| spelling | Farkašovský, P. Jurečková, J. 2019-06-16T15:11:48Z 2019-06-16T15:11:48Z 2015 Effects of correlated hopping on electronic ferroelectricity in the extended Falicov-Kimball model in two dimensions / P. Farkašovský, J. Jurečková // Condensed Matter Physics. — 2015. — Т. 18, № 3. — С. 33704: 1–8. — Бібліогр.: 23 назв. — англ. 1607-324X PACS: 71.27.+a, 71.28.+d, 71.30.+h DOI:10.5488/CMP.18.33704 arXiv:1510.06562 https://nasplib.isofts.kiev.ua/handle/123456789/155263 We use the Hartree-Fock (HF) approximation with the charge-density-wave (CDW) instability to study the influence of correlated hopping on the stability of electronic ferroelectricity in the extended Falicov-Kimball model (FKM) in two dimensions. Ми використовуємо наближення Гартрi-Фока (ГФ) iз нестiйкiстю щодо хвиль зарядової густини до дослiдження впливу корельованого переносу на стiйкiсть електронної сегнетоелектрики в розширенiй моделi Фалiкова-Кiмбала (ФК) у двох вимiрах. This work was supported by the Slovak Grant Agency VEGA under Grant No. 2/0077/13, Slovak Research and Development Agency (APVV) under Grant APVV–0097–12 and EU Grants No. ITMS26210120002
 and ITMS26110230097. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Effects of correlated hopping on electronic ferroelectricity in the extended Falicov-Kimball model in two dimensions Вплив корельованого переносу на електронну сегнетоелектрику в розширенiй моделi Фалiкова-Кiмбала у двох вимiрах Article published earlier |
| spellingShingle | Effects of correlated hopping on electronic ferroelectricity in the extended Falicov-Kimball model in two dimensions Farkašovský, P. Jurečková, J. |
| title | Effects of correlated hopping on electronic ferroelectricity in the extended Falicov-Kimball model in two dimensions |
| title_alt | Вплив корельованого переносу на електронну сегнетоелектрику в розширенiй моделi Фалiкова-Кiмбала у двох вимiрах |
| title_full | Effects of correlated hopping on electronic ferroelectricity in the extended Falicov-Kimball model in two dimensions |
| title_fullStr | Effects of correlated hopping on electronic ferroelectricity in the extended Falicov-Kimball model in two dimensions |
| title_full_unstemmed | Effects of correlated hopping on electronic ferroelectricity in the extended Falicov-Kimball model in two dimensions |
| title_short | Effects of correlated hopping on electronic ferroelectricity in the extended Falicov-Kimball model in two dimensions |
| title_sort | effects of correlated hopping on electronic ferroelectricity in the extended falicov-kimball model in two dimensions |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/155263 |
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