Formation of charge and spin ordering in strongly correlated electron systems
In this review we present results of our theoretical study of charge and spin ordering in strongly correlated electron systems obtained within various generalizations of the Falicov-Kimball model. The primary goal of this study was to identify crucial interactions that lead to the stabilization of v...
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Čenčariková, H. Farkašovský, P. 2017-06-10T18:59:52Z 2017-06-10T18:59:52Z 2011 Formation of charge and spin ordering in strongly correlated electron systems/ H. Čenčariková, P. Farkašovský // Condensed Matter Physics. — 2011. — Т. 14, № 3. — С. 42701:1-66. — Бібліогр.: 129 назв. — англ. 1607-324X PACS: 75.10.Lp, 71.27.+a, 71.28.+d, 71.30.+h, 05.30.Fk DOI:10.5488/CMP.14.42701 arXiv:1202.4565 https://nasplib.isofts.kiev.ua/handle/123456789/120030 In this review we present results of our theoretical study of charge and spin ordering in strongly correlated electron systems obtained within various generalizations of the Falicov-Kimball model. The primary goal of this study was to identify crucial interactions that lead to the stabilization of various types of charge ordering in these systems, like the axial striped ordering, diagonal striped ordering, phase-separated ordering, phase-segregated ordering, etc. Among the major interactions that come into account, we have examined the effect of local Coulomb interaction between localized and itinerant electrons, long-range and correlated hopping of itinerant electrons, long-range Coulomb interaction between localized and itinerant electrons, local Coulomb interaction between itinerant electrons, local Coulomb interaction between localized electrons, spin-dependent interaction between localized and itinerant electrons, both for zero and nonzero temperatures, as well as for doped and undoped systems. Finally, the relevance of resultant solutions for a description of rare-earth and transition-metal compounds is discussed. У цьому оглядi ми представляємо результати наших теоретичних дослiджень зарядового i спiно-вого впорядкування в сильно скорельованих електронних системах, що отриманi в рамках рiзних узагальнень моделi Фалiкова-Кiмбала. Основною метою цього дослiдження було iдентифiкувати ви-рiшальнi взаємодiї, що приводять до стабiлiзацiї рiзних типiв зарядового впорядкування в цих систе-мах, таких як осьове стрiчкове впорядкування, дiагональне стрiчкове впорядкування, фазове роз-шарування, фазова сегрегацiя i т.п. З помiж основних взаємодiй, якi враховують, нами розглядався вплив локальних кулонiвських взаємодiй мiж локалiзованими i колективними електронами, далек-осяжного i скорельованого переносу колективних електронiв, далекосяжної кулонiвської взаємодiї мiж локалiзованими i колективними електронами, локальної кулонiвської взаємодiї мiж колективни-ми електронами, локальної кулонiвської взаємодiї мiж локалiзованими електронами, спiнозалежної взаємодiї мiж локалiзованими i колективними електронами, як при нульовiй так i при ненульовiй температурах, а також для легованих i нелегованих систем. На завершення, обговорюється засто-совнiсть отриманих розв’язкiв для опису сполук рiдкiсноземельних i перехiдних елементiв. This work was supported by Slovak Grant Agency VEGA under Grant No. 2/0175/10 and by the ERDF EU grant, under the contract No. ITMS26220120005. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Formation of charge and spin ordering in strongly correlated electron systems Формування зарядового i спiнового впорядкування в сильно скорельованих електронних системах Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
Formation of charge and spin ordering in strongly correlated electron systems |
| spellingShingle |
Formation of charge and spin ordering in strongly correlated electron systems Čenčariková, H. Farkašovský, P. |
| title_short |
Formation of charge and spin ordering in strongly correlated electron systems |
| title_full |
Formation of charge and spin ordering in strongly correlated electron systems |
| title_fullStr |
Formation of charge and spin ordering in strongly correlated electron systems |
| title_full_unstemmed |
Formation of charge and spin ordering in strongly correlated electron systems |
| title_sort |
formation of charge and spin ordering in strongly correlated electron systems |
| author |
Čenčariková, H. Farkašovský, P. |
| author_facet |
Čenčariková, H. Farkašovský, P. |
| publishDate |
2011 |
| language |
English |
| container_title |
Condensed Matter Physics |
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Інститут фізики конденсованих систем НАН України |
| format |
Article |
| title_alt |
Формування зарядового i спiнового впорядкування в сильно скорельованих електронних системах |
| description |
In this review we present results of our theoretical study of charge and spin ordering in strongly correlated electron systems obtained within various generalizations of the Falicov-Kimball model. The primary goal of this study was to identify crucial interactions that lead to the stabilization of various types of charge ordering in these systems, like the axial striped ordering, diagonal striped ordering, phase-separated ordering, phase-segregated ordering, etc. Among the major interactions that come into account, we have examined the effect of local Coulomb interaction between localized and itinerant electrons, long-range and correlated hopping of itinerant electrons, long-range Coulomb interaction between localized and itinerant electrons, local Coulomb interaction between itinerant electrons, local Coulomb interaction between localized electrons, spin-dependent interaction between localized and itinerant electrons, both for zero and nonzero temperatures, as well as for doped and undoped systems. Finally, the relevance of resultant solutions for a description of rare-earth and transition-metal compounds is discussed.
У цьому оглядi ми представляємо результати наших теоретичних дослiджень зарядового i спiно-вого впорядкування в сильно скорельованих електронних системах, що отриманi в рамках рiзних узагальнень моделi Фалiкова-Кiмбала. Основною метою цього дослiдження було iдентифiкувати ви-рiшальнi взаємодiї, що приводять до стабiлiзацiї рiзних типiв зарядового впорядкування в цих систе-мах, таких як осьове стрiчкове впорядкування, дiагональне стрiчкове впорядкування, фазове роз-шарування, фазова сегрегацiя i т.п. З помiж основних взаємодiй, якi враховують, нами розглядався вплив локальних кулонiвських взаємодiй мiж локалiзованими i колективними електронами, далек-осяжного i скорельованого переносу колективних електронiв, далекосяжної кулонiвської взаємодiї мiж локалiзованими i колективними електронами, локальної кулонiвської взаємодiї мiж колективни-ми електронами, локальної кулонiвської взаємодiї мiж локалiзованими електронами, спiнозалежної взаємодiї мiж локалiзованими i колективними електронами, як при нульовiй так i при ненульовiй температурах, а також для легованих i нелегованих систем. На завершення, обговорюється засто-совнiсть отриманих розв’язкiв для опису сполук рiдкiсноземельних i перехiдних елементiв.
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1607-324X |
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https://nasplib.isofts.kiev.ua/handle/123456789/120030 |
| citation_txt |
Formation of charge and spin ordering in strongly correlated electron systems/ H. Čenčariková, P. Farkašovský // Condensed Matter Physics. — 2011. — Т. 14, № 3. — С. 42701:1-66. — Бібліогр.: 129 назв. — англ. |
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Condensed Matter Physics, 2011, Vol. 14, No 4, 42701: 1–66
DOI: 10.5488/CMP.14.42701
http://www.icmp.lviv.ua/journal
Review article
Formation of charge and spin ordering in strongly
correlated electron systems
H. Čenčariková, P. Farkašovský
Institute of Experimental Physics, Slovak Academy of Sciences, Watsonová 47, 040 01 Košice, Slovakia
Received July 7, 2011, in final form November 7, 2011
In this review we present results of our theoretical study of charge and spin ordering in strongly correlated
electron systems obtained within various generalizations of the Falicov-Kimball model. The primary goal of
this study was to identify crucial interactions that lead to the stabilization of various types of charge ordering
in these systems, like the axial striped ordering, diagonal striped ordering, phase-separated ordering, phase-
segregated ordering, etc. Among the major interactions that come into account, we have examined the effect
of local Coulomb interaction between localized and itinerant electrons, long-range and correlated hopping of
itinerant electrons, long-range Coulomb interaction between localized and itinerant electrons, local Coulomb
interaction between itinerant electrons, local Coulomb interaction between localized electrons, spin-dependent
interaction between localized and itinerant electrons, both for zero and nonzero temperatures, as well as for
doped and undoped systems. Finally, the relevance of resultant solutions for a description of rare-earth and
transition-metal compounds is discussed.
Key words: charge and spin ordering, metal-insulator transitions, valence transitions, Falicov-Kimball model,
strongly correlated systems
PACS: 75.10.Lp, 71.27.+a, 71.28.+d, 71.30.+h, 05.30.Fk
Contents
1 Introduction 2
2 Methods 7
2.1 Exact diagonalization technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Approximate method based on the reduction of the total energy . . . . . . . . . . 8
3 Charge ordering in the spinless Falicov-Kimball model 10
3.1 The effect of local Coulomb interaction . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 The effect of long-range hopping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3 The effect of nonlocal Coulomb interactions . . . . . . . . . . . . . . . . . . . . . . 21
3.3.1 The effect of correlated hopping . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3.2 The effect of nearest-neighbour Coulomb interaction . . . . . . . . . . . . . 25
3.4 The effect of lattice geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4 Charge and spin ordering in the spin-1/2 Falicov-Kimball model 32
4.1 Spin-1/2 Falicov-Kimball model without the Ising interaction . . . . . . . . . . . . 32
4.2 Spin-1/2 Falicov-Kimball model with the Ising interaction . . . . . . . . . . . . . . 37
5 Applications of the Falicov-Kimball model to a description of realmaterials 45
5.1 Ground-state properties of NaxCoO2 . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.2 Magnetization processes in rare-earth tetraborides . . . . . . . . . . . . . . . . . . 48
5.3 Doping-induced valence changes in rare-earth compounds . . . . . . . . . . . . . . 51
c© H. Čenčariková, P. Farkašovský, 2011 42701-1
http://dx.doi.org/10.5488/CMP.14.42701
http://www.icmp.lviv.ua/journal
H. Čenčariková, P. Farkašovský
6 Stability of charge and spin ordering at finite temperatures 55
7 Conclusion 61
1. Introduction
The problem of inhomogeneous charge and magnetic ordering in strongly interacting electron
systems is certainly one of the most intensively studied problems of the contemporary
solid state physics. The reason is that the inhomogeneous charge ordering (e.g., the striped phases)
were experimentally observed in many rare-earth and transition-metal compounds [1–4], like
La1.6Nd0.4SrxCuO4, YBa2Cu3O6+x, Bi2Sr2Cu2O8+x, La1.5Sr0.5NiO4, NaxCoO2, some of which
exhibit a high temperature superconductivity. This phenomenon was most frequently studied in
the literature within the Hubbard and t−J model [5–12]. Theoretical studies based on these mod-
els pointed to two possible mechanisms of forming the inhomogeneous charge ordering in these
materials: (i) Kivelson, Emery, and co-workers [7] proposed that strongly correlated systems have
a natural tendency toward phase separation and the inhomogeneous spatial charge ordering arises
from a competition between this tendency to phase separate and the long-range Coulomb inter-
action which does not allow the electron density to stray too far from its average; and (ii) White
and Scalapino [9–12] proposed that the stripe order arises from a competition between kinetic and
exchange energies in a doped antiferromagnet which does not require long-range Coulomb forces
to stabilize the stripes.
Moreover, shortly after the introduction of the spinless Falicov-Kimball model in 1986 by
Kennedy and Lieb [13] and Brandt and Schmidt [14] it was found that this probably the sim-
plest model of correlated electrons on a lattice is capable of describing various types of charge
ordering including the periodic as well as phase-separated and segregated configurations [15–28].
This opened a new route of studying this important phenomenon. The model describes a system
of itinerant and localized particles on the lattice, which interact only through the local Coulomb
interaction. The Hamiltonian of the spinless Falicov-Kimball model can be written in the form
H =
∑
ij
tijc
+
i cj + U
∑
i
wic
+
i ci , (1)
where c+i (ci) are the creation (anihilation) operators of the itinerant spinless particles at site i, tij
describes the hopping probability from site i to j and wi is the occupation number of the localized
particles taking the value 1 or 0 according to whether the site i is occupied or unoccupied by the
localized particle.
Accordingly, as the itinerant and localized particles are interpreted, we get different interpreta-
tions of the model. Historically, the first interpretation of the model: itinerant particles = electrons
with spin up and localized particles = electrons with spin down, was already used in the original
work by Hubbard [29] as the approximative solution of the original Hubbard model, in which one
type of particles, e.g., electrons with spin down, are immobilized. Therefore, this approximation is
sometimes referred to as the static Hubbard model. The second interpretation: itinerant particles
= itinerant electrons and localized particles = immobile ions, comes from Kennedy and Lieb [13]
and represents a very simple model of crystallization in solids. With respect to the fact that the
Falicov-Kimball model was originally proposed to describe valence transitions and metal-insulator
transitions in rare-earth compounds, one of the most frequent interpretations is to consider the
conduction d electrons instead of itinerant particles and the valence f electrons instead of localized
particles.
The greatest advantage of the spinless Falicov-Kimball model is its relative simplicity, which
makes the model more accessible to analytical and numerical studies compared with e.g., the
Hubbard or periodic Anderson model. Two diametrically different ways have been used to solve
the spinless Falicov-Kimball model. The first direction represents analytical studies in the limit of
infinite dimensions (D = ∞) and other analytical and numerical studies in reduced dimensions
(D = 1 and D = 2).
42701-2
Formation of charge and spin ordering in strongly correlated electron systems
Regarding the works devoted to the study of the Falicov-Kimball model in the infinite dimen-
sional limit, it is worth noting that there are two good reasons to study the model in this limit. The
first reason is that for D → ∞ the self-energy becomes local [30], which simplifies the analytical
calculations to the extent that they permit to solve many problems exactly. The second reason
for studying the model in the limit of infinite dimension is the fact that some physical quantities
calculated for D = ∞ reproduce the three-dimensional results better than the one-dimensional
solutions. A milestone in this direction is the work by Brandt and Mielsch [31], in which the exact
solution of the spinless Falicov-Kimball model is presented for the symmetric case (the number of
d electrons = the number of f electrons = L/2, where L is the number of lattice sites). The main
result of this study was a determination of the critical transition temperature for a transition from
the high temperature disordered phase to the low-temperature ordered (checkerboard) phase. In
subsequent articles, Brandt et al. [31–35] built the foundations of physics of the Falicov-Kimball
model in the limit of infinite dimensions, on which many other theoretical physicists built both the
spinless and spin-one-half Falicov-Kimball model [36–46]. The results of these studies are summa-
rized in an excellent review article by Freericks and Zlatic [47] devoted to exact solutions of the
Falicov-Kimball model in the dynamic mean field theory.
Regarding the second direction, the analytical solutions of the Falicov-Kimball model in the
limit of reduced dimensions, it should be noted that despite the huge efforts of theorists and relative
simplicity of the model, so far only very few exact results for the ground state and thermodynamics
of the spinless model have been obtained. In addition to the aforementioned evidence of the long-
range arrangement at low temperatures and dimensions D > 2, the following has been proven:
(i) the absence of spontaneous hybridization at finite temperatures [48], (ii) the phase separation
and periodic arrangement in the limit of strong Coulomb interactions and D = 1 [49], (iii) the
phase separation in the two-dimensional model for selected values of electron concentrations and
sufficiently large Coulomb interactions [21, 27, 28, 50], (iv) the phase separation in the limit of
strong Coulomb interactions for all dimensions [51, 52].
Under these circumstances, numerical calculations seem to be very valuable. Although they do
not always provide definitive answers to the questions searched, they are capable of pointing out the
fundamental trends in the system, and thus help to understand the physics of the Falicov-Kimball
model. Basically, there are two main directions of the numerical study of the Falicov-Kimball model.
The first is an exact diagonalization of the model Hamiltonian on finite lattices over a complete
set of accessible configurations of localized particles. As the number of configurations increases
as 2L, this method is severely limited by the size of clusters, which are eligible for the numerical
study (L ∼ 36–40). In the one dimensional case such cluster sizes are sufficient to extrapolate the
results obtained to the thermodynamic limit (L → ∞), yielding definite results on the ground
state or thermodynamics of the model, at least for a certain area of model parameters (e.g., strong
interactions). In the two-dimensional and three-dimensional cases, however, such cluster sizes are
insufficient and, therefore, it is necessary to use approximate methods that somehow reduce the
number of investigated configurations. This procedure was first used in the work by Freericks and
Falicov [18] in the study of one-dimensional phase diagram of the spinless Falicov-Kimball model for
selected concentrations of localized particles. Since the same formalism with minor modifications
was used later by many other authors in the study of the ground states of the one-dimensional
and two-dimensional models, we briefly summarize the main points of this algorithm. Consider
some configuration w = {w1, w2, . . . , wL}, which corresponds to a specific distribution of particles
localized on the lattice consisting of L sites, while the classical variable wi = 1, 0 indicates whether
or not the site i is occupied by a localized particle. The energy of the system with Nd itinerant
electrons corresponding to the selected configuration is then given by
E(U,Nd, w) =
EF
∫
−∞
n(E,w)EdE, (2)
where n(E,w) is the density of states corresponding to the configuration w and EF is the Fermi
42701-3
H. Čenčariková, P. Farkašovský
energy, which can be determined from the condition
Nd =
EF
∫
−∞
n(E,w)dE. (3)
The density of states n(E,w) in the aforementioned expression is given by
n(E,w) = −
1
πL
Im lim
ε→0
∑
j
Gj(E + iε) , (4)
where Gj(E) are the local Green’s functions for which Freericks and Falicov [18] found the following
expression
Gj(E) =
1
E − Uwj −∆+
j (E)−∆−
j (E)
, (5)
where
∆±
j =
1
E − Uwj±1 −
1
E − Uwj±2 −
1
E − Uwj±3 − . . .
. (6)
The problem is thus reduced to the calculation of infinite fractions ∆±
j [53]. For the concentration
of localized particles equal to nf = 0.5, Freericks and Falicov [18] investigated ten periodic config-
urations of ions (their list is given in table 1), which represent all different physical configurations
Table 1. The complete list of all physically different periodic configurations of ions with the
period p < 9 and nf = 1/2 [18].
k configuration period k configuration period
1 10 6 11101000
2 1100 7 11100100
3 111000 8 11011000
4 110100 9 11010100
5 11110000 10 11010010
of localized particles with the period less than 9 and a segregated phase (an incoherent mixture of
completely empty and fully occupied lattice) with a density of states
nseg(E) = (1− nf )n
empty(E) + nfn
fully(E) . (7)
Using the numerical analysis they obtained (for the case nf = 1/2) the coherent ground-
state phase diagram (see figure 1) of the model that exhibits some general features: (a) The
alternating phase w1 = {1010 . . .10} is the ground state at nd = 1/2 for all values of U as stated
by previous investigations [13]. (b) The phase diagram tends to be simplified as the interaction
strength increases indicating that many-body effects stabilize the system (this is a consequence of
the segregation principle). (c) There is a trend for phases that disappear from the phase diagram as
U increases to reappear as phase islands at even larger values of U (e.g., the configuration No. 3).
(d) Phase islands of configurations not present at U = 0 may be formed at larger values of U (e.g.,
the configuration No. 8). (e) Some configurations are not the ground state for any value of U or
nd (e.g., the configurations {11101000 . . .} and {11010100 . . .} do not appear).
A similar method was used later on by Gruber et al. [22] for the study of the one-dimensional
phase diagram of the spinless Falicov-Kimball model for the so-called neutral case, where the
42701-4
Formation of charge and spin ordering in strongly correlated electron systems
concentration of itinerant electrons nd is equal to the concentration of localized electrons/ions nf .
Unlike the previous work [18], Gruber et al. did not focus on studying the model only for selected
values of nf , but they studied the comprehensive phase diagram of the model in the nf − U
plane. Similarly to the previous case, the set of input configurations is not complete, since only
periodic configurations with small periods and mixtures of these periodic configurations with an
empty lattice have been taken into account1. The main result of their numerical and analytical
studies is that the ground states of the spinless Falicov-Kimball model (nf 6 0.5) are either the
most homogeneous distributions of localized particles (for U and nf large enough) or mixtures
of periodic configurations and the empty lattice (U and nf small). Since the Fermi level for the
mixtures of periodic configurations and the empty lattice lies in the conduction band, while for the
most homogeneous configurations lie in the energy gap, the boundary between these two domains, is
a boundary of metal-insulator transitions, which may be induced either by the Coulomb interaction
or by changing the concentration of localized electrons.
Figure 1. The phase diagram of the one-dimensional spinless Falicov-Kimball model in the
nd −U plane calculated for nf = 1/2 [18]. The set of ground-state configurations consists of the
segregated configuration and the periodic configurations with the smallest periods (see table 1).
For the two-dimensional case, this method was generalized by Watson and Lemanski [23] and
later it was used by Lemanski, Freericks and Banach to study the ground-state phase diagram
of the model in D = 2 [54]. In this case, the set of input configurations includes all periodic
configurations with a unit cell having a smaller number of sites than the selected critical value Nc
and all possible mixtures thereof. The most interesting result of these studies was the observation
of axial and diagonal striped phases of localized particles (see figure 2), suggesting that in the
system of itinerant and localized particles, a sufficient mechanism leading to the formation of
inhomogeneous charge arrangement is the local Coulomb interaction between these two electron
subsystems. This is a much simpler mechanism of forming inhomogeneous charge stripes than the
one considered earlier within the Hubbard model [5, 6], respectively, within the t−J model [7–12].
Due to the incomplete input set of investigated configurations, an open question remains whether
these results persist if the complete set of configurations will be considered, and especially, whether
they persist for the more realistic three-dimensional case.
1The difference between the methods presented in [18] and [22] is that in reference [18] the canonical phase
diagrams were constructed, where only the simplest periodic configurations and the segregated phase are taken into
account, but not a mixture thereof. However, in reference [22] the grand canonical phase diagrams were constructed
first and only then they were transformed into the canonical phase diagrams. This procedure ensures both the
simplest periodic configurations, the segregated phase and all possible mixtures thereof are included.
42701-5
H. Čenčariková, P. Farkašovský
Figure 2. The ground-state phase diagram of the spinless Falicov-Kimball model in D = 2
calculated by the method of restricted set diagrams [54]. The typical ground-state configurations
detected in the phase diagram. The large dots correspond to the occupied sites and the small
dots correspond to the vacant sites.
One of the major shortcomings of the Falicov-Kimball model is that it does not include any spin
interactions between electrons, and, therefore, it is not capable of describing the magnetic super-
structures, which in many of rare-earth and transition-metal compounds coexist with charge order-
ing. This phenomenon was observed, for example, not only in nickelates [55–57], manganates [58],
0 0.5 1 1.5 2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
D1
D2
D3
D4
D5
| |
F FAF1 2 3 4 1234
E
n
d
n f
↑ ↑
↑ ↑
↑ ↑
↑ ↑
↑ ↑
↑ ↑
↓ ↓ ↓ ↓
↓ ↓ ↓ ↓
↓ ↓ ↓ ↓
↓ ↓ ↓ ↓
↓ ↓ ↓ ↓
↓ ↓ ↓ ↓
1
↑ ↑
↑ ↑
↑ ↑
↑ ↑
↑ ↑
↑ ↑
↓ ↓ ↓ ↓
↓ ↓ ↓ ↓
↓ ↓ ↓ ↓
↓ ↓ ↓ ↓
↓ ↓ ↓ ↓
↓ ↓ ↓ ↓
2
↑ ↑ ↑
↑ ↑ ↑
↑ ↑ ↑
↑ ↑ ↑
↑ ↑ ↑
↑ ↑ ↑
↓ ↓ ↓
↓ ↓ ↓
↓ ↓ ↓
↓ ↓ ↓
↓ ↓ ↓
↓ ↓ ↓
3
↑ ↑ ↑ ↑
↑ ↑ ↑ ↑
↑ ↑ ↑ ↑
↑ ↑ ↑ ↑
↑ ↑ ↑ ↑
↑ ↑ ↑ ↑
↓ ↓
↓ ↓
↓ ↓
↓ ↓
↓ ↓
↓ ↓
4
↑ ↑ ↑
↑ ↑ ↑ ↑ ↑ ↑
↑ ↑ ↑
↑ ↑ ↑ ↑ ↑ ↑
↑ ↑ ↑
↑ ↑ ↑ ↑ ↑ ↑
D1
↑ ↑
↑ ↑
↑ ↑
↑ ↑
↑ ↑
↑ ↑
↓ ↓
↓ ↓
↓ ↓
↓ ↓
↓ ↓
↓ ↓
D2
↑ ↑ ↑
↑ ↑ ↑
↑ ↑ ↑
↓ ↓ ↓
↓ ↓ ↓
↓ ↓ ↓
D3
↑ ↑
↑ ↑
↑ ↑
↑ ↑
↑ ↑
↑ ↑
D4
↑ ↑ ↑
↑ ↑ ↑
↑ ↑ ↑
D5
Figure 3. The restricted ground-state phase diagram of the spin-1/2 Falicov-Kimball model
with the Ising interaction for the Coulomb interaction U = 8 and the Ising interaction J = 0.5
(the canonical phase diagram in the plane nf − nd). F, AF and E refer to the ferromagnetic,
antiferromagnetic and empty phases, respectively. Periodic phases with nf=1 are denoted by
the numbers 1–4 (right hand panel), and those with nf + nd/2= 1 by the symbols D1–D5. The
phases are ground states only on the bold straight-line segments or at single points. Outside
these straight line segments or points there are mixtures of periodic phases that have lower
energy than any periodic phase taken from the restricted set. The small vertical straight line
segments, crossing the nf= 1 line, mark the limits of F. The diagonal line nf +nd/2 = 1 is only
a visual guide [63].
42701-6
Formation of charge and spin ordering in strongly correlated electron systems
cobaltates [59, 60], but also in materials exhibiting high-temperature superconductivity [1, 61, 62].
At present, there are still intensive discussions on possible mechanisms of the formation of inho-
mogeneous charge and spin ordering and its relation to the physical properties of the systems, e.g.,
high-temperature superconductivity. The easiest way of introducing spin interactions in the system
of itinerant and localized electrons is to bind them by the Ising interaction. This idea was first used
by Lemanski [63], who found that turning on the Ising interaction between itinerant and localized
electrons leads to the stabilization of different types of charge and spin arrangement, including the
axial and diagonal striped phases (see figure 3). Moreover, a number of simple rules of formation of
various sorts of ground-state phases have been presented in reference [64]. Since these results were
obtained on a restricted set of configurations, an open challenge for further theoretical studies was
whether the nature of the ground state would remain unchanged after taking into account the com-
plete set of configurations. For this reason we have decided to perform a systematic study (within
an exact diagonalization and well-controlled approximate method described in the next section) of
the ground state of the spinless as well as generalized spin-1/2 Falicov-Kimball model, in order to
find the fundamental mechanisms of the formation of inhomogeneous charge and spin ordering in
strongly correlated systems. Apart from the above mentioned local Coulomb interaction between d
and f electrons, we have also investigated the effect of nonlocal Coulomb interaction [65, 66], the
correlated hopping [67–71], the lattice geometry [72, 73], the dimension of the system [74, 75], the
anisotropic spin-dependent interaction [76–78] and the Hubbard interaction [76]. In what follows,
we state a brief overview of the main results we have reached in our numerical studies.
2. Methods
To study the ground-state properties of the model Hamiltonians based on the spinless/ spin-1/2
Falicov-Kimball model we have used the method of exact diagonalization on finite clusters, where
diagonalizations are performed over all possible distributions of localized particles, as well as the
approximate method developed by us, in which the acceptance of configuration is realized using
the principle of reducing the total energy of the system. Periodic boundary conditions are used in
all the examined cases, since the fastest convergence of numerical results to the thermodynamic
limit is observed for this type of boundary conditions.
2.1. Exact diagonalization technique
Although in the next major steps, the exact diagonalization method (EDM) will be illustrated
for the spinless Falicov-Kimball model, the applicability of the method is much broader and with
minor modifications it can be directly extended to the spin-1/2 Falicov-Kimball model, as well
as the spin-1/2 Falicov-Kimball model with the Ising interaction between localized and itinerant
electrons. The method is flexible with regard to the changes of the hopping matrix tij and so it
may be used, without any additional numerical complications, to study the effects of long-range
and correlated hopping of electrons on the ground-state properties of the model.
Hereinafter we will use solely the interpretation of the spinless Falicov-Kimball model in which
the itinerant particles are d electrons and the localized particles are f electrons from localized 4f
or 5f states of rare-earth ions. Then, spinless Falicov-Kimball model can be written in the form
H =
∑
ij
tijd
+
i dj + U
∑
i
wid
+
i di , (8)
where wi (wi = 1, 0) now describes the occupancy of the f orbital at lattice site i.
It is important to note that for any distribution of f electrons w = {w1, w2, . . . , wL}, the
Hamiltonian (8) is a single particle Hamiltonian in the representation of the second quantization
H =
∑
ij
hij(w)d
+
i dj , (9)
where hij(w) = tij + Uwiδij . Thus, the solution of the model (8) reduces to the problem of
determining the spectra of matrix h(w) for different distributions of f electrons on the lattice of
42701-7
H. Čenčariková, P. Farkašovský
the size L. Since the problem is analytically solvable only for special types of configurations (e.g.,
the periodic configurations with the smallest periods), the only way to exactly solve this problem
is to use the numerical diagonalization on finite clusters. Then, a fundamental task is to find a
distribution of f electrons, for which the system has the lowest energy. The numerical algorithm
for finding the configuration w0, which minimizes the energy of the system is as follows: (i) Having
w, U and tij fixed, we find all eigenvalues λk of h(w). (ii) For a given Nf =
∑
i wi we determine
the ground-state energy E(w) =
∑Nd
k=1 λk of a particular f -electron configuration w by filling in
the lowest Nd one-electron levels. (iii) We find w0 (examining all possible distributions of localized
electrons), for which E(w,U) has a minimum. Repeating this procedure for different values of
model parameters one can immediately study the ground-state phase diagrams of the model.
Such exact calculations can be performed at present up to L ∼ 36–40 sites, which in some cases
(the one-dimensional case and strong Coulomb interactions) is sufficient for an extrapolation of
the results to the thermodynamic limit. In general, however, such cluster sizes are insufficient to
obtain reliable conclusions about the behaviour of macroscopic systems in higher dimensions. Un-
der these circumstances, the only way seems to be to use approximate methods. When selecting an
appropriate approximate method one should have in mind the fact that charge and spin ordering
as well as valence and metal-insulator transitions are very sensitive to the type of approximation
used [79, 80], and thus their description can only be successful within the approximations that
introduce only small simplifications of the model system. Instead of searching for an appropriate
method among the existing approximations, comparing them and excluding the least accurate
candidates, we decided to develop a new numerical method, which would be sewn from the be-
ginning on the Falicov-Kimball model, while retaining some degree of variability due to possible
generalizations of this model.
2.2. Approximate method based on the reduction of the total e nergy
The natural starting point in building a new approximate method (AM) seemed to us to
be the method of exact diagonalization. As stated above, within this method the single particle
Hamiltonian h(w) is exactly diagonalized over all possible (2L) distributions of localized particles
in order to find the only configuration w0, which minimizes the total energy of the system. This
procedure is necessary, but not efficient. Much more efficient than passing through the complete
set of configurations could be regulating the choice of configurations from the initial configuration
w to the final configuration w0, under some criterion that would significantly reduce the number
of configurations that should be examined. The most natural criterion seems to be a reduction
of the total energy in a sequence of configurations from w to w0. This is the basic idea of our
AM, the algorithm of which can be described as follows [74]: (i) Choose a trial configuration w =
{w1, w2, . . . , wL}. (ii) Having w, U fixed, find all eigenvalues λk of h(w). (iii) For a givenNf =
∑
i wi
determine the ground-state energy E(w) =
∑Nd
k=1 λk of a particular f -electron configuration w by
filling in the lowest Nd one-electron levels. (iv) Generate a new configuration w′ by moving a
randomly chosen electron to a new position which is also chosen as random. (v) Calculate the
ground-state energy E(w′). If E(w′) < E(w), the new configuration is accepted, otherwise w′ is
rejected. Then, the steps (ii)–(v) are repeated until the convergence (for a given U) is reached.
Of course, one can move instead of one electron (in step (iv)) simultaneously two or more
electrons, thereby improving the convergence of method. Indeed, the tests that we have performed
for a wide range of model parameters showed that the subsequent implementation of the method,
in which 1 < p < pmax electrons (p should be chosen at random) are moved to new positions, better
overcomes the local minima of the ground-state energy. As usual, we have performed calculations
with pmax = Nf . The main advantage of this implementation is that in any iteration step the
system has a chance to lower its energy (even if it is in a local minimum), thereby the problem
of local minima is strongly reduced (in principle, the method becomes exact if the number of
iteration steps goes to infinity). On the other hand, a disadvantage of this selection is that the
method converges slower than for pmax = 2 and pmax = 3. To speed up the convergence of the
method (for pmax = Nf ) and still to hold its advantage we generate instead the random number
p (in step (iv)) the pseudo-random number p that probability of choosing decreases (according to
42701-8
Formation of charge and spin ordering in strongly correlated electron systems
the power law) with increasing p. Such a modification considerably improves the convergence of
the method.
Apart from the number of the moved electrons, the method was also tested on the optimum
length of the iteration cycle M . It is obvious that if M → ∞, the method is exact. Unfortunately,
with respect to the time factor, such a choice is not possible and we must consider only a finite
number of iteration steps. The test process was realized on different finite clusters in the one-,
two- and three-dimensional cases. We have observed that it is very convenient to divide the whole
iteration process into several smaller independent cycles (from 5 to 10), among which the ground-
state configuration with the lowest energy is selected. This also minimizes the problem of local
minima. In table 2 we compare our numerical results on the one-dimensional cluster of L = 60
Table 2. The difference of the ground-state energies of the one-dimensional spinless Falicov-
Kimball model calculated exactly and by our numerical method (∆ = |Eexact −Emet.|) for three
different Coulomb interactions (U = 2, U = 4 and U = 8) for M = 200, 400 and 600 iterations.
Nf M=200 M=400 M=600
∆(U=2) ∆(U=4) ∆(U=8) ∆(U=2) ∆(U=4) ∆(U=8) ∆(U=2) ∆(U=4) ∆(U=8)
1 0 0 0 0 0 0 0 0 0
2 0 0 0 0 0 0 0 0 0
3 0 0 0 0 0 0 0 0 0
4 0 0 0 0 0 0 0 0 0
5 0 0 0 0 0 0 0 0 0
6 0 0 0 0 0 0 0 0 0
7 0 0 0 0 0 0 0 0 0
8 1, 5.10−8
0 0 0 0 0 0 0 0
9 0 0 0 0 0 0 0 0 0
10 8, 7.10−4
0 5, 7.10−8
0 0 1, 2.10−7
0 0 0
11 0 0 0 0 0 0 0 0 0
12 0 0 3, 0.10−6
0 0 0 0 0 0
13 0 0 0 0 0 0 0 0 0
14 0 0 0 0 0 0 0 0 0
15 0 0 0 0 0 0 0 0 0
16 4, 3.10−7 5, 5.10−9
0 0 0 0 0 0 0
17 0 0 0 0 0 0 0 0 0
18 0 5, 2.10−8
0 0 0 0 0 0 0
19 0 0 0 0 0 0 0 0 0
20 0 0 0 0 0 0 0 0 0
21 1, 6.10−7
0 0 0 0 0 0 0 0
22 0 0 0 0 0 0 0 0 0
23 6, 4.10−6
0 0 0 0 0 0 0 0
24 0 9, 1.10−4 1, 8.10−5
0 0 1, 8.10−5
0 0 0
25 1, 6.10−5
0 0 0 0 0 0 0 0
26 0 0 0 0 0 0 0 0 0
27 9, 7.10−9
0 0 0 0 0 0 0 0
28 6, 6.10−7
0 0 0 0 0 0 0 0
29 0 0 0 0 0 0 0 0 0
30 0 0 0 0 0 0 0 0 0
sites, obtained with 10 iteration cycles for M = 200, M = 400 and M = 600 iterations per site
with the exact results. It should be noted that the exact results were obtained by the EDM on a
set of the most homogeneous configurations, which are the ground states of the one-dimensional
Falicov-Kimball model for all nf and U > 1.2. Therefore, the comparative tests were made for U =
2, U = 4 and U = 8. As seen in table 2 already relatively small number of iterations (M = 600)
was sufficient to obtain the exact ground states for every Nf on a sufficiently large cluster of
L = 60. The same test was repeated on a twice larger lattice, where at 500 iterations, there were
still observed small differences in the ground-state energy. These differences ranged in the order of
about 10−4 − 10−9, and their number decreased with the increasing number of iteration steps M .
For M = 2000, we have found full consistency between our and exact results.
42701-9
H. Čenčariková, P. Farkašovský
3. Charge ordering in the spinless Falicov-Kimball model
3.1. The effect of local Coulomb interaction
One-dimensional case
We have started the study of ground-state properties of the spinless Falicov-Kimball model,
in the limit U > 1 [79]. This case was chosen for the reason that the analytical calculations
showed [49] that the ground states of the model (at sufficiently large U) can be only the most
homogeneous distributions, when f electrons are as far apart as possible, taking into account the
periodic boundary conditions. The knowledge of the ground states in the limit of strong interactions
make possible a direct comparison between our results obtained on finite clusters with the results
obtained in the thermodynamic limit (L → ∞) and at the same time it permits to specify more
precisely the area of stability of these configurations. Numerical calculations were performed using
the EDM, which permits to find the ground state of the model on the finite cluster of size L
for any model parameters: nd, nf and U . Although nd and nf can be considered as independent
parameters, we have bound them with the condition nf + nd = 1, since we are also interested in
valence transitions, i.e., transitions induced by migration of valence f electrons to the conduction
band. To minimize the finite-size effects, the model was studied on finite clusters from L = 4 to
L = 24 sites. We have chosen the values of the Coulomb interaction from 1 to 10 with a unit step.
The main result of our theoretical study was the finding that the most homogeneous configurations
are the ground states of the spinless Falicov-Kimball model not only in the limit of strong Coulomb
interactions, but for all the investigated values of U > 1, and for all f -electron concentrations. Since
the Fermi level for the most homogeneous configurations always lies within the energy gap [22], all
ground states for U > 1 are insulating, and thus the spinless Falicov-Kimball model is not capable
of describing the metal-insulator transitions in this limit.
For this reason we have turned our attention to the case U < 1. Using the same method, we have
performed a systematic study of the model for the selected value of Coulomb interaction (U = 0.6)
and all even lattices from L = 16 to L = 48 lattice sites [80]. The main result of our study was
the finding that for every L there exists a critical value of the f -electron occupation number Nc
below which the ground states are no longer the most homogeneous configurations, but the phase-
separated configurations that may be formally presented as an incoherent mixture of a configuration
w and the empty lattice (w&w0). A complete list of such configurations together with critical values
Nc is summarized in table 3. Thus, in accordance with the results by Gruber et al. [22] obtained
Table 3. The critical values of the f -electron occupation number Nc, f -level position Ec, and the
ground-state configurations below Nc calculated for U = 0.6 and different values of L. Here the
lower index denotes the number of consecutive sites occupied (unoccupied) by f electrons [80].
L Nc Ec GSC
16 2 1.4149 12014
18 2 1.2922 12016, 120411011
20 3 1.3440 12018, 120511014
22 3 1.3711 12020, 120411015
24 3 1.3957 12022, 120511016
26 4 1.3675 12024, 120611017, 120312019
28 5 1.3091 12026, 120611019, 120412020, 1203120311017
30 5 1.3399 12028, 120711020, 120412022, 1203120411018
32 5 1.3532 12030, 120711022, 120412024, 1203120411020
34 5 1.3678 12032, 120711024, 120412026, 1204120411021
36 6 1.3485 12034, 120811025, 120512027, 1204120511022, 1203120312024
48 8 1.3434 12046, 13045, 120612038, 1205120611032, 1204120412034
1204120412034, 12041204120411029, 12031203120312028
42701-10
Formation of charge and spin ordering in strongly correlated electron systems
0 0.5 1 1.5
0
0.05
0.1
0.15
0.2
0.25
0.3
MHD
F
1
+F
2
F
1
+F
2
F
1
+F
2
F
2
+F
3
F
3
F
3
+F
4
F
4
F
2
+F
3
F
3
S
F
2
3
F
2
4
F
2
5
F
2
6F
2
7
F
2
8
F
2
9−F
2
13
U
n f
L=120, D=1
Figure 4. Phase diagram of the one-dimensional spinless Falicov-Kimball model. Different phases
are discussed in detail in the text.
on an incomplete set of configurations (the periodic configurations and the mixtures of periodic
configurations with the empty lattice), we have found that the ground states of the spinless Falicov-
Kimball model for U < 1 may be incoherent mixtures of the type w&w0, with a small difference,
and namely, that the configuration w is not necessarily periodic.
To reveal the effect of Coulomb interaction U on the formation of charge ordering in these
phase-separated configurations, we have performed an exhaustive numerical study of the model for
a wide range of Coulomb interactions (U = 0.01, 0.02, . . . , 2) using our new AM that permits to
treat much larger clusters. Thereby the finite size effects are considerably reduced. In figure 4 we
present numerical results obtained for the ground-state phase diagram in the nf −U plane on finite
clusters of L = 120 sites. We have found that the phase separation takes place for all Coulomb
interaction U . 1.2 for both small (nf < 0.25) and large (nf > 0.75) electron filling (the nf − U
phase diagram is symmetric around nf = 0.5 line, and thus here we present only the results for
nf 6 0.5) and that this domain, except a few isolated points/lines (denoted by ∗) is continuous.
0 20 40 60 80 100 120
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
i
U
n
f
=1/20
0 20 40 60 80 100 120
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
i
U
n
f
=1/8
Figure 5. Distributions of f electrons on the finite cluster of L = 120 sites calculated for
U = 0.01, 0.02, . . . , 1.2 and two different values of nf .
42701-11
H. Čenčariková, P. Farkašovský
This is an obvious difference between our phase diagram and one obtained by Gruber et al. [22] for
periodic configurations and mixtures of these configurations with empty lattices, where large islands
of the most homogeneous phase are observed in the phase-separated region. The second important
difference is the existence of a narrow intermediate region (in our phase diagram) between the
phase-separated configurations with regular (quasi-regular) distributions of f electrons (within w)
and the most homogeneous domain (MHD). In this region (the medium gray area) the ground states
are mixtures of an empty lattice and the aperiodic configurations w with two-molecule distributions
in the middle of w and atomic distributions (a single occupied site) at the beginning and at the end
of w (F1 +F2). This fact is clearly demonstrated in figure 5, where all ground-state configurations
for U from 0.01 to 1.2 are displayed for two selected values of f -electron concentration nf = 1/20
and nf = 1/8. Analysing our numerical results we have found the following trends in the system:
(i) In the weak coupling and low concentration limit there is an obvious tendency to form phase-
segregated configurations (S) or mixtures of regularly (quasi-regularly) distributed n-molecules
with the empty lattice (the phases Fn with n = 3 and 4). (ii) With increasing U and nf , large
n-molecules split into smaller ones, but their regular distribution persists. (iii) The largest region
of the phase-separated domain corresponds to two-molecule distributions F2. This domain exhibits
a relatively simple internal structure. There are pure regions Fm
2 (m = 3, 4, . . . , 13), where the
ground states are regular distributions of two-molecules with the distance between them being
equal to m and the mixtured regions (the light gray areas) where the ground states are mixtures
of Fm
2 and Fm+1
2 (m = 3, 4, . . . , 8) phases with regular (quasi-regular) distribution of distances m
and m+ 1, and an obvious tendency to reduce the distance m with an increasing nf .
Figure 6. The ground-state phase diagram of the one-dimensional spinless Falicov-Kimball
model in the Ef − U plane obtained for L = 240. All 120 phases corresponding to f -electron
densities between nf = 0 and nf = 1/2 are displayed. The largest regions of stability correspond
to the periodic configurations with the smallest periods and the rational f -electron densities:
nf = 1/2, 3/7, 3/8, 2/5, 1/3, 1/4, 1/5, 1/6 and nf = 0. The 3/7 phase has been calculated for
L = 420. The inset shows the regions of stability of the metallic (M) and insulating (I) phase
for nf > 0 [74].
From the viewpoint of rare-earth compounds where the role of itinerant particles is played by
the d electrons and the role of immobile particles is played by the f electrons localized on the
energy level Ef (the term EfNf should be added to the model Hamiltonian (8)) it is interesting
to transform the nf − U phase diagram into Ef − U coordinates since the cuts of the Ef − U
phase diagram in the nf direction represent the valence transition at a given U . Since there is a
42701-12
Formation of charge and spin ordering in strongly correlated electron systems
direct parametrization between Ef and the external pressure p [81], the nf(Ef ) behaviour is de-
scribed (at least qualitatively) by the pressure induced valence changes in rare-earth compounds.
The results of our numerical calculations for the Ef − U phase diagram are summarized in fig-
ure 6. One can see that the phases with the largest area of stability in the Ef − U phase diagram
correspond to the periodic configurations with the smallest periods (p < 9) and the rational f -
electron concentrations. The number of phases with the relevant width is strongly reduced with
increasing U , and thus only a few relevant phases (with p 6 5) form the basic structure of the
phase diagram in the strong coupling limit. A detailed analysis of the model performed for U = 10
(L = 240 and L = 420) showed that some periodic phases with larger periods also persist in the
strong-coupling regime, but their width is considerably smaller. A complete set of phases (with
width wD > 10−10) that have been determined numerically as the ground states of the model for
U = 10 is shown in table 4. The phases with the smallest periods also persist in the weak coupling
Table 4. A complete set of phases (with width wD > 10−10) that have been determined numer-
ically as the ground states of the one-dimensional Falicov-Kimball model for U = 10 [74].
nf wD
1/2 1.8712578× 10−1
1/3 1.0677360× 10−2
1/4 2.1048106× 10−4
2/5 2.1852806× 10−5
1/5 3.4509125× 10−6
2/7 1.1864030× 10−8
3/7 3.7646341× 10−8
1/6 5.0851783× 10−8
3/8 1.0913297× 10−9
1/7 6.9578354× 10−10
limit, but with decreasing U they are gradually suppressed by the periodic configuration with
p > 10 for Ef < Ec
f (U) and by phase-separated configurations for Ef > Ec
f (U). The corresponding
picture of valence transition based on this phase diagram is displayed in figure 7. It is seen that
the valence transitions have a staircase structure, where different phases correspond to different
areas of stability. The largest stability regions correspond to the most homogeneous configurations
with the smallest periods (nf=1/2, 1/3, 1/4, . . . ). These phases form a primary structure of the
−2 −1 0 1 2
0
0.2
0.4
0.6
0.8
1
E
f
n f
L=120
U=0.6
U=1.5
U=5
U=12
−2 −1 0 1 2
0
0.2
0.4
0.6
0.8
1
E
f
n f
L=240
U=0.6
U=1.5
U=5
U=12
Figure 7. Dependence of the f -electron occupation number nf on the f -level position Ef for
U = 0.6, 1.5, 5, 12 and L = 120 (left panel) and L = 240 (right panel).
42701-13
H. Čenčariková, P. Farkašovský
valence transition. Its characteristic feature is that it does not change with an increasing lattice
size and therefore it can be used to represent the behaviour of macroscopic systems. The remaining
phases form the secondary structure that unlike the previous one is very sensitive to the lattice
size. This secondary structure is observed only for small values of the Coulomb interaction U and
with an increasing U it rapidly disappears. Consequently, the valence transitions for intermediate
values of Coulomb interactions consist of only a few valence steps, whose number is further re-
duced with an increasing U . For example, for 5 < U < 10 the valence transition consists of only
four relevant transitions, and namely, from nf = 1 to nf = 2/3, from nf = 2/3 to nf = 1/2,
from nf = 1/2 to nf = 1/3 and from nf = 1/3 to nf = 0 and for U > 10 there are even two
relevant valence transitions: the first from nf = 1 to nf = 1/2 and the second from nf = 1/2 to
nf = 0. Thus, we can conclude that the spinless Falicov-Kimball model is capable of describing
two basic types of valence transitions, and namely, the transition from the integer-valence ground
state into the inhomogeneous intermediate-valence state and transition from one inhomogeneous
intermediate-valence state into another inhomogeneous intermediate-valence state. Moreover, our
numerical results confirmed that the crucial role in the mechanism of valence transitions is played
by the Coulomb interaction between itinerant and localized electrons.
Two-dimensional case
The combination of EDM and AM has been also used for the study of the charge ordering
in a two-dimensional spinless Falicov-Kimball model [74]. In this case we were forced to limit
ourselves only to the area of intermediate and strong Coulomb interactions (U > 1), whereas in
the opposite limit U < 1 lattice effects were still strong, even on lattices with L = 400 sites.
Figure 8 represents a cut of the phase diagram in the nf −Ef plane (the valence transition) for the
n
f
=1/4n
f
=1/2
Figure 8. Left: Dependence of the f -electron occupation number nf on the f -level position Ef
calculated for U = 2 and two different clusters (8 × 8 and 12 × 12). Right: The ground-state
configurations of the two-dimensional spinless Falicov-Kimball model for U = 2 and two different
f -electron densities: nf = 1/2 and nf = 1/4 [74].
intermediate value of the Coulomb interaction U = 2 and L = 8× 8 and L = 12× 12. Similarly to
the one-dimensional case, the largest regions of stability again correspond to configurations with
the rational f -electron concentrations (nf = 1/2, 1/3, 1/4, . . . ) and similar are also the charge
distributions (see figure 8) with the difference that the periodic one-dimensional distributions are
reflected now in the diagonal charge stripes regularly repeated with the same periodicity like in
the one-dimensional case (nf = 1/2 and nf = 1/4)2. Below a certain critical value nc
f the ground
states are phase separated. This means that f electrons occupy only one part of the lattice, while
the remaining part is empty. Typical examples of such phase-separated configurations are shown
in figure 9 (left hand panel).
2For nf = 1/2 and nf = 1/4 our results are fully consistent with the previous results of Watson and Lemanski
obtained within the method of restricted phase diagrams [23].
42701-14
Formation of charge and spin ordering in strongly correlated electron systems
Figure 9. Left: The ground-state configurations of the two-dimensional Falicov-Kimball model
for U = 2 and several f -electron densities nf < nc
f . Right: The region of phase separation (PS)
of the two-dimensional Falicov-Kimball model calculated for two different clusters (12× 12 and
16× 16) [74].
Similarly to the one-dimensional case, these configurations are metallic, and thus the bound-
ary between the phase-separated domain and the rest of the phase diagram is the boundary of
metal-insulator transitions. For U > 1, where the lattice effects are negligible, we have specified
N
f
=2 N
f
=4 N
f
=6 N
f
=8 N
f
=10 N
f
=12 N
f
=14 N
f
=16
N
f
=18 N
f
=20 N
f
=22 N
f
=24 N
f
=26 N
f
=28 N
f
=30 N
f
=32
N
f
=34 N
f
=36 N
f
=38 N
f
=40 N
f
=42 N
f
=44 N
f
=46 N
f
=48
N
f
=50 N
f
=52 N
f
=54 N
f
=56 N
f
=58 N
f
=60 N
f
=62 N
f
=64
Figure 10. The ground-state configurations of the two-dimensional Falicov-Kimball model ob-
tained for L = 16× 16 and U = 2 (Nf 6 L/4).
42701-15
H. Čenčariková, P. Farkašovský
the region of stability of this domain very precisely (figure 9). A surprising finding was that in
the two-dimensional case the stability region of phase-separated domain shifts very significantly
to higher values of U (U ∼ 3.3). This result is very important from the point of view of pos-
sible applications of the model for a description of metal-insulator transitions in rare-earth and
transition-metal compounds. It is generally assumed that in these materials the values of the local
Coulomb interaction U are much larger than the hopping integrals tij , and, therefore, for a correct
description of metal-insulator transitions in real materials one should consider the limit of U > t
rather than U < t.
For this reason we have performed additional numerical studies of the model in the intermediate
(U = 2) and strong (U = 4) coupling limit on the larger cluster of L = 16× 16 sites. The complete
list of ground-state configurations obtained in the intermediate coupling limit for even values of Nf
are displayed in figure 10 (Nf 6 L/4) and figure 11 (Nf > L/4). Going with Nf from zero to L/2 we
N
f
=66 N
f
=68 N
f
=70 N
f
=72 N
f
=74 N
f
=76 N
f
=78 N
f
=80
N
f
=82 N
f
=84 N
f
=86 N
f
=88 N
f
=90 N
f
=92 N
f
=94 N
f
=96
N
f
=98 N
f
=100 N
f
=102 N
f
=104 N
f
=106 N
f
=108 N
f
=110 N
f
=112
N
f
=114 N
f
=116 N
f
=118 N
f
=120 N
f
=122 N
f
=124 N
f
=126 N
f
=128
Figure 11. The ground-state configurations of the two-dimensional Falicov-Kimball model ob-
tained for L = 16× 16 and U = 2 (L/4 < Nf 6 L/2).
have observed the following configuration types. For small f -electron concentrations (Nf 6 4) the
ground states are phase segregated. Then there follows the region of phase separation (Nf . 18)
when two-molecule clusters of f electrons are distributed only over one part of lattice leaving
another part of lattice free of f electrons. The two-molecule distributions disappear at Nf = 20.
This is also the point of phase transition from the phase separated to the homogeneous/quasi-
homogeneous phase, where the single f electrons are distributed regularly/quasi-regularly over
the whole lattice. This region ends at Nf ∼ 64 where the axial stripes of empty and alternating
configurations are observed. Above Nf = 64 the axial bands of width wD = 2 are formed with
the chessboard distribution of f electrons separated by empty lines. At Nf ∼ 84 the chessboard
structure starts to develop, first in the form of small clusters of four molecules and then in the
form of larger and larger clusters with fully developed chessboard ordering separated by diagonal
lines of empty sites.
42701-16
Formation of charge and spin ordering in strongly correlated electron systems
We have observed the same picture for larger values of Coulomb interactions. The larger values
of U only modify the stability regions of some phases, but no new configuration types appear. In
particular, for U = 4, the region of phase segregation/separation is fully suppressed and the region
of regular/quasi-regular distributions extends up to nf → 0.
Three-dimensional case
In principle, the same procedure as the one used in D = 1 and D = 2 can be also used in
D = 3. Due to the numerical complexity of the problem in D = 3 we have performed numerical
calculations only for selected values of the Coulomb interactionU representing the typical behaviour
of the model at the weak (U = 1), intermediate (U = 2) and strong (U = 8) interactions [75]. In
order to reveal the finite-size effects, numerical calculations were made on two different clusters of
4× 4× 4 and 6× 6 × 6 sites. A direct comparison of numerical results obtained on 4× 4× 4 and
6× 6× 6 clusters showed that the ground-state configurations fall into several different categories
whose stability regions are practically independent of L. For this reason we present here only the
results obtained on L = 6 × 6 × 6 (figure 12 and figure 13). The largest number of configuration
types is observed in the weak-coupling limit. Going with Nf from zero to half-filling (Nf = L/2)
we have observed the following configuration types for U = 1. At low f -electron concentrations,
the ground-states are the phase-segregated configurations (f electrons clump together while the
remaining part of lattice is free of f electrons). Typical examples of ground-state configurations
from this region are depicted in figure 12 (left hand panel).
Above the region of phase segregation, we have observed the region of stripe formation (Nf =
10, . . . , 20). In this region the f electrons form the one-dimensional charge lines (stripes) that can be
perpendicular or parallel. This result shows that the crucial mechanism leading to the formation of
stripes in strongly correlated systems should be a competition between the kinetic and short-range
Coulomb interaction.
Going with Nf to higher values, the stripes vanish and again appear at Nf = 26, though
in a fully different distribution. While at smaller values of Nf the stripes have been distributed
inhomogeneously (only over one half of the lattice), the stripes in the region Nf = 26, . . . , 31 are
distributed regularly.
Above this region a new type of configurations starts to develop. We call them diagonal charge
planes with an incomplete chessboard structure, since the f electrons prefer to occupy the diagonal
Figure 12. Left: Typical examples of phase-segregated (a) and striped (b) configurations ob-
tained for U = 1 and L = 6 × 6 × 6. Right: Typical examples of striped configurations with
regular distribution (a) and diagonal charge planes with an incomplete chessboard structure (b)
obtained for U = 1 and L = 6× 6× 6 [75].
42701-17
H. Čenčariková, P. Farkašovský
Figure 13. Left: Ground-state configurations for intermediate f -electron concentrations. (a)
The formation of a chessboard structure. (b) The examples of ground-state configurations that
can be considered as mixtures of configuration types with smaller nf (U = 1, L = 6 × 6 × 6).
Right: Examples of an incomplete chessboard structure obtained for U = 1 and L = 6× 6× 6.
(a) The chessboard structure is fully developed in some regions (planes) that are separated by
planes with incompletely developed chessboard structure. (b) Both regions with complete and
incomplete chessboard structure have a three-dimensional character [75].
planes with slope 1, and within these planes they form a chessboard structure. This region is
relatively broad and extends up to Nf ∼ 50. Then there follows the region in which the chessboard
structure starts to develop. As illustrated in figure 13, the f electrons begin to preferably occupy
the sites of sublattice A, leaving the sublattice B free of f electrons. Furthermore, the configurations
that can be considered as mixtures of previous configuration types are also observed in this region.
However, with increasing Nf the configurations of chessboard type become dominant. Analysing
these configurations we have found that the transition to the purely chessboard configuration
is realized through several steps. The first step, the formation of the chessboard structure has
been illustrated in figure 12. The second step is shown in figure 13. It is seen that the chessboard
structure is fully developed in some regions (planes) that are separated by planes with an incomplete
developed chessboard structure. Such type of distribution is replaced for larger values of Nf by a
new type of distributions (step three), where both regions with complete and incomplete chessboard
structure have a three-dimensional character.
2 3 4 5 6
0
0.05
0.1
0.15
PS
U
n f
L=6×6×6
L=8×8×8
Figure 14. The region of phase separation (PS) of the three-dimensional Falicov-Kimball model
calculated for two different clusters (6× 6× 6 and 8× 8× 8) [82].
42701-18
Formation of charge and spin ordering in strongly correlated electron systems
We have observed the same picture for intermediate values of Coulomb interactions (U = 2).
Larger values of U only slightly modify the stability regions of some phases, but no new configura-
tion types appear. In particular, the domain of phase segregation, as well as the domain of stripe
formation are reduced while the domain of diagonal planes with chessboard structure increases.
This trend is also observed for larger values of U . In the strong coupling limit (U = 8) the phase
segregated and striped phases are absent and the region of stability of the diagonal planes extends
to relatively small values of Nf ∼ 20. Below this value a homogeneous distribution of f electrons
is observed.
For the same reasons as discussed above for the two-dimensional case we have investigated
the stability of phase-separated (metallic) domain in the three-dimensional case and found (see
figure 14) that the phase-separated region (and thus the metal-insulator transitions) extends up
to U ∼ 5.5, which is a much larger value than in the two-dimensional case [82].
3.2. The effect of long-range hopping
Since the model including the electron hopping solely to the nearest neighbours may seem at
first glance a very crude approximation, in order to have a more realistic description of electron
processes in rare-earth compounds, we have generalized this model by taking into account the
transitions to other neighbours [83, 84]. Basically, there were two possible ways of performing such
a generalization. The first way was to assign independent transition amplitudes for the first (t1),
second (t2), third (t3), . . . nearest neighbours, while the second way was to describe the electron
hopping by a one-parametric formula with power decaying transition amplitudes tij ∼ q|i−j|,
where q 6 1. From the practical point of view, the second method is more suitable because
it does not expand the model parameter space and has a clearer physical meaning, since the
atomic wave functions have also the power law decay with an increasing distance. For this reason,
for a description of electron hopping in the generalized model we have chosen the long-range
hopping with power decreasing amplitudes. Explicit expressions of matrix elements tij for the case
of periodic boundary conditions in the one-dimensional case have the form:
tij = −tqL/2−||i−j|−L/2|−1(1− δij) , (10)
from which it follows that the case q → 0 approximates the nearest neighbours hopping (tij =
−t), while the q = 1 case corresponds to an unconstrained hopping, when the model is solvable
exactly [85]. The results of our numerical simulations obtained within the AM are summarized in
figure 15 in the form of nf − U diagrams. Figure 15 presents the one-dimensional ground-state
0 0.5 1 1.5
0
0.05
0.1
0.15
0.2
0.25
0.3
MHD
F
1
+F
2
F
1
+F
2
F
2
+F
3
F
3
F
3
+F
4
F
4
F
2
+F
3
F
3
S
F
2
3
F
2
4
F
2
5
F
2
6
F
2
7F
2
8
F
2
9−F
2
12
U
n f
L=120, D=1, q=0.1
0 0.5 1 1.5
0.7
0.75
0.8
0.85
0.9
0.95
1
MHDF
1
+F
2
F
1
+F
2
F
2
+F
3
F
3
F
3
+F
4
F
4 F
2
+F
3
F
3
S
F
2
3
F
2
4
F
2
5
F
2
6
F
2
7
F
2
8F
2
9F
2
10−F
2
13
U
n f
L=120, D=1, q=0.1
Figure 15. The ground-state phase diagram of the one-dimensional spinless Falicov-Kimball
model with long-range hopping (q = 0.1) for nf < 0.3 (the left hand panel) and nf > 0.7 (the
right hand panel).
42701-19
H. Čenčariková, P. Farkašovský
0 0.5 1 1.5 2 2.5 3 3.5 4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
U
n f
L=120, D=1, q=0.3
PSD
1
PSD
2
PSD
3
MH
G
1
G
2
Figure 16. The ground-state phase diagram of the one-dimensional spinless Falicov-Kimball
model with long-range hopping calculated for q = 0.3 and L = 120.
phase diagram of the model for q = 0.1 obtained on the finite cluster of L = 120 sites. Comparing
this phase diagram with its q = 0 counterpart one can find obvious similarities. The largest part
of the nf − U phase diagram corresponds to the most homogeneous domain, while the phase-
separated domains appear only for U small and nf . 0.25 or nf & 0.75. Practically the same
is the internal structure of the phase-separated domains corresponding to ground-state phases of
the model for q = 0 and q = 0.1. The term of long-range hopping only renormalizes the size of
the phase-separated domains (for nf . 0.25 the phase-separated domain is shifted to smaller U ,
while for nf & 0.75 it is shifted to higher U and higher nf), but practically no new phases are
generated for q = 0.1. However, for q = 0.3 the situation changes dramatically. One can see
(figure 16) that the largest region of stability still corresponds to the most homogeneous domain.
However, in addition to this domain and two small phase-separated domains near nf = 0 (PSD1)
and nf = 1 (PSD2), there appear two new large domains in the limit of intermediate and strong
interactions. In the first region (PSD3) located at U > Uc ≈ 2.783 and nf 6 0.5 the ground
0 0.5 1 1.5 2 2.5 3 3.5 4
0
0.5
1
1.5
2
2.5
3
3.5
U
∆
n
f
=1/12
n
f
=1/6
n
f
=1/4
0 0.5 1 1.5 2 2.5 3 3.5 4
0
0.5
1
1.5
2
2.5
3
3.5
U
∆
n
f
=17/30
n
f
=3/4
n
f
=5/6
n
f
=9/10
0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
L → ∞
U
∆
n
f
=5/6
n
f
=9/10
Figure 17. The U -dependence of the energy gap ∆ for the one-dimensional spinless Falicov-
Kimball model with long-range hopping (q = 0.3) calculated for L = 120 and nf < 0.5 (left
panel) and nf > 0.5 (right hand panel). The inset shows the extrapolated data (L → ∞) for
nf = 5/6 and nf = 9/10.
42701-20
Formation of charge and spin ordering in strongly correlated electron systems
states are configurations that can be considered as mixtures of the empty configuration and the
alternating configuration wa(Nf ) = {1010 . . .10} with Nf = 1, 2, . . . , L/2. The second domain
located above Uc and nf > 0.5 consists of two subdomains G1 and G2, where the ground states
are configurations of the following types:
G1: w = 110[1010110]k[10]L−7k−3
2
, with k=3, 7, 11, 15 , (11)
G2: w = 110[10110]k[10]L−5k−3
2
, with k=7, 11, 15, 19 . (12)
Analysing the energy gaps of ground-state configurations in all the above mentioned domains we
have found (see figure 17) that only domains PSD1 and PSD2 are metallic, while all the remaining
domains, including PSD3 are insulating. Thus, in accordance with the q = 0 case, only the phase
boundary between the small phase-separated domains PSD1 and PSD2 and the most homogeneous
domain is a boundary of the metal-insulator transitions induced by Coulomb interaction (the f -
electron concentration).
3.3. The effect of nonlocal Coulomb interactions
One of the shortcomings of the basic variant of the spinless Falicov-Kimball model is that
it neglects all nonlocal interactions between electrons, which immediately evokes the question of
possible instability of numerical solutions discussed above with respect to the case when some of
these nonlocal interactions are turned on. To answer this question we have examined the effects of
two nonlocal interactions, and namely, the correlated hopping
Ht′ = t′
∑
〈ij〉
(f+
i fi + f+
j fj)d
+
i dj , (13)
and the nearest-neighbour Coulomb interaction between d a f electrons
Hnon = Unon
∑
〈i,j〉
f+
i fid
+
j dj . (14)
3.3.1. The effect of correlated hopping
Let us first describe the effect of the first term. This term is in the literature usually referred
to as a term of correlated hopping, since it can be interpreted as a single-particle Hamiltonian
describing the hopping of d electrons between the neighbouring d orbitals with an amplitude
that explicitly depends on the occupancy of f orbitals. The selection of this term was motivated
by earlier works [86–88], which showed its importance in describing the properties of strongly
correlated systems, e.g., the superconducting state [89]. We have focused our attention on examining
the effects of this term on charge ordering and valence and metal-insulator transitions [67]. A
fundamental result of our numerical study is presented in figure 18, where we have displayed the
phase diagram of the generalized model in the t′ − U plane for the half-filled band case nf =
nd = 0.5. One can see that already very small values of the correlated hopping term lead to the
instability of alternating phase w1 = {1010 . . .10} (which is for t′ = 0 the ground state of the
model for all values of U > 0). For t′ < 0, this phase transforms to the alternating phase with
a double period w2 = {1100 . . .1100} and for t′ > 0 on w2 (U < 1), respectively, the segregated
configuration w3 = {11 . . .100 . . .0} (U > 1). Since the ground states corresponding to alternating
phases w1 and w2 are insulating, while the ground state corresponding to w3 is metallic, the phase
boundary between w1 and w3 as well as between w2 and w3 is the boundary of metal-insulator
transitions induced by the correlated hopping term. Similar instabilities were observed outside
the symmetric case nf = nd = 0.5, which ultimately led to a completely different picture of
valence and metal-insulator transitions for t′ 6= 0. Nonzero values of t′ reduce the total width of
valence transitions (figure 18) as well as the width of stairs and above some critical value, the
phase transition becomes continuous, initially only in certain areas (e.g., nf < 0.5 for t′ = 0.6),
and finally, in the whole area (t′ > 0.9). The parts of valence transition with staircase structure
42701-21
H. Čenčariková, P. Farkašovský
Figure 18. Left: t′-U phase diagram of the one-dimensional Falicov-Kimball model with corre-
lated hopping at half-filling (Ef = 0, nf = nd = 0.5). Three different phases correspond to the
alternating configuration w1 = {10 . . . 10}, the alternating configuration w2 = {1100 . . . 1100}
with double period and the segregated configuration w3 = {1 . . . 10 . . . 0}. For L = 24 the phase
diagram has been obtained over a full set of f -electron configurations, while for L = 600 only the
restricted set of configurations w1, w2 and w3 has been used to determine the phase boundaries.
The inset shows t′-dependence of energy gaps corresponding to the ground-state configurations
for different values of U . Right: Dependence of f -electron occupation number nf on the f -level
position Ef for different values of correlated hopping t′ at U = 1.5 [67].
correspond to the most homogeneous (insulating) phases, while the continuous parts correspond
to the segregated (metallic) phases. The border points between these two phases are, therefore,
critical points of pressure (p ∼ Ef ) induced metal-insulator transitions.
We have performed the same study in the two-dimensional case, where we have used our AM.
We have found [68] that all characteristics discussed above, including the phase diagram for the
half-filled band (figure 19) as well as the picture of metal-insulator transitions remain unchanged
Figure 19. Left: The t′-U phase diagram of the two-dimensional Falicov-Kimball model with
correlated hopping at half-filling (Ef = 0, nf = nd = 0.5). The inset shows the t′-U phase
diagram for nf =1/4 and U > 2. Right: The ground-state configurations for nf =1/2 (w1, w2
and w3) and nf =1/4 (w4) [68].
42701-22
Formation of charge and spin ordering in strongly correlated electron systems
in the two-dimensional case. A new and very interesting result is the observation of the charge
striped ordering of the type w2 in the half-filled band case. Since the mechanism of formation
of inhomogeneous striped ordering in strongly correlated electron systems is a very intensively
discussed topic in recent years, this result is also valuable from the cognitive perspective, because
it opens up a new way to the study of this certainly interesting phenomenon.
(a1):U=0.5, t′=0.2
N
f
=72
(a2):U=0.5, t′=0.2
N
f
=66
(a3):U=2, t′=−0.4
N
f
=80
(b1):U=2, t′=−0.4
N
f
=12
(b2):U=0.5, t′=−0.6
N
f
=18
(b3):U=2, t′=0
N
f
=30
(c1):U=0.5, t′=0.2
N
f
=28
(c2):U=1, t′=−0.6
N
f
=118
(c3):U=2, t′=−0.4
N
f
=126
(d1):U=0.5, t′=0.2
N
f
=60
(d2):U=2, t′=−0.4
N
f
=96
(d3):U=1, t′=−1
N
f
=82
(d4):U=0.5, t′=−0.6
N
f
=78
(d5):U=0.5, t′=0.5
N
f
=72
(d6):U=1, t′=0.5
N
f
=78
(d7):U=0.5, t′=−0.6
N
f
=50
(e1):U=0.5, t′=−0.2
N
f
=96
(e2):U=2, t′=−0.6
N
f
=116
(f1):U=0.5, t′=0.5
N
f
=32
(f2):U=2, t′=−0.2
N
f
=10
Figure 20. Representative ground-state configurations that form the basic structure of the
phase diagrams in the nf−t′ plane: (a1) the chessboard phase, (a2, a3) the perturbed chessboard
phases, (b1–b3) the diagonal stripes and perturbed diagonal stripes, (c1–c3) n-molecular phases,
(d1–d7) the axial stripes and perturbed axial stripes, (e1, e2) the mixed phases and (f1, f2) the
segregated and phase-separated configurations. The large dots correspond to occupied sites and
the small dots correspond to vacant sites [70].
For this reason we have performed the exhaustive numerical studies of the model outside the
half-filled band case. The primary goal of these studies was to identify all possible types of charge
ordering induced by the term of correlated hopping. To fulfill this goal we have performed an ex-
haustive numerical study of the two-dimensional spinless Falicov-Kimball model for a wide range of
Coulomb interactions U and correlated hopping t′ [70]. For each selected U and t′ the ground-state
configurations for Nf = 0, 1, . . . , L are calculated using our AM. To minimize the finite-size effects
the same procedure is repeated on several different clusters. Of course, such a procedure demands
in practice a considerable amount of CPU time, which imposes severe restrictions on the size of
clusters that can be studied using this method (L = 4×4, 6×6, 8×8, 10×10, 12×12). Fortunately,
we have found that the main features of the phase diagrams hold on all the examined lattices and
thus can be used satisfactorily to represent the behaviour of macroscopic systems. In particular,
42701-23
H. Čenčariková, P. Farkašovský
we have found that for each L there is a finite number of basic types of ground-state configurations
that form the basic structure of the phase diagram. This structure depends only very weakly on
the size of clusters and covers practically the whole area of the phase diagram in the nf − t′ plane.
Let us start a discussion of these phase diagrams (for U = 0.5, 1 and 2) with a description of config-
uration types that form their basic structure (see figure 20). (a) The chessboard phase (nf = 1/2).
The f electrons occupy the A sublattice of the bipartite lattice and the B sublattice is empty
(a1). The perturbed chessboard phases (nf close 1/2), denote the chessboard structure decorated
by two-dimensional patterns of occupied or empty sites (e.g., a2, a3). (b) The diagonal stripes and
perturbed diagonal stripes. The mentioned phases could be divided into three principal categories
that are represented by examples b1, b2 and b3. (c) n-molecular phases (c1–c3). (d) The axial
stripes and perturbed axial stripes. This group consists of several subgroups. In the first subgroup,
ground states are configurations that can be constructed by spaced lines of occupations (vacancies)
aligned with the lattice axes, into the chessboard structure (d1, d2). The second subgroup includes
perpendicular axial stripes (d3, d4). The third subgroup is formed by simple axial and perturbed
axial stripes (d5, d6). The last subgroup consists of n-molecular axial stripes (d7). (e) The mixed
phases. They can be considered as mixtures of chessboard and fully occupied (empty) lattice (e1,
e2). (f) The segregated and phase-separated phases (f1, f2). The f electrons clump together, or
they are distributed only over one half of the lattice, leaving another part of the lattice free of f
electrons. (g) Unspecified phases (various mixtures of previous configuration types). The stability
regions of all the above described phases are displayed in figure 21, where the comprehensive phase
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
correlated hopping
0
0.2
0.4
0.6
0.8
1
n
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
correlated hopping
0
0.2
0.4
0.6
0.8
1
n
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
correlated hopping
0
0.2
0.4
0.6
0.8
1
n
- chessboard & perturbed chessboard phases
- diagonal & perturbed diagonal phases
- n-molecular phases
- axial stripes (the first subgroup of type )
- axial & perturbed axial stripes
- mixed phases
- segregated phases
- separated phases
- unspecified phases
U = 2
U = 0.5 U = 1
ff
f
a) b)
c)
d
(the remaining subgroups of type ) d
Figure 21. nf−t′ phase diagrams of the two-dimensional Falicov-Kimball model with correlated
hopping for U = 0.5 (a), U = 1 (b) and U = 2 (c) [70].
diagrams of the Falicov-Kimball model with correlated hopping are presented for weak, interme-
diate and strong interactions. A direct comparison of these results reveals one general trend, and
namely that the structure of phase diagrams is gradually simplified with increasing U and becomes
very simple in the strong-coupling limit. In this case the basic structure of the phase diagram (in
the nf−t
′ plane) is formed by large segregated domains and several horizontal (band) domains cor-
42701-24
Formation of charge and spin ordering in strongly correlated electron systems
responding to the separated phases, n-molecular phases, diagonal and perturbed diagonal stripes,
axial stripes (the first subgroup discussed above), mixed phases and finally perturbed chessboard
configurations. Small deviations from the horizontal structure are observed for n-molecular phases
for small (nf 6 0.2) and large (nf > 0.7) f -electron concentrations. Comparing the phase diagram
with conventional Falicov-Kimball model (t′ = 0) it is seen that the positive values of t′ do not
essentially effect the ground states up to some critical value t′c(nf ). However, at t′c(nf ) the system
exhibits a steep transition to the segregated phase that is stable for all nf and t′ > t′c. A slightly
different picture occurs for negative values of t′. In this case the correlated hopping term induces
new regions of axial stripes (the type d1, d2). The strong effect of negative t′ is apparent for f -
electron concentrations close to 1, where the diagonal configuration type (the type b) gradually
disappears, while the segregated region is stabilized.
As was discussed above, the phase diagrams become more complicated when the Coulomb
interaction decreases. The simple band structure observed in the strong-coupling limit persists
only for f -electron concentrations close to half-filling and it is suppressed gradually by axial-stripe
configurations (the type d3–d7) for both positive and negative values of t′. It should be noted
that these axial stripe configurations have an arrangement principally different from the axial
stripes observed in the conventional Falicov-Kimball model (d1, d2). The appearance of new types
of axial stripes is one of the most interesting effects of correlated hopping on the ground-state
properties of the two-dimensional Falicov-Kimball model. At the same time, this result positively
answers the question whether the correlated hopping term can or cannot stabilize the stripe phases.
Our results show that already relatively small values of t′ (positive as well as negative) stabilize
this inhomogeneous charge ordering. Moreover, it was found (see figure 21) that the capability of
correlated hopping to generate stripe ordering increases with a decreasing Coulomb interaction
between localized and itinerant electrons. This opens up a new route towards the understanding
of the nature of stripe formations in strongly correlated electron systems.
3.3.2. The effect of nearest-neighbour Coulomb interactio n
Let us now discuss the effects of another nonlocal interaction term, i.e., the nearest-neighbour
Coulomb interaction between the localized f and itinerant d electrons, that is of the same order as
the term of correlated hopping. To study the effect of nonlocal Coulomb interaction Unon on ground-
state properties of the one- and two-dimensional Falicov-Kimball model, we have performed an
exhaustive numerical study of the model for weak (U = 0.5), intermediate (U = 2) and strong (U =
8) on-site Coulomb interactions and for a wide range of nonlocal Coulomb interaction Unon [65].
To determine the ground states of the model we used the EDM (up to L = 36 lattice sites) in
combination with AM (up to L = 120 sites). We have started our study with the one dimensional
case and U large (U = 8), which are relatively simple for a description.
Firstly, we have studied the ground-state phase diagram of the model in the nf − unon plane
(unon = Unon/U changes from 0 to 1 with step 0.001). To reveal the finite-size effects on the ground
states of the model, we have performed numerical calculations for three different clusters of L =
12, 24 and 30 sites. The numerical results obtained for L = 30 are displayed in figure 223. Our
numerical results clearly demonstrate that already relatively small changes of unon can produce
large changes in the ground-state f -electron distributions. Indeed, we have found that already
values of Unon around 60 times smaller than U (ucnon = 0.015) are capable of fully destroying
the most homogeneous distributions of f -electrons (that are the ground states of the conventional
Falicov-Kimball model in the strong coupling limit for all nf ). It is interesting that only two
configuration types are stabilized above the critical value of nonlocal interaction ucnon. The first
configuration type (wh) is formed by the homogeneous distribution of [1100] and [10] clusters.
A complete set of these configurations that are stable only in a very narrow region of unon (for
1/3 < nf < 2/3) are listed in figure 22 (the first panel below). The second type of configurations
determined in the phase diagram of the Falicov-Kimball model with nonlocal Coulomb interaction
3We have found that the ground-state phase diagrams depend on L only very weakly and thus already the results
obtained for L = 30 can be used satisfactorily to represent the behaviour of macroscopic systems.
42701-25
H. Čenčariková, P. Farkašovský
in the strong coupling are the segregated configurations that are preferred as a ground state for
all nf started at unon = 0.016.
In figure 22 we present the ground-state phase diagrams of the extended Falicov-Kimball model
for intermediate (U = 2) and weak (U = 0.5) interactions. One can see that the main feature of
the phase diagram found for U = 8, the nonlocal Coulomb interaction induced transition from the
regular f -electron distributions to the phase-segregated distributions, also holds for smaller values
of U , although the phase boundaries of different ground-state configuration types are now not so
obvious due to the finite-size effects. A detailed analysis of the phase diagram for intermediate
couplings showed that besides the most homogeneous configurations only three other configuration
types enter the ground-state phase diagram, and namely: the segregated configurations (◦), the
weakly perturbed segregated configurations (△) and wh(Nf ) distributions (+). Contrary to U = 8,
wh(Nf ) configurations occur rarely at nf = 1/2, while weakly perturbed segregated configurations
(△) are observed on relatively large unon intervals. As shown in figure 22, in the weak interaction
limit, the configurations wh (+) fully vanish and the set of ground-state configurations is much
richer. Apart from the most homogeneous configurations (·) and the segregated configurations (◦)
we have determined a number of phase-separated configurations (denoted by △). A complete list
of these configurations is given in figure 22.
0 0.005 0.01 0.015 0.02 0.025 0.03
0
0.1
0.2
0.3
0.4
0.5
U=8,L=30
u
non
n f
0.12 0.13 0.14 0.15 0.16 0.17
0
0.1
0.2
0.3
0.4
0.5
U=2,L=30
u
non
n f
0 0.05 0.1 0.15 0.2 0.25 0.3
0
0.1
0.2
0.3
0.4
0.5
U=0.5,L=30
u
non
n f
− the most homogeneous configurations
− the segregated configurations
+ (N
f
=11)
+ (N
f
=12)
+ (N
f
=13)
+ (N
f
=14)
+ (N
f
=15)
− the most homogeneous configurations
− the segregated configurations
+ (N
f
=13)
+ (N
f
=14)
(N
f
=5)
(N
f
=8)
(N
f
=12)
(N
f
=14)
− the most homogeneous configurations
− the segregated configurations
(N
f
=4)
(N
f
=5)
(N
f
=6)
(N
f
=6)
(N
f
=6)
(N
f
=7)
(N
f
=8)
(N
f
=8)
(N
f
=8)
(N
f
=8)
(N
f
=10)
(N
f
=10)
(N
f
=14)
(N
f
=14)
Figure 22. Up: nf − unon ground-state phase diagrams of the spinless Falicov-Kimball model
extended by nonlocal Coulomb interaction for L = 30 in the strong (U = 8), intermediate (U =
2) and weak (U = 0.5) interaction limit, calculated for nf +nd = 1. The one-dimensional exact-
diagonalization results. Down: The complete lists of ground-state configurations corresponding
to different interaction limits. The large (small) dots correspond to occupied (vacant) sites [65].
We have performed the same calculations in two dimensions. To minimize the finite-size effects,
the numerical calculations have been performed on three different clusters of 4× 4, 6× 6 and 8× 8
sites. On the 4×4 cluster, the calculations have been performed by the EDM, and on larger clusters
our AM was used.
Similarly to the one-dimension we have started our two-dimensional studies with U = 8. The
ground-state phase diagram (calculated for 8×8) is shown in figure 23 (the first panel). Comparing
this phase diagram with its one-dimensional counterpart one can find obvious similarities. In both
cases the basic structure of the phase diagram only very weakly depends on L and consists of only
three configuration types. Again one can see that the relatively small values of nonlocal interaction
lead to changes of ground-state configurations from the regular distributions (·) to the segregated
arrangements (◦) straightforwardly, or through some n-molecular distributions (+) (usually diag-
42701-26
Formation of charge and spin ordering in strongly correlated electron systems
0 0.005 0.01 0.015 0.02 0.025 0.03
0
0.1
0.2
0.3
0.4
0.5
L=64,U=8
u
non
n f
0 0.05 0.1 0.15 0.2 0.25
0
0.1
0.2
0.3
0.4
0.5
L=8x8,U=2
u
non
n f
0 0.2 0.4 0.6 0.8
0
0.1
0.2
0.3
0.4
0.5
L=8x8,U=0.5
u
non
n f
Figure 23. nf − unon ground-state phase diagrams of the spinless Falicov-Kimball model ex-
tended by nonlocal Coulomb interaction for L = 64 in the strong (U = 8), intermediate (U = 2)
and weak (U = 0.5) interaction limit, calculated for nf + nd = 1. The two-dimensional ap-
proximative results. Three different regions of stability corresponding to regular distributions,
phase-segregated distributions and “other phases” are denoted as ·, ◦ and + [65].
onal n-molecules but also the mixture of regular and diagonal 2-molecular distributions). Typical
examples of n-molecular distributions (for L = 8 × 8) are displayed in figure 24 (the first row).
While in the one-dimension the area of these configurations is stable only in isolated points of unon
(unon = 0.015) and for 2/3 > nf > 1/3, in two dimensions these phases also persist for smaller
nf and on the wider unon interval. On the other hand, it is interesting to note that the critical
value of ucnon = 0.018 above which all ground-state configurations are only the segregated phases
is almost identical to 1D case (ucnon = 0.016). Similarities between phase diagrams of 1D and 2D
U=8,N
f
=6 U=8,N
f
=30U=8,N
f
=16 U=8,N
f
=18 U=8,N
f
=21 U=8,N
f
=32
U=2,N
f
=5 U=2,N
f
=13U=2,N
f
=6 U=2,N
f
=9 U=2,N
f
=10 U=2,N
f
=18
U=0.5,N
f
=8 U=0.5,N
f
=9 U=0.5,N
f
=14 U=0.5,N
f
=17 U=0.5,N
f
=19 U=0.5,N
f
=27
Figure 24. Typical examples of ground states of the spinless Falicov-Kimball model extended
by nonlocal Coulomb interaction from the area depicted by symbol (+) for strong (U = 8),
intermediate (U = 2) and weak (U = 0.5) interaction limit. The large (small) dots correspond
to occupied (vacant) sites [65].
42701-27
H. Čenčariková, P. Farkašovský
cases, can be also observed for intermediate and weak interactions. For both cases one can see a
transition from ground states corresponding to unon=0 to phase-segregated distributions, due to
the nonlocal Coulomb interaction. For U = 2, the obtained results show that the ground-state
phase diagram consists of three different configuration types, denoted as regular distributions (·),
phase-segregated distributions (◦) and other types (+) including many n-molecular distributions
(usually arranged to the “ladders” or to the blocks). Typical examples of these distributions (for
L = 8 × 8) are displayed in figure 24 (the second row). Similarly, in the weak interaction limit
the nonlocal Coulomb interaction unon prefers only a few types of ground-state configurations. We
again observed regular distributions (·), phase segregated distributions (◦) and some specific ar-
rangements (+) discussed below. As was shown in figure 23 the critical value of unon, above which
phase-segregated configurations are ground-states (for all nf ), shifts to higher values of unon. On
the other hand, the critical value of unon, where the ground states of conventional Falicov-Kimball
model are changed into the other ones, are observed already for unon ∼ 0.05 (for L = 8 × 8). Be-
tween these two boundaries one can find various f -electron distributions. In particular, there exist
regular n-molecules, n-molecular “ladders”, mixtures of chessboard structure and phase-segregated
distributions, some periodic structures as well as the stripe formations (see figure 24, the third
row). This clearly shows that nonlocal interaction can stabilize various types of inhomogeneous
charge ordering in strongly correlated electron systems.
3.4. The effect of lattice geometry
There exists a large group of rare-earth and transition-metal compounds (e.g., GdI2, NaxCoO2,
etc.) in which atoms instead of a square or cubic lattice decorate a triangular lattice (see figure 25)
and they exhibit a number of anomalous physical characteristics [1, 55, 58, 59, 90, 91]. From this
point of view it is interesting to perform the same numerical study for a two-dimensional spinless
Falicov-Kimball model on a triangular lattice. To reveal the effects of the lattice geometry on the
ground-state properties of the Falicov-Kimball model we have started with the half-filled band
case for which the nature of the ground state, its structural and energetic properties are quite
understandable on the square lattice [73]. In this case the localized f electrons fill up one of two
a
c
b
I
Gd
a b
Gd
Figure 25. Crystallographic structure of GdI2 (P6/mmc group).
sublattices of the square lattice (the checkerboard structure) and the corresponding ground state
is insulating for all U > 0. Thus, for finite interaction strength U there is no correlation-induced
phase or metal-insulator transition at half-filling.
The numerical calculations that we have performed in the half-filled band case on the triangular
lattice revealed a completely different behaviour of the model for nonzero U in comparison with
the square variant. Indeed, with increasing U we have observed a sequence of two correlation-
42701-28
Formation of charge and spin ordering in strongly correlated electron systems
(a): U=2 (b): U=10 (c): U=20
Figure 26. The ground-state configurations of the triangular Falicov-Kimball model in the
half-filled band case [73].
induced phase transitions, indicating strong effects of the lattice geometry on the ground-state
characteristics of correlated electron systems. In the weak and intermediate coupling, the f electrons
preferably form the closed lines that surround clusters of empty sites, without apparent long-
range order (see figure 26 (a)). At U ∼ 4, the system undergoes a correlation induced phase
transition to the ordered phase (figure 26 (b)) characterized by a diagonal distribution of f -electron
pairs (stripes). This phase persists up to relatively large values of U (U ∼ 12) where the system
undergoes the second-phase transition into the phase-separated phase (figure 26 (c)) formed by
a mixture of nf = 1/3 and nf = 2/3 phase. The dramatic effects of the lattice geometry on the
ground-state characteristics of the Falicov-Kimball model at half-filling indicate that the picture
of valence and metal-insulator transitions on the triangular lattice should be significantly changed
in comparison with the square variant. To verify this conjecture we have performed an exhaustive
numerical study of the model outside the half-filled case for a wide range of the Coulomb interaction
U = 1, 2, . . . , 9, 10, 12, 16 and 20 and several different cluster sizes (L = 6×6, 8×8, 10×10, 12×12).
For each selected U the ground-state configurations for all Nf = 0, 1, . . . , L have been calculated
using our AM. Analysing the ground-state configurations we have found that for each L there is a
finite number of basic types of distributions that form the basic structure of a phase diagram in the
nf − U plane. Including the above described configurations we have identified the following basic
configuration types (depicted in figure 27): (a) The regular or quasi-regular distributions. This
type of distributions is dominant in the phase diagram and can be found for all the investigated
U . (b) The bounded phases, where small regions of empty sites are encircled by occupied sites.
(c) The diagonal striped configurations and (d) the mixtures of 4-molecular distributions and the
(a):U=16, N
f
=108 (b):U=2, N
f
=80 (c):U=9, N
f
=72
(d):U=10, N
f
=54 (e):U=2, N
f
=22 (f):U=1, N
f
=30
(g1):U=1, N
f
=22 (g2):U=8, Nf =6 (g3):U=20, N
f
=72
Figure 27. Representative ground-state configurations that form the basic structure of the phase
diagram of the triangular Falicov-Kimball model in the nf −U plane. Large dots correspond to
occupied sites and small dots correspond to vacant sites [73].
42701-29
H. Čenčariková, P. Farkašovský
5 10 15 20
U
0
0.2
0.4
0.6
0.8
1
n
f
- phase separations,
- bounded phases,
- regular or quasiregular distributions,
- the most homogeneous with 2D pattern,
- double axial or diagonal stripes,
- diagonal striped configurations, - mixed 4-molecular phases,
Figure 28. The nf − U phase diagram of the two-dimensional Falicov-Kimball model with
triangular lattice [73].
most homogeneous phase for nf = 1/3. (e) The double axial or diagonal stripes, observed only for
small nf similarly to (f) mixtures of the most homogeneous distribution with nf = 1/3 and the
empty lattice. (g) The phase-separated configurations. This group consists of several subgroups. In
the first subgroup (g1) the f -electrons clump to n-molecules (usually 3-molecules) and they are
distributed only over one half of the lattice, leaving another part of the lattice free of f -electrons.
This configuration type is observed only for nf → 0 and U < 5. The second type (g2), observed
for nf → 0/nf → 1, is a typical phase-separated distribution leaving at least one half of the lattice
free of f -electrons while the single f electrons are distributed over the remaining part of the lattice.
The last subgroup (g3) is formed by mixtures of nf = 1/3 and nf = 2/3 phases.
The stability regions of all the above described phases are displayed in figure 28. It is seen that
the phase diagram of the triangular Falicov-Kimball model still keeps the band structure similarly
to its square equivalent [70], with dominant area corresponding to the regular and quasi-regular
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
E
f
n f
L=144
U=10
U=20
Figure 29. The valence transition nf − Ef depicted for L = 144 and two different values of
Coulomb interaction U = 10 and U = 20 [73].
42701-30
Formation of charge and spin ordering in strongly correlated electron systems
0 50 100
−20
−15
−10
−5
0
triangular lattice
L=144
U=20
N
f
E
g
0 50 100
−20
−15
−10
−5
0
square lattice
L=144
U=20
N
f
E
g
Figure 30. Dependences of the total energy Eg on the number of localized electrons for triangular
as well as square lattices with L = 144 lattice sites in the strong coupling limit (U = 20) [73].
distributions. The central band consists of the bounded phases, that are replaced with increasing U
by other new phases, namely the diagonal stripes, the mixtures of 4-molecular configurations, the
phase-separated configurations (around nf ∼ 1/2), while the central band of the phase diagram of
the square Falicov-Kimball model consists of only the most homogeneous configurations decorated
by 2D pattern [70]. Moreover, with decreasing nf and decreasing Coulomb interaction, the new
types of f -electron distributions (missing in the square Falicov-Kimball model) have occurred (the
type e and f ).
Having a complete set of ground-state configurations we have tried to construct the picture
of valence transitions in different interaction limits. The resultant behaviours obtained for inter-
mediate and strong values of Coulomb interactions are depicted in figure 29. Comparing these
behaviours with their square counterparts [70] one can find significant differences. Indeed, while
the valence transitions for square Falicov-Kimball model is symmetric with the largest step at
nf = 1/2, the valence transitions on the triangular lattice are asymmetric and without any step
at nf = 1/2. Instead of a significant step at nf = 1/2 there are now two large steps at nf = 1/3
and nf = 2/3. The origin of this different behaviour relates with the behaviour of the total energy
Eg (see figure 30), which instead of the total minimum at nf = 1/2 (the square Falicov-Kimball
model) exhibits the total minimum at nf = 2/3 with the linear behaviour between nf = 1/3 and
nf = 2/3. For this reason the valence transition for all finite clusters has a staircase structure, that
follows the sequence nf = 1 → nf = 2/3 → nf = 1/3 to nf = 0 and it is practically indepen-
dent of L for intermediate and strong interactions. For U → ∞, the total energy Eg = 0 (for all
f -electron concentrations), the steps at nf = 2/3 and nf = 1/3 vanish and the valence transition
0 0.2 0.4 0.6 0.8 1
14
15
16
17
18
19
20
n
f
∆
U=20
L=6x6
L=12x12
L=18x18
Figure 31. Dependence of the energy gap ∆ at the Fermi level on the f -electron density nf for
U = 20 and different clusters [73].
42701-31
H. Čenčariková, P. Farkašovský
0 1 2 3 4
0
0.5
1
1.5
2
2.5
3
n
f
=2/3
U
∆
L=12x12
L=24x24
0 1 2 3 4
0
0.2
0.4
0.6
0.8
n
f
=8/9
U
∆
L=9x9
L=18x18
Figure 32. Dependences of the energy gap ∆ at the Fermi level on the Coulomb interaction U
for nf = 2/3 (L = 12× 12, L = 24× 24) and nf = 8/9 (L = 9× 9, L = 18× 18) [73].
is discontinuous from nf = 1 to nf = 0.
Although in the strong coupling limit there exists a special type of phase separation, the depen-
dence of the energy gap ∆ at the Fermi level on the f -electron concentration exhibits an insulating
behaviour in the whole region. One can see (figure 31) that outside the interval nf ∈ (1/3, 2/3) the
energy gap ∆ depends on the cluster size only very weakly and always has a finite value indicating
an insulating behaviour. Inside the mentioned interval the energy gap slightly falls down and even
depends on L, but the insulating behaviour is evident. The detailed analysis performed in the weak
coupling limit at selected values of nf (nf = 2/3 and 8/9) showed (see figure 32), that there exists a
critical value of U (Uc ∼ 1 for nf = 2/3 and Uc ∼ 3 for nf = 8/9) below which the Falicov-Kimball
model on the triangular lattice also exhibits a metallic behaviour. From this point of view, the
principal question occurring in the literature, namely, whether the systems with triangular lattice
are capable of exhibiting the Mott-Hubbard transition, has been answered positively.
4. Charge and spin ordering in the spin-1/2 Falicov-Kimball model
4.1. Spin-1/2 Falicov-Kimball model without the Ising inte raction
The transition from spinless to spin version of the Falicov-Kimball model is formally trivial,
and it is sufficient to add the spin variable to the creation and annihilation operators of itinerant
and localized electrons:
H =
∑
ijσ
tijd
+
iσdjσ +
∑
iσσ′
d+iσdiσf
+
iσ′fiσ′ + Ef
∑
iσ
f+
iσfiσ . (15)
It should be noted, however, that from the physical point of view the spin Falicov-Kimball model
describes a fully different physical reality. While in the spinless model, all states with double
occupancy are projected out (the Coulomb interaction between d electrons with opposite spins
Udd as well as between f electrons with opposite spins Uff are infinitely large) in the spin model,
such states are permitted (Udd = 0, Uff = 0). The total omission of Coulomb interactions between
the itinerant and localized electrons with opposite spins is, however, a too crude simplification of
physical reality in real systems, and therefore as the first step of our study we have generalized the
model Hamiltonian (15) by the term:
Uff
∑
i
f+
i↑fi↑f
+
i↓fi↓ (16)
describing the Coulomb repulsion of two f electrons with oppositely oriented spins localized at the
same position, and studied its effect on the valence and metal-insulator transitions [92].
Since (16) does not violate the commutativity of f+
iσfiσ with the total Hamiltonian of the
system, one can again replace f+
iσfiσ by the classical variable wiσ = 0, 1 and use for the study
of the generalized spin-1/2 Falicov-Kimball model the same procedures and methods as for the
42701-32
Formation of charge and spin ordering in strongly correlated electron systems
spinless model (the exact diagonalization on finite clusters followed by extrapolation of the results
to the thermodynamic limit). First we have investigated, the spin-1/2 Falicov-Kimball model for
small finite clusters (up to 24 sites) and for all possible configurations of the localized f electrons.
The small-cluster exact-diagonalization calculations have been performed for the following set of
Uff and U values: Uff = 0, 0.1, 0.2, . . . , 1.5, U = 1.5, 2, 3, 4, 5, 10. We summarize our results with
some observations. (i) For Uff < 2, the results do not sensitively depend upon d-f interaction
strength U . (Next the value U = 3 is chosen to represent the typical behaviour of the model in a
strong coupling limit.) (ii) The ground state for Uff = 0 is a segregated configuration of the local
f pairs (wp = {22 . . .200 . . .0} for Nf even and wp = {22 . . .2100 . . .0} for Nf odd). (iii) For a
given Nf the segregated configuration wp persists as the ground state for Uff < Uff1 . (iv) For
Uff1 < Uff < Uff2 , the number of local f pairs is reduced with increasing Uff . For Uff > Uff2 ,
the ground state is the segregated configuration with singly occupied sites (ws = {11 . . .100 . . .0}).
Furthermore, we have found that the transition from wp to ws is realized through the following
steps
wp = {2 . . . 20 . . .0} → {12 . . .210 . . .0}
→ {112 . . .2110 . . .0} → · · · → {1 . . . 10 . . .0} = ws (17)
or
wp = {2 . . .210 . . .0} → {12 . . .2110 . . .0}
→ {112 . . .21110 . . .0} → · · · → {1 . . .10 . . . 0} = ws . (18)
The last observation is very important for the extrapolation of small-cluster exact-diagonali-
zation calculations since it allows us to avoid technical difficulties associated with a large number of
configurations and consequently to study much larger systems. Figure 33 presents numerical results
Figure 33. Critical interaction strengths Uff1 and Uff2 as a function of nf calculated for
L = 500 and U = 3 [92].
for critical interaction strengths Uff1 and Uff2 as functions of the f -electron occupation number
nf obtained for U = 3 and L = 500. It is seen that there is a relative large region of Uff values
where the configurations with a nonzero number of local f pairs are the ground states. The fact
that the f electrons form the local f pairs, in spite of a relatively large repulsive interaction Uff ,
indicates that there is an attractive interaction that is capable of overcoming this direct repulsion.
One of the most important results for the spin-1/2 Falicov-Kimball model is that the interaction of
the localized f electrons with the itinerant d-band electrons leads to an effective on-site attraction
between the localized f electrons. It is interesting to study whether this feature changes the picture
of valence and metal-insulator transitions found for the spinless version of the model. The numerical
42701-33
H. Čenčariková, P. Farkašovský
Figure 34. Left: Dependence of the f -electron occupation number nf on the f -level position Ef
for L = 300, U = 3 and five different values of Uff : (a) Uff = 0, (b) Uff = 0.2, (c) Uff = 0.4,
(d) Uff = 0.8, (e) Uff = 1.5. Inset: dependence of energy gap ∆ on the f -level position Ef
calculated for L = 300, U = 3 and three different values of Uff . Right: Dependence of the
f -electron occupation number nf on the f -level position Ef for U = 0.5 and Uff = 1. The
behaviour for L = 24 has been calculated for all possible f -electron configurations, for L = 480
only the most homogeneous configurations have been considered, and the MF curve represents
the mean-field result obtained using the exact density of states [92].
results for Ef dependence of nf (calculated for configurations of type (17) or (18)) are plotted in
figure 34 for U = 3 and different values of Uff . They lead to the following conclusions. (i) The
transition is continuous for Uff = 0. (ii) For 0 < Uff < U c
ff (U c
ff ∼ 1.273) there are discontinuous
transitions from an integer-valence state nf = 1 into an inhomogeneous intermediate-valence state
nf 6= 1 at Ef = Ec
f (Uff ). (iii) For Uff > U c
ff the transitions are discontinuous from nf = 1 to
nf = 0. They take place at Ec = −U c
ff independently of Uff .
In the weak coupling limit, we have found strong finite-size effects and, therefore, we have turned
our attention to the case Uff = ∞ that permits to reduce the total number of the investigated
f -electron configurations from 4L to 2L, and thus to greatly increase the size of the clusters studied,
which is very important for the correct analysis of structural properties of the model [93].
Figure 35. The ground-state phase diagram of the spin-1/2 Falicov-Kimball model obtained
over the full set of f -electron configurations. For 1/4 < nf < 3/4 numerical calculations have
been done on the lattice with L = 28 while for nf 6 1/4 and nf > 3/4 on the lattice with
L = 36. Four different regions of stability corresponding to mixtures wa&we, wb&we, wc&we
and wd&wf are denoted as ◦,×,+ and △ [93].
42701-34
Formation of charge and spin ordering in strongly correlated electron systems
To reveal the basic structure of the phase diagram in the nf -U plane (Ef = 0) we have per-
formed an exhaustive study of the model on finite (even) clusters up to 36 sites. For fixed L, the
numerical calculations have been performed along the lines discussed above with a step ∆Nf = 2
and ∆U = 0.05. The results of numerical computations are summarized in figure 35. These results
show that the phase diagram of the spin-1/2 Falicov-Kimball model consists of three main do-
mains: the most homogeneous domain MHD (in figure 35 denoted as ·) and two phase separation
domains PSD1 (denoted as ◦,× and +) and PSD2 (denoted as △). In the MHD the ground states
are configurations in which the atomic or n-molecule clusters of f electrons are distributed in such
a manner that the distances between two consecutive clusters are either d or d+ 2. Furthermore,
distribution of the distances of d and d+2 has to be the most homogeneous. Two basic types of the
ground state configurations that fill up practically the whole MHD are displayed in table 5. In the
PSD1 the ground states are configurations in which all f electrons are distributed only in one part
of the lattice (w) while another (connecting) part of the lattice (we) is free of f electrons (phase
separation). In accordance with Gruber et al. [22] we refer to such configurations as mixtures and
denote them by w&we. We have found three basic types of configurations w which form these
mixtures: (i) aperiodic atomic configurations wa = {10k1
10k2
. . . 10ki
1 . . . 0k2
10k1
1} (with ki > 0),
(ii) aperiodic ni-molecule configurations wb = {1n1
0k1
1n2
0k2
. . . 1ni
0ki
1ni
. . . 0k2
1n2
0k1
1n1
} (with
1 6 ni 6 Nf/2 and ki > 0), and (iii) Nf -molecule (segregated) configurations wc = {11 . . .1}.
Three different regions of stability corresponding to mixtures wa&we, wb&we and wc&we are de-
noted in figure 35 as ◦,× and +. It is seen that the mixtures of the atomic configuration wa
and the empty configuration are stable only at low f -electron concentrations and the Coulomb
interactions U < 2.2. A direct comparison of the results obtained for the spin-1/2 and spinless
Table 5. Two basic types of the most homogeneous configurations that fill up practically the
whole MHD for L = 24 [93].
Nf wa
h wb
h
4 100001000000100001000000 110000000000110000000000
6 100100001001000010010000 110000001100000011000000
8 100100100100100100100100 110000110000110000110000
10 110010011001001001100100 110011000011001100110000
12 110011001100110011001100 110011001100110011001100
14 111001110011100111001100
16 111100111100111100111100
18 111111001111110011111100
20 111111111100111111111100
Falicov-Kimball model [22] shows that this region roughly corresponds to a region of phase sep-
aration in the spinless Falicov-Kimball model. Outside these regions, the phase diagrams of the
spin-1/2 and spinless Falicov-Kimball model are, however, strongly different. While the phase sep-
aration in the spinless Falicov-Kimball model takes place only for weak interactions (U < 1.2) the
spin-1/2 Falicov-Kimball model exhibits the phase separation for all Coulomb interactions. Even
with an increasing U , the phase separation shifts to higher f -electron concentrations. Particularly,
in the region 2.5 < U < 2.7 where the ground states are the mixtures of n-molecule configurations
wb with the empty configuration, the phase separation takes place for all Nf < L/2 and in the
region U > 2.7 where the ground states are the segregated configurations wS = wc&we even for
all Nf < L. At large f -electron concentrations nf but in the opposite limit (U < 0.4) there exists
another small domain of a phase separation PSD2 (denoted by △). The numerical results on finite
lattices up to 36 sites revealed only one type of configurations that can be the ground state con-
figurations in this domain, and namely, the mixtures of the periodic n-molecule configurations wd
with the fully occupied lattice wf = {11 . . .1} (the length of a connected cluster of the occupied
sites in these mixtures is at least L/2).
42701-35
H. Čenčariková, P. Farkašovský
Figure 36. Ground-state phase diagram of the spin-1/2 Falicov-Kimball model obtained on
the extrapolated set of f -electron configurations for L = 100. Four different regions of stability
correspond to mixtures wa&we (a), wb&we (b), wc&we (c) and wd&we (d) and MHD denotes
the region of the most homogeneous configurations [93].
The second step in our numerical studies has been the extrapolation of small-cluster exact
diagonalization results on large lattices. In figure 36 we present the ground state phase diagram of
the spin-1/2 Falicov-Kimball model obtained for L = 100 on the extrapolated set of configurations
that includes practically all possible types of the ground-state configurations found on finite lattices
up to 36 sites. In particular, we have considered (i) the most homogeneous configurations wh of the
type a and b (see table 5), (ii) all mixtures wa&we with ki smaller than 6, (iii) all mixtures wb&we
with ni and ki smaller than 6, (iv) all mixtures wd&wf with periods smaller than 12, and (v) all
segregated configurations. One can see that all fundamental features of the phase diagram found
on small lattices hold on much larger lattices too. Of course, the phase boundaries of different
regions corresponding to mixtures wa&we (a), wb&we (b), wc&we (c) and wd&wf (d) are now
more obvious. Since the mixtures wa&we, wb&we, wc&we and wd&wf are metallic [22] one can
expect that the phase boundary between the MHD and PSD is also the boundary of the correlation
induced metal-insulator transition. To confirm this conjecture it is necessary to show that the most
homogeneous configurations from the MHD are insulating, or, in other words, that there is a finite
Figure 37. Dependence of the f -electron occupation number nf on the f -level position Ef
calculated for U = 5 (the first panel), U = 2.5 (the second panel) and U = 1 (the third panel)
and L = 400, 800 a 1200. Insets show the behaviour of nf and ∆ close to the insulator-metal
transition point [93].
42701-36
Formation of charge and spin ordering in strongly correlated electron systems
energy gap ∆ at the Fermi energy in the spectra of these configurations4. The numerical results for
the most-homogeneous configurations of the type a and b that fill up practically the whole MHD
are displayed in figure 37. The insulating character of these configurations is clearly demonstrated
by the finite ∆ that exists for all nonzero f -electron concentrations nf and Coulomb interactions U .
Thus we can conclude that the spin-1/2 Falicov-Kimball model undergoes (on the phase boundary
between the MHD and PSD) the correlation induced metal-insulator transition that is accompanied
by a discontinuous change of the energy gap ∆. This result is similar to what was found for the
spinless Falicov-Kimball model by numerical [22] and analytical calculations in the strong [49] and
weak-coupling limit [42].
Of course, this fact has to lead to a different picture of valence and metal-insulator transitions
(figure 37) induced by pressure (increasing Ef ). We have found that: (i) In the strong-coupling
limit (U > 4), the model exhibits a pressure induced discontinuous insulator-metal transition from
an integer-valence state (nf = 1) into another integer-valence state (nf = 0). (ii) For intermediate
values of U (U ∼ 2.5) the Falicov-Kimball model undergoes a few discontinuous intermediate-
valence transitions. There are several discontinuous insulator-insulator transitions from nf = 1
to nf = 1/2 and discontinuous insulator-metal transition from nf = 1/2 to nf = 0. (iii) In the
weak-coupling limit (U < 2), the model undergoes a few consecutive discontinuous and continuous
intermediate-valence transitions as well as a discontinuous metal-insulator transition.
4.2. Spin-1/2 Falicov-Kimball model with the Ising interac tion
Despite the unquestionable success of the spinless Falicov-Kimball model in describing the
charge ordering in strongly correlated systems, this relatively simple model was not capable of
accounting for all aspects of real experiments, and namely, that the charge superstructure is often
accompanied by a magnetic superstructure. In order to more realistically describe electronic and
spin processes in real materials, the original spin-1/2 Falicov-Kimball model has been extended by
the spin-dependent interaction (of the Ising type) between localized and itinerant electrons [63, 64,
76]
H =
∑
ijσ
tijd
+
iσdjσ + U
∑
iσσ′
f+
iσfiσd
+
iσ′diσ′ + J
∑
iσ
(f+
i−σfi−σ − f+
iσfiσ)d
+
iσdiσ (19)
and, in addition, the local Coulomb interaction between the f electrons in the limit of Uff → ∞ has
been included. Thus, from the major interaction terms that come into account for the interacting
d and f -electron subsystems, only the Hubbard type interaction
Hdd = Udd
∑
i
d+i↑di↑d
+
i↓di↓ (20)
between the spin-up and spin-down d electrons has been omitted in the Hamiltonian (19).
In the work [63] Lemanski presented a simple justification for the omission of this term based
on an intuitive argument: the longer time electrons occupy the same site, the more important
becomes the interaction between them. According to this rule the interaction between the itinerant
d electrons (Udd) is smaller than the interaction between the localized f electrons (Uff ) as well as
smaller than the spin-independent interaction between the localized and itinerant electrons U .
We have tried to verify the validity of these intuitive statements numerically and we have
found that their validity is restricted only to the region of intermediate and strong Coulomb
interactions U between the localized and itinerant electrons. Our numerical proof is based on the
exact diagonalization of the model Hamiltonian (19) extended by Hdd on small finite clusters.
Such a study is numerically very exhaustive, since it is necessary for each configuration of localized
electrons/spins to perform the Lanczos diagonalization [94] of the Hamiltonian over the full Hilbert
space of the model and this process should be repeated
(
L
Nf
)
times. Of course, such a procedure
demands in practice a considerable amount of CPU time, which imposes severe restrictions on the
4Since we consider the case Nf +Nd = L, the Fermi level EF and the energy gap ∆ of some configuration w are
presented by EF = λL−Nf
and ∆ = λL−Nf+1 − λL−Nf
, respectively.
42701-37
H. Čenčariková, P. Farkašovský
0 5 10 15
0.2
0.3
0.4
0.5
0.6
0.7
0.8
U
dd
n
f
a) U=4
L=8
L=10
L=12
0 5 10 15
0.2
0.3
0.4
0.5
0.6
0.7
0.8
U
dd
n
f
b) U=8
L=8
L=10
L=12
Figure 38. The ground-state phase diagrams of the spin-1/2 Falicov-Kimball model extended
by the Hubbard interaction between the itinerant electrons calculated for two different values of
U on small finite clusters of L = 8, 10 and 12 sites. Below Uc
dd the ground states are the ground-
state configurations of the conventional spin-1/2 Falicov-Kimball model (Udd = 0). Above Uc
dd
these ground states become unstable. The one-dimensional exact-diagonalization results [76].
size of clusters that can be studied within the exact diagonalization calculations. For this reason
we were able to exactly investigate only the clusters up to L = 12. Fortunately, it was found that in
some parameter regimes the ground-state characteristics of the model are practically independent
of L and thus already such small clusters can be satisfactorily used to represent the behaviour of
macroscopic systems. In particular, we have studied the stability of the ground-state configuration
w0(Nf ) (obtained for Udd = 0 and fixed Nf) at finite values of Udd. The results of numerical
calculations obtained for U = 4 and U = 8 are summarized in figure 38 in the form of nf − Udd
phase diagrams (the half-filled band case nf +nd = 1 is considered). One can see that the ground-
state configuration w0(Nf ) found for Udd = 0 persists as a ground state up to relatively large
values of Udd (U c
dd ∼ 2, for U = 4 and U c
dd ∼ 6, for U = 8), revealing small effects of the Udd term
on the ground state of the model in the strong U interaction limit. Contrary to the strong coupling
case, for small (U = 1) and intermediate (U = 2) values of the Coulomb interaction between the
localized and itinerant electrons, very strong effects of Udd term on the ground states of the model
0 0.2 0.4 0.6 0.8 1
0
4
8
12
16
20
24
28
32
f−electrons
J
N
d
0 0.2 0.4 0.6 0.8 1
0
8
16
24
32
d−electrons
J
N
d
FP:
↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑
PP:
↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↓↓
↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↓↓↓
↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↓↓↓↓
↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↓↓↓↓↓↓
↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↓↓↓↓↓↓↓
↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓
↑↑↑↑↑↑↑↑↑↑↑↓↓↓↑↑↑↓↓↓↑↑↑↓↓↓↑↑↑↓↓↓
↑↑↓↓↓↓↓↓↑↑↓↓↓↓↓↓↑↑↓↓↓↓↓↓↑↑↓↓↓↓↓↓
↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↓↓↑↑↓↓↓↑↑↓↓
↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↓↓↑↑↓↓
↑↑↑↑↑↓↓↑↑↑↑↑↓↓↑↑↑↑↑↓↓↑↑↑↑↑↓↓↑↑↓↓
↑↑↑↑↑↑↑↑↑↑↑↑↑↑↓↓↑↑↓↓↑↑↓↓↑↑↓↓↑↑↓↓
↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↓↓↑↑↓↓↑↑↓↓↑↑↓↓
↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↓↓↑↑↓↓↑↑↓↓
FP:
↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑
PP:
↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↓↓
↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↓↓↓
↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↓↓↓↓
↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↓↓↓↓↓↓
↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↓↓↓↓↓↓↓
↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓
↑↑↑↑↑↑↑↑↑↑↑↓↓↓↑↑↑↓↓↓↑↑↑↓↓↓↑↑↑↓↓↓
↑↑↓↓↓↓↓↓↑↑↓↓↓↓↓↓↑↑↓↓↓↓↓↓↑↑↓↓↓↓↓↓
↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↓↓↑↑↓↓↓↑↑↓↓
↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↓↓↑↑↓↓
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↑↑↑↑↑↑↑↑↑↑↑↑↑↑↓↓↑↑↓↓↑↑↓↓↑↑↓↓↑↑↓↓
↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↓↓↑↑↓↓↑↑↓↓↑↑↓↓
↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↓↓↑↑↓↓↑↑↓↓
NP:
(N
d
=6) ↑↑↑↑↑↓↓↓↓↓↓↑↑↑↑↑↓↓↓↓↓↑↑↑↑↑↑↓↓↓↓↓
(N
d
=8) ↑↑↑↑↓↓↓↓↑↑↑↑↓↓↓↓↑↑↑↑↓↓↓↓↑↑↑↑↓↓↓↓
(N
d
=10) ↑↑↑↓↓↓↑↑↑↑↓↓↓↑↑↑↓↓↓↑↑↑↓↓↓↓↑↑↑↓↓↓
(N
d
=12) ↑↑↑↓↓↑↑↑↓↓↓↑↑↓↓↓↑↑↑↓↓↑↑↑↓↓↓↑↑↓↓↓
(N
d
=14) ↑↑↑↓↓↑↑↑↓↓↓↑↑↓↓↓↑↑↑↓↓↑↑↑↓↓↓↑↑↓↓↓
(N
d
=14) ↑↑↓↓↓↓↓↑↑↓↓↓↓↓↑↑↓↓↑↑↑↑↑↓↓↑↑↑↑↑↓↓
(N
d
=16) ↑↑↓↓↑↑↓↓↑↑↓↓↑↑↓↓↑↑↓↓↑↑↓↓↑↑↓↓↑↑↓↓
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d
=18) ↑↑↓↓↑↓↓↑↑↓↓↑↓↓↑↑↓↓↑↑↓↑↑↓↓↑↑↓↑↑↓↓
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d
=20) ↑↑↓↑↑↓↑↑↓↓↑↓↓↑↓↓↑↑↓↑↑↓↑↑↓↓↑↓↓↑↓↓
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d
=22) ↑↓↓↑↓↓↑↓↓↑↓↓↑↓↓↑↓↑↑↓↑↑↓↑↑↓↑↑↓↑↑↓
(N
d
=24) ↑↓↓↑↓↑↑↓↑↓↓↑↓↑↑↓↑↓↓↑↓↑↑↓↑↓↓↑↓↑↑↓
(N
d
=26) ↑↓↑↑↓↑↓↑↑↓↑↓↑↑↓↑↓↑↓↓↑↓↑↓↓↑↓↑↓↓↑↓
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d
=26) ↑↓↑↑↓↑↓↑↑↓↑↓↑↓↓↑↓↑↑↓↑↓↑↓↓↑↓↑↓↓↑↓
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d
=28) ↑↓↑↓↓↑↓↑↓↑↓↑↑↓↑↓↑↓↑↓↓↑↓↑↓↑↓↑↑↓↑↓
(N
d
=30) ↑↓↑↓↑↓↑↓↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↑↓↑↓↑↓↑↓
(N
d
=32) ↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓
NP:
(N
d
=6) ↑↑↑↑↑↓↓↓↓↓↓↑↑↑↑↑↓↓↓↓↓↑↑↑↑↑↑↓↓↓↓↓
(N
d
=8) ↑↑↑↑↓↓↓↓↑↑↑↑↓↓↓↓↑↑↑↑↓↓↓↓↑↑↑↑↓↓↓↓
(N
d
=10) ↑↑↑↓↓↓↑↑↑↑↓↓↓↑↑↑↓↓↓↑↑↑↓↓↓↓↑↑↑↓↓↓
(N
d
=12) ↑↑↑↓↓↑↑↑↓↓↓↑↑↓↓↓↑↑↑↓↓↑↑↑↓↓↓↑↑↓↓↓
(N
d
=14) ↑↑↑↓↓↑↑↑↓↓↓↑↑↓↓↓↑↑↑↓↓↑↑↑↓↓↓↑↑↓↓↓
(N
d
=14) ↑↑↓↓↓↓↓↑↑↓↓↓↓↓↑↑↓↓↑↑↑↑↑↓↓↑↑↑↑↑↓↓
(N
d
=16) ↑↑↓↓↑↑↓↓↑↑↓↓↑↑↓↓↑↑↓↓↑↑↓↓↑↑↓↓↑↑↓↓
(N
d
=18) ↑↑↓↓↑↓↓↑↑↓↓↑↓↓↑↑↓↓↑↑↓↑↑↓↓↑↑↓↑↑↓↓
(N
d
=20) ↑↑↓↑↑↓↑↑↓↓↑↓↓↑↓↓↑↑↓↑↑↓↑↑↓↓↑↓↓↑↓↓
(N
d
=22) ↑↓↓↑↓↓↑↓↓↑↓↓↑↓↓↑↓↑↑↓↑↑↓↑↑↓↑↑↓↑↑↓
(N
d
=24) ↑↓↓↑↓↑↑↓↑↓↓↑↓↑↑↓↑↓↓↑↓↑↑↓↑↓↓↑↓↑↑↓
(N
d
=26) ↑↓↑↑↓↑↓↑↑↓↑↓↑↑↓↑↓↑↓↓↑↓↑↓↓↑↓↑↓↓↑↓
(N
d
=26) ↑↓↑↑↓↑↓↑↑↓↑↓↑↓↓↑↓↑↑↓↑↓↑↓↓↑↓↑↓↓↑↓
(N
d
=28) ↑↓↑↓↓↑↓↑↓↑↓↑↑↓↑↓↑↓↑↓↓↑↓↑↓↑↓↑↑↓↑↓
(N
d
=30) ↑↓↑↓↑↓↑↓↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↑↓↑↓↑↓↑↓
(N
d
=32) ↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓
Figure 39. The f -electron ground-state phase diagram of the spin-1/2 Falicov-Kimball model ex-
tended by the spin-dependent interaction calculated for Nf = L (L = 32). The one-dimensional
exact-diagonalization results. Inset: The d-electron ground-state phase diagram of the model
calculated at the same values of model parameters. (·): the NP phase, (+): the FP phase, (◦):
the PP phase [76].
42701-38
Formation of charge and spin ordering in strongly correlated electron systems
2 6 10 14 18 22 26 30 34 38 42 46
4
8
12
16
20
24
L = 24
J = 0.5
U = 4
(I)
(II)
(III)
f−electrons
N
d
N
f
Figure 40. The f -electron skeleton phase diagram of the spin-1/2 Falicov-Kimball model ex-
tended by the spin-dependent interaction calculated for Nf 6 L, J = 0.5 and L = 24. (·): the
NP phase, (+): the FP phase, (◦): the PP phase. See the text for a definition of configuration
types that are ground states in the areas I, II and III. The one-dimensional exact-diagonalization
results [76].
have been observed. In these cases the typical values of U c
dd are of the order of 0.5 but for some Nf
even much smaller values were found. Thus, we can conclude that the Hubbard type interaction
between the spin-up and spin-down d electrons can be neglected in the strong interaction limit
between the localized and itinerant electrons (U > 4). We have obtained the same conclusion
for the spin-dependent interaction J > 0 [78]. For this reason, all subsequent calculations on the
spin-1/2 Falicov-Kimball model with spin-dependent Coulomb interaction J between the f and d
electrons have been done at U = 4.
We have started the study of the effect of anisotropic, spin-dependent interaction between
localized and itinerant electrons on the ground-state properties of the model in the simplest case
Nf = L (D = 1), which is accessible to exact numerical calculations on relatively large clusters
(L = 32). The results of our numerical study are summarized in figure 39 in the form ofNd−J phase
diagram. Various phases that enter the phase diagram are classified according to Sz
f =
∑
i(w
0
i↑−w
0
i↓)
2 10 18 26 34 42 50 58 66 74 82 90
2
6
10
14
18
22
26
30
34
38
42
46
L = 48
J = 0.5
U = 4
(I)
(II)
(III)
f−electrons
N
d
N
f
2 10 18 26 34 42 50 58 66 74 82 90
2
6
10
14
18
22
26
30
34
38
42
46
L = 48
J = 0.5
U = 4
d−eledtrons
N
d
N
f
Figure 41. The f and d-electron phase diagrams of the spin-1/2 Falicov-Kimball model extended
by the spin-dependent interaction calculated for Nf < L, J = 0.5 and L = 48. (·): the NP phase,
(+): the FP phase, (◦): the PP phase. The one-dimensional approximative results [76].
42701-39
H. Čenčariková, P. Farkašovský
and Sz
d = Nd↑ − Nd↓: the fully polarized (FP) phase is characterized by |Sz
f | = Nf , |Sz
d | = Nd,
the partially polarized (PP) phases are characterized by 0 < |Sz
f | < Nf , 0 < |Sz
d | < Nd and the
non-polarized (NP) phases are characterized by |Sz
f | = 0, |Sz
d | = 0.
Comparing numerical results obtained for |Sz
f | and |Sz
d | one can find a nice correspondence
between the magnetic phase diagrams of localized (f) and itinerant (d) subsystems. Indeed, with
the exception of several isolated points at J = 0.05, the corresponding FP, PP and NP phases
perfectly coincide over the remaining part of diagrams showing the strong coupling between the
magnetic subsystems of localized and itinerant electrons for nonzero values of J .
In general, the spin-dependent interaction J stabilizes the FP and PP phases, while the NP
phase is gradually suppressed with increasing J . In figure 39 we present a complete set of ground-
state configurations from the NP region. Among them one can find different types of periodic
and nonperiodic configurations, but the most interesting examples represent configurations formed
0 0.5 1 1.5 2 2.5 3 3.5 4
0
4
8
12
16
20
24
28
32
36 f−electrons
J
N
d
L=36
N
f
=L
0 0.5 1 1.5 2 2.5 3 3.5 4
0
4
8
12
16
20
24
28
32
36 d−electrons
J
N
d
L=36
N
f
=L
(1) (2) (3) (4) (5) (6) (7) (8)
(9) (10) (11) (12) (13) (14) (15)
Figure 42. Up: The f and d-electron ground-state phase diagrams of the spin-1/2 Falicov-
Kimball model extended by the spin-dependent interaction calculated for Nf = L (L = 36). The
two-dimensional approximative results. Down: Typical examples of ground states of the spin-
1/2 Falicov-Kimball model extended by the spin-dependent interaction obtained for Nf = L
(L = 36). To visualize the spin distributions we use △ for the up spin electrons and H for the
down spin electrons. The two-dimensional approximative results [76].
42701-40
Formation of charge and spin ordering in strongly correlated electron systems
by antiparallel ferromagnetic domains, that convincingly illustrate the cooperative effects of spin-
dependent interaction J between the localized and itinerant electrons.
As the next step, we have performed numerical studies of the model for the case Nf 6= L.
From the numerical point of view this case is considerably exact, since now we have to minimize
the ground-state energy not only over all different spin configurations but also over all different
f -electron distributions. This takes a considerable amount of CPU time and for this reason we
were able to investigate exactly only the clusters up to L = 24. We have found that the spectrum
of magnetic solutions that yields the model for the NP and PP phases is very rich. Indeed, for
L = 24 and J = 0.5 we have found 140 different NP phases and 20 different PP phases that enter
the Nf − Nd phase diagram. The main configuration types, with the largest stability regions are
presented in figure 40 in the form of a skeleton phase diagram. From the NP phases, the largest
stability region (denoted by I) corresponds to configurations of the type ↑n↓n 0L−2n. The second
largest region (denoted by II) corresponds to NP configurations of the type [↑ 0n ↓ 0n]k0L−2k(n+1).
Typical examples of the NP ground states from the central region of the phase diagram represent
periodic configurations of the type ↑n 0m ↓n 0m (below the main diagonal) and configurations of
the type ↑2 [↑↓]k1
↓2 0m[↓ 0p ↑ 0p]k2
, or ↑2 [↑↓]k1
↓2 0m[↓ 0p ↑ 0p−1]k2
, above the main diagonal.
Between these configurations and the FP region, the ground states are the segregated configurations
of the type ↑2 [↓↑]k ↓2 0m, or [↑↓]k1
↑2 [↓↑]k2
↓2 [↑↓]k3
0m, or their modifications (the region denoted
by III). In the PP region, the typical examples of ground states represent configurations of the
type ↑n [0 ↓ 0 ↑]k0 ↓ 0.
In order to minimize the finite-size effects we have performed the same study using our AM on
(2)(1) (3) (4) (5) (6)
(7) (8) (9) (10) (11) (12)
(13) (14) (15) (16) (17) (18)
(19) (20) (21) (22) (23) (24)
(25) (26) (27) (28) (29) (30)
Figure 43. Typical examples of ground states of the spin-1/2 Falicov-Kimball model extended by
the spin-dependent interaction obtained for Nf < L, J = 0.5 and L = 64. The two-dimensional
approximative results [76].
42701-41
H. Čenčariková, P. Farkašovský
two times larger clusters (L = 48). Our numerical results showed that all the main results obtained
on small clusters also hold on larger clusters. Again we have observed a strong coupling between two
magnetic subsystems and a coincidence of corresponding magnetic phases, that stability regions
are practically unchanged with increasing L (figure 41).
However, from the experimental point of view, the two-dimensional case is much more inter-
esting. Numerical results obtained for this case and Nf = L are displayed in figure 42. Similarly
to the one-dimensional case, the basic structure of the phase diagram in D = 2 is formed by three
large FP, PP and NP domains that are accompanied by secondary phases (points in which the
d and f -electron phase diagrams do not coincide). However, while in the one dimension the sec-
ondary phases are stable only in isolated points at very small values of J , in two dimensions these
secondary phases also persist for large J . Calculations that we have performed on different clusters
(4× 4, 6× 6, and 8× 8) showed that the secondary structure depends very strongly on the cluster
size and with increasing L it is gradually suppressed.
The typical examples of ground-state configurations (that represent the most frequently ap-
n
f
=1/16
L=8x8
n
f
=1/8 n
f
=1/4 n
f
=1/2 n
f
=3/4
n
f
=7/8 n
f
=15/16
L=10x10
n
f
=1/10 n
f
=1/5 n
f
=1/2 n
f
=4/5 n
f
=9/10
L=12x12
n
f
=1/12 n
f
=1/8 n
f
=1/6 n
f
=1/4 n
f
=1/3
n
f
=1/2 n
f
=2/3 n
f
=3/4 n
f
=5/6 n
f
=7/8
n
f
=11/12
Figure 44. Typical ground-state configurations of the two-dimensional generalized Falicov-
Kimball model obtained for selected values of nf on finite clusters of L = 8 × 8, L = 10 × 10
and L = 12× 12 sites at U = 4, J = 0.5 and nf + nd = 2 [78].
42701-42
Formation of charge and spin ordering in strongly correlated electron systems
pearing types of ground states in the Nd − J phase diagram) are displayed in figure 42. Again one
can see that the spectrum of magnetic solutions that yields the Falicov-Kimball model extended
by spin-dependent interaction is very rich. In addition to the FP phase (that the stability region
shifts to higher d-electron concentrations when J increases) there are various types of NP and
PP structures like the antiparallel ferromagnetic domains (2–3), the axial magnetic stripes (4–7),
the diagonal magnetic stripes (8–11) and the perturbed diagonal magnetic stripes (12–15). This
again demonstrates strong effects of the spin-dependent interaction on the formation of magnetic
superstructures in the extended Falicov-Kimball model and its importance for a correct description
of correlated electron systems.
Finally, we have performed the same numerical study for the caseNf < L. The typical examples
of the resulting charge and spin ordering are displayed in figure 43. Among them one can find various
types of phase-segregated (1), phase-separated (16) and n-molecular (3) configurations with FP,
PP and NP ground states as well as various types of axial (9) and diagonal (5) magnetic/charge
stripes. In general, we have observed that the system shows tendency towards phase segregation
for small and large d-electron concentrations, while near the nd = 1 point the system prefers to
form various types of axial and diagonal stripes.
For the most physically interesting cases Nf + Nd = L and Nf + Nd = 2L we have also
performed a detailed analysis of charge and magnetic ordering. We have started our study with the
case Nf +Nd = 2L, which is slightly simpler for a description. As shown in figure 44 the ground
states for Nf +Nd = 2L are antiferromagnetic (AF) with alternating pattern, where the electrons
(for nf < 1/2) or holes (for nf > 1/2) form axial distributions. Our calculations showed that these
inhomogeneous stripe distributions are stable for large Coulomb interactions, while decreasing U
U=4, L=8x8, n
f
=1/4 U=4, L=12x12, n
f
=1/4 U=4, L=16x16, n
f
=1/4
U=2, L=8x8, n
f
=1/4 U=2, L=12x12, n
f
=1/4 U=2, L=16x16, n
f
=1/4
U=4, L=8x8, n
f
=1/8 U=4, L=12x12, n
f
=1/8 U=4, L=16x16, n
f
=1/8
U=2, L=8x8, n
f
=1/8 U=2, L=12x12, n
f
=1/8 U=2, L=16x16, n
f
=1/8
Figure 45. Ground-state configurations of the two-dimensional generalized Falicov-Kimball
model obtained for different U (U = 4 and U = 2) and nf (nf = 1/4, nf = 1/8 at nf + nd =
2) [78].
42701-43
H. Čenčariková, P. Farkašovský
leads to their destruction and prefers a homogeneous electron arrangement (see figure 45).
For Nf + Nd = L the situation is fully different. The first fundamental difference is that for
sufficiently small f -electron concentrations nf the ground state could be ferromagnetically (F)
ordered (see figure 46).
n
f
=1/16
L=8x8
n
f
=1/8 n
f
=1/4 n
f
=1/2 n
f
=3/4
n
f
=7/8 n
f
=15/16
L=10x10
n
f
=1/10 n
f
=1/5 n
f
=1/2 n
f
=4/5 n
f
=9/10
L=12x12
n
f
=1/12 n
f
=1/8 n
f
=1/6 n
f
=1/4 n
f
=1/3
n
f
=1/2 n
f
=2/3 n
f
=3/4 n
f
=5/6 n
f
=7/8
n
f
=11/12
Figure 46. Typical ground-state configurations of the two-dimensional generalized Falicov-
Kimball model obtained for selected values of nf on finite clusters of L = 8 × 8, L = 10 × 10
and L = 12× 12 sites at U = 4, J = 0.5 and nf + nd = 1 [78].
The second difference is that although with increasing f -electron concentration the ground
states are the AF, these AF arrangements are formed by F ordered clusters (domains). In addition,
for nf = 1/4 and nf = 1/3 (L > 144) a new type of stripes (known as the ladders) occurs. And
finally, a detailed analyses showed that there exists a critical f -electron concentration nc
f ∼ 1/4
below which the ground-states are phase separated.
42701-44
Formation of charge and spin ordering in strongly correlated electron systems
5. Applications of the Falicov-Kimball model to a descripti on of real
materials
5.1. Ground-state properties of Na xCoO2
The NaxCoO2 system, which forms the basis for a quasi-two-dimensional transition metal oxide
superconductor (Tc = 4.5 K) when hydrated [95] shows a wide variety of unexplained behaviours
in the accessible range 0 < x < 1. It consists of alternate stacks of electronically active triangular
CoO2 layers (with edge sharing CoO6 octahedra) separated by Na layers that act not only as
spacers, leading to electronic two-dimensionality (figure 47), but also as charge reservoirs [96–
98]. With increasing Na content x, the phase diagram nonhydrated NaxCoO2 system exhibits a
Co
Na
O
b
c
a
a b
Co
O above Co
O below Co
Figure 47. Crystal structure of NaxCoO2 (left hand panel). The (a)–(b) planes, showing the
orientation of the oxygen tetrahedron around a central Co ion (right hand panel).
succession of ground states [98] changing from a paramagnetic metal at x = 0.3 through a charge-
ordered insulator at x = 0.5, a “Curie-Weiss metal” for x = 0.75, and finally a magnetically ordered
state for x > 0.75. At x = 0.5, detailed electron and neutron diffraction measurements [99, 100]
revealed an ordering of the Na ions along one crystallographic direction which decorates the chains
of Co ions with different amounts of charge [100].
It is suggested that there are strong correlations between the Na ions and electrons (holes)
from the CoO2 layers, even though they occupy separate layers, and that a charge ordering in the
CoO2 layers is induced by sodium ion ordering. Unfortunately, numerical calculations performed
at the Density Function Theory (DFT) and DFT+U level by Li et al. [101] did not confirm this
conjecture. In the spirit of these results we have tried to describe the conducting properties and a
formation of charge ordering in the CoO2 layers within a relatively simple model that takes into
account both the interplane interactions between the ordered Na ions and electrons (holes) form
the CoO2 planes as well as the intraplane interaction between electrons within CoO2 layers [72].
The system is modelled by the spinless Falicov-Kimball Hamiltonian that automatically projects
out the states with two holes on the 3d (dz2) orbitals. Thus, our starting Hamiltonian can be
written as a sum of three terms:
H =
∑
〈ij〉
tijd
+
i dj −
∑
i
ωid
+
i di + V
∑
〈ij〉
ninj . (21)
The first term of (21) describes hopping of 3d-electrons on the triangle network of Co4+ ions. These
intersite hopping transitions are described by the matrix elements tij which are −t if i and j are
the nearest neighbours and zero otherwise. The second term describes the interplane interaction
42701-45
H. Čenčariková, P. Farkašovský
−10 −5 0 5
0
0.02
0.04
0.06
0.08
E
d−
D
O
S
U
1
=2, V=0, x=0.5
L=100 × 100
−15 −10 −5 0
0
0.02
0.04
0.06
0.08
E
d−
D
O
S
U
1
=5.6, V=0, x=0.5
L=100 × 100
−15 −10 −5 0
0
0.02
0.04
0.06
0.08
E
d−
D
O
S
U
1
=5.8, V=0, x=0.5
L=100 × 100
−15 −10 −5 0
0
0.02
0.04
0.06
0.08
E
d−
D
O
S
U
1
=8, V=0, x=0.5
L=100 × 100
Figure 48. The d-electron density of states of the model (21) calculated for x = 0.5, V = 0,
L = 100× 100 and four different values of the nearest neighbour interplane interaction U1. The
second (U2) and third (U3) nearest neighbour interplane interactions are modelled by U2 = 0.2U1
and U3 = 0.2U2 [72].
between the Na ions and the d electrons from the CoO2 layers, where ωi represents the static
potential of the first, second and third nearest Na ions to site i. The strengths of the Coulomb
interactions corresponding to the first (U1), second (U2) and third (U3) nearest neighbours are
considered as the model parameters. The third term represents the nearest-neighbour Coulomb
repulsion between two d electrons on the Co triangle network.
We have started our numerical study with the case x = 0.5 and V = 0. As mentioned above, at
x = 0.5, the ordered Na ions form one-dimensional zig-zag chains and so, one can easily reconstruct
the static potential ωi for any selected values of U1, U2 and U3. Having ωi, the Hamiltonian (21) can
n
i
=0.93
n
i
=0.07
x=0.5
a
b
a
b
a
b
a
b
Figure 49. The d-electron on-site occupation ni = 〈d+i di〉 calculated for x = 0.5, V = 0, U1 = 8
(U2 = 0.2U1, U3 = 0.2U2) on L = 8 × 8 cluster. The on-site occupation ni is represented by
filled-circles (•), where the radius of the circle on site i is proportional to ni. The open symbols
represent the Na positions above (circles) and below (squares) the CoO2 layer [72].
42701-46
Formation of charge and spin ordering in strongly correlated electron systems
Table 6. The d-electron on-site occupation ni on stripes a and b calculated for x = 0.5, U1 = 8
(U2 = 0.2U1, U3 = 0.2U2) and four different values of intraplane interaction V [72].
ni V = 0 V = 2 V = 4 V = 8
stripes a 0.92936153 0.9664959 0.9811269 0.99188500
stripes b 0.07063846 0.03350408 0.0188730 0.00811499
be diagonalized by standard numerical routines and used directly in the analysis of desired zero-
temperature characteristics like the d-electron density of states, the on-site occupation ni = 〈d+i di〉,
etc. The numerical results for the d-electron density of states obtained for four selected values of the
nearest interplane Coulomb interaction U1 (in the rest of the paper the second and third nearest
neighbour interplane interactions are modelled by U2 = 0.2U1, U3 = 0.2U2) are shown in figure 48.
One can see that for small and intermediate values of U1 the system is metallic, but with increasing
−10 −5 0 5
0
0.01
0.02
0.03
0.04
0.05
0.06
E
d−
D
O
S
x=0.3
U
1
=8, V=0
−20 −15 −10 −5 0
0
0.01
0.02
0.03
0.04
0.05
0.06
E
d−
D
O
S
x=0.71
U
1
=8, V=0
−15 −10 −5 0
0
0.01
0.02
0.03
0.04
0.05
0.06
E
d−
D
O
S
x=0.75
U
1
=8, V=0
n
i
=0.316
n
i
=0.104
x=0.3
n
i
=0.987
n
i
=0.501
x=0.71
n
i
=0.665
n
i
=0.974
x=0.75
Figure 50. Up: The d-electron density of states of the model (21) calculated for V = 0, U1 = 8
(U2 = 0.2U1, U3 = 0.2U2) and three different Na ion distributions taken from experiments.
Down: The d-electron on-site occupation ni calculated for three different Na ion distributions
x = 0.3, x = 0.71 and x = 0.75 taken from experiment. The on-site occupation ni is represented
by filled-circles (•) whose radius is proportional to ni [72].
42701-47
H. Čenčariková, P. Farkašovský
U1 the charge gap at the Fermi level opens, and the system undergoes the metal-insulator transition
at U1 ∼ 5.8. This confirms a supposition that the interplane interactions between the ordered Na
ions and the d electrons from Co layers play a crucial role in the stabilization of the insulating
state in the Na0.5CoO2 material.
From this point of view it is interesting to ask if these interactions could also stabilize the charge
ordering within the Co layers. To answer this question we have calculated the on-site occupation
ni = 〈ψG|d
+
i di|ψG〉, where |ψG〉 is the ground state of Hamiltonian (21) obtained directly from the
exact numerical diagonalization solution. The typical results for ni within the insulating phase are
shown in figure 49 for a finite cluster of L = 8× 8 sites.
Our results clearly demonstrate that an inhomogeneous charge ordering, of the stripe-like form,
develops within the Co layers. For the sites below (above) the Na ions the occupation number ni is
equal to 0.93 (the stripes a), while for the remaining sites ni = 0.07 (the stripes b). Thus, we arrive
at the conclusion that interplane interactions between the ordered Na ions and the d electrons
from the CoO2 layers not only stabilize the insulating ground state in Na0.5CoO2, but they are
also capable of describing the formation of inhomogeneous charge ordering within the CoO2 layers.
Since the Coulomb interactions between d electrons inside CoO2 planes are not negligible in
comparison with interplane interactions, it was necessary to examine what happens if these in-
teractions are switched on (V 6= 0). We have solved this task in a manner analogous to the case
V = 0, i.e., by a direct calculation of the d-electron density of states and the d-electron on-site
occupation ni, with only one technical complication, namely, that the diagonalization of the full
Hamiltonian is performed by the Lanczos method, because for V 6= 0, the Hamiltonian (21) is no
longer a single particle. The results of our numerical calculations for ni obtained for four different
values of the intraplane interaction V are summarized in table 6. One can see that for nonzero
interplane interactions, the nearest-neighbour intraplane interaction does not destroy the charge
ordering induced by Na ions, but on the contrary, increasing V further stabilizes the inhomogeneous
charge ordering. This effect is very strong and for sufficiently large V , practically all d electrons are
localized on sites below (above) Na ions (the stripes a). Taking into account the fact that ni = 0 (in
our notation) corresponds to Co4+ and ni = 1 to Co3+, these results are consistent with the model
proposed recently by Choy [102] for a description of the sign change of the Hall-coefficient (as a
function of temperature), in which the rows of Co4+ ions alternate with rows of Co3+ ions. This
indicates that our model, in spite of its relative simplicity, may still contain the relevant physics of
cobaltate systems.
We have used the same procedure for the case of x 6= 0.5. Our main motivation in this case was
to answer the question whether our simple model is capable of describing correctly the conducting
properties of NaxCoO2 for x < 0.5 as well as for x > 0.5. In figure 50 we present numerical
results for d-electron density of states for three selected experimentally confirmed Na structures
with x = 0.3, x = 0.71 a x = 0.75, representing the typical examples from the region of x < 0.5
and x > 0.5 (the experimentally proposed Na ion distributions are shown in figure 50, together
with our results for ni).
It is seen that in all three investigated cases the Fermi level lies inside the conduction band, and
thus all three cases correspond to a metallic state which is perfectly consistent with experimental
measurements [98].
5.2. Magnetization processes in rare-earth tetraborides
A spin system is frustrated when all local interactions between spin pairs cannot be satisfied
at the same time. Frustration can arise from the competing interactions or/and from a particular
geometry of the lattice, as seen in the triangular lattice. The Shastry-Sutherland lattice was con-
sidered more than 20 years ago by Shastry and Sutherland [103] as an interesting example of a
frustrated quantum spin system with an exact ground state. It can be described as a square lattice
with antiferromagnetic couplings J between nearest neighbours and additional antiferromagnetic
couplings J ′ between next-nearest neighbours in every second square (see figure 51). This lattice
attracted much attention after its experimental realization in the SrCu2(BO3)2 compound [104].
The observation of a fascinating sequence of magnetization (m/ms =1/2, 1/3, 1/4 and 1/8 of the
42701-48
Formation of charge and spin ordering in strongly correlated electron systems
J
J ′
a
b
Figure 51. The Shastry-Sutherland lattice with magnetic couplings J bonds along the edges of
the squares and J ′ along the diagonals (left hand panel), and the topologically identical structure
realized in the (001) plane of SrCu2(BO3)2 and rare-earth tetraborides (right hand panel).
saturated magnetization ms) in this material [104] stimulated further theoretical and experimental
studies of the Shastry-Sutherland lattice [105, 106].
Similar phenomena of magnetization plateaus is also observed in the rare-earth tetraborid
TmB4 [107]. Since fully polarized state can be reached for experimentally accessible magnetic
fields, this compound permits exploration of its complete magnetization process. It was found that
the magnetization diagram of TmB4 consists of magnetization plateaus located at small fractional
values ofm/ms =1/9, 1/8, 1/7 of the saturated magnetization, followed by the major magnetization
plateau located at m/ms = 1/2. Due to strong crystal field effects, the effective spin model for
TmB4 has been suggested to be described by the spin-1/2 Shastry-Sutherland model under strong
Ising (or easy-axis) anisotropy in a magnetic field h [107]
HJJ′ = J
∑
〈i,j〉
Sz
i S
z
j + J ′
∑
〈〈i,j〉〉
Sz
i S
z
j − h
∑
i
Sz
i , (22)
where Sz
i = ±1/2 denotes the z-component of a spin-1/2 degree of freedom on site i of a square
lattice and J , J ′ are the antiferromagnetic exchange couplings between all nearest neighbour bonds
(J) and next-nearest neighbour bonds in every second square (J ′), as indicated in figure 51 (left
hand panel).
Numerical simulations obtained within the Monte-Carlo [108] and tensor renormalization group
methods [109] on large systems showed that the Ising model on the Shastry-Sutherland lattice ex-
hibits, in the presence of the magnetic field, a magnetization plateau only at 1/3 of the saturated
magnetization. The existence of the magnetization plateau at only 1/3 of the saturated magneti-
zation and its absence at 1/2 indicates that it is necessary to go beyond the classical Ising limit
to reach the correct description of the magnetization process in TmB4 and other rare-earth tetra-
borides. The first attempt of this kind has been made by Meng and Wessel [108] who studied the
spin-1/2 easy-axis Heisenberg model on the Shastry-Sutherland lattice with ferromagnetic trans-
verse spin exchange using quantum Monte-Carlo and degenerate perturbation theory. Besides the
magnetization plateau at 1/3 of the saturated magnetization they found a further plateau at 1/2,
which persists only in the quantum regime.
It should be noted that the behaviour similar to TmB4 has been also observed for other rare-
earth tetraborides. For example, for ErB4 the magnetization plateau has been found at m/ms =
1/2 [110, 111], for TbB4 at m/ms = 1/2, 4/9, 1/3, 2/9 and 7/9 [112] and for HoB4 at m/ms =
1/3, 4/9 and 3/5 [111].
Quite recently we have proposed an alternative model [113] of stabilizing the magnetization
plateaus in the rare-earth tetraborides based on the fact that these materials, in contrast to
SrCu2(BO3)2, are metallic. Thus, for a correct description of ground-state properties of rare-earth
tetraborides one should take into account both spin and electron subsystems as well as the coupling
between them. Supposing that electron and spin subsystems interact only via the spin dependent
42701-49
H. Čenčariková, P. Farkašovský
Ising interaction Jz, the Hamiltonian of the system can be written as
H =
∑
ijσ
tijd
+
iσdjσ + Jz
∑
i
(ni↑ − ni↓)S
z
i − h
∑
i
(ni↑ − ni↓) +HJJ′ . (23)
The model described by (23) is a straightforward extension of the spin-1/2 Falicov-Kimball model
with anisotropic spin-dependent interaction discussed in detail in [63]. The only differences are
that we consider here a direct spin interaction (of the Ising type) between the localized spins and
that the underlying lattice is of the Shastry-Sutherland type.
To examine the magnetization curve corresponding to the model Hamiltonian we have used our
AM. Using this method we have performed exhaustive numerical studies of the model (23) for a
wide range of model parameters h, Jz, t (the hopping integral between the nearest-neighbours),
t′ (the hopping integral between the next-nearest neighbours) and J/J ′ = 1 selected based on
the experimental measurements [107]. To exclude the finite-size effects, the numerical calculations
have been performed for several different Shastry-Sutherland clusters consisting of L = 8 × 8,
10× 10 and 12× 12 sites. The most important result obtained from these calculations is that the
switching on of Jz and t leads to a stabilization of the uniform ground-state spin arrangement
consisting of parallel antiferromagnetic bands separated by ferromagnetic stripes. A complete list
of the ground-state spin arrangements (for 0 < msp/msp
s < 1) that are stable on finite intervals of
magnetic field values is depicted in figure 52. The second important observation is that the width
L=8×8
1/4
1/2
L=10×10
1/5
1/2
L=12×12
1/6
1/3
1/2
Figure 52. The complete list of the ground-state spin configurations that are stable on finite
intervals of magnetic field for L = 8× 8, L = 10× 10 and L = 12× 12. The big dots correspond
to the up-spin orientation and the small dots correspond to the down-spin orientation [113].
w of the antiferromagnetic bands cannot be arbitrary, but fulfills severe restrictions. Indeed, we
have found that with the exception of the case msp/msp
s = 1/2, in all the remaining cases the
permitted width of the antiferromagnetic band is only w or w + 2, where w is an even number.
This fact is very important from the numerical point of view, since it allows us to perform the
numerical calculations on much larger clusters with the extrapolated set of configurations of the
above described type. The resulting magnetization curves obtained on the extrapolated set of
ground-state spin configurations consisting of parallel antiferromagnetic bands of width w (w and
w + 2) separated by ferromagnetic stripes are shown in figure 53 for the selected values of model
parameters that represent the typical behaviour of the model. One can see that the switching
on of the spin-dependent interaction Jz and taking into account the electron hopping t on the
42701-50
Formation of charge and spin ordering in strongly correlated electron systems
1.5 2 2.5 3 3.5 4 4.5 5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
h
m
sp
/m
sp s
L=60×60
L=120×120
0 5 10 15
0
0.2
0.4
0.6
0.8
1
h
m
α /m
α s
L=120×120
spin
electrons
1.5 2 2.5 3 3.5 4 4.5 5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
h
m
sp
/m
sp s
L=60×60
L=120×120
0 5 10 15
0
0.2
0.4
0.6
0.8
1
h
m
α /m
α s
L=120×120
spin
electrons
Figure 53. (Color online) The magnetization curves calculated for J ′/J = 1, Jz = 2, t = 4,
t′ = 0 (left hand panel) and for J ′/J = 1, Jz = 4, t = 4, t′ = 0 (right hand panel). Insets present
the behaviour of magnetization curves for the spin and the electron subsystem calculated on
L = 120× 120 (α = el or sp) [113].
nearest lattice sites of the Shastry-Sutherland lattice leads to a stabilization of new magnetization
plateaus (the next-nearest hopping integrals t′ only renormalize the width of the magnetization
plateaus). In addition to the Ising magnetization plateau at msp/msp
s = 1/3 we have found two
new magnetization plateaus located at msp/msp
s = 1/2 and msp/msp
s = 1/5. The ground-state spin
arrangements corresponding to these magnetization plateaus have the same structure consisting
of parallel antiferromagnetic bands of a width w (where w = 1 for msp/msp
s = 1/2, w = 2 for
msp/msp
s = 1/3 and w = 4 for msp/msp
s = 1/5) separated by ferromagnetic stripes. Thus, our
numerical results show that besides the pure spin mechanism (e.g., the easy-axis Heisenberg model
on the Shastry-Sutherland lattice [108]) of stabilization the magnetization plateaus in rare-earth
tetraborides, there also exists an alternative mechanism based on the coexistence of electron and
spin subsystems that are present in these materials. From this point of view it is interesting to
compare in detail the ground states obtained within these two different approaches. For msp/msp
s =
1/3 our results are identical to the ones obtained within the Ising [107, 109] as well as easy-axis
Heisenberg [108, 114] model on the Shastry-Sutherland lattice. The accordance between our and
the easy-axis Heisenberg solution [108] is surprisingly found for msp/msp
s = 1/2 as well. In this
case both approaches predict the sequence of parallel antiferromagnetic and ferromagnetic stripes.
For msp/msp
s = 1/5 our results postulate a new type of spin ordering.
Finally, it should be noted that more exhaustive studies of the model performed on much larger
lattices have revealed the existence of magnetization plateaus at msp/msp
s = 1/7, 1/9 and 1/11 (in
accordance with experimental measurements in TmB4) but the stability regions of these phases
are much narrower in comparison with the ones of 1/2, 1/3 and 1/5 plateau phases.
5.3. Doping-induced valence changes in rare-earth compoun ds
The substitution of one type of atoms/ions by another type changes electronic relations in a
given material which permits to directly study manifestations of electron correlations on physical
properties of the system. From the perspective of the Falicov-Kimball model, which was origi-
nally introduced to describe valence and metal-insulator transitions in rare-earth compounds, it is
therefore particularly interesting to examine what predictions the model is capable to provide for
the case when we replace the rare-earth ions by other ions. Since previous theoretical works have
shown [80, 115] that the Falicov-Kimball model can provide good qualitative predictions for the
transport, magnetic and spectroscopic properties of some mixed valence compounds (e.g., SmB6,
SmS, etc.), we have focused on studying the effects of doping just for this type of materials. For
SmB6 it has been found, for example, that the substitution of Sm by nonmagnetic divalent ions
(e.g., Sr2+, Yb2+) increases the average samarium valence (the average occupancy of f orbitals
42701-51
H. Čenčariková, P. Farkašovský
0 0.2 0.4 0.6 0.8 1
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
N
f
/L
E
g/
L
a)
n=1
L=120
L=240
0 0.2 0.4 0.6 0.8
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
N
f
/L
E
g/
L
b)
1/6
1/3
1/2
2/3
5/6
L=120
L=240
Figure 54. (a) The ground-state energy Eg of the one dimensional Falicov-Kimball model as
a function of Nf/L calculated for two finite clusters of L = 120 and L=240 sites at n = 1
(n = N/L = (Nd +Nf )/L). (b) The ground-state energy Eg as a function of Nf/L calculated
for two finite clusters of L = 120 and L=240 sites at n = 1/6, 1/3, 1/2, 2/3, 5/6. In both cases
Ef = 0 and U = 0.5 [118].
decreases), while the substitution of Sm by nonmagnetic trivalent ions (e.g., Y3+, La3+) produces
the opposite effect [116, 117].
In our previous papers [118, 119] we have examined theoretically both cases: (i) the substitution
of rare-earth Sm ions by non-magnetic trivalent ions (e.g. Y3+) which introduce conduction elec-
trons into the d-conduction band (one electron per dopant) and (ii) the substitution of rare-earth
ions by non-magnetic divalent ions (e.g. Sr2+), which play a dilution role and reduce the number
of conduction electrons in the d-conduction band (no additional electrons are introduced to the
system). We have started our study with the first case which is slightly simpler. In this case the
0 0.2 0.4 0.6 0.8
0
0.2
0.4
0.6
0.8
1
x
n f
trivalent ions
divalent ions
D=1
U=0.5
L=120
L=240
0 0.2 0.4 0.6 0.8
0
0.2
0.4
0.6
0.8
1
x
n f
trivalent ions
divalent ions
D=1
U=0.5
L=120
L=240
Figure 55. The average occupancy of f orbitals as a function of x for divalent and trivalent
dopants. The one-dimensional case [118].
42701-52
Formation of charge and spin ordering in strongly correlated electron systems
total number of electrons N = Nf +Nd is equal to the total number of lattice sites L (the case of
one electron per rare-earth atom is considered) and does not depend on the number of dopants X .
Consequently, the ground-state energy Eg(Nf ) =
∑L−Nf
k=1 λk is also independent of X and can be
calculated directly using our AM for arbitrary Nf from the interval [0, L−X ]. The results of nu-
merical calculations for the one dimensional lattices of L = 120 and L = 240 sites are summarized
in figure 54 (a) for representative model parameters (Ef = 0, U = 0.5). It is seen that finite-size
effects are negligible and thus these results can be satisfactorily used to represent the behavior of
macroscopic systems. From the Nf dependence of the ground-state energy one can easily deduce
(we remember that Nf = 0, 1, . . . , L−X) that Eg has a minimum at N0
f = L/2 for Nf > L/2 and
at N0
f = L−X for Nf < L/2. Thus, the average occupancy of the f -electron orbitals nf =
N0
f
L−X
can be finally written as
nf =
{ L
2(L−X) for X < L/2
1 for X > L/2.
(24)
The situation is more complicated for the substitution of rare-earth ions by non-magnetic divalent
ions that yield no additional electrons in the d-electron conduction band. In this case the total
number of electrons N = L−X as well as the number of d-electrons Nd = L−X−Nf depend on the
number of dopants X and thus the ground-state energy Eg(Nf , X) =
∑Nd
k=1 λk has to be calculated
many times for the selected values of X . The numerical results obtained in the one-dimensional
case for several selected values of n = N/L are presented in figure 54 (b). The finite-size effects
are small again and so the results can be satisfactorily extrapolated to the thermodynamic limit
(L → ∞). Comparing these results with the previous case (doping by trivalent ions) one can see
a fully different behavior. For small values of N = L−X , the ground-state energy Eg(Nf , X) has
s minimum at N0
f = 0 and then monotonously increases. This type of behavior holds up to the
critical value Nc = L −Xc ∼ L/2. Above this value Eg reaches a minimum at N0
f > 0 and with
increasing N (decreasing X) the position of this minimum shifts to L/2. The resultant behavior
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
n f
U=0.5
D=1
E
f
=0
trivalent ions
divalent ions
a b
c d
a
b
c
d
L=12
L=16
L=20
0 0.2 0.4 0.6 0.8 1
2
2.2
2.4
2.6
2.8
3
x
S
m
v
al
en
ce
Sr
Yb
Y
La
Figure 56. The average occupancy of f orbitals as a function of x for doping by divalent and
trivalent ions calculated for L = 12, 16, 20 and several different temperatures τ . Curve a, τ = 0.1;
curve b, τ = 0.3; curve c, τ = 0.6; curve d, τ = 1 [119]. Inset: The average samarium valence in
the Sm1−xMxB6 systems (M=Y3+, La3+, Sr2+, Yb2+) measured at 300 K (reference [116]).
42701-53
H. Čenčariková, P. Farkašovský
of the average occupancy of the f -electron orbitals nf =
N0
f
L−X on the concentration of dopants
x = X/L is plotted in figure 55. For comparison in figure 55 we have displayed the dependence
of nf on x for doping by trivalent ions. One can see that the substitution of rare-earth ions by
trivalent and divalent ions produces altogether different effects. While in the first case the average
occupancy of f -orbitals increases, in the second case nf decreases. These results are qualitatively
in agreement with experimental measurements [116] of the average samarium valence v = 3 − nf
performed on Sm1−xMxB6 (M=Y3+, La3+, Sr2+, Yb2+) despite the fact that our calculations
have been done at T = 0, while the experiments have been done at room temperatures. However,
it is not expected that the increasing temperature could change dramatically this picture. From
the theoretical investigation of the spinless Falicov-Kimball model at non-zero temperatures it
is known [120] that the effect of temperature is such that a finite temperature only smears the
behavior found for T = 0. Thus, one can expect that for non-zero temperatures the accordance of
theoretical and experimental results could be even better.
0 0.2 0.4 0.6 0.8 1
2
2.2
2.4
2.6
2.8
3
x
v=
3−
n f
a)
E
f
=0.3, U=0.5, D=1, τ=1
0 0.2 0.4 0.6 0.8 1
2
2.2
2.4
2.6
2.8
3
x
v=
3−
n f
b)
E
f
=0.6, U=0.5, D=1, τ=1
0 0.2 0.4 0.6 0.8 1
2
2.2
2.4
2.6
2.8
3
x
v=
3−
n f
c)
E
f
=0.8, U=0.5, D=1, τ=1
0 0.2 0.4 0.6 0.8 1
2
2.2
2.4
2.6
2.8
3
x
v=
3−
n f
d)
E
f
=1, U=0.5, D=1, τ=1
St
Yb
Y
La
St
Yb
Y
La
St
Yb
Y
La
St
Yb
Y
La
Figure 57. Comparison of the one-dimensional theoretical (dashed lines) and experimental
results for the average valence v = 3− nf as a function of x. The theoretical results have been
calculated for U = 0.5, τ = 1 and several different values of the f -level position Ef . The finite-
size effects are negligible, and on the linear scale it is not possible on the drawing to distinguish
the behaviors obtained for L = 12, 16 and 20 over the whole range of x plotted [119]. The
experimental results have been obtained for T=300 K (reference [116]).
To verify this conjecture, we have explicitly calculated the dependence of the f -state occupancy
nf on the dopant concentration x for nonzero-temperatures using small-cluster exact-diagonaliza-
tion technique at finite temperatures (this method is described in detail in reference [120]). The
results of our numerical calculations are summarized in figure 56 for the selected values of temper-
ature τ . It is seen, at a glance, that the spinless Falicov-Kimball model despite its simplicity can
provide a qualitatively correct description of the effects of doping on thermodynamic properties
of rare-earth systems for both divalent and trivalent ions. In addition, optimizing the nf (x) be-
haviour with respect to τ and Ef one can obtain a nice quantitative correspondence between the
theoretical and experimental results (see figure 57). Since no significant effects of the increasing
dimension D (D = 2, 3) on the behaviour of nf (x) has been observed for both zero and non-zero
temperatures [118, 119], our results can be satisfactorily used to describe the behaviour of real
three dimensional materials.
42701-54
Formation of charge and spin ordering in strongly correlated electron systems
6. Stability of charge and spin ordering at finite temperatur es
Examples of charge and spin ordering discussed in previous sections concerned exclusively
the case of T = 0. Real experiments are, however, always performed at finite temperatures, and
therefore getting an answer to the question about the stability of zero temperature solutions at
finite temperatures seems to be the task of fundamental importance. A positive answer to this
question is available at least for the chessboard charge ordering that is the ground state of the
conventional Falicov-Kimball model at half-filling for all Coulomb interactions U > 0. For this case,
there exists an exact proof [13] of the existence of a phase transition from the low-temperature
ordered phase (the chessboard phase) to the high-temperature disordered phase at finite critical
temperature τc (for dimensions D > 2) which strongly depends on the local Coulomb interaction
U . In addition, the numerical simulations within the grand-canonical Monte-Carlo showed that the
phase transitions are of the first order for small and intermediate values of the Coulomb interaction
U and of the second order for strong interactions [121, 122]. In our paper [123] we have extended
the numerical study of the temperature induced phase transitions to the case of phase segregated
and striped phases. Moreover, we have considered a more general situation
H = HFKM +Ht′ , (25)
where HFKM is the conventional spinless Falicov-Kimball model (8) and Ht′ is the term of the
correlated hopping (13). As was discussed in section 3.3.1, all three above mentioned phases (the
chessboard phase, the segregated phase and the axial striped phase) are the ground states of the
model Hamiltonian (25) at the symmetric band point, where both Nf and Nd are fixed to L/2 and,
therefore, all numerical calculations at nonzero temperatures are done exclusively in the canonical
ensemble. In this formalism, the partition function and the internal energy corresponding to the
model Hamiltonian (25) can be written as:
Z =
∑
wf ,wd
e−E/τ , E =
∑
i
εi(w
f )wd
i , (26)
〈E〉 =
∑
wf ,wd
Ee−E/τ , (27)
where τ = kBT and the summation goes over all possible
(
L
Nf
)
distributions wf of f electrons
on L lattice sites and
(
L
Nd
)
distributions wd of d electrons on L single-particle energy levels εi
corresponding to the matrix h(wf ) with elements hij(w
f ) = tij + t′ij(w
f
i + wf
j ) + Uwf
i δij . In the
next step, the summation over all f and d distributions is replaced by the Monte-Carlo summation
with the statistical weight e−E/τ/Z.
To identify the transition temperatures from low-temperature ordered phases to high-tempera-
ture disordered phase and the type of the phase transition, we have numerically calculated the
specific heat C = (〈E2〉−〈E〉2)/(Lτ2), the thermal average of the f -electron occupation ws = 〈wf 〉
and the energy distribution P (E). The numerical calculations are performed exclusively at U = 0.5,
since the ground-state phase diagram exhibits a richer spectrum of solutions in the weak and in-
termediate coupling regions in comparison to the strong coupling limit. To verify the capability of
our method to describe the phase transitions at finite temperatures we have started with the con-
ventional two-dimensional Falicov-Kimball model (t′ = 0) at half-filling. As was mentioned above,
the physical picture of temperature-induced phase transitions within this relatively simple model is
quite understandable at present. For all finite Coulomb interaction U > 0, the ground state of the
model is the chessboard phase that persists up to critical temperature τc(U), where the system un-
dergoes a phase transition to the homogeneous phase. The phase transition is of the first order for
U < 1 and of the second order for U > 1 [121]. Our numerical results obtained within the canonical
Monte-Carlo method for C, ws and P (E) fully confirm this picture (see figure 58) [123]. The specific
heat curves exhibit a sharp low-temperature peak at τc ∼ 0.028 that is obviously connected with
the phase transition from the chessboard phase to the homogeneous phase, as can be seen from the
behaviour of the average f -electron occupation ws for temperatures slightly lower or slightly higher
42701-55
H. Čenčariková, P. Farkašovský
Figure 58. The specific heat (a), the energy distribution (b) and the thermal average of the
f -electron occupation (c)–(d) for the conventional Falicov-Kimball model (t′ = 0) in two dimen-
sions [123].
Figure 59. The specific heat (a), the energy distribution (b) and the thermal average of the
f -electron occupation (c)–(d) for the two-dimensional Falicov-Kimball model with correlated
hopping t′ = −0.3 [123].
than τc. Moreover, the energy distribution function P (E) exhibits an apparent two-peak structure
near the critical point τc (it can be considered as a superposition of two Gaussians), which in accor-
dance with the theory of Challa, Landau and Binder [124] points on the first order phase transition
at τc. Let us now discuss how this picture is changed when the correlated hopping term is added.
42701-56
Formation of charge and spin ordering in strongly correlated electron systems
Figure 60. The specific heat (a), the energy distribution (b) and the thermal average of the
f -electron occupation (c)–(d) for the two-dimensional Falicov-Kimball model with correlated
hopping t′ = 0.3 [123].
Firstly, we have examined the case of small values of |t′| for which the ground state of the model
is still the chessboard phase [68]. The typical examples of C, ws and P (E) from the positive and
negative region of t′ are displayed in figure 59 and in figure 60 for t′ = −0.3 and t′ = 0.3. One can
see that the correlated hopping term (in the limit of small |t′|) does not qualitatively change the
picture of temperature induced phase transitions found for t′ = 0. For both, positive and negative
t′, there is the first order phase transition from the low-temperature ordered phase to the high-
temperature disordered phase, similarly to t′ = 0, and the only difference between these cases is
that the correlated hopping term reduces slightly the critical temperature τc of the phase transition.
Therefore, in the next step we have turned our attention to the physically much less explored
type of configurations, namely, the axial striped configurations that are ground states of the Falicov-
Kimball model for the intermediate values of t′ (|t′| ∼ 0.5). Note, that for the axial striped phase
even the fundamental question concerning the temperature stability of this phase has remained
unanswered so far. This is due to the fact that it is very difficult to find this phase in a pure
form. For example, in the conventional Falicov-Kimball model (t′ = 0), the axial striped phases
are stable for a relatively wide range of model parameters [54], but solely in mixtures with other
phases (e.g., the empty configuration). In addition, strong finite-size effects have been observed on
the stability of these mixtures and therefore it is practically impossible to make any conclusions
concerning their stability at finite temperatures from the numerical calculations on finite clusters.
However, in the Falicov-Kimball model with correlated hopping, the axial striped phase exists in
a pure form for a wide range of model parameters t′ and U , the finite-size effects on the stability
of this phase at τ = 0 are negligible, and so the corresponding numerical study of the temperature
stability of the axial striped phase can be performed straightforwardly.
In figure 61 and figure 62 we present our canonical Monte-Carlo results for C, ws and P (E)
obtained for two different values of t′ (t′ = 0.5 and t′ = 0.55) from the region where the ground-
state of the model is just the axial striped phase. Again, the specific heat curves exhibit a sharp
low-temperature peak, the existence of which indicates a phase transition from the axial striped
phase to the homogeneous phase. This was independently verified by calculating the average f -
electron occupation ws and the energy distribution P (E) near the transition point τc, which clearly
42701-57
H. Čenčariková, P. Farkašovský
Figure 61. The specific heat (a), the energy distribution (b) and the thermal average of the f -
electron occupation (c)–(d) for the two-dimensional Falicov-Kimball model with correlated hop-
ping t′ = 0.5. The inset shows the critical temperature τc as a function of the cluster size L [123].
Figure 62. The specific heat (a), the energy distribution (b) and the thermal average of the
f -electron occupation (c)–(d) for the two-dimensional Falicov-Kimball model with correlated
hopping t′ = 0.55. The inset shows the critical temperature τc as a function of the cluster
size L [123].
demonstrates the presence of the first order phase transition at τc. Since the critical temperature
τc of the phase transition for both values of t′ shifts to smaller values with increasing L, we have
performed a detailed finite-size scaling analysis of the τc(L) dependence to exclude the possibility
42701-58
Formation of charge and spin ordering in strongly correlated electron systems
of τc vanishing in the thermodynamic limit L→ ∞. The resultant τc(L) dependencies are plotted
as insets in figure 61 and in figure 62. It is clearly seen that the critical temperatures τc for both
t′ = 0.5 and t′ = 0.55 also persist in the thermodynamic limit, meaning that the axial striped
phase remains stable at finite temperatures as well. Moreover, our numerical results show that
critical temperatures for an axial striped phase are considerably higher in comparison with critical
temperatures for a chessboard phase. We have observed the same behaviour for negative values of
t′ (t′ = −0.7). However, the critical temperature in this case was only slightly larger than the one
corresponding to t′ = 0.
With increasing t′, the half-filled Falicov-Kimball model with correlated hopping exhibits (at
τ = 0) a phase transition from the axial striped phase to the segregated phase [68] that takes place
at t′ ∼ 0.6. Since both the chessboard phase and the axial striped phase are insulating while the
segregated phase is metallic [68], one can expect a fully different thermodynamic behaviour of the
model for the last case. To verify this conjecture we have performed exhaustive numerical studies
of the temperature dependence of C, ws and P (E) for t′ = 1. This study is also important from
the point of view that the thermodynamics of the metallic phase has been so far examined only
in a few cases [125, 126], while for the insulating phase (usually the chessboard phase) there is a
number of analytical and numerical results [13, 84, 127].
Figure 63. The specific heat (a), the energy distribution (b) and the thermal average of the
f -electron occupation (c)–(f) for the two-dimensional Falicov-Kimball model with correlated
hopping t′ = 1. The inset shows the specific heat C in the low-temperature region for L =
16× 16 [123].
42701-59
H. Čenčariková, P. Farkašovský
↑
↑
↑
↑
↑
↑
↑
↑
↑↓
↓
↓
↓
↓
↓
↓
↓
↓
(a)
↑
↑
↑
↑
↑
↑
↑
↑
↑
↓
↓
↓
↓
↓
↓
↓
↓
↓
(b)
Figure 64. Ground states of the spin-1/2 Falicov-Kimball model extended by spin-dependent
interaction J and local Coulomb interaction Uff : (a) for U = 2, J = 1, Uff = 4, Ef = 0 and
N = 2L, (b) for U = 4, Jz = 0.8, Uff = 8, Ef = −2 and N = 3L/2 [128].
The results of our numerical calculations obtained for the specific heat C are shown in figure 63.
To reveal the finite-size effects, the calculations for C have been done on several different clusters
of L=6× 6, 8× 8, 10× 10, 12× 12 and 16× 16 sites. We have found that the specific heat curves,
in the low-temperature region, strongly depends on the cluster sizes, and, therefore, a very careful
analysis has to be performed to find the correct behaviour of the model in the thermodynamic
limit L → ∞. On small finite clusters (L = 6 × 6 and L = 8 × 8), the specific heat exhibits only
one-peak structure in the low-temperature region (τ ∼ 0.15). With the increasing cluster size L,
an additional peak is stabilized at slightly higher temperatures (τ ∼ 0.23), while the first peak is
gradually suppressed and probably fully disappears in the thermodynamic limit. The behaviour of
the average f -electron occupation shows (see figure 63) that the second peak in the specific heat
corresponds to the phase transition from the low-temperature ordered (segregated) phase to the
high-temperature disordered phase.
The nature of this phase transition is, however, different in comparison to the previous cases.
While the energy distribution function P (E) is double peaked for the chessboard and the axial
striped phase near the transition temperature τc (the first order phase transition), P (E) exhibits
a single-peak structure for the segregated phase, which points to the second order phase transition
at τc. Comparing the thermodynamic behaviour of the model in the chessboard, axial striped and
segregated regions one can find two other important differences, namely, (i) the critical temperature
of the second order phase transition is approximately ten times higher than critical temperatures
of the first order phase transitions, and (ii) the specific heat (in the low-temperature region)
exponentially decreases for the chessboard and axial striped phase, while in the segregated phase
the specific heat C(τ) seems to show a linear behaviour indicating the Fermi-liquid behaviour for
τ < 0.08 (see the inset in figure 63 (a)). The observation of the linear contribution to the specific
heat in the low-temperature region (τ < 0.08) is consistent with the behaviour of the average
f -electron occupation in this region (see figure 63 (c)). One can see that despite the increasing
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
c v
/k
B
τ
(a) U=2, J=1, Uff=4, Ef=0, N=2L
L=10
L=16
L=20
L=32
L=36
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
c v
/k
B
τ
(b) U=4, J=0.8, Uff=8, Ef=-2, N=3L/2
0
0.2
0.4
0.6
0.8
1
0 0.1 0.2 0.3 0.4 0.5
C
τ
,
S
τ
τ
Cτ
Sτ
0
0.2
0.4
0.6
0.8
1
0 0.1 0.2 0.3 0.4 0.5
C
τ
,
S
τ
τ
Cτ
Sτ
Figure 65. Specific heat as a function of τ = kBT/t for the phase (a) and phase (b) from
figure 64. Insets present the temperature dependences of Cτ and Sτ defined in the text [128].
42701-60
Formation of charge and spin ordering in strongly correlated electron systems
temperature (from 0 to 0.08) the f -electrons preferably occupy only one half of the lattice leaving
another part empty. Due to the on-site Coulomb interaction between the f and d electrons, the
itinerant d electrons preferably occupy the empty part of the lattice, where they can move as free
particles yielding the linear contribution to the specific heat.
We have performed the same numerical study for the spin-1/2 Falicov-Kimball model extended
by the spin-dependent Coulomb interaction J between the localized f and itinerant d electrons
as well as the on-site Coulomb interaction (Uff ) between the localized f -electrons [128]. The
coexistence of charge and spin degrees of freedom for such type of a system evokes the question
concerning a possible sequence of two phase transitions corresponding to the breaking of different
types of charge/spin ordering. We have tested the possibility of such a scenario independently
for two configuration types displayed in figure 64, which are the ground states of the generalized
Falicov-Kimball model for U = 2, J = 1, Uff = 4, Ef = 0 and N = 2L or U = 4, J = 0.8, Uff = 8,
Ef = −2 and N = 3L/2. To identify the type of phase transition, we have used the the specific
heat and the structure factor of the charge and spin ordering defined by [128]:
Cτ =
1
L2
L
∑
j,k
exp(i ~Q(~Rj − ~Rk))〈(wj↑ + wj↓)(wk↑ + wk↓)〉 (28)
and
Sτ =
1
L2
L
∑
j,k
exp(i ~Q(~Rj − ~Rk))〈(wj↑ − wj↓)(wk↑ − wk↓)〉 . (29)
The temperature behaviours of these quantities are displayed in figure 65. It is seen that in the
first case both types of ordering are simultaneously destroyed at the same critical temperature
τc ∼ 0.05, while in the second case the spin ordering disappears already at very low temperatures
(τc < 0.01), and the charge ordering persists up to τc ∼ 0.255.
7. Conclusion
In this review we have presented the results of our theoretical study of charge and spin order-
ing in strongly correlated electron systems obtained within various generalizations of the Falicov-
Kimball model. The primary goal of this study was to identify crucial interactions that lead to the
stabilization of various types of charge ordering in these systems such as the axial striped order-
ing, diagonal striped ordering, phase separated ordering, phase segregated ordering, etc. Among
the major interactions that come into account, we have examined the effect of local Coulomb
interaction between localized and itinerant electrons, long-range and correlated hopping of itin-
erant electrons, long-range Coulomb interaction between localized and itinerant electrons, local
Coulomb interaction between itinerant electrons, local Coulomb interaction between localized elec-
trons, spin-dependent interaction between localized and itinerant electrons, both for zero and
nonzero temperatures, as well as for doped and undoped systems.
We have started our study of charge ordering in the strongly correlated systems with the spinless
Falicov-Kimball model. First, we have focused on a description of the ground-state properties of the
model at T = 0. Since the previous theoretical studies have shown that the ground-state properties
of the Falicov-Kimball model are very sensitive to the type of approximation used, to study the
charge ordering, valence and metal-insulator transitions in the Falicov-Kimball model we have used
the EDM on finite clusters with subsequent extrapolation of the results to the thermodynamic limit
(infinitely large system). Our results showed that the spinless Falicov-Kimball model is capable of
describing homogeneous as well as phase-separated distributions of f electrons and thereby discon-
tinuous valence and metal-insulator transitions induced by pressure. The major driving interaction
of these transitions is the Coulomb interaction between itinerant and localized electrons.
5It should be noted that similar calculations of finite temperature properties of the model have been recently
performed by Wrzodak and Lemanski [129] on the square 4× 4 cluster at half-filling.
42701-61
H. Čenčariková, P. Farkašovský
In order to elaborate the most realistic description of charge ordering and valence and metal-
insulator transitions in real materials (rare-earth and transition-metal compounds), we have gen-
eralized the original type of electron hopping (only to the nearest neighbours) to a much more
realistic type of hopping (the long-range hopping with power decreasing hopping amplitudes) and
in addition we have taken into account the term of correlated hopping, which modifies the hopping
amplitudes of itinerant electrons from one site to another according to the occupancy of these sites
by localized electrons. We have found that the term of correlated hopping has a significant effect
on the dynamics of valence and metal-insulator transitions in the spinless Falicov-Kimball model
and should, therefore, be taken into account for the correct description of these phenomena. More-
over, we have shown that the correlated hopping term leads to the stabilization of the axial charge
stripes, suggesting an alternate and a much simpler mechanism (than the one which was previously
considered within the Hubbard and t− J model) of forming an inhomogeneous charge ordering in
strongly correlated systems. Regarding the effect of long-range hopping term on metal-insulator
transitions, we have found that it is not very significant, at least for q 6 0.3, which physically turns
out to be the most interesting case.
The next step towards a comprehensive description of cooperative phenomena in the Falicov-
Kimball model was to study the model for dimensions D = 2 and D = 3. To fulfill this task, it
was necessary to develop a new numerical method, since the EDM is capable of analysing only
clusters up to L ∼ 40 sites, which does not offer any possibility to extrapolate the results to the
thermodynamic limit (L = ∞). This goal has been fully fulfilled. Our numerical method (based
on a modification of an exact diagonalization algorithm) is very accurate, even for cluster sizes
of several hundred sites (this has been tested in the limiting cases where the exact solutions are
known). Moreover, it is very flexible, and so it permits to study various generalizations of the model.
The study of the Falicov-Kimball model by this method showed that the basic picture of charge
ordering and valence and metal-insulator transitions remains unchanged for higher dimensions.
The only important difference is that the stability area of metallic phases shifts to higher values
of the Coulomb interaction U . This fact is very important from the practical point of view, since
real materials are just in this limit.
The fact that we were able to analyse the model for cluster sizes of several hundred lattice sites
allowed us to study the formation of charge ordering in two- and three-dimensional systems for
arbitrary values of U , which was very valuable, since the previous results were limited either to the
area of U ≪ 1 or U ≫ 1, or were limited to the type of the investigated configurations (e.g., the
periodic configurations with the small periods and their mixtures). We have found that the spinless
Falicov-Kimball model is capable (in D = 2 and D = 3) of describing a very wide range of charge
orderings involving periodic charge ordering, phase-segregated and phase-separated ordering as
well as axial and diagonal striped ordering. In the the three-dimensional case, our results represent
the first and so far the only attempt to systematically describe the formation of inhomogeneous
charge ordering in the spinless Falicov-Kimball model, which was to a large extent possible due to
a new elaborated numerical method.
Moreover, the inclusion of spins and the spin interactions of the Hubbard type between itiner-
ant electrons and of the Ising type between itinerant and localized electrons clearly demonstrated
the enormous potential of the model in describing different types of charge and spin superstruc-
tures, including the segregated phase and phase-separated orderings, as well as axial and diagonal
striped (band) orderings with ferro-, ferri-, or antiferromagnetic ground state. Furthermore, the
spin model in the presence of interband Coulomb interaction between localized and itinerant elec-
trons maintains advantages of a spinless model, namely, the capability of describing valence and
metal-insulator transitions induced by pressure, temperature and alloying.
Acknowledgements
This work was supported by Slovak Grant Agency VEGA under Grant No. 2/0175/10 and by
the ERDF EU grant, under the contract No. ITMS26220120005.
42701-62
Formation of charge and spin ordering in strongly correlated electron systems
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Формування зарядового i спiнового впорядкування в сильно
скорельованих електронних системах
Г. Ченчарiкова, П. Фаркашовскi
Iнститут експериментальної фiзики, Словацька академiя наук, Кошицi, Словаччина
У цьому оглядi ми представляємо результати наших теоретичних дослiджень зарядового i спiно-
вого впорядкування в сильно скорельованих електронних системах, що отриманi в рамках рiзних
узагальнень моделi Фалiкова-Кiмбала. Основною метою цього дослiдження було iдентифiкувати ви-
рiшальнi взаємодiї, що приводять до стабiлiзацiї рiзних типiв зарядового впорядкування в цих систе-
мах, таких як осьове стрiчкове впорядкування, дiагональне стрiчкове впорядкування, фазове роз-
шарування, фазова сегрегацiя i т.п. З помiж основних взаємодiй, якi враховують, нами розглядався
вплив локальних кулонiвських взаємодiй мiж локалiзованими i колективними електронами, далек-
осяжного i скорельованого переносу колективних електронiв, далекосяжної кулонiвської взаємодiї
мiж локалiзованими i колективними електронами, локальної кулонiвської взаємодiї мiж колективни-
ми електронами, локальної кулонiвської взаємодiї мiж локалiзованими електронами, спiнозалежної
взаємодiї мiж локалiзованими i колективними електронами, як при нульовiй так i при ненульовiй
температурах, а також для легованих i нелегованих систем. На завершення, обговорюється засто-
совнiсть отриманих розв’язкiв для опису сполук рiдкiсноземельних i перехiдних елементiв.
Ключовi слова: зарядове i спiнове впорядкування, переходи метал-дiелектрик, переходи зi
змiною валентностi, модель Фалiкова-Кiмбала, сильно скорельованi системи
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http://dx.doi.org/10.1103/PhysRevB.82.195118
Introduction
Methods
Exact diagonalization technique
Approximate method based on the reduction of the total energy
Charge ordering in the spinless Falicov-Kimball model
The effect of local Coulomb interaction
The effect of long-range hopping
The effect of nonlocal Coulomb interactions
The effect of correlated hopping
The effect of nearest-neighbour Coulomb interaction
The effect of lattice geometry
Charge and spin ordering in the spin-1/2 Falicov-Kimball model
Spin-1/2 Falicov-Kimball model without the Ising interaction
Spin-1/2 Falicov-Kimball model with the Ising interaction
Applications of the Falicov-Kimball model to a description of real materials
Ground-state properties of NaxCoO2
Magnetization processes in rare-earth tetraborides
Doping-induced valence changes in rare-earth compounds
Stability of charge and spin ordering at finite temperatures
Conclusion
|