Page 1

The effects of the next-nearest-neighbour density-density interaction

in the atomic limit of the extended Hubbard model

Konrad Kapcia∗and Stanisław Robaszkiewicz

Electron States of Solids Division, Faculty of Physics,

Adam Mickiewicz University, Umultowska 85, PL-61-614 Poznań, Poland

(Dated: January 22, 2011)

We have studied the extended Hubbard model in the atomic limit. The Hamiltonian analyzed

consists of the effective on-site interaction U and the intersite density-density interactions Wij (both:

nearest-neighbour and next-nearest-neighbour). The model can be considered as a simple effective

model of charge ordered insulators. The phase diagrams and thermodynamic properties of this

system have been determined within the variational approach, which treats the on-site interaction

term exactly and the intersite interactions within the mean-field approximation. Our investigation

of the general case taking into account for the first time the effects of longer-ranged density-density

interaction (repulsive and attractive) as well as possible phase separations shows that, depending

on the values of the interaction parameters and the electron concentration, the system can ex-

hibit not only several homogeneous charge ordered (CO) phases, but also various phase separated

states (CO-CO and CO-nonordered). One finds that the model considered exhibits very interesting

multicritical behaviours and features, including among others bicritical, tricritical, critical-end and

isolated critical points.

PACS numbers: 71.10.Fd Lattice fermion models (Hubbard model, etc.); 71.45.Lr Charge-density-wave

systems; 71.10.-w Theories and models of many-electron systems

Keywords: charge orderings, phase separation, phase diagrams, phase transitions, extended Hubbard model

I.INTRODUCTION

Electron charge orderings phenomena in strongly cor-

related electron systems are currently under intense in-

vestigations.Charge orderings (COs) are relevant to

a broad range of important materials, including mangan-

ites, cuprates, magnetite, several nickel, vanadium and

cobalt oxides, heavy fermion systems (e. g. Yb4As3) and

numerous organic compounds [1–18].

Various types of COs have been also observed in a great

number of experimental systems with local electron

pairing (for review see [14–16] and references therein),

in particular in the compounds that contain cations

in two valence states differing by 2e (on-site pairing)

– valence skipping, “negative-U” centers [17], and in

the transition metal oxides showing intersite bipolarons

e. g.Ti4−xVxO7, WO3−x, with double charge fluc-

tuations on the molecular (rather than atomic) units

[(Ti4+-Ti4+),(Ti3+-Ti3+)], etc.

COs are often found in broad ranges of electron dop-

ing (e. g.doped manganites [8–13], nickelates [1, 2],

Ba1−xKxBiO3[16–18]). In several of these systems many

experiments showed phase separations involving charge

orderings [10–13].The CO transitions at T > 0 take

place either as first order or continuous.

some of CO systems exhibit also a tricritical behaviour

(e. g. (DI-DCNQI)2Ag [6]).

An important, conceptually simple model for study-

ing correlations and for description of charge orderings

(and various other types of electron orderings) in nar-

Moreover,

∗corresponding author; e-mail: kakonrad@amu.edu.pl

row energy band systems is the extended Hubbard model

taking into account both the on-site (U) and the in-

tersite (Wij) density-density interactions (the t-U-Wij

model [16, 19–24]).

In this paper we focus on the atomic limit (tij = 0

limit) of the t-U-Wijmodel. The Hamiltonian considered

has the following form:

ˆH = U

?

i

ˆ ni↑ˆ ni↓+W1

2

?

?

?i,j?1

ˆ niˆ nj+

+W2

2

?

?i,j?2

ˆ niˆ nj− µ

i

ˆ ni,

(1)

where ˆ c+

with spin σ =↑,↓ at the site i, ˆ ni=?

sum over nearest-neighbour (m = 1) and next-nearest-

neighbour (m = 2) sites i and j independently. zm de-

notes the number of m-th neighbours. U is the on-site

density interaction, W1and W2are the intersite density-

density interactions between nearest neighbours (nn) and

next-nearest neighbours (nnn), respectively. The chemi-

cal potential µ depends on the concentration of electrons:

iσdenotes the creation operator of an electron

σˆ niσ, ˆ niσ= ˆ c+

?i,j?mindicates the

iσˆ ciσ.

µ is the chemical potential.

?

n =

1

N

?

i

?ˆ ni?,

(2)

with 0 ≤ n ≤ 2 and N – the total number of lattice sites.

?ˆ ni? denotes the average value of the ˆ nioperator.

The model (1) can be considered as a simple model of

charge ordered insulators. The interactions U and Wij

can be treated as the effective ones and assumed to in-

clude all the possible contributions and renormalizations

arXiv:1101.4287v3 [cond-mat.str-el] 22 Feb 2011

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2

like those coming from the strong electron-phonon cou-

pling or from the coupling between electrons and other

electronic subsystems in solid or chemical complexes. In

such a general case arbitrary values and signs of U and

Wijare important to consider.

Notice that the model (1) can be viewed as the classical

gas with four possible states at each site and it is equiva-

lent to a special kind of the Blume-Capel model i. e. the

S = 1 Ising model with single-ion anisotropy and eigen-

value ‘zero’ doubly degenerate in an effective magnetic

field given self-consistently by a value of fixed magneti-

zation [27, 33, 37].

In the analysis we have adopted a variational approach

(VA) which treats the on-site interaction U exactly and

the intersite interactions Wij within the mean-field ap-

proximation (MFA).

Within the VA the phase diagrams of (1) have been

investigated till now for the case W2= 0 [25–27] and the

stability conditions of states with phase separation have

not been discussed. Some preliminary results for the case

W2?= 0 have been presented by us in [28].

We perform extensive study of the phase diagrams and

thermodynamic properties of the model (1) within VA for

arbitrary electron concentration n, arbitrary strength of

the on-site interaction U and the nn repulsion W1> 0,

taking into account the effects of interactions between

nnn W2 (repulsive and attractive). Our comprehensive

investigation of the general case finds that, depending

on the values of the interaction parameters and the elec-

tron concentration, the system can exhibit the charge

ordered and nonordered homogeneous phases as well as

(for attractive W2) at least two types of phase separation

involving charge orderings. Transitions between different

states and phases can be continuous and discontinuous,

what implies existence of different critical points on the

phase diagrams. We present detailed results concerning

the evolution of phase diagrams as a function of the in-

teraction parameters and the electron concentration.

Our studies of the Hamiltonian (1) are exact for at-

tractive W2 in the limit of infinite dimensions.

are important as a test and a starting point for a per-

turbation expansion in powers of the hopping tijand as

a benchmark for various approximate approaches (like

dynamical mean field approximation, which is exact the-

ory for fermion system in the limit of infinite dimensions

for tij?= 0 [20]) analyzing the corresponding finite band-

width models. They can also be useful in a qualitative

analysis of experimental data for real narrow bandwidth

materials in which charge orderings phenomena are ob-

served.

In the limit W2= 0 the model (1) has been analyzed in

detail (for review see [27, 30] and references therein). In

particular, the exact solutions were obtained for the one-

dimensional (d = 1) case (T ≥ 0) employing the method

based on the equations of motion and Green’s function

formalism [30] or the transfer-matrix method [31, 32].

In [33] the phase diagram of (1) as a function of µ for

W2= 0 has been derived at T = 0 and confirmed in [34].

They

These studies were based on the Pirogov and Sinai meth-

ods [35]. In two dimensional case d = 2 (W2?= 0) exact

ground state diagrams as a function of µ have also been

obtained [36] using the metod of constructing ground

state phase diagrams by the reflection positivity prop-

erty with respect to reflection in lattice planes.

A number of numerical simulation has also been done

for W2= 0. In particular, the critical behaviour near the

tricritical point have been analyzed using Monte Carlo

(MC) simulation and MFA [22]. A study of the model in

finite temperatures using MC simulation has also been

done for a square lattice (d = 2) [37, 38]. In particu-

lar, the possibility of phase separation and formation of

stripes in finite systems was evidenced there.

In the following we will restrict ourselves to the case of

repulsive W1> 0, which favours charge orderings, and

z1W1> z2W2.For the sake of simplicity we consider

mainly two-sublattice orderings on the alternate lattices,

i. e. the lattices consisting of two interpenetrating sub-

lattices (every nearest neighbour of every site in one sub-

lattice is a site in the other sublattice), such as for ex-

ample simple cubic (SC) or body-center cubic lattices.

Some preliminary studies of ground state beyond the

two-sublattice assumption at half-filling (n = 1) are also

performed, which show that for repulsive W2there is pos-

sibility of occurrence of the multi-sublattice orderings.

The paper is organized as follows. In section II we

describe the metod used in this work. There are also

derived explicit formulas for the free energies of homoge-

neous phases and states with phase separation as well as

equations determining the charge-order parameters and

the chemical potential in homogeneous phases. In sec-

tion III we analyze the properties of the system at zero

temperature and present ground state diagrams. Sec-

tion IV is devoted to the study of the finite temperature

phase diagrams for W2≥ 0 and W2< 0. Some partic-

ular temperature dependencies of the charge-order pa-

rameter are discussed in section V. Section VI contains

ground state results for half-filling beyond two-sublattice

assumption. Finally, section VII reports the most impor-

tant conclusions and supplementary discussion including

the validity of the approximation used and the compar-

ison with real materials. The appendix presents explicit

expressions of site-dependent self-consistent VA equa-

tions.

II.THE METHOD

The free energy of the system and the self-consistent

equations for the average number of electrons on sites

are derived within site-dependent VA in the Appendix.

Restricting analysis to the two-sublattice orderings the

explicit formula for the free energy per site obtained in

the VA has the following form

f(n) =F

N= µn −1

2W0n2−1

2WQn2

Q−

1

2βln[ZAZB],

(3)

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3

where

Zα= 1 + 2exp[β(µ − ψα)] + exp[β(µ − 2ψα− U)],

ψA= nW0+ nQWQ,

W0= z1W1+ z2W2,WQ= −z1W1+ z2W2,

and β = 1/(kBT). The charge-order parameter is defined

as nQ= (1/2)(nA− nB), where nα=

average electron density in a sublattice α = A,B.

The condition for the electron concentration (2) and

a minimization of f(n,T) with respect to the charge-

order parameter lead to a set of self-consistent equations

(for homogeneous phases):

ψB= nW0− nQWQ,

2

N

?

i∈α?ˆ ni? is the

n = (1/2)(nA+ nB),

nQ= (1/2)(nA− nB),

(4)

(5)

where

nα=

2

Zα

double

?

{exp[β(µ − ψα)] + exp[β(2µ − 2ψα− U)]}.

occupancyper

i?ˆ ni↑ˆ ni↓? has the following form:

D = (1/2)(DA+ DB),

The

D =

sitedefinedas

1

N

(6)

where Dα= exp[β(2µ − 2ψα− U)]/Zα.

The equations (4)–(5) are solved numerically for T ≥ 0

and we obtain nQ and µ when n is fixed. The charge-

ordered (CO) phase is characterized by non-zero value of

nQ, whereas nQ= 0 in the non-ordered (NO) phase.

Let us notice that the free energy (3) is an even func-

tion of nQ so we can restrict ourselves to solutions of

the set (4)–(5) in the range 0 ≤ nQ≤ 1. It is the result

of the equivalence of two sublattices. Moreover, (5) is

only the necessary condition for an extremum of (3) thus

the solutions of (4)–(5) can correspond to a minimum or

a maximum (or a point of inflection) of (3). In addition

the number of minimums can be larger than one, so it is

very important to find the solution which corresponds to

the global minimum of (3).

Phase separation (PS) is a state in which two domains

with different electron concentration: n+ and n− exist

in the system (coexistence of two homogeneous phases).

The free energies of the PS states are calculated from the

expression:

fPS(n+,n−) = mf+(n+) + (1 − m)f−(n−),

where f±(n±) are values of a free energy at n± corre-

sponding to the lowest energy homogeneous solution and

m = (n − n−)/(n+− n−) is a fraction of the system with

a charge density n+(n−< n < n+). The minimization

of (7) with respect to n+and n−yields the equality be-

tween the chemical potentials in both domains:

(7)

µ+(n+) = µ−(n−)

(8)

(chemical equilibrium) and the following equation (so-

called Maxwell’s construction):

µ+(n+) =f+(n+) − f−(n−)

n+− n−

.

(9)

In

µ = µ+(n+) = µ−(n−)

tron concentration, i.e. ∂µ/∂n = 0.

In the model considered only the following types of PS

states can occur: (i) PS1 is a coexistence of CO and NO

phases and (ii) PS2 is a coexistence of two CO phases

with different concentrations.

In the paper we have used the following conven-

tion.A second (first) order transition is a transition

between homogeneous phases with a (dis-)continuous

change of the order parameter at the transition temper-

ature. A transition between homogeneous phase and PS

state is symbolically named as a “third order” transi-

tion. During this transition a size of one domain in the

PS state decreases continuously to zero at the transi-

tion temperature. We have also distinguished a second

(first) order transition between two PS states, at which

a (dis-)continuous change of the order parameter in one

of domains takes place. In the both cases the order pa-

rameter in the other domain changes continuously.

Second order transitions are denoted by solid lines on

phase diagrams, whereas dotted and dashed curves de-

note first order and “third order” transitions, respec-

tively.

The phase diagrams obtained are symmetric with re-

spect to half-filling because of the particle-hole symmetry

of the hamiltonian (1) [16, 30, 39], so the diagrams will

be presented only in the range 0 ≤ n ≤ 1.

thePSstatesthe

independent

chemicalpotential

theisofelec-

III.THE GROUND STATE

A.

W2 > 0

In the case of nnn repulsion, i. e.

(k = z2W2/z1W1), the system at T = 0 can exhibit two

types of CO: “high CO” (HCO), involving the on-site

pairing of electrons and “low CO” (LCO), which is the

ordering without on-site pairs.

For this case the ground state (GS) diagram de-

rived within VA as a function of n and U/(−WQ) is

shown in figure 1a.At T = 0 HCO can be stable

phase only if U/(−WQ) < 1 and LCO if U > 0.

0 < U/(−WQ) < 1 both types of order can be realized

depending on n. In the GS, one obtains the following

results for the ordering parameter nQ, the chemical po-

tential µ and the double occupancy per site D: (i) LCO

phases: for LCOA (only sublattice A is filled by elec-

trons without double occupancy nA= 2n, sublattice B is

empty): nQ= n, µ = 2z2W2n, D = 0 and for LCOB(ev-

ery site in sublattice A is singly occupied nA= 1, whereas

nB= 2n − 1): nQ= 1 − n, µ = 2z2W2n − WQ, D = 0,

(ii) HCO phases (electrons are located only in sublattice

A, nA= 2n, nB= 0): for HCOA (every site in sublat-

tice A is doubly occupied): nQ= n, µ = U/2 + 2z2W2n,

D = n/2 and for HCOB(at every site in sublattice A at

least one electron is located): nQ= n, µ = U + 2z2W2n,

D = n − 1/2. For n = 1 the transition at U/(−WQ) = 1

0 < k < 1

For

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0.00.20.40.60.81.0

0

1

2

(a)

NO

HCO

LCO

LCOB

LCOA

HCOA

U/(-WQ)

n

HCOB

0.00.20.40.60.81.0

0

1

2

(b)

PS1B

LCO

PS1C

PS2*

PS1A

U/|z2W2| <1

U/(-WQ)

n

U/|z2W2| >1

NO

HCO

FIG. 1. Ground state phase diagrams U/(−WQ) vs. n: (a) for

1 > k > 0 and (b) for k < 0 (k = −1). Details in text and

Table I.

TABLE I. PS states occurring in the ground state.

State

PS1A

PS1B

PS1C

PS2∗

Domain

HCO

LCOa

NO

HCO

n+

1

0.5

1

1

Domain

NO

NO

LCOa

LCOa

n−

0

0

0.5

0.5

aThe value of µ in the homogeneous phase is irrelevant here.

is from the HCO (nQ= 1, D = 0.5) to the NO (Mott

state, nQ= 0, D = 0). In both phases at n = 1 the chem-

ical potential is µ = U/2 + W0= U/2 + 2z2W2− WQ.

At quarter-filling (n = 0.5) in LCO (nQ= 0.5, D = 0)

the value of the chemical potential is µ = U/2 + z2W2for

0 < U/(−WQ) < 1 and µ = (1/2)W0for U/(−WQ) > 1.

For fixed U/(−WQ) the chemical potential µ changes

discontinuously (except LCO–HCOBand LCO–LCOB),

whereas for fixed n it is continuous at the phase bound-

aries.The double occupancy D changes continuously

at transitions with fixed U/(−WQ), while transitions

with fixed n are associated with discontinuous change of

D (except LCO–LCO for n = 1/2 and U/(−WQ) = 1).

Apart from HCO–NO (for n = 1) and HCOB–LCOB

transitions, the charge-order parameter nQchanges con-

tinuously in all transitions.

B.

W2 < 0

In the GS diagram as a function of n for W2< 0 (fig-

ure 1b) one finds also simple linear boundaries between

various states but in this case the free energies of PS

states are lower than those of homogeneous phases, apart

from particular values of concentration: (i) for n = 0.5

and U/|z2W2| > 1 the LCO with nQ= 0.5 is stable,

(ii) for n = 1 the HCO with nQ= 1 if U/(−WQ) < 1 and

the NO with nQ= 0 if U/(−WQ) > 1 are stable. One can

also check that the first derivative of chemical potential

∂µ/∂n in homogeneous phases is negative (apart from the

ranges mentioned above) what implies that these phases

are not stable. Definitions of all PS states occurring in

the GS are collected in Table I. For U/|z2W2| < 1 only

the PS1Astate (with D = n/2, µ = U/2 + z2W2) occurs.

When 0.5 < n < 1, U/|z2W2| > 1 and U/(−WQ) < 1 the

PS2∗state with D = n − 1/2 and µ = U + (3/2)z2W2is

stable. For U/|z2W2| > 1 and n < 0.5 the PS1B with

D = 0 and µ = z2W2/2 has the lowest energy, whereas

for U/|WQ| > 1 and 0.5 < n < 1 the PS1C with D = 0

and µ = (3/2)z2W2− WQis stable.

All transitions in GS for W2< 0 are associated with

discontinuous change of the chemical potential µ. The

double occupancy D changes continuously at transitions

with fixed U/(−WQ), while transitions with fixed n are

connected with discontinuous change of D (except LCO–

LCO for n = 1/2 and U/(−WQ) = 1).

Notice that for W1> 0 and W2< 0 the condition

U/(−WQ) = 1 implies that U/|z2W2| > 1, so the line

U/(−WQ) = 1 is above the line U/|z2W2| = 1 and the

GS phase diagram for any W2< 0 has always the form

shown in figure 1b.

One should stress that for W2< 0 the phase stability

condition is not fulfilled (i. e. ∂µ/∂n < 0) in homoge-

neous phases except n = 0.5 for U/|z2W2| > 1 and n = 1.

It means that the homogeneous phases are not stable.

For W2= 0 the free energies of homogeneous phases

and PS states are degenerated at T = 0 and for such

a case ∂µ/∂n = 0 in homogeneous phases. This degener-

ation is removed in any finite temperatures and at T > 0

homogeneous phases have the lowest energy. Our dia-

grams obtained for W2= 0 are consistent with results

presented in [27].

IV. FINITE TEMPERATURES

The behaviours of the system for repulsive W2> 0 and

attractive W2< 0 are qualitatively different.

Oneobtains from numerical

z1W1> z2W2≥ 0

sublattice orderings, the PS states are unstable at

any T > 0 and the homogeneous phases are stable

(∂µ/∂n > 0 for any T > 0, even for W2= 0). The finite

temperature phase diagrams have the forms determined

in [27], with the replacements: z1W1→ −WQ> 0. Thus

we describe obtained results in this case shortly, direct-

ing the reader for detailed analyses to [27]. One can con-

clude that transition temperatures between homogeneous

phases decrease with increasing W2> 0 and U/(−WQ).

For the on-site interaction U/(−WQ) < (2/3)ln2

and U/(−WQ) > 1 only the second order CO–NO

transitions occur with increasing temperature.

(2/3)ln2 < U/(−WQ) < 0.62 the first order CO–NO

transition appears near n = 1 with a tricritical point

TC

connected with a change of transition order

(e. g. figure 3c).The TC for n = 1 is located at

kBT/(−WQ) = 1/3 and U/(−WQ) = 2/3ln2.

range 0.62 < U/(−WQ) < 1 the first order CO–CO line

appears, which is ended at an isolated critical point IC

of the liquid-gas type (cf. figure 4a).

analysis

account

that,

only

for

takingintotwo-

For

In the

In this case we

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0.00.20.40.60.81.0

0.0

0.1

0.2

0.3

0.4

0.5

U/(-WQ) = 0

k = - 0.2

B

F

E

PS2A

PS1A

NO

CO

kBT/(-WQ)

n

(a)

0.00.20.40.60.81.0

0.0

0.1

0.2

0.3

0.4

0.5

(b)

U/(-WQ) = 0

k = - 1

T

PS1A

NO

CO

kBT/(-WQ)

n

FIG. 2. (Color online) Phase diagrams kBT/(−WQ) vs. n for U/(−WQ) = 0, W1 > 0 and different values of k = z2W2/z1W1:

k = −0.2 (a) and k = −1 (b). Solid and dashed lines indicate second order and “third order” boundaries, respectively.

have also a critical end point CE, where three bound-

ary lines (one of second order:

first order: CO–NO and CO–CO) connect together.

The two CO phases: HCO and LCO are distinguish-

able only in the neighbourhood of the first order line

CO–CO (HCO–LCO). This line is associated with the

HCOB–LCOB transition in GS. The first order transi-

tion CO–NO can exist inside the region 0.79 < n < 1

and 0.5 < n < 1. The lines consisting of TC, IC and

CE meet at a new multicritical point, which coordinates

are n = 0.79, U/(−WQ) = 0.62 and kBT/(−WQ) = 0.24,

approximately.

CO–NO and two of

The phase diagrams obtained for attractive W2< 0 are

essentially different from those for W2> 0. The main dif-

ference is that at sufficiently low temperatures PS states

are stable. In the ranges of PS states occurrence the ho-

mogeneous phases can be metastable (if ∂µ/∂n > 0) or

unstable (if ∂µ/∂n < 0). In the homogeneous phases oc-

curring at higher temperatures (above the regions of PS

occurrence) the stability condition ∂µ/∂n > 0 is fulfilled.

For clarity of the presentation we will discuss the be-

haviour of the system at T > 0 for W2< 0 distinguish-

ing three regimes of U/(−WQ): on-site attraction (sec-

tion IVA), strong on-site repulsion (section IVB) and

weak on-site repulsion (section IVC).

We also distinguish three different PS1 states and four

different PS2 states (labeled by subscripts A, B, C or

superscript ∗). All states in a particular group are states

with the same type of phase separation (i. e. CO–NO or

CO–CO), however they occur in different regions of the

phase diagram and such distinction has been introduced

to clarify the presented diagrams (cf. especially figure 5).

A similar distinction has been done for all critical points

connected with the phase separation (four B-type points:

B, B?, B??, B∗, three T-type points: T, T?, T??and several

points of H-, E-, F- types). The lines M-N-O and X-Y

indicate the first order transitions between PS states on

the kBT/(−WQ) vs. n diagrams (cf. figures 3 and 4).

A.The case of on-site attraction

For any on-site attraction (U ≤ 0) the phase diagrams

are qualitatively similar, all (second order and “third or-

der”) transition temperatures decrease with increasing

U and for U = 0 the transition temperatures account for

a half of these in the limit U → −∞. In a case of W2< 0

beyond half-filling the PS states can be stable also at

T > 0. Examples of the kBT/(−WQ) vs. n phase dia-

grams evaluated for U/(−WQ) = 0 and various ratios of

k = z2W2/z1W1≤ 0 are shown in figure 2. If 0 ≤ |k| ≤ 1

the (homogeneous) CO and NO phases are separated by

the second order transition line.

When −0.6 < k < 0 (figure 2a) a “third order” transi-

tion takes place at low temperatures, leading first to the

PS into two coexisting CO phases (PS2A), while at still

lower temperatures CO and NO phases coexist (PS1A).

The critical point for this phase separation (denoted as

B, we shall call this point a bicritical endpoint, BEP)

is located inside the CO phase. The E-F solid line (we

shall refer to E-point as a critical endpoint, CEP) is asso-

ciated with continuous transition between two different

PS states (PS1A–PS2A, the second order CO–NO tran-

sition occurs in the domain with lower concentration).

For k < −0.6 (figure 2b) the transition between PS

states does not occur, the area of PS2Astability vanishes

and the critical point for the phase separation (denoted

as T, which is a tricritical point, TCP) lies on the sec-

ond order line CO–NO. As k → −∞ the T-point occurs

at n = 1 and the homogeneous CO phase does not exist

beyond half-filling.

When k = −0.6 the lower branch of the “third or-

der” curve approaches the critical point (H) parabol-

ically.The tricritical behaviour for k < −0.6 changes

into the bicritical behaviour for k > −0.6. The H-point

is a higher order critical point (HCP) and in this point

the lines consisting of B E, F, and T points connect to-

gether (for fixed U/(−WQ)). Similar scenario takes place

also for the on-site repulsion apart from the bicritical be-

haviour connected with the PS2∗state, which can exist

also for k > −0.6).

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0.02

0.04

0.06

0.08

0.10

0.12

0.14

(a)

E’

F’

PS2B

PS2*

PS1B

CO

kBT/(-WQ)

n

U/(-WQ) = 0.2

k = - 0.2

NO

B*

B’

0.00.20.40.6 0.81.0

0.02

0.04

0.06

0.08

0.10

0.12

0.14

(b)

O

N

M

F’

E’

PS2B

PS2*

PS1B

PS1A

CO

NO

U/(-WQ) = 0.2

k = - 0.4

kBT/(-WQ)

n

B’

B*

0.00.20.40.60.81.0

0.05

0.10

0.15

0.20

0.25

0.30

(c)

B*

PS1B

PS2B

PS2*

NO

CO

U/(-WQ) = 0.6

k = - 0.2

kBT/(-WQ)

n

TC

B’

F’

E’

0.00.20.40.60.81.0

0.05

0.10

0.15

0.20

0.25

0.30

(d)

T’

TC

U/(-WQ) = 0.6

k = - 1

PS1B

PS2*

CO

NO

kBT/(-WQ)

n

B*

FIG. 3. (Color online) Phase diagrams kBT/(−WQ) vs. n for W1 > 0 and several values of U/(−WQ) and k = z2W2/z1W1:

(a) U/(−WQ) = 0.2, k = −0.2; (b) U/(−WQ) = 0.2, k = −0.4; (c) U/(−WQ) = 0.6, k = −0.2; and (d) U/(−WQ) = 0.6, k = −1.

Dotted, solid and dashed lines indicate first order, second order and “third order” boundaries, respectively.

One should notice that a type of the critical point for

separation (which can be BEP, TCP or HCP) is modi-

fied only by a change of the strength of the nnn attrac-

tion. The location of the transition lines between ho-

mogeneous phases on the kBT/(−WQ) vs. n diagrams is

not affected by the value of W2 (the transitions are at

the same kBT/(−WQ) as in the previous case of W2> 0,

which only depends on the on-site interaction U for fixed

n). Effectively, the transition temperatures increase with

increasing strength of attractive W2.

The labels of the critical points for phase separations

are given with a correspondence to those in [29], where

the Ising model with nn and nnn interactions was con-

sidered. Notice that our model is equivalent to the Ising

one in the U → ±∞ limits.

B.The case of strong on-site repulsion

For any U/(−WQ) ≥ 1 the structure of the phase di-

agrams and the sequences of transitions are similar as

those in the previous case (for corresponding values of

k), but now the double occupancy of sites is strongly

reduced due to repulsive U and the phase diagrams are

(almost) symmetric with respect to n = 0.5 (cf. figure 1b

and table I). B?, H?, T?, E?and F?points (as well as B??,

H??, T??, E??and F??points) appear (cf. [28]), which corre-

spond to B, H, T, E and F points, respectively. Critical

behaviours at A?and A??points are the same as at A

points (A = B,H,T).

The exact symmetry occurs at U → +∞.

limit the phase diagrams are the diagrams for U → −∞

with re-scaled axes: kBT/(−WQ) → kBT/(−4WQ) and

n → (1/2)n.

Transition temperatures are only weakly dependent on

the on-site repulsion and one finds a small decrease of

them with increasing U for U/(−WQ) ≥ 1.

In this

C.The case of small on-site repulsion

The range 0 < U/(−WQ) < 1 is the most interesting

one and the phase diagrams are more complicated than

those in previous cases. Due to the variety of the be-

haviour in this regime of on-site interaction we only

present some particular examples of the phase diagrams

(figures 3 and 4).

In this range of U/(−WQ) an occurrence of the PS2∗

state for 0.5 < n−< n < n+< 1 is possible and the crit-

ical point for this phase separation (BEP, denoted as

B∗) is located inside the CO phase also for |k| ≥ 0.6.

It contrasts with BEPs mentioned previously (i. e. B,

B?and B??), which occur only for |k| < 0.6.

gion of PS2∗occurrence extends from the ground state if

U/|z2W2| > 1.

When 0 < U/(−WQ) < 2/3ln2 one can see on the

phase diagrams a new behaviour, i. e. discontinuous tran-

sitions between PS states connected with the M-N-O

The re-

Page 7

7

0.00.20.40.60.81.0

0.05

0.10

0.15

0.20

0.25

0.30

(a)

E’

F’

U/(-WQ) = 0.8

k = - 0.2

NO

CO

PS2B

PS1B

HCO

LCO

PS2*

kBT/(-WQ)

n

CE

IC

B*

B’

0.00.20.40.60.8 1.0

0.05

0.10

0.15

0.20

0.25

0.30

(b)

Y

X

U/(-WQ) = 0.8

k = - 0.8

B*

PS2*

T’’

T’

NO

CO

PS1C

PS1B

kBT/(-WQ)

n

0.0 0.20.40.6 0.8 1.0

0.05

0.10

0.15

0.20

0.25

(c)

Y

X

F’’

F’

E’’

E’

PS2C

PS2B

B’’

CO

PS1C

PS2*

PS1B

NO

kBT/(-WQ)

n

U/(-WQ) = 0.94

k = - 0.4

B’

0.00.2 0.4 0.60.8 1.0

0.05

0.10

0.15

0.20

0.25

(d)

Y

X

T’’

T’

CO

PS2*

PS1B

PS1C

NO

U/(-WQ) = 0.94

k = - 1

kBT/(-WQ)

n

FIG. 4. (Color online) Phase diagrams kBT/(−WQ) vs. n for W1 > 0 and several values of U/(−WQ) and k = z2W2/z1W1:

(a) U/(−WQ) = 0.8, k = −0.2; (b) U/(−WQ) = 0.8, k = −0.8; (c) U/(−WQ) = 0.94, k = −0.4; and (d) U/(−WQ) = 0.94,

k = −1. Dotted, solid and dashed lines indicate first order, second order and “third order” boundaries, respectively. Near

n = 1, on the right of the PS2∗occurrence region, the CO phase is stable, what is not shown on the diagrams (c) and (d)

explicitly.

0.0 0.10.2 0.30.4

U/(-WQ)

0.50.6 0.7 0.80.91.0

0.00

0.02

0.04

0.06

0.08

0.18

0.20

0.22

(a)

PS2B

PS1B

CO

NO

PS2A

PS1A

kBT/(-WQ)

n = 0.25

k = - 0.2

0.00.10.20.70.80.91.0

PS1C

1.1

0.00

0.05

0.10

0.15

0.20

0.25

0.45

(b)

IC

NO

LCO

HCO

CO

PS1A

PS2C

PS2*

PS2A

n = 0.75

k = - 0.2

kBT/(-WQ)

U/(-WQ)

0.00.10.20.3 0.4

U/(-WQ)

0.50.6 0.70.80.91.0

0.00

0.05

0.10

0.15

0.20

0.25

(c)

PS1B

PS1A

CO

kBT/(-WQ)

NO

n = 0.25

k = -1

0.00.20.4 0.60.81.0

0.0

0.1

0.2

0.3

0.4

(d)

PS1C

PS2*

PS1A

CO

NO

n = 0.75

k = -1

kBT/(-WQ)

U/(-WQ)

FIG. 5. (Color online) Phase diagrams kBT/(−WQ) vs. U/(−WQ) for several values of n and k = z2W2/z1W1: (a) k = −0.2,

n = 0.25; (b) k = −0.2, n = 0.75; (c) k = −1, n = 0.25; and (d) k = −1, n = 0.25. Dotted, solid and dashed lines indicate first

order, second order and “third order” boundaries, respectively. One should notice that some axes are broken.

Page 8

8

line. On figure 3b two such transitions are presented,

i. e. PS1A–PS1Band PS1A–PS2∗. There are also pos-

sible PS2A–PS2Band PS2A–PS2∗transitions, which are

not shown there. All transitions between homogeneous

phases are second order in this range.

In the range 2/3ln2 < U/(−WQ) < 1, besides the be-

haviours mentioned previously, the critical points con-

nected with transitions between homogeneous phases

can appear on the phase diagrams:

U/(−WQ) < 0.62 and the first order CO–NO boundary

near n = 1 (cf. figures 3c and 3d) and (ii) CE and IC for

U/(−WQ) > 0.62 with the first order HCO–LCO tran-

sition slightly dependent on the electron concentration

(cf. figures 4). It leads to an appearance of isolated ar-

eas of PS1Cstate stability (only for k ≤ −0.6, figure 4b)

and to discontinuous transitions between two PS states:

(i) PS2∗–PS1C (figures 4c and 4d) and (ii) PS2∗–PS2C

(only for |k| < 0.6, cf.

ranges of U/(−WQ).

with the X-Y line, which is linked with the first order

border lines CO–NO and CO–CO.

above the X-Y line a second order PS1C–PS2C transi-

tion also occurs. The temperature associated with X-Y

line is independent of k and depends only on U/(−WQ).

The finite temperature phase diagrams as a func-

tion of U/(−WQ) at fixed n for k = −0.2 and k = −1

are shown in figure 5.The first order boundaries:

(i) PS1A–PS1B and PS2A–PS2B (on the diagrams for

n = 0.25) and (ii) PS1A–PS2∗and PS2A–PS2∗(on the

diagrams for n = 0.75) are associated with the M-N-O

line, whereas the first order boundaries: (iii) PS2∗–PS1C

and PS2∗–PS2C(on the diagrams for n = 0.75) are con-

nected with the X-Y lines. For k = −0.2 and n = 0.25

at higher temperatures the PS1Aand the PS1Bare not

distinguishable and on the diagram (figure 5c) the first

order boundary line ends at a critical point of the liquid-

gas type (similar to IC). In figure 5d for n = 0.75 one can

also see following sequence of transitions with increasing

temperature: PS2∗–CO–PS1C–CO–NO. It is interest-

ing to notice that the PS1C state exists here at higher

temperatures than the homogeneous CO phase (see also

figure 4b).

One should notice that the “third order” boundaries

in figure 5 are not the lines of BEPs (nor TCPs). For

considered ranges of U/(−WQ) and k the n-coordinates

of these points fulfill the following conditions: (i) T and B

points: n < 0.5, (ii) T?and B?points: n < 0.25, (iii) T??,

B??and B∗points: n > 0.75. One should also remember

that E-F, E?-F?, E??-F??, M-N-O and X-Y lines in their

ranges of occurrence are independent of n.

(i) TC

for

also figure 5b) in a restricted

These behaviours are connected

For 0 > k > −0.6

V.CHARGE-ORDER PARAMETER VS.

TEMPERATURE

In this section we present two representative tempera-

ture dependencies of the charge-order parameter for fixed

model parameters.

0.050.100.150.20

0.0

0.2

0.4

0.6

0.8

1.0

CO

NO

n = 0.2

U/(-WQ) = 0.2

k = -0.4

PS2B

nQ

kBT/(-WQ)

PS1B

PS1A

(a)

0.040.08

kBT/(-WQ)

0.120.16 0.20

0.0

0.2

0.4

0.6

0.8

1.0

(b)

n = 0.75

U/(-WQ) = 0.94

k = -0.4

PS2C

PS1C

PS2*

CO

NO

nQ

FIG. 6.

charge-order parameter nQ for (a) U/(−WQ) = 0.2, n = 0.2

and k = −0.4;

k = −0.4.

(Color online) Temperature dependencies of the

and (b) U/(−WQ) = 0.94, n = 0.75 and

In figure 6a we have plotted the charge-order parame-

ter nQas a function of kBT/(−WQ) for U/(−WQ) = 0.2,

n = 0.2 and k = −0.4. At kBT/(−WQ) = 0.058 one ob-

serves a discontinuous change of nQ> 0 in one domain

(in the other nQ= 0), connected with the first-order

transition PS1A–PS1B.For kBT/(−WQ) = 0.073 the

continuous transition PS1B–PS2Boccurs and nQraises to

non-zero value in the domain with lower electron concen-

tration (cf. figure 3b). At kBT/(−WQ) = 0.079 the do-

main with higher electron concentration vanishes contin-

uously, a “third order” transition PS2B–CO occurs and

the whole system is characterized by one value of the

charge order-parameter. Next, at kBT/(−WQ) = 0.167

nQgoes to zero, i. e. one has the second order CO–NO

transition.

Finally, let us comment on the temperature depen-

dence of nQfor U/(−WQ) = 0.94, n = 0.75 and k = −0.4

(figure 6b).At kBT/(−WQ) = 0.043 one observes

a first order PS2∗–PS1C transition, which is connected

with discontinuous change of nQ in the domain with

higher electron concentration (cf. figure 4c). Next, at

kBT/(−WQ) = 0.069 a continuous PS1C–PS2C transi-

tion occurs (now one has continuous change of nQin the

domain with higher electron concentration). At higher

temperatures one can notice a “third order” PS2C–CO

transition (at kBT/(−WQ) = 0.075) and a second order

CO–NO transition (at kBT/(−WQ) = 0.196).

VI.BEYOND THE TWO-SUBLATTICE

ORDERINGS FOR REPULSIVE W2

The nn repulsion W1> 0, as well as the nnn attrac-

tion W2< 0, favour two-sublattice ordering and in such

a case no other types of long-range order can occur.

On the other hand, the repulsive W2> 0 compete with

W1> 0 reducing stability of two-sublattice orderings and

can yield the appearance of multi-sublattice orderings. In

this section we consider charge-orderings on the regular

lattices taking into account not only two-sublattice or-

derings. We will consider them as follows.

More general, we can define the charge-order param-

eter as n? q=?

iniexp(i? q ·?Ri/a), where?Ri determine

a location of i site in the space and a is a hypercubic

Page 9

9

lattice constant.

The four-sublattice orderings can be considered on the

alternate lattices, in which both interpenetrating sublat-

tices are also alternate lattices. Examples of such lattices

are 1D-chain and 2D-square (SQ) lattice.

When we consider four-sublattice orderings in the GS,

the homogeneous phases not mentioned in section III are

found to occur for W2> 0 in some definite ranges of k

and U/z1W1.

For example at half-filling a so-called island charge or-

dered phase (ICO, ...2200..., in d = 1) or stripe charge

ordered phase (SCO, in d = 2, ? q = (0,π)) can occur for

z2W2/z1W1> 0.5. In HCO phase we have ? q = π in d = 1

and ? q = (π,π) in d = 2. The GS phase diagram taking

into account the four-sublattice orderings on 2D-square

lattice for n = 1, W1> 0 is shown in figure 7a. In the

case of 1D-chain the range of ICO occurrence is the same

as that of SCO in 2D. In both cases the VA results (for

n = 1) are consistent with exact results [33, 34, 36, 40].

On the SC lattice we cannot consider four-sublattice

orderings, because the two interpenetrating sublattices

are fcc sublattices and they are not alternate lattices. In

case of such a lattice at half-filling the following three

types of commensurate charge orderings should be con-

sidered: (i) ? q = (0,0,π) (plane charge ordered phase,

PCO), (ii) ? q = (0,π,π) (SCO) and (iii) ? q = (π,π,π)

(HCO). The GS phase diagram for SC lattice and n = 1

is shown in figure 7b. The PCO phase does not occur in

GS and a region of the NO phase stability is extended in

comparison to the lower dimension cases.

In all CO phases mentioned previously the number of

electrons on the particular site can be ni= 0 or ni= 2

and the charge order parameter in each phase has a max-

imum possible value, i. e. n? q= 1.

(ni= 1 at every site).

One should notice that a region of the HCO phase oc-

currence (on the k vs. U/(z1W1) diagram) does not de-

pend on the lattice dimension. This result is in agreement

with GS phase diagrams obtained in section III. The

discontinuous transition HCO–NO is at k = 1 − U/z1W1

what is equivalent to U/(−WQ) = 1 (for k < 0.5 and

n = 1).

Let us stress that we have not analyzed the four-

sublattice orderings at T > 0.

sufficiently low temperatures, such as SCO near half-

filling for k ? 0.5 [41] or some states with phase sepa-

ration between different CO phases or between CO and

NO phases (for n ?= 1) [42]. Thus, the finite temperature

phase diagrams for W2> 0 with taking into account the

four-sublattice orderings can be in general more involved

than those discussed in section IV.

In the NO n? q= 0

They can be stable at

VII.CONCLUDING REMARKS

In this paper we studied atomic limit of the ex-

tended Hubbard model with intersite nn repulsion W1.

By taking into account for the first time the effects of

-0.50.00.51.01.5

-0.5

0.0

0.5

1.0

1.5

(a)

(0,0)

(p,p)

(0,p)

SCO

NO

HCO

k

U/z1W1

-0.5 0.00.51.01.5

-0.5

0.0

0.5

1.0

1.5

(b)

(0,0,0)

SCO

(0,p,p)

(p,p,p)

NO

HCO

k

U/z1W1

FIG. 7. Ground state phase diagrams for n = 1 with consid-

eration of the multi-sublattice orderings: (a) for 2D-square

lattice (for 1D-chain the region of SCO is replaced by that of

ICO), (b) for SC lattice in d = 3. The phases are also labeled

by vector ? q (k = z2W2/z1W1, W1 > 0).

nnn density-density interaction (attractive and repulsive

W2≶ 0) and including into consideration the states with

phase separation (involving CO) our paper substantially

extends and generalizes the results of previous works con-

cerning the model considered [26, 27]. Let us summarize

the most important conclusions of our work:

(i) Depending on the values of the interaction param-

eters and the electron concentration, the system can ex-

hibits not only several charge ordered and nonordered

homogeneous phases, but also (for attractive W2) var-

ious phase separated states involving charge orderings:

PS1(CO-NO) and PS2(CO-CO).

(ii) Obtained phase diagrams have a very rich struc-

ture with multicritical behaviours. The regions of PS

states (both PS1 and PS2) stability expand with increas-

ing of the next-nearest-neighbour attraction. Moreover,

the transitions between PS states can be both continuous

and discontinuous such as those between homogeneous

phases.

(iii) The value of W2< 0 determines a type of the

multicritical point associated with PS states (which we

mentioned as bicritical, tricritical or high-order critical

point).

(iv) For repulsive W2> 0 there is a possibility of oc-

currence of multi-sublattice orderings, e. g. stripe (or

island) structures. In particular, these types of ordering

have the lowest energy at T = 0 for n = 1 (cf. section VI)

if k = z2W2/z1W1> 0.5 and U/z1W1> 1 as well as for

n = 0.5 if k > 0.5 and U/z1W1→ +∞ and in such cases

they can be stable also at sufficiently low temperatures.

Let us stress that for W1> 0 and W2< 0 the derived

results are exact in the limit of infinite dimensions, where

the MFA treatment of intersite interactions becomes the

rigorous one.

Charge ordered phases are stable (for W2= 0) when

the quantum perturbation by finite bandwidth is intro-

duced [34, 43–46]. Thus one can conclude that the PS1

and PS2 states, which involve the CO phases, should also

occur in the presence of hopping term. The stability of

the phase separated CO–NO state in finite temperature

Page 10

10

(for tij?= 0 and W2= 0) in a definite range of the electron

concentration was confirmed by using dynamical mean

field approximation [20]. If tij?= 0 for U < 0 the on-site

superconducting states can occur, whereas for U > 0 it is

necessary to consider also magnetic orderings [16, 46–49].

In such a case various phase separation states involving

superconducting and (or) magnetic orderings can also be

stable.

For the case W1< 0, which was not analysed in the

present work, the model (1) (W2= 0) exhibits a phase

separation NO–NO (electron droplet states) at low tem-

peratures [50, 51]. In this PS state different spatial non-

ordered regions have different average electron concen-

trations. In such a case, at higher temperatures only the

homogeneous NO phase occurs.

The VA results can be exact in the limit of infinite

dimensions only. Below we discuss briefly some results

obtained for W2= 0 within other approaches. In par-

ticular, the Bethe–Peierls–Weiss (BPW) treatment of

the W1 term predicts the following ranges of the exis-

tence of long-range CO at T = 0 for SQ and SC lat-

tices in the limits: (a) U/z1W1→ +∞ (LCO phase):

1/z1< n < (2z1− 1)/z1 and (b) U/z1W1→ −∞ (HCO

phase): 2/z1< n < 2(z1− 1)/z1. In particular:

(i) d = 2 (SQ lattice):

1.25 < n < 1.75 (LCO), (b) 0.5 < n < 1.5 (HCO);

(ii) d = 3 (SC lattice):

1.17 < n < 1.83 (LCO), (b) 0.33 < n < 1.67 (HCO);

(iii) d = +∞ (hypercubic lattice): (a) 0 < n < 1 and

1 < n < 2 (LCO), (b) 0 < n < 2 (HCO).

The Monte Carlo calculations performed for SQ lat-

tice [37] yields for the case U → +∞ even more re-

stricted ranges for long-range CO (LCO) at T = 0:

0.37 < n < 0.63 and 1.37 < n < 1.63. In this particular

case the existence of percolations of the effective clus-

ters has been confirmed. These percolations vanish at

transition temperature.

The phase diagrams for W1> 0 and W2= 0 obtained

by exact solution for the Bethe lattice [51] (which

is equivalent to BPW approximation) have a similar

structure as the VA diagrams (even for small z1= 3).

The main difference is a reentrant behaviour found in

the case of Bethe lattice for U < 0 if n < 2/z1 and

n > 2(z1− 1)/z1, where the sequence of phase transi-

tions: NO→CO→NO can occur with increasing tempera-

ture. The transition temperatures determined in [51] are

in general lower than those obtained in VA, but obviously

in the limit of large coordination number the rigorous re-

sults for Bethe lattice reduce to those of VA.

Comparing the GS diagram obtained for W2= 0 in

VA (figure 1a) with the exact one for 1D-chain [30], we

notice that all the border lines are the same, although

in the exact solution for d = 1 the long-range charge or-

derings in GS exist only in the ranges (i) 0.5 < n < 1

and 0 ≤ U/(−WQ) ≤ 1, (ii) n = 0.5 and U/(−WQ) ≥ 0,

(iii) n = 1 and U/(−WQ) ≤ 1 (what corresponds to the

regions of HCOB, LCO and HCO phases existence in fig-

ure 1a, respectively). The values of µ and D obtained

(a) 0.25 < n < 0.75 and

(a) 0.17 < n < 0.83 and

in VA are consistent with exact ones for arbitrary n and

U/(−WQ). Moreover, the GS phase diagrams as a func-

tion of µ for W2= 0 derived within VA agree exactly

with the corresponding rigorous solutions in d = 1 and

d = 2 [33–36].

The above discussion implies that VA in the case

W2= 0 can give qualitatively reasonable results beyond

the percolation thresholds also for lattices of finite di-

mensionality and this statement should also be true for

W2?= 0, at least for small attractive W2.

The electron concentration n and chemical potential

µ are (thermodynamically) conjugated variables in the

bulk systems [30]. However, one can fit the concentra-

tion rather than the potential in a controlled way ex-

perimentally. In such a case µ is a dependent internal

parameter, which is determined by the temperature, the

value of n, and other model parameters (cf. (2)). Thus

the obtained phase diagrams as a function of the con-

centration are quite important because in real systems

n can vary in a large range and charge orderings are of-

ten found in extended ranges of electron doping (e. g. in

doped manganites [8–13], nickelates [1, 2] and the doped

barium bismuthates [16–18]). In Bechgaard salts the con-

centration is n = 1/2 [5–7]. In charge transfer salts n

changes, dependent on the pressure, in the vicinity of

n = 2/3, whereas for several complex TCNQ salts n is

near n = 1/2 [3–5, 52]. In cuprates ([15, 16] and refer-

ences therein) and in conducting polymers [53] n is near

half-filling in the insulating state and it strongly changes

under doping.

Although our model is (in many aspects) oversimpli-

fied, it can be useful in qualitative analysis of experi-

mental data for real narrow-band materials and it can

be used to understand better properties of several CO

systems mentioned above and in section I.

In particular, our results predict existence of the phase

separation (CO-NO, CO-CO) generated by the effective

nnn attractive interactions and describe their possible

evolutions and phase transitions with increasing T and

a change of n. The electron phase separation involving

COs is shown experimentally in several systems quoted

above, e. g. in R1−xCaxMnO3 (R=La, Bi, Nd, etc.), at

dopings ranging from x = 0.33 to x = 0.82 [10–13, 20].

Among the materials for which the on-site local electron

pairing (valence skipping) has been either established or

suggested (cf. section I) the best candidates to exhibit

the phase separation phenomena are the doped barium

bismuthates (BaPb1−xBixO3and Ba1−xKxBiO3) [16–18,

48]. For these systems, being oxide perovskites, a very

large dielectric constant strongly weakens the long-range

Coulomb repulsion, which is the main factor preventing

the phase separation [54].

Our results show that also the transitions at T > 0 be-

tween various homogeneous phases (HCO and LCO) and

nonordered states can be either first order or continuous

ones and both these types of the CO transitions are ex-

perimentally observed in real narrow-band materials [10–

13]. Moreover, the theory predicts that with a change of

Page 11

11

the model parameters (U/(−WQ), W2, n) the system can

exhibits various types of multicritical behaviour (includ-

ing TCP, BCP, etc.) resulting from the competition of

the on-site repulsion (U > 0) and the effective intersite

repulsion WQ< 0. In fact, some of charge ordered sys-

tems are found to exhibit the multicritical behaviour, e. g.

in organic conductor (DI-DCNQI)2Ag (Tc= 210 K) the

temperature vs. pressure phase diagram shows continu-

ous and first order boundaries with a tricritical point [6].

The increasing pressure changes first the order of transi-

tion, resulting in a tricritical point, then it yields a com-

plete suppression of charge orderings at any T.

ACKNOWLEDGMENTS

The authors wish to thank R. Micnas and T. Kostyrko

for helpful discussions and a careful reading of the

manuscript.

Appendix: Site-dependent self-consistent VA

equations

Within the VA the on-site interaction term is treated

exactly and the intersite interactions are decoupled

within the MFA (site-dependent):

ˆ niˆ nj→ ?ˆ ni? ˆ nj+ ?ˆ nj? ˆ ni− ?ˆ ni??ˆ nj?.

(A.1)

A variational Hamiltonian for the model (1) has a form

ˆH0=

?

i

ˆHi=

?

i

?

Uˆ ni↑ˆ ni↓− µiˆ ni−1

2niψi

?

,

(A.2)

where ψi=?

ˆHi is diagonal in the base consisting of |0?, | ↑?, | ↓?,

| ↑↓? at i site with eigenvalues: 0, doubly degenerated

−µi, and U − 2µi, respectively) and a general expression

for the free energy F in the grand canonical ensemble in

the VA is

j?=iWijnj, µi= µ − ψiand ni= ?ˆ ni?. ˆH0

is diagonal in representation of occupancy numbers (i. e.

F = −1

βln

?

Tr

?

exp(−βˆ

H0)

??

+ µ?ˆ Ne?,

where β =

of electrons in the system. The average value of operator

1

kBT, ˆ

Ne=?

iˆ ni, ?ˆ

Ne? = nN is the number

ˆA is defined as ?ˆA? =

of any operatorˆB and it is calculated in the Fock space.

The explicit formula for the free energy obtained in the

VA has the following form

?

Tr[exp(−βˆ H0)ˆ

Tr[exp(−βˆ H0)]. TrˆB means a trace

A]

F =

?

i

ni(µ −1

2ψi) −1

βlnZi

?

,

(A.3)

where

Zi= 1 + 2exp[β(µ − ψi)] + exp[β(2µ − 2ψi− U)],

while the expression for the average number of electrons

at i-site is given by

ni=

2

Zi{exp[β(µ − ψi)] + exp[β(2µ − 2ψi− U)]},

(A.4)

so one has a set of N + 1 self-consistent equations to solve

consisting of N equations in form of (A.4) (for every site

from N sites) and the condition (2) in the form:

?

The double occupancy Di of the site i is determined

by the following equation:

n −

i

ni= 0.

(A.5)

Di= ?ˆ ni↑ˆ ni↓? =

1

Zi

exp[β(2µ − 2ψi− U)].

(A.6)

The solutions of the set (A.4)–(A.5) can correspond to

a minimum, a maximum or a point of inflection of the

free energy (A.3) on the (N − 1)-dimensional manifold

in N-dimensional space {ni}N

(A.5). To find the solutions corresponding to stable (or

metastable) states of the system, one should find a con-

ditional minimum of F with respect to all ni with the

condition (A.5).

One can prove that for two-sublattice orderings, if

F = F(nA,nB) have a conditional minimum with respect

to nAand nB with condition (A.5), then F = F(n,nQ)

have also a minimum with respect to nQ= (nA− nB)/2

(if n is fixed). So the procedure used in section II does

not lose stable (metastable) solutions. In this instance,

F is the free energy only of homogeneous phases (as

an assumption) and cannot describe any phase separated

states, which energies are calculated from (7). One needs

to check the stability condition ∂µ/∂n > 0 for the ho-

mogeneous phases, which is one of the sufficient con-

ditions for the conditional minimum of F. In the case

of two-sublattice orderings on the alternate lattices the

Eqs. (A.3)–(A.4) reduce to Eqs. (3)–(5) obtained in sec-

tion II.

i=1defined by the condition

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