Two-orbital Hubbard model vs spin S=1 Heisenberg model: studies on clusters
We perform exact numeric calculations for the two-orbital Hubbard model on the four-site cluster. In the limit
 of large on-site coupling the model becomes equivalent to the spin S “ 1 Heisenberg model. By comparing
 energy spectra of these two models, we quantified the range of inte...
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| Cite this: | Two-orbital Hubbard model vs spin S=1 Heisenberg model: studies on clusters / R. Lemański, J. Matysiak // Condensed Matter Physics. — 2018. — Т. 21, № 3. — С. 33301: 1–8. — Бібліогр.: 15 назв. — англ. |
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| citation_txt | Two-orbital Hubbard model vs spin S=1 Heisenberg model: studies on clusters / R. Lemański, J. Matysiak // Condensed Matter Physics. — 2018. — Т. 21, № 3. — С. 33301: 1–8. — Бібліогр.: 15 назв. — англ. |
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| description | We perform exact numeric calculations for the two-orbital Hubbard model on the four-site cluster. In the limit
of large on-site coupling the model becomes equivalent to the spin S “ 1 Heisenberg model. By comparing
energy spectra of these two models, we quantified the range of interaction parameters for which the Heisenberg
model satisfactorily reproduces the two-orbital Hubbard model. Then we examined how the spectrum evolves
when we are outside of this region, focusing especially on checking of how it is modified when various ways of
interatomic hoppings of electrons between different orbitals are taken into account. We finally show how these
modifications affect the dependence of specific heat on temperature.
Нами проведено точнi числовi розрахунки для двоорбiтальної моделi Хаббарда на чотиривузловому кластерi. В границi великої одновузлової взаємодiї ця модель стає еквiвалентною спiн S “ 1 моделi Гайзенберга. З порiвняння енергетичних спектрiв цих двох моделей нами визначено кiлькiсно дiапазон значень параметрiв взаємодiї, для якого модель Гайзенберга задовiльно вiдтворює двоорбiтальну модель
Хаббарда. Потiм ми розглянули змiни спектру, коли ми знаходимося за межами цього дiапазону, зосереджуючись, зокрема, на перевiрцi того, як вiн модифiкується, коли враховуються рiзнi способи мiжатомних перескокiв електронiв мiж рiзними орбiталями. Насамкiнець, нами показано як цi змiни впливають
на температурну залежнiсть питомої теплоємностi.
|
| first_indexed | 2025-12-07T19:00:08Z |
| format | Article |
| fulltext |
Condensed Matter Physics, 2018, Vol. 21, No 3, 33301: 1–8
DOI: 10.5488/CMP.21.33301
http://www.icmp.lviv.ua/journal
Two-orbital Hubbard model vs spin S “ 1 Heisenberg
model: studies on clusters
R. Lemański, J. Matysiak
Institute of Low Temperature and Structure Research, Polish Academy of Science,
ul. Okólna 2, 50-422 Wrocław, Poland
Received June 15, 2018, in final form August 13, 2018
We perform exact numeric calculations for the two-orbital Hubbard model on the four-site cluster. In the limit
of large on-site coupling the model becomes equivalent to the spin S “ 1 Heisenberg model. By comparing
energy spectra of these twomodels, we quantified the range of interaction parameters for which the Heisenberg
model satisfactorily reproduces the two-orbital Hubbard model. Then we examined how the spectrum evolves
when we are outside of this region, focusing especially on checking of how it is modified when various ways of
interatomic hoppings of electrons between different orbitals are taken into account. We finally show how these
modifications affect the dependence of specific heat on temperature.
Key words:multi-orbital Hubbard model, Heisenberg model, magnetic molecules
PACS: 31.15.vq, 75.10.Jm, 75.10.Pq
1. Introduction
Multi-orbital Hubbard models (MOHMs) are suitable for studying correlated electron materials with
orbital degeneracies, such as transition metal compounds [1–3]. In particular, they seem to be relevant
in the theoretical analysis of single molecular magnets (SMMs). However, magnetic molecules have
been usually described by phenomenological spin model Hamiltonians with the dominant Heisenberg
interaction term (e.g., [4, 5]). This is because the low-energy excited states observed in these systems
are often similar to the energy spectrum of the Heisenberg model. Then, attempts were made to describe
these systems from the first principles using the DFT method and to determine the exchange parameters
entering the Heisenberg model from the comparison of energies of various magnetic configurations [6].
The resulting energy spectrum obtained in these calculations slightly deviates from the spectrum of the
simplest version of the Heisenberg model, but it can be well tuned to the spectrum of the Falicov-Kimball
model [6]. Unfortunately, the calculations reported in [6] did not take into account the spin flip processes,
which made the exchange integrals too large to be compatible with the experimental data.
It has been recently proposed to describe SMMs using the MOHM in the large interaction limit
combined with DFT calculations [7]. In this work, the values of exchange integrals were obtained within
the perturbation theory. They turned out to be several times smaller than those obtained in [6], thus better
suited to experimental data. The point is that in [7], the DFT was not used to calculate the energies of the
system with different magnetic configurations, but it was focused on determining the local Coulomb and
Hund interactions and the amplitudes of inter-ionic hoppings of electrons. Then, the exchange integral
was determined by putting these data into the formula obtained from the perturbation theory in the limit
of large values of the local Coulomb interaction, when the MOHM reduces to the Heisenberg model.
In fact, the transition from the Hubbard model to the spin model by applying the perturbation theory is
a commonly used procedure in correlated electron systems not only in relation to magnetism, but also
relative to other phenomena, such as superconductivity [8–14].
Although the approach used in [7] improved the procedure of finding the integrals of the exchange,
still the system was described only using the Heisenberg model. Therefore, nothing was known what
This work is licensed under a Creative Commons Attribution 4.0 International License . Further distribution
of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
33301-1
https://doi.org/10.5488/CMP.21.33301
http://www.icmp.lviv.ua/journal
http://creativecommons.org/licenses/by/4.0/
R. Lemański, J. Matysiak
exactly we could gain when instead of the Heisenberg model we examine the system using the MOHM.
Indeed, the procedure applied there did not allow one to determine the range of the model parameters,
beyond which the energy spectra of the MOHM and the Heisenberg model clearly differ from each other.
Moreover, regardless of the orbitals between which the electrons jump from site to site, the perturbation
theory always leads to the Heisenberg model, modifying only the value of the exchange constant. In
other words, the perturbative theory does not distinguish whether the electron jumps between the same
or between different orbitals (‘hybridization’).
To find out what is the advantage of the description of a SMM using the microscope model MOHM
over the phenomenological description using the Heisenberg model, and in particular to see how the
energy spectra of these two models differ from each other, here we analyze the four-site ring with two
orbitals per site. Such a small ring was chosen to make it possible to perform exact numerical calculations
efficiently and rather easily. Therefore, our analysis does not focus on a particular material, but rather
on capturing a role played by various hopping amplitudes of electrons, when the system goes out of the
range where the second order perturbation theory is justified. For a better illustration of our results, in
this work we also present a set of curves of the specific heat versus temperature for various relationships
between amplitudes of electron hoppings between different orbitals.
2. Two-orbital Hubbard model (TOHM)
Studies of the two-orbital case require an inclusion into the single-orbital Hubbardmodel of additional
on-site couplings between various orbitals: the direct and exchange Coulomb interactions that ensure
the fulfillment of the Hund’s rule. An additional complication that arises here is the possibility of
electron jumping between more than one orbital and also between different orbitals (mixing terms or
‘hybridization’) belonging to the neighboring sites (see figure 1). Taking into account all these factors
ensures not only an increase of the number of model parameters, but above all a significant increase of
the number of states of the system, which obviously complicates diagonalization of the Hamiltonian.
The Hamiltonian HHM of the TOHM that we study in this paper is as follows:
HHM “ H0 ` H1 ,
H0 “ U
ÿ
im
nimÓnimÒ `
1
2
ÿ
i,m‰m1,σ
`
U1nimσnim1σ̄ `U2nimσnim1σ
˘
`
1
2
ÿ
i,m‰m1,σ
´
Jc:imσc:
im1σ̄cimσ̄cim1σ ` Jc:imσc:imσ̄cim1σ̄cim1σ
¯
,
H1 “
ÿ
i‰j,m,m1,σ
tmm1 c:imσcjm1σ ,
(2.1)
where i and j denote nearest-neighbour sites, m,m1 label orbitals and σ, σ̄ label spins of electrons
(σ̄ “ ´σ). U, U1 and U2 describe the Coulomb type on-site interactions between two electrons: U —
on the same orbital and U1 (U2) — on different orbitals with opposite (parallel) spins, respectively.
J represents the on-site exchange coupling, but it also enters the interaction constants in the TOHM.
Figure 1. Labelling of the hopping amplitudes tmm1 between the sites i and j in the TOHM. m,m1 indicate
orbital indices.
33301-2
Two-orbital Hubbard model vs spin S “ 1 Heisenberg model
Figure 2. A part of energy spectrum of the half-filled TOHM for the four-site ring in the atomic limit.
The interaction parameters are U “ 10 eV, J “ 0.5 eV, Sz “ 0.
Indeed, we take U1 “ U ´ 2J, U2 “ U ´ 3J, which is valid for T2g orbitals in an octahedral crystal
field [15].
The Hamiltonian (2.1) preserves the total magnetisation. Therefore, in the beginning we consider only
the states with total magnetisation equal to zero, which constitutes the largest subspace. The total number
of these states is 4900. It is helpful to analyse first the spectrum of the TOHM in the atomic limit, i.e.,
when all tmm1 “ 0 (the lowest part of the spectrum is shown in figure 2). We have chosen the parameters
in our calculations to be U “ 10 eV, J “ 0.5 eV. The spectrum consists of 26 degenerate energy levels.
Then, we are interested in the region of small values of the hopping constants tmm1 , and it is understood
that the overall structure of the spectrum in figure 2 is preserved, but each of the degenerate levels splits
in a specific way when tmm1 becomes non-zero. Here, we focus mainly on finding the splitting of the
lowest energy level.
To obtain the exact spectrum of the model (2.1) for our small system, we use the exact diagonalization
method.
3. Computation details
In the atomic limit, the eigenstates of the Hamiltonian (2.1) for a single site together with their
representations and energies are shown in table 1. As indicated in the first column, they are also the
eigenstates of the total spin S and its z component Sz . The full Hilbert space base for the whole 4-site
system is formed from the tensor products of the states from table 1.
For a site occupied by two electrons, the lowest energy refers to the eigenstates 1–3 from table 1.
However, obviously, after one of the electrons jumps from one site to another, the first site will contain
only one electron and the other one three electrons. Then, the lowest energy eigenstate for the site with
one electron is one of the eigenstates 7–10 from table 1., and for the site with three electrons, it is one of
the eigenstates 11–14 from table 1.
In the case of a ring with 4 sites and 2 orbitals per site, we have the total number of orbitals r “ 8.
Then, at half filling, i.e., when the number of electrons is equal to the namber of orbitals, the numbers of
the states that are needed to diagonalize the Hamiltonian are: 12870 for the TOHM, but only 34 “ 81 for
the Heisenberg model.
3.1. Perturbation theory
In the perturbation theory, H1 is taken as the perturbation and H0 is the Hamiltonian for which the
solution is known. Contrary to the single orbital model, in the multi-orbital case, the states with one
electron per orbital and with different spin configurations have, in general, different energies even in the
limit when tmm1 “ 0 for all hopping amplitudes. Then, in order to obtain an effective Hamiltonian with
33301-3
R. Lemański, J. Matysiak
Table 1. Eigenstates and eigenvalues of the model (2.1) for a single site with two orbitals per site. Sections
separated by double horizontal lines correspond to the different numbers of electrons n on the site.
nr |n, S, Szy State representation Energy
1 |2, 1, 1y pÒ | Òq U ´ 3J
2 |2, 1, 0y
1
?
2
´
pÒ | Óq ` pÓ | Òq
¯
U ´ 3J
3 |2, 1,´1y pÓ | Óq U ´ 3J
4 |2, 0, 0y
1
?
2
´
pÒ | Óq ´ pÓ | Òq
¯
U ´ J
5 |2, 0a, 0y
1
?
2
´
pÖ |´q ` p´| Öq
¯
U ´ J
6 |2, 0b, 0y
1
?
2
´
pÖ |´q ´ p´| Öq
¯
U ` J
7 |1, 1{2a, 1{2y pÒ |´q 0
8 |1, 1{2a,´1{2y pÓ |´q 0
9 |1, 1{2b, 1{2y p´| Òq 0
10 |1, 1{2b,´1{2y p´| Óq 0
11 |3, 1{2a, 1{2y pÒ | Öq 3U ´ 5J
12 |3, 1{2a,´1{2y pÓ | Öq 3U ´ 5J
13 |3, 1{2b, 1{2y pÖ | Òq 3U ´ 5J
14 |3, 1{2b,´1{2y pÖ | Óq 3U ´ 5J
15 |0, 0, 0y p´|´q 0
16 |4, 0, 0y pÖ | Öq 6U ´ 10J
only spin degrees of freedom, we restrict our considerations to a single subspace, where all unperturbed
states are degenerate. In particular, we focus on the lowest energy subspace.
Here, we only provide the final result without going through the calculations. A detailed derivation
of the formulae reported below can be found in [7]. The result, which is valid only for the subspace with
the lowest energy, is that the effective Hamiltonian turns out to be the standard Heisenberg Hamiltonian
with the effective Heisenberg constant Γeff:
Heff “ Γeff
ÿ
i‰j
`
SiSj ´ 4
˘
,
Γeff “
4|teff |2
U ` J
.
(3.1)
Γeff is determined by a single effective hopping constant teff, which in a simple way depends on an
initial hopping constants tmm1 :
teff “
b
ř
m,m1 t2
mm1
2
. (3.2)
4. The lowest part of energy spectra
We start the presentation of our results by specifying the conditions for the model parameters for
which the perturbation theory is justified. Our calculations are illustrated in figure 3 where the differences
between the energies of the ground states, of the first excited states and of the second excited states were
determined from the exact diagonalization and by using the Heisenberg model with the effective exchange
integral calculated from the perturbation theory. From figure 3 it is clear that these differences increase
more steeply than linearly with an increase of the ratio teff{pU ` Jq. Interestingly, the energies of the
33301-4
Two-orbital Hubbard model vs spin S “ 1 Heisenberg model
Figure 3. (Colour online) Comparison of the difference between the energies calculated for the TOHM
and for the Heisenberg model as a function of teff{pU ` Jq (Sz “ 0, t12 “ t21 “ 0, t11 “ t22). The data
are shown for the lowest three eigenvalues. The interaction parameters are: U “ 10 eV, J “ 0.5 eV.
ground state differ the most, almost twice less differ the energies of the first excited state and even less
do the ones of the second excited state.
Obviously, the limit value of teff{pU ` Jq above which the Heisenberg model ceases to describe
this system adequately, can be fixed arbitrarily. Here, we assumed that the difference between the state
energies should not exceed 0.03 eV. This corresponds to teff{pU ` Jq value equal to 0.05.
Next, we present the evolution of the lowest part of the energy spectrum of the TOHM depending on
the ratio t12{t11 between the hopping amplitudes of electrons, while the constant value of teff is maintained.
In the case presented in figure 4, we have chosen teff “ 0.5. When teff is constant, then the perturbation
theory gives the energy spectrum typical of the Heisenberg model, which does not depend on the ratio
t12{t11 (see the first column in figure 4). However, the exact solution shows that the energy spectrum
of the TOHM clearly depends on the ratio t12{t11, which is illustrated by the other columns in figure 4.
Thus, the theory of perturbation is too imprecise to distinguish the cases with different ways of electron
jumping between neighboring magnetic ions. In fact, in figure 4, we present only the energy levels with
the total Sz “ 0.
Figure 4. (Colour online) Comparison of the energy spectrum of the Heisenberg model with the lowest
parts of the energy spectra of the TOHM for the set of the following relationships between the hopping
amplitudes: t12{t11 “ 0, 0.2, 0.4, 0.6, 0.8, 1. Here, t21 “ t12, t22 “ 0, U “ 10 eV and J “ 0.5 eV,
Sz “ 0. In all these cases, the same teff “ 0.5 is kept. For the Heisenberg model, the exchange constant
Γeff “ 4t2
eff{pU ` Jq.
33301-5
R. Lemański, J. Matysiak
Figure 5. (Colour online) Comparison of the difference between energies of the second excited state
calculated exactly for the MOHM and for the Heisenberg model as a function of t12{t11, Sz “ 0. The
interaction parameters are: U “ 10 eV, J “ 0.5 eV.
The difference between the exact solution and that based on the Heisenberg model is clearly seen by
comparing the energy of the second excited state depending on the ratio t12{t11 for several different values
of teff, as it is shown in figure 5. For that particular state, the difference decreases with an increasing
t12{t11 ratio. The same trend takes place for all teff values, though themagnitude of the difference increases
exponentially.
It is obvious that the distribution of the lowest energy levels determines the dependence of specific
heat on temperature as it is illustrated in figures 6 and 7. Figure 6 illustrates the CvpTq changes with an
increase of the t12{t11 ratio for the fixed value teff “ 0.5 for the TOHM, as well asCvpTq for the Heisenberg
model for comparison. In turn, in figure 7, the differences between the values of CvpTq calculated for the
TOHM and the Heisenberg model with an increase of the ratio t12{t11 for the same value of teff “ 0.5
are displayed. Additionally, table 2 contains numeric values of the maxima of the curves from figure 7.
Based on the drawings given in figures 6 and 7, and especially in the table 2, it can be supposed that from
a course of the dependency Cv you can draw conclusions about the degree of hybridization expressed by
the ratio t12{t11.
Figure 6. (Colour online) Specific heat as a function of temperature for the Heisenberg model with
Γeff “ 4teff{pU` Jq “ 0.095 eV and for various versions of the MOHMmodel (for all of them teff “ 0.5
eV is kept). The curves for the MOHMmodel differ by ratios of the hopping constants t12{t11 (t21 “ t12,
t22 “ 0), as indicated on the plot legend. The parameters of the model are: U “ 10 eV, J “ 0.5 eV.
33301-6
Two-orbital Hubbard model vs spin S “ 1 Heisenberg model
Figure 7. (Colour online) The difference between specific heat of the Heisenberg model with Γ “
4teff{pU ` Jq “ 0.095 eV and the different cases of the MOHM model (for all of them teff “ 0.5 eV
is kept). The curves for the MOHM model differ by ratios of the hopping constants t12{t11 (t21 “ t12,
t22 “ 0), as indicated on the plot legend. The parameters of the model are: U “ 10 eV, J “ 0.5 eV.
Table 2.Maxima of the differences between the values of specific heat calculated for the TOHM and for
the Heisenberg model calculated for various t12{t11 ratios.
t12{t11 ratio 0 0.2 0.4 0.6 0.8 1
maximum ∆Cv [meV/K] 0.051 0.049 0.046 0.036 0.024 0.018
5. Summary and conclusions
We carried out exact numerical calculations for the four-site ring with two orbitals and two electrons
per site, using the TOHM and we compared the results with those obtained for the respective Heisenberg
model. On the basis of this comparison, we estimated the range of the interaction parameters, and, more
specifically, the range of the teff{pU` Jq ratio, for which the TOHM is well represented by the Heisenberg
model.
However, the main achievement of our work is a demonstration that the energy spectrum of the
system depends in a significant way on how an electron can jump from one site to another. Indeed,
the case when an electron can only jump between the same type of orbital is different from the case
when the electron can jump both between the same type and between different types of orbitals. This
difference manifests itself through a modification of the energy spectrum, which evidently evolves with
an increase in the degree of hybridization between different orbitals. This, in turn, brings a change, e.g.,
into the dependence of specific heat on temperature CvpTq. It can, therefore, be supposed that precise
measurements of the specific heat would enable us to determine the degree of hybridization between
different orbitals located on neighboring sites. However, to make this possible, the ratio of the effective
amplitude teff of the electron jump to the on-site interaction constant U ` J should be not very small,
because only in that case the TOHM is significantly different from the Heisenberg model.
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Порiвняння двоорбiтальної моделi Хаббарда та спiн S “ 1
моделi Гайзенберга: дослiдження на кластерах
Р. Лєманьскi, Я.Матисяк
Iнститут низьких температур та структурних дослiджень Польської академiї наук,
вул. Окульна 2, 50-422 Вроцлав, Польща
Нами проведено точнi числовi розрахунки для двоорбiтальної моделi Хаббарда на чотиривузловому кла-
стерi. В границi великої одновузлової взаємодiї ця модель стає еквiвалентною спiн S “ 1 моделi Гай-
зенберга. З порiвняння енергетичних спектрiв цих двох моделей нами визначено кiлькiсно дiапазон зна-
чень параметрiв взаємодiї, для якого модель Гайзенберга задовiльно вiдтворює двоорбiтальну модель
Хаббарда. Потiм ми розглянули змiни спектру, коли ми знаходимося за межами цього дiапазону, зосере-
джуючись, зокрема, на перевiрцi того, як вiн модифiкується, коли враховуються рiзнi способи мiжатом-
них перескокiв електронiв мiж рiзними орбiталями. Насамкiнець, нами показано як цi змiни впливають
на температурну залежнiсть питомої теплоємностi.
Ключовi слова: багатоорбiтальна модель Хаббарда, модель Гайзенберга, магнiтнi молекули
33301-8
https://doi.org/10.1103/PhysRevB.95.075115
https://doi.org/10.1103/PhysRevB.68.104403
https://doi.org/10.1166/jnn.2011.4301
https://doi.org/10.1007/s11051-013-1528-2
https://doi.org/10.1103/physrevlett.110.157204
https://doi.org/10.1016/0921-4534(89)90456-5
https://doi.org/10.1103/PhysRevB.37.9753
https://doi.org/10.1119/1.10537
https://doi.org/10.1016/0375-9601(77)90702-2
https://doi.org/10.1103/physrevb.18.3453
https://doi.org/10.1103/revmodphys.62.113
https://doi.org/10.1103/revmodphys.63.1
https://doi.org/10.1103/physrevb.56.12909
Introduction
Two-orbital Hubbard model (TOHM)
Computation details
Perturbation theory
The lowest part of energy spectra
Summary and conclusions
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| id | nasplib_isofts_kiev_ua-123456789-157113 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1607-324X |
| language | English |
| last_indexed | 2025-12-07T19:00:08Z |
| publishDate | 2018 |
| publisher | Інститут фізики конденсованих систем НАН України |
| record_format | dspace |
| spelling | Lemański, R. Matysiak, J. 2019-06-19T15:11:15Z 2019-06-19T15:11:15Z 2018 Two-orbital Hubbard model vs spin S=1 Heisenberg model: studies on clusters / R. Lemański, J. Matysiak // Condensed Matter Physics. — 2018. — Т. 21, № 3. — С. 33301: 1–8. — Бібліогр.: 15 назв. — англ. 1607-324X PACS: 31.15.vq, 75.10.Jm, 75.10.Pq DOI:10.5488/CMP.21.33301 arXiv:1809.09483 https://nasplib.isofts.kiev.ua/handle/123456789/157113 We perform exact numeric calculations for the two-orbital Hubbard model on the four-site cluster. In the limit
 of large on-site coupling the model becomes equivalent to the spin S “ 1 Heisenberg model. By comparing
 energy spectra of these two models, we quantified the range of interaction parameters for which the Heisenberg
 model satisfactorily reproduces the two-orbital Hubbard model. Then we examined how the spectrum evolves
 when we are outside of this region, focusing especially on checking of how it is modified when various ways of
 interatomic hoppings of electrons between different orbitals are taken into account. We finally show how these
 modifications affect the dependence of specific heat on temperature. Нами проведено точнi числовi розрахунки для двоорбiтальної моделi Хаббарда на чотиривузловому кластерi. В границi великої одновузлової взаємодiї ця модель стає еквiвалентною спiн S “ 1 моделi Гайзенберга. З порiвняння енергетичних спектрiв цих двох моделей нами визначено кiлькiсно дiапазон значень параметрiв взаємодiї, для якого модель Гайзенберга задовiльно вiдтворює двоорбiтальну модель
 Хаббарда. Потiм ми розглянули змiни спектру, коли ми знаходимося за межами цього дiапазону, зосереджуючись, зокрема, на перевiрцi того, як вiн модифiкується, коли враховуються рiзнi способи мiжатомних перескокiв електронiв мiж рiзними орбiталями. Насамкiнець, нами показано як цi змiни впливають
 на температурну залежнiсть питомої теплоємностi. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Two-orbital Hubbard model vs spin S=1 Heisenberg model: studies on clusters Порiвняння двоорбiтальної моделi Хаббарда та спiн S “ 1 моделi Гайзенберга: дослiдження на кластерах Article published earlier |
| spellingShingle | Two-orbital Hubbard model vs spin S=1 Heisenberg model: studies on clusters Lemański, R. Matysiak, J. |
| title | Two-orbital Hubbard model vs spin S=1 Heisenberg model: studies on clusters |
| title_alt | Порiвняння двоорбiтальної моделi Хаббарда та спiн S “ 1 моделi Гайзенберга: дослiдження на кластерах |
| title_full | Two-orbital Hubbard model vs spin S=1 Heisenberg model: studies on clusters |
| title_fullStr | Two-orbital Hubbard model vs spin S=1 Heisenberg model: studies on clusters |
| title_full_unstemmed | Two-orbital Hubbard model vs spin S=1 Heisenberg model: studies on clusters |
| title_short | Two-orbital Hubbard model vs spin S=1 Heisenberg model: studies on clusters |
| title_sort | two-orbital hubbard model vs spin s=1 heisenberg model: studies on clusters |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/157113 |
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