# Electron-hole asymmetry in electron systems with orbital degeneracy and
correlated hopping

Yu. Skorenkyy, O. Kramar, Yu. Dovhopyaty

(Received June 15, 2020, in final form September 15, 2020)

###### Abstract

Microscopic models of electronic subsystems with orbital degeneracy of energy
states and non-diagonal matrix elements of electron interactions (correlated
hopping) are considered within the configuration-operator approach. Equations
for arbitrary-temperature numerical calculation of the doublon concentration
for the integer band filling $n=1$ at different forms of the model density of
states are derived. The energy spectra obtained within the Green function
method are discussed with special emphasis on the role of correlated hopping
in transition from itinerant to localized behavior observed in vanadium
Magneli phases VnO2n-1, transition-metal dichalcogenides NiS2-xSex, fulleride
AnC60 systems.

Key words: electron correlations, energy spectrum, orbital degeneracy

###### Abstract

Ó ðîáîòi äîñëiäæåíî ìiêðîñêîïiíi ìîäåëi åëåêòðîííèõ ñèñòåì ç îðáiòàëüíèì
âèðîäæåííÿì åíåðãåòèíèõ ðiâíiâ òà íåäiàãîíàëüíèìè ìàòðèíèìè åëåìåíòàìè
ìiæåëåêòðîííî¿ âçàìîäi¿ (êîðåëüîâàíîãî ïåðåíîñó) çà äîïîìîãîþ êîíôiãóðàöiéíîãî
ïðåäñòàâëåííÿ. Îòðèìàíî ðiâíÿííÿ äëÿ èñëîâîãî ðîçðàõóíêó ïðè äîâiëüíèõ
òåìïåðàòóðàõ êîíöåíòðàöi¿ ïîëÿðíèõ ñòàíiâ äëÿ ñèñòåì ç âèðîäæåííÿì òà
öiëîèñåëüíèì çàïîâíåííÿì çîíè $n=1$ ïðè çàñòîñóâàííi ðiçíèõ ôîðì íåçáóðåíî¿
ãóñòèíè åëåêòðîííèõ ñòàíiâ. Åíåðãåòèíi ñïåêòðè, îòðèìàíi â ïiäõîäi ðiâíÿíü
ðóõó äëÿ ôóíêöié ðiíà, çàñòîñîâàíi äëÿ çÿñóâàííÿ ðîëi êîðåëüîâàíîãî ïåðåíîñó â
åôåêòàõ ëîêàëiçàöi¿ åëåêòðîíiâ, ÿêi ñïîñòåðiãàþòüñÿ â ôàçàõ Ìàãíåëi VnO2n-1,
äèõàëüêîãåíiäàõ ïåðåõiäíèõ ìåòàëiâ NiS2-xSex, ôóëåðèäàõ AnC60.

Ключов слова: åëåêòðîííi êîðåëÿöi¿, åíåðãåòèíèé ñïåêòð, îðáiòàëüíå âèðîäæåííÿ

##  1 Introduction

The idea of interaction-driven electron localization in a paramagnet due to
Coulomb forces (the Mott transition [1, 2]) and attempts to explain
ferromagnetism of transition metals have led to the elaboration of extremely
simple and insightful models of quantum statistics, namely Anderson model [3]
and Hubbard model [4]. The Hubbard model, describing a single non-degenerate
band of electrons with the on-site Coulomb interaction, provided the
mechanisms of the metal-to-insulator transition (MIT) and magnetic orderings.
The model Hamiltonian contains just two energy parameters: the matrix element
$t$ being the hopping integral of an electron to a neighboring site and the
parameter $U$ of the intra-atomic Coulomb repulsion of two electrons on the
same site. This model is used for a description of a variety of phenomena and
is intensively studied [5, 6, 7]. Albeit apparent simplicity, the Hubbard
model does not allow for exact solution except for the very special cases of
$d=1$ (one spatial dimension [8]) or $d\to\infty$ (infinite spatial
dimensionality [9]). The limit $d\to\infty$ seems to explain physics of the
real $3D-$materials quite well but the wide variety of compounds in which
strong electron correlations govern localization phenomena requires more than
just two-parameter model to account for all of the distinctive features. To
describe the peculiarities of real materials, the matrix elements of electron
interactions neglected in the original form of the Hubbard model as well as
orbital degeneracy of atomic energy levels [10, 11] are to be taken into
account.

Classic materials to manifest the Mott transition are a family of vanadium
Magneli phases, which have recently seen renewed attention [12]. In these
compounds not only the metal-to-insulator transition at increasing
temperatures is observed but also doping of cation subsystem or application of
the external pressure determine electrical properties. The transitions in VO2
and V2O3 can be explained within the framework of non-degenerated Hubbard
model [1, 13]. On the contrary, Mott-Hubbard physics of transition metal
dichalcogenides [14], which may find new applications in next generation
nanoelectronics [15], is the best described when the double orbital degeneracy
of $e_{g}$-electrons is explicitly incorporated into the Hamiltonian [16].
Another class of materials with inherent orbital degeneracy and promising
physical properties are fullerides [17], solid-state systems with doped
fullerene molecules C60. Unusual metal-insulator transition is observed in
fullerides AxC60 doped with alkali metals A. Among these compounds, A3C60 is
metallic, and the other phases AC60, A2C60 and A4C60 are insulators [18] while
the band structure calculations predict a purely metallic behavior [19]. The
orbital degeneracy of energy levels is responsible for the noted disagreement
of theory and experiment. The Gutzwiller variational approach [20] predicts
MIT for all integer band fillings $n=1,2,3,4,5$. Experimentally measured
electrical resistivity of polycrystals C60 [21] monotonously changes with
temperature and the energy gap has a monotonous dependence on the external
pressure.

A breakthrough in solving the puzzle of strong correlation happened due to the
development of the dynamical mean-field theory (DMFT) in the limit of infinite
dimensionality [9, 22]. It was quickly realized that DMFT is not applicable to
the system with the hopping amplitudes dependent on the site occupancies [23].
Studies of Falicov-Kimball model [24, 25], a partial case of the Hubbard model
with two sorts of particles, one localized and the other itinerant, shed light
on the peculiarities of the systems with correlated hopping of electrons
(dependency of hopping amplitudes on lattice site occupancies). Within DMFT
for Falicov-Kimball model, an exact two-pole structure was obtained for Green
function, while for a general case of Hubbard model, a finite-temperature
perturbation theory with electron hopping term as a perturbation parameter was
built [26].On this basis, a rigorous approach for description of correlated
hopping within the Falicov-Kimball model was built, thermodynamics of the
model and temperature-induced MIT were studied [27, 28]. Within the elaborated
scheme, considerable progress was achieved in theoretical description of
inelastic Raman and X-ray scattering in the vicinity of MIT [29, 30] and in
charge-density wave phase [31, 32]. The model with correlated hopping was
shown to manifest a peculiar optical conductivity [33] and thermoelectric
properties [34]–[36] due to the correlated hopping of electrons.

The present study is devoted to investigation of a system close to Mott-
Hubbard transition within the models taking into account the orbital
degeneracy of energy levels, strong Coulomb interaction and correlated hopping
of electrons. The structure of this paper is as follows. Section 2 is devoted
to the model formulation and model Hamiltonians for non-degenerated, doubly
and triply degenerated models. The general representations of the Hamiltonians
in the electron and Hubbard operators is given. In section 3 the
concentrations of empty sites (holes) and doubly occupied sites close to
metal-insulator transition at the electron concentration $n=1$ for models with
non-degenerated, doubly- and triply-degenerated energy levels are calculated.
For calculation of quasiparticle energy spectra, the nonperturbative approach,
generalized mean-field approximation [37, 38] is used. Section 4 presents our
conclusions on the role of correlated hopping in models with orbital
degeneracy.

##  2 The Hamiltonians of electronic subsystem with orbitally degenerated
energy levels

Within the second quantization formalism, the Hamiltonian of interacting
electron systems can be written [39] as

$\displaystyle
H=-\mu\sum_{i\lambda\sigma}a_{i\lambda\sigma}^{+}a_{i\lambda\sigma}+{\sum_{ij\lambda\sigma}}^{\prime}t_{ij}a_{i\lambda\sigma}^{+}a_{j\lambda\sigma}+\frac{1}{2}{\sum_{ijkl}}{\sum_{\alpha\beta\gamma\delta}}{\sum_{\sigma\sigma^{\prime}}}J^{\alpha\beta\gamma\delta}_{ijkl}a_{i\alpha\sigma}^{+}a_{j\beta\sigma^{\prime}}^{+}a_{l\delta\sigma^{\prime}}a_{k\gamma\sigma}\,,$
(2.1)

where $\mu$ stands for chemical potential, $a_{i\lambda\sigma}^{+}$,
$a_{i\lambda\sigma}$ are operators of spin-$\sigma$ electron creation and
annihilation in orbital state $\lambda$ on lattice site $i$, respectively, the
second sum with matrix element

$\displaystyle t_{ij}=\int{\mathrm{d}^{3}}r{\phi}_{\lambda}^{*}({\bf r}-{\bf
R}_{i})\left[-\frac{\hbar^{2}}{2m}\Delta+V^{\text{ion}}({\bf
r})\right]\phi_{\lambda}({\bf r}-{\bf R}_{j})$ (2.2)

describes translations (hopping) of electrons in the crystal field
$V^{\text{ion}}(\bf{r})$. The third sum in equation (2.1) is the general
expression for pair electron interactions described by matrix elements

$\displaystyle J^{\alpha\beta\gamma\delta}_{ijkl}$ $\displaystyle=$
$\displaystyle\int{\int{{\phi}_{\alpha}^{*}({\bf r}-{\bf
R}_{i}){\phi}_{\beta}({\bf r}-{\bf
R}_{j})}}\frac{\mathrm{e}^{2}}{|r-r^{\prime}|}{\phi}_{\delta}^{*}({\bf
r^{\prime}}-{\bf R}_{l}){\phi}_{\gamma}({\bf r^{\prime}}-{\bf
R}_{k})\mathrm{d}r\mathrm{d}r^{\prime}.$ (2.3)

In the above formulae, indices $\alpha$, $\beta$, $\gamma$, $\delta$,
$\lambda$ denote orbital states, ${\phi}_{\lambda i}$ is wave-function in
Wannier (site) representation, other notations are standard.

Hamiltonian (2.1) is hard to treat mathematically due to its non-diagonal form
in both Wannier- and wave-vector representation. The model is greatly
simplified if all matrix elements are neglected except on-site Coulomb
correlation (the Hubbard parameter $U$):

$\displaystyle U=\int{\int{|{\phi}_{\lambda}^{*}({\bf r}-{\bf
R}_{i})|^{2}\frac{\mathrm{e}^{2}}{|r-r^{\prime}|}|\phi_{\lambda}({\bf
r^{\prime}}-{\bf R}_{i})|^{2}\mathrm{d}r\mathrm{d}r^{\prime}}}.$ (2.4)

In this case, however, we fail to describe the influence of lattice site
occupation on the electron hopping (correlated hopping of electrons) which
removes the electron-hole symmetry for real correlated electron systems [16]
as well as plays an important role in the onset of itinerant ferromagnetism
[40, 41, 42, 43, 44]. Besides, for a system with orbital degeneracy of energy
states, the on-site exchange integral (Hund’s rule coupling)

$\displaystyle J_{H}$ $\displaystyle=$
$\displaystyle\int{\int{\phi}_{\lambda}^{*}}({\bf r}-{\bf
R}_{i})\phi_{\lambda^{{}^{\prime}}}({\bf r}-{\bf
R}_{i}){\mathrm{e}^{2}\over|{r}-{r}^{{}^{\prime}}|}\phi^{*}{{}_{\lambda^{{}^{\prime}}}}({\bf
r}^{{}^{\prime}}-{\bf R}_{i})\phi_{\lambda}({\bf r}^{{}^{\prime}}-{\bf
R}_{i})\mathrm{d}{\bf r}\mathrm{d}{\bf r}^{{}^{\prime}},$ (2.5)

which makes the configurations with parallel spins energetically more
favorable, should be taken into account.

The Hamiltonian of the generalized Hubbard model with correlated hopping of
electrons then reads as

$\displaystyle H_{s}$ $\displaystyle=$
$\displaystyle-\mu\sum_{i\sigma}n_{i\sigma}+U\sum_{i}n_{i\uparrow}n_{i\downarrow}+{\sum_{ij\sigma}}^{\prime}t_{ij}(n)a_{i\sigma}^{+}a_{j\sigma}+{\sum_{ij\lambda\sigma}}^{\prime}T^{{}^{\prime}}_{ij}\left(a_{i\sigma}^{+}a_{j\sigma}n_{i\bar{\sigma}}+h.c.\right),$
(2.6)

where $n_{i\sigma}=a_{i\sigma}^{+}a_{i\sigma}$ is the site occupancy operator,
hopping integrals $t_{ij}(n)$, $T^{{}^{\prime}}_{ij}$ taking into account two
types of correlated hopping of electrons [37] in non-degenerate model are
introduced. Such a generalized model is not invariant under particle-hole
transformation $a_{i\sigma}^{+}\to a_{i\sigma}$, in distinction from the
Hubbard model [45].

The Hamiltonian of the orbitally degenerated electronic system with strong
electron correlation and correlated hopping of electrons can be represented in
the form

$\displaystyle H_{\text{deg}}$ $\displaystyle=H_{\text{loc}}+H_{\text{tr}}\,,$
$\displaystyle H_{\text{loc}}$
$\displaystyle=-\mu\sum_{i\lambda\sigma}n_{i\lambda\sigma}+U\sum_{i\lambda}n_{i\lambda\uparrow}n_{i\lambda\downarrow}+\frac{U^{\prime}}{2}\sum_{i\lambda\sigma}n_{i\lambda\sigma}n_{i\lambda^{\prime}\bar{\sigma}}+\frac{U^{\prime}-J_{H}}{2}\sum_{i\lambda\lambda^{\prime}\sigma}n_{i\lambda\sigma}n_{i\lambda^{\prime}\sigma}\,,$
$\displaystyle H_{\text{tr}}$
$\displaystyle={\sum_{ij\lambda\sigma}}^{\prime}t_{ij}(n)a_{i\lambda\sigma}^{+}a_{j\lambda\sigma}+{\sum_{ij\lambda\sigma}}^{\prime}t^{{}^{\prime}}_{ij}\left(a_{i\lambda\sigma}^{+}a_{j\lambda\sigma}n_{i\bar{\lambda}}+h.c.\right)+{\sum_{ij\lambda\sigma}}^{\prime}t^{{}^{\prime\prime}}_{ij}\left(a_{i\lambda\sigma}^{+}a_{j\lambda\sigma}n_{i\lambda\bar{\sigma}}+h.c.\right),$

where $n_{i\lambda\sigma}=a_{i\lambda\sigma}^{+}a_{i\lambda\sigma}$,
$U^{\prime}=U-2J_{H}$ and hopping integrals $t_{ij}(n)$,
$t^{{}^{\prime}}_{ij},t^{{}^{\prime\prime}}_{ij}$ are introduced taking into
account different types of correlated hopping of electrons [16]. In a model of
doubly degenerate band, every site can be in one of 16 configurations [16]
while for triple degeneracy, inherent to fullerides, the number of possible
configurations rises to 64 [46].

In fullerenes, which have triply degenerated lowest unoccupied molecular
orbital of $t_{1u}$ symmetry, the competition between intra-site Coulomb
interaction and electron hoppings defines the electrical properties, with
insulator or metallic behavior realized [10]. Fullerides (fullerene crystals)
are semiconductors with an energy gap of $1.2-1.9$ eV [47, 48]. Intra-atomic
repulsion energy estimates [49] give $2.7$ eV. Calculation with a screening
effect taken into account gives $U=2.7$ eV [50, 51]. The energy cost of
electron configurations with two spins aligned in parallel on different
orbitals is considerably less than for anti-parallel alignment due to Hund’s
rule coupling.

To consider the case of strong electron correlation, it is reasonable to pass
from electron operator to Hubbard operators $X^{pl}$ of site transition from
state $|l\rangle$ to state $|p\rangle$ with anticommutation relations
$\\{X_{i}^{pl};X_{j}^{kt}\\}=\delta_{ij}(\delta_{lk}X_{i}^{pt}+\delta_{pt}X_{i}^{kl})$,
and normalizing condition $\sum\limits_{i}X_{i}^{p}=1$, for number operators
$X_{i}^{p}=X_{i}^{pl}X_{i}^{lp}$ of $|p\rangle$–state on site $i$.

In the configuration-operator representation, the model Hamiltonian takes the
form

$\displaystyle
H=H_{\text{loc}}+\sum_{\lambda=\alpha,\beta,\gamma}\left(H_{b}^{(\lambda)}+H_{h}^{(\lambda)}\right),$
(2.8)

where the interaction part $H_{\text{loc}}$ is diagonal, the processes forming
Hubbard subbands described by $H_{b}^{(\lambda)}$ and the hybridization of
these subbands $H_{h}^{(\lambda)}$ are clearly separated in the translational
part of the Hamiltonian, and which constitute a distinctive feature of
configuration-operator representation. Different hopping integrals correspond
to transitions in (or between) the different subbands.

In the partial case of band filling $n=1$, strong Coulomb correlation and
strong Hund’s coupling (parameter $U-3J_{H}$ is much greater than the
bandwidth), the states with three and more electrons on the same site are
excluded. Then, the influence of correlated hopping can be described by three
different hopping integrals. The bare band hopping integral $t_{ij}$ is
renormalized to take into account the band narrowing caused by concentration
dependent correlated hopping as $t_{ij}(n)=t_{ij}(1-\tau_{1}n)$. This hopping
integral characterizes the lower Hubbard subband. Parameter $\tau_{1}$ is
usually neglected, but it is of principal importance for a consistent
description of correlation effects in narrow band systems. The hopping
integral for the upper Hubbard subband is
$\tilde{t}_{ij}(n)=t_{ij}(n)+2t^{\prime}_{ij}=(1-\tau_{1}n-2\tau_{2})t_{ij}$
and $\bar{t}_{ij}(n)=t_{ij}(n)+t^{\prime}_{ij}=(1-\tau_{1}n-\tau_{2})t_{ij}$
describes a hybridization of lower and upper Hubbard subbands, correlated
hopping parameter $\tau_{2}$ describes the effect of occupancy of the sites
involved in the electron transfer and $\tau_{1}$ describes that of the
nearest-neighbor sites.

##  3 Results and discussion

Green functions technique allows us to calculate the energy spectrum of the
model in the case of electron concentration $n=1$. One can rewrite the single-
particle Green function $\langle\langle
a_{i\lambda\sigma}|a_{j\lambda\sigma}^{+}\rangle\rangle$ on the basis of the
relation between electronic operators and Hubbard’s $X$-operators describing
electron hoppings between holes, single-electron configurations and Hund’s
doublons. Probabilities of the processes involving another type of doubly
occupied states, states with three or more electrons tend to zero in the
considered limit of strong intra-atomic correlation.

Substantial extension of the Hilbert space for the cases with orbital
degeneracy makes the direct application of standard methods quite complicated
[52]. One may considerably simplify the calculation of quasiparticle energy
spectrum by projecting the terms arising from commutation with $H_{\text{tr}}$
in the Green function equation of motions onto a set of basic operators which
describe processes responsible for Hubbard subbands formation. The
configuration representation of Hubbard operators is especially well-suited
for such procedures and makes physical nature of the metal-to-insulator
transition and the influence of correlated hopping relatively transparent. In
the limit of strong Coulomb correlation and strong Hund’s coupling, for the
doubly degenerated band, such set of operators is $X^{0,\lambda\sigma}_{i}$,
$X^{\lambda\sigma,2}_{i}$, and for triply degenerated model it consists of
$X^{0,\lambda\sigma}_{i}$ and $Y^{\lambda\sigma,2}_{i}$ where $Y$-operator is
introduced to take into account that for the triply degenerate orbitals two
equivalent states are involved, for example,
$Y^{\alpha\uparrow,2}_{i}=X^{0\uparrow 0,\uparrow\uparrow
0}_{i}+X^{00\uparrow,\uparrow 0\uparrow}_{i}$.

To obtain a closed system of equations for Green functions $\langle\langle
X_{p}^{0,\alpha\uparrow}|X_{p^{\prime}}^{\alpha\uparrow 0}\rangle\rangle$ and
$\langle\langle Y_{p}|X_{p^{\prime}}^{\alpha\uparrow,0}\rangle\rangle$, we use
the projection procedure [37]:

$\displaystyle[X_{p}^{0,\alpha\uparrow};\sum_{\lambda}{H_{b}^{(\lambda)}}]$
$\displaystyle=$
$\displaystyle\sum_{i}\varepsilon_{pi}^{b}X_{i}^{0,\uparrow};\qquad[Y_{p};\sum_{\lambda}{H_{b}^{(\lambda)}}]=\sum_{i}\tilde{\varepsilon}_{pi}^{b}Y_{i};$
(3.1)
$\displaystyle[X_{p}^{0,\alpha\uparrow};\sum_{\lambda}{H_{h}^{(\lambda)}}]$
$\displaystyle=$
$\displaystyle\sum_{i}\varepsilon_{pi}^{h}Y_{i};\qquad\quad[Y_{p};\sum_{\lambda}{H_{h}^{(\lambda)}}]=\sum_{i}\tilde{\varepsilon}_{pi}^{h}X_{i}^{0,\uparrow}.$

This procedure realizes the generalized mean-field approximation which allows
one to easily break off the sequence of Green function equations and results
in two-pole quasiparticle Green function, with the spectrum describing two
Hubbard subbands with renormalized widths and hybridization. To obtain the
single-electron Green function, however, one should solve two systems of
equations (for $\langle\langle
X_{p}^{0,\alpha\uparrow}|X_{p^{\prime}}^{\alpha\uparrow 0}\rangle\rangle$,
$\langle\langle Y_{p}|X_{p^{\prime}}^{\alpha\uparrow,0}\rangle\rangle$ and
another for $\langle\langle X_{p}^{0,\alpha\uparrow}|Y^{+}_{p}\rangle\rangle$,
$\langle\langle Y_{p}|Y^{+}_{p}\rangle\rangle$), neglecting the irreducible
parts. Through the non-operator coefficients $\varepsilon^{b}({\bf
k}),\tilde{\varepsilon}^{b}({\bf k}),\varepsilon^{h}({\bf
k}),\tilde{\varepsilon}^{h}({\bf k})$, the spectrum becomes dependent on the
mean numbers of sites in particular electron configurations (for example, Hund
doublons in the doubly orbitally degenerated case) and thus dependent on
temperature.

Solving analytically the system of equations for $\langle\langle
X_{p}^{0,\alpha\uparrow}|X_{p^{\prime}}^{\alpha\uparrow 0}\rangle\rangle$ and
$\langle\langle Y_{p}|X_{p^{\prime}}^{\alpha\uparrow,0}\rangle\rangle$ after
Fourier transformation, we obtain the quasi-particle energy spectrum

$\displaystyle E_{1,2}({\bf k})=$ $\displaystyle-$
$\displaystyle\mu+\frac{\Delta}{2}+\frac{\varepsilon^{b}({\bf
k})+\tilde{\varepsilon}^{b}({\bf k})}{2}$ (3.2) $\displaystyle\mp$
$\displaystyle\frac{1}{2}\sqrt{(\Delta-\varepsilon^{b}({\bf
k})+\tilde{\varepsilon}^{b}({\bf k}))^{2}+4\varepsilon^{h}({\bf
k})\tilde{\varepsilon}^{h}({\bf k})}$

with $\Delta=U$ corresponding to the non-degenerated Hubbard model and
$\Delta=U-3J_{H}$ being the energy cost of creating a Hund’s doublon — a pair
of electrons with parallel spins in different orbitals of the same lattice
site [16, 11] for the models with orbital degeneracy. In a more robust
approach of work [26], the two-pole structure of the Green function was found
to realize only for Falicov-Kimball model while in general four state,
subspaces (electron, hole and resonant-valence bond) were considered. In
terminology of papers [26, 53], we study here only the low-energy sector of
the model to compare the effects produced by the degeneracy of the energy
levels and the correlated hopping, both removing the particle-hole symmetry.
Concentrations of doublons appear to be an important parameter in this
approach that allows one to naturally introduce a dependence of the energy
spectrum (3.2) on temperature or doping. In the degenerate model, in
distinction from the original Hubbard model, a doublon is not just charge
excitation but may carry a spin as well. For sufficiently high temperatures,
in the absence of any type of magnetic order, we denote the concentration of
empty lattice sites (holes) by $c$, concentration of sites occupied by a
single electron of any spin projection $\sigma$ in any orbital state $\lambda$
by $s$, Hund’s doublons concentration by $d$ and neglect all other
configurations. For the case of strong Hund’s coupling, the high energy
doublon configurations (spin-singlet Hubbard doublons and spin-triplet non-
Hund doublons) are excluded. We make use of the completeness condition for the
$X$-operator set to obtain constraints $c+2s+d=1$ for a non-degenerate model,
$c+4s+6d=1$ for a model with double degeneracy, $c+6s+6d=1$ for a model with
triple degeneracy at strong Coulomb correlation and at strong Hund’s coupling.

The non-operator coefficients $\varepsilon^{b}({\bf
k}),\tilde{\varepsilon}^{b}({\bf k}),\varepsilon^{h}({\bf
k}),\tilde{\varepsilon}^{h}({\bf k})$ can be obtained by a procedure described
in detail in paper [38] in the paramagnetic case at $n=1$ for the doubly
degenerated model as

$\displaystyle\varepsilon^{b}=\left((1-\tau_{1})\frac{1+4d+12(1-4d)^{2}}{4(1+4d)}+(1-\tau_{1}-2\tau_{2})\frac{8d^{2}}{1+4d}\right)\varepsilon;$
(3.3) $\displaystyle\varepsilon^{h}=(1-\tau_{1}-\tau_{2})d(3+4d)\varepsilon,$
(3.4)
$\displaystyle\tilde{\varepsilon}^{b}=\left(-8d^{2}(1-\tau_{1})+\frac{(1-\tau_{1}-2\tau_{2})}{8}(1-16d+32d^{2})\right)\varepsilon,$
(3.5)
$\displaystyle\tilde{\varepsilon}^{h}=(1-\tau_{1}-\tau_{2})\frac{d(3+4d)}{1+4d}\varepsilon$
(3.6)

and for the triply degenerated paramagnet at $n=1$ as

$\displaystyle\varepsilon^{b}=\left((1-\tau_{1})\frac{216d^{2}-12d+1}{24d+1}+(1-\tau_{1}-2\tau_{2})\frac{72d^{2}}{24d+1}\right)\varepsilon;$
(3.7)
$\displaystyle\varepsilon^{h}=(1-\tau_{1}-\tau_{2})\frac{8d-12d^{2}}{1-6d}\varepsilon,$
(3.8)
$\displaystyle\tilde{\varepsilon}^{b}=\left(-(1-\tau_{1})\frac{36d^{2}}{1-6d}+(1-\tau_{1}-2\tau_{2})\frac{1-16d+84d^{2}}{2(1-6d)}\right)\varepsilon,$
(3.9)
$\displaystyle\tilde{\varepsilon}^{h}=(1-\tau_{1}-\tau_{2})\frac{24d+1-216d^{2}}{3(24d+1)}\varepsilon.$
(3.10)

Corresponding expressions for non-degenerated model [38] are

$\displaystyle\varepsilon^{b}=\left[(1-\tau_{1})(1-2d+2d^{2})-2(1-\tau_{1}-2\tau_{2})d^{2}\right]\varepsilon;$
(3.11)
$\displaystyle\tilde{\varepsilon}^{b}=\left[(1-\tau_{1}-2\tau_{2})(1-2d+2d^{2})-2(1-\tau_{1})d^{2}\right]\varepsilon;$
(3.12)
$\displaystyle\varepsilon^{h}=\tilde{\varepsilon}^{h}=-2(1-\tau_{1}-\tau_{2})d\varepsilon.$
(3.13)

One can see that in degenerated models there is no symmetry between
expressions describing hole and doublon sectors.

For the generalized Hubbard model with correlated hopping [38], the doublon
concentration is given by

$\displaystyle
d=C\int^{w}_{-w}\rho(\varepsilon){\left[\frac{A(\varepsilon)}{\exp(\frac{E_{1}(\varepsilon)}{\Theta}+1)}+\frac{B(\varepsilon)}{\exp(\frac{E_{2}(\varepsilon)}{\Theta}+1)}\right]}\mathrm{d}\varepsilon,$
(3.14)

where $C={1\over 2}$, $\Theta=k_{\text{B}}T$,

$\displaystyle A({\bf k})={1\over
2}\left(1+\frac{\Delta+\tilde{\varepsilon}^{b}-\varepsilon^{b}}{\sqrt{(\Delta-\varepsilon^{b}+\tilde{\varepsilon}^{b})^{2}+4\varepsilon^{h}\tilde{\varepsilon}^{h}}}\right),$
$\displaystyle B({\bf k})=1-A({\bf k}).$ (3.15)

The equation for holes for non-degenerated model is completely identical to
the above equation as $c=d$, so $\mu$ is obtained from condition $n=1$.

For the doubly degenerated model with correlated hopping, the doublon
concentration is determined by the condition (3.14) with $C={1\over 4}$ to be
solved together with the condition for the chemical potential $c=2d$, which
for the doubly degenerated model can be written as

$\displaystyle
8d=(1+4d)\int^{w}_{-w}{\rho(\varepsilon)\left(\frac{B(\varepsilon)}{\exp(\frac{-E_{1}(\varepsilon)}{\Theta}+1)}+\frac{A(\varepsilon)}{\exp(\frac{-E_{2}(\varepsilon)}{\Theta}+1)}\right)}d\varepsilon.$
(3.16)

In analogous way we obtain equation (3.14) with $C={1\over 12}$ for triply
degenerated model with correlated hopping. To complete a set of equations, the
condition for the chemical potential $c=6d$ for the triply degenerated model
is also used

$\displaystyle
36d=(1-24d)\int^{w}_{-w}{\rho(\varepsilon)\left[\frac{B(\varepsilon)}{\exp(\frac{-E_{1}(\varepsilon)}{\Theta}+1)}+\frac{A(\varepsilon)}{\exp(\frac{-E_{2}(\varepsilon)}{\Theta}+1)}\right]}\mathrm{d}\varepsilon,$
(3.17)

Solving these equations numerically with appropriate model density
$\rho(\varepsilon)$ of electronic states (DOS), we obtain the doublon
concentration as functions of the model parameters and may apply the gap
criterion for the spectra (3.2) to study a metal-insulator transition (MIT)
[2, 14]. In the ground state, at the point of MIT, the polar states (holes and
doublons) concentrations tend to zero.

In distinction from the non-degenerated model, where it is the correlated
hopping which destroys the particle-hole symmetry, in doubly and triply-
degenerated models, there is no symmetry even at $\tau_{1}=\tau_{2}=0$ due to
non-equivalence of subspaces for empty states and doublon states (see [27]).

Dependencies of the doublon concentration on the correlation strength, shown
in figures 2, 2, allow us to illustrate the influence of correlated hopping
and asymmetry of the DOS form.

Figure 1: (Colour online) Dependence of doublon concentration on the energy
parameter $\Delta/w$ for different correlated hopping values at $\Theta/w=0.2$
for non-degenerated model. Solid curve corresponds to the Hubbard model
$\tau_{1}=\tau_{2}=0$, short-dashed curve corresponds to
$\tau_{1}=\tau_{2}=0.1$ and long-dashed curve corresponds to
$\tau_{1}=\tau_{2}=0.2$. Semi-elliptic DOS is used in calculations.

Figure 2: (Colour online) Dependence of doublon concentration in doubly
orbitally degenerated model at $\Theta/w=0.4$ on the energy parameter
$\Delta/w$ for DOS with asymmetry on a band edge at different values of the
asymmetry parameter $a=0,0.3,0.5,0.99$ from upper to lower curve.

Figure 3: (Colour online) Dependence of doublon concentration in doubly
orbitally degenerated model on temperature at fixed values of the energy
parameter $\Delta/w=0.6$ for the upper group of curves, $\Delta/w=1.2$ for the
middle group of curves and $\Delta/w=1.8$ for the lower group of curves. Solid
curves correspond to the semi-elliptic DOS, long-dashed curves correspond to
simple cubic lattice DOS and short-dashed curves correspond to DOS with
asymmetry.

Figure 4: (Colour online) Dependence of energy gap on the interaction strength
parameter for non-degenerated and triply degenerated models. In each pair, the
lower curve corresponds to non-degenerated model and the upper one corresponds
to the triply-degenerated model: for solid curves $\tau_{1}=\tau_{2}=0$, for
long-dashed curves $\tau_{1}=\tau_{2}=0.1$, for short-dashed curves
$\tau_{1}=\tau_{2}=0.2$.

One can see from figures 2–4 that the correlated hopping substantially
influences the doubly occupied states concentration in a system with orbital
degeneracy of the energy levels. Quite naturally, an increase of the
correlated hopping suppresses the conductance through electron localization,
and the doublon concentration decreases, whose effect for non-degenerated
model can be seen from figure 2. In a way, the role of the correlated hopping
does not change when the models with orbital degeneracy are considered, but
those models offer more mechanisms for particle-hole asymmetry than just the
correlated hopping. The DOS form can also affect the probability of doublon
and hole pair creation. To study an effect of bare-band density of states
asymmetry on the energy spectrum of a model with orbital degeneracy, we make
use of a model DOS with asymmetry on the band edge [41]. If the density of
states is not symmetrically distributed with respect to the subband center,
the doublon concentration is smaller than for a semi-elliptical DOS (see
figure 2) and $\rho(\varepsilon)$ makes a more prominent effect for weaker
correlations. DOS form asymmetry was shown to make more preferable conditions
for ferromagnetism stabilization [41]. From our results we conclude that both
DOS asymmetry and correlated hopping favor electron delocalization and
ferromagnetism (see also [44] where a similar effect was observed in the case
of week correlation). Figure 4 visualizes a temperature dependence of the
doublon concentration. Three families of dependencies are shown, each
corresponding to a fixed value of the correlation strength $\Delta=U-3J_{H}$.
The dependency for DOS form which corresponds to a simple cubic lattice of
transition metal compounds with double orbital degeneracy of $e_{g}$ band is
depicted by long-dashed curves. One can see that the model DOS with tunable
asymmetry parameter provides lower (at high asymmetries) and upper (for the
limiting case of semi-elliptical DOS) bounds for doublon concentration with
sc-lattice DOS. Qualitatively, the temperature dependencies monotonously
increase in the whole temperature interval, but are flattened at the greater
values of the interaction parameter. Both the occupancy of the sites involved
into the hopping processes (through the correlated hopping of the first type)
and the neighbor sites (through the second type correlated hopping), have the
effect of a gap in the energy spectrum and stabilization of the insulator
state. The energy gap, however, will not open up until a relatively large
increase of correlated hopping parameters takes place, not achievable in real
systems. At an increase of intra-site interaction parameter over a critical
value (dependent on the correlated hopping strength), the energy gap occurs
and the metal-insulator transition takes place (see figure 4 for zero
temperature). An increase of the polar states (holes and doublons)
concentration with temperature change modifies the energy spectrum, the energy
gap opens up and the insulating state stabilizes. On the contrary, the
application of the external pressure or doping in systems
(V1-xCr${}_{x})_{2}$O3 and NiS2-xSex leads to metalization [1] by increasing
the energy subbands width and reducing $\frac{\Delta}{w}$ below the critical
value. The critical values for the partial case when there is no correlated
hopping are in agreement with the results of paper [38] for non-degenerated
Hubbard model and paper [46] for a triply degenerated model.

##  4 Conclusions

Continuing the studies of non-denenerate Hubbard model generalized with taking
into account the correlated hopping of electrons, quasiparticle energy spectra
for doubly- and triply degenerated model are calculated to show that both the
orbital degeneracy and the correlated hopping remove the particle-hole
symmetry and have a strong effect on the electron localization. The energy
spectra of the lower and upper subband depend on the polar states
concentration and are essentially non-equivalent.

Equations are derived for an arbitrary-temperature numerical calculation of
the doublon concentration for the integer band filling $n=1$ at different
forms of the model density of states. Within the models of strongly correlated
electron systems with orbitally degenerated energy levels, both the correlated
hopping and the degeneracy of energy levels are crucial to describe the
effects of particle-hole asymmetry. The use of Hubbard $X$-operators
representation appears to be useful to structure the model Hamiltonian
according to the occupancies of the state subspaces and allows one to single
out the quasiparticle subbands, relevant for metal-insulator transition. The
expression for the energy gap which is temperature dependent through the polar
state concentration, which also depends on the DOS form, alows one to describe
the phase transition under the external actions. Critical values of the
correlation strength parameter for correlation-driven metal-insulator
transition in models with the degeneracy of energy levels and correlated
hopping substantially differ from that of Hubbard model. Taking into account a
number of microscopic parameters can be a leverage for tuning the analytical
expressions to reveal the relevant microscopic mechanisms of electron
localization in materials with non-equivalent subbands.

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