Article

Some Properties of the Model of a Superconductor with Pair Hopping and Magnetic Interactions at Half-Filling

Acta Physica Polonica Series a (Impact Factor: 0.53). 11/2011; 121(4):733. DOI: 10.12693/APhysPolA.121.733
Source: arXiv
ABSTRACT
We present our preliminary studies of an effective model of a superconductor
with short coherence length involving magnetic interactions. The Hamiltonian
considered consists of (i) the effective on-site interaction U, (ii) the
intersite magnetic exchange interactions (Jz, Jxy) between nearest-neighbors
and (iii) the intersite charge exchange term I, determining the hopping of
electron pairs between nearest-neighbor sites. In the analysis of the phase
diagrams and thermodynamic properties of this model for half-filling (n=1) we
have adopted the variational approach, which treats the on-site interaction
term exactly and the intersite interactions within the mean-field
approximation. One finds that the system considered can exhibit very
interesting multicritical behaviors (including tricritical, critical-end and
bicritical points) caused by the competition between magnetism and
superconductivity, even for n=1. Our investigations show that, depending on the
values of interaction parameters, the system at half-filling can exhibit three
homogeneous phases: superconducting (SS), (anti-)ferromagnetic (F) and
nonordered (NO). The transitions between ordered phases (SS, F) and the NO
phase can be first order as well as second order ones, whereas SS-F transition
is first order one. Temperature dependencies of the order parameters and
thermodynamic properties of the system at the sequence of transitions: SS-F-NO
with increasing temperature for J/I=0.3, U/I0 = 0.69 and n=1 are also
presented.

Full-text

Available from: Konrad Jerzy Kapcia, Jan 05, 2014
Some properties of the model of a superconductor
with pair hopping and magnetic interactions at half-filling
Konrad Kapcia
Electron States of Solids Division, Faculty of Physics,
Adam Mickiewicz University, Umultowska 85, 61-614 Poznań, Poland
(Dated: November 25, 2011)
We present our preliminary studies of an effective model of a superconductor with short coher-
ence length involving magnetic interactions. The Hamiltonian considered consists of (i) the effective
on-site interaction U, (ii) the intersite magnetic exchange interactions (J
z
, J
xy
) between nearest-
neighbors and (iii) the intersite charge exchange term I, determining the hopping of electron pairs
between nearest-neighbor sites. In the analysis of the phase diagrams and thermodynamic proper-
ties of this model for half-filling (n = 1) we have adopted the variational approach, which treats the
on-site interaction term exactly and the intersite interactions within the mean-field approximation.
One finds that the system considered can exhibit very interesting multicritical behaviors (includ-
ing tricritical, critical-end and bicritical points) caused by the competition between magnetism and
superconductivity, even for n = 1. Our investigations show that, depending on the values of interac-
tion parameters, the system at half-filling can exhibit three homogeneous phases: superconducting
(SS), (anti-)ferromagnetic (F) and nonordered (NO). The transitions between ordered phases (SS,
F) and the NO phase can be first order as well as second order ones, whereas SS–F transition is first
order one. Temperature dependencies of the order parameters and thermodynamic properties of
the system at the sequence of transitions: SSFNO with increasing temperature for J/I = 0.3,
U/I
0
= 0.69 and n = 1 are also presented.
PACS numbers: 71.10.Fd, 71.10.-w, 74.20.-z, 74.81.-g, 75.30.Fv
I. INTRODUCTION
There has been much interest in superconductivity
with very short coherence length. This interest is due
to its possible relevance to high temperature supercon-
ductors (the cuprates, doped bismuthates, fullerenes and
iron-based) and also to the several other exotic super-
conducting materials (for a review, see Refs. 1 and 2 and
references therein). It can also give relevant insight into
a behavior of strongly bounded fermion pairs on the op-
tical lattices.
The interplay and competition between superconduc-
tivity and magnetic orderings is currently under intense
investigations (among others in iron chalcogenides and
cuprates, e. g. Refs. 3–5 and references therein). A con-
ceptually simple model for studying that competition will
be studied in this report.
The Hamiltonian considered has the following form:
ˆ
H = U
X
i
ˆn
i
ˆn
i
I
X
hi,ji
ˆρ
+
i
ˆρ
j
+ ˆρ
+
j
ˆρ
i
+ (1)
2J
X
hi,ji
ˆs
z
i
ˆs
z
j
µ
X
i
ˆn
i
,
where ˆn
i
=
P
σ
ˆn
, ˆn
= ˆc
+
ˆc
, ˆρ
+
i
= (ˆρ
i
)
= ˆc
+
i
ˆc
+
i
,
ˆs
z
i
= 1/2(ˆn
i
ˆn
i
). ˆc
and ˆc
+
denote annihilation and
creation operators of an electron with spin σ =, at the
site i, which satisfy canonical anticommutation relations
{ˆc
, ˆc
+
jσ
0
} = δ
ij
δ
σσ
0
, {ˆc
, ˆc
jσ
0
} = {ˆc
+
, ˆc
+
jσ
0
} = 0, (2)
e-mail: kakonrad@amu.edu.pl
where δ
ij
is the Kronecker delta.
P
hi,ji
indicates the
sum over nearest-neighbor sites i and j independently.
U is the on-site density interaction, I is the intersite
charge exchange interaction between nearest neighbors
and J is the Ising-like magnetic interaction between near-
est neighbors. µ is the chemical potential, depending on
the concentration of electrons:
n =
1
N
X
i
hˆn
i
i, (3)
with 0 n 2 and N is the total number of lattice sites.
There are two competitive interaction parameters of
the model: (i) the pair hopping interaction I, deter-
mining the electron pair mobility and responsible for
the long-range superconducting order (nonlocal pairing
mechanism) and (ii) the Ising-like interaction J between
nearest neighbors responsible for magnetism in the sys-
tem. The on-site density-density interaction U con-
tributes (together with I) to the pair binding energy by
reducing (U > 0) or enhancing (U < 0) its value. More-
over, repulsive U > 0 favors magnetic ordering. To sim-
plify our analysis we do not include in Hamiltonian (1)
the single electron hopping term (
P
i,j
t
ij
ˆc
+
ˆc
jσ
) as well
as other inter-site interaction terms. This assumption
corresponds to the situation when single particle mobil-
ity is much smaller than the pair mobility and can be
neglected.
The interactions U, I and J will be treated as the
effective ones and will be assumed to include all the pos-
sible contributions and renormalizations like those com-
ing from the strong electron-phonon coupling or from the
coupling between electrons and other electronic subsys-
arXiv:1111.7296v1 [cond-mat.str-el] 30 Nov 2011
Page 1
2
tems in solid or chemical complexes [1].
Ferromagnetic XY-order of pseudospins
ˆ
~ρ
i
(for I > 0)
corresponds to the SS phase (s-pairing superconduct-
ing), whereas the antiferromagnetic XY-order (for I < 0)
to the Sη phase (η-pairing superconducting). For
t
ij
= 0 there is a well known isomorphism between the
SS and Sη cases (with an obvious redefinition of the
order parameter: ∆ = ∆
SS
=
1
N
P
i
hˆρ
i
i, for I > 0 and
ηS
=
1
N
P
i
exp (i
~
Q ·
~
R
i
)hˆρ
i
i, for I < 0,
~
Q is half of the
smallest reciprocal lattice vector) for lattices consisting of
two interpenetrating sublattices such as for example SC
or BCC lattices. One should also notice that, in the ab-
sence of the single electron hopping term, ferromagnetic
(J > 0) interactions are simply mapped onto the antifer-
romagnetic cases (J < 0) by redefining the spin direction
on one sublattice in lattices decomposed into two inter-
penetrating sublattices. Thus, we restrict ourselves to
the case of I > 0 and J > 0.
We have performed extensive study of the phase dia-
grams of the model (1) for arbitrary n and µ [6]. In the
analysis we have adopted a variational approach (VA),
which treats the onsite interaction term (U ) exactly and
the intersite interactions (I, J) within the mean-field ap-
proximation (MFA). In this paper we present our prelim-
inary results for the half-filling (n = 1).
The model (1) have been analyzed within VA only for
particular cases: (i) J = 0 [7–11] and (ii) I = 0 [12, 13]
till now. The rigorous results for I = 0 in ground state
have been also obtained [14]. Some preliminary study of
the I = 0 case in finite temperatures using Monte Carlo
simulations has also been done for a square lattice [13].
The ferromagnetic (F) phase is characterized by
nonzero value of the magnetic order parameter (magneti-
zation) defined as m = (1/N)
P
i
hˆs
z
i
i (and ∆ = 0), in the
superconducting (SS) phase the order parameter 6= 0
(and m = 0) and in the nonordered (NO) phase m = 0
and ∆ = 0.
Within the VA the intersite interactions are decou-
pled within the MFA, what let us find a grand canon-
ical potential per site ω(µ) (or free energy per site
f(n) = ω(µ) + µn) in the grand canonical ensemble. One
can also calculate the averages: n, and m, what gives
a set of three non-linear self-consistent equations (for ho-
mogeneous phases). This set for T > 0 is solved numer-
ically and one obtains , m, and n (or µ) when µ (or
n) is fixed. It is important to find a solution correspond-
ing to the lowest ω(µ) (or f(n)). For n = 1 one obtains
µ = U/2 and two equations for and m need to be solved
numerically.
We also introduce the following denotation: I
0
= zI,
J
0
= zJ, where z is the number of nearest neighbors.
II. RESULTS AND DISCUSSION
There are two well defined limits of the model (1):
(i) U −∞ favoring superconductivity and
FIG. 1. (Color online) Phase diagrams k
B
T/I
0
vs. U/I
0
at half-filling (n = 1) for (a) J/I = 0.3, (b) J/I = 0.51 and
(c) J/I = 3. Dotted and solid lines indicate first order and
second order boundaries, respectively. T , E and B denote
tricritical, critical-end and bicritical points, respectively.
(ii) U +, where only magnetic orderings can
appear in the system.
For U −∞ (states with single occupancy are ex-
cluded and only local pairs can exists in the system)
the model is equivalent with the hard-core charged bo-
son model on the lattice [9, 15, 16]. In this limit the
Page 2
3
SS–NO transition is second order one and is to the NO
phase being a state of dynamically disordered local pairs.
The SS–NO transition temperatures increase monotoni-
cally with decreasing |n 1|. The maximum value of the
transition temperature is k
B
T/I
0
= 1 for n = 1 [9, 11].
In the opposite limit (i. e. U +) the double oc-
cupied sites are excluded and only the magnetic states
can occur on the phase diagram [12, 13]. At sufficiently
low temperatures the homogeneous phases are not states
with the lowest free energy and the PS state are stable (if
n 6= 1). On the phase diagram there is a second order line
at high temperatures, separating the F and NO phases,
whereas first order transition takes place at lower tem-
peratures, leading to a phase separation of the F and NO
phases. The critical point for the phase separation (tri-
critical point) lies on the second order F–NO line and it
is located at k
B
T/J
0
= 1/3 and n = 1/3 [12]. The F–NO
(second order) transition temperature decreases with in-
creasing |n 1| and its maximum value is k
B
T/J
0
= 1
for n = 1 [13].
A. The phase diagrams at half-filling
A few representative k
B
T/I
0
vs. U/I
0
phase diagrams
of the model (1) evaluated for various ratios of J/I at
half-filling (n = 1) are presented in Fig. 1.
The phase diagram k
B
T/I
0
vs. U/I
0
for J/I = 0.3 and
n = 1 is shown in Fig. 1a. Two ordered phases: the
SS phase and the F phase are separated by first order
boundary on the diagram. Both order parameters change
discontinuous at the SS–F transition. With increasing
U/I
0
the SS–NO transition temperature decreases from
k
B
T/I
0
= 1 at U/I
0
−∞. At U/I
0
=
2
3
ln(2) ' 0.462
and k
B
T/I
0
= 1/3 the transition changes its type from
second order one to first order one resulting in the tri-
critical point T on the phase diagram. The F–NO tran-
sition temperature is slightly dependent on U/I
0
and in-
creases to k
B
T/I
0
= 0.3 (k
B
T/J
0
= 1) at U/I
0
+.
The F–NO second order line ends at critical-end point E
on the first order boundary of the SS phase occurrence.
The possible sequences of transitions with increasing
temperatures and the transition orders of them are listed
below (for J/I = 0.3):
(i) SSNO: second order, for U/I
0
< 0.46 and first
order, for 0.46 < U/I
0
< 0.63,
(ii) SSFNO: first order and second order, respec-
tively, for 0.63 < U/I
0
< 0.7,
(iii) FNO: second order, for U/I
0
> 0.7.
The phase diagram for J/I = 0.51 is qualitatively dif-
ferent than that for J/I = 0.3. For J/I = 0.51 the sys-
tem exhibits bicritical behavior (Fig. 1b) in contrary to
the tricritical behavior (and occurrence of E-point) for
J/I = 0.3. Similarly as for J/I = 0.3, the SS–F tran-
sition is first order one while the SS–NO and F–NO
FIG. 2. (Color online) Temperature dependence of (a) su-
perconducting order parameter || and (b) magnetic order
parameter m for J/I = 0.3, U/I
0
= 0.69 and n = 1.
transitions are second order ones. The two second or-
der boundaries and the first order boundary merge at
bicritical point B.
The system exhibits the tricritical behavior for
|J/I| < 0.5, whereas the bicritical behavior occurs for
0.5 < |J/I| < 2. For |J/I| > 2 the system exhibits tri-
critical behavior again, however the tricritical point T is
located at the F–NO line at U/J
0
=
2
3
ln(2) ' 0.462
and k
B
T/J
0
= 1/3 (cf. Fig. 1c). For |J/I| > 2 the F–NO
transition can be second order (for U/J
0
> 0.46) as well
as first order (for U/J
0
< 0.46). Notice that the axis in
Fig. 1 are normalized by I
0
, not by J
0
.
One should notice that, for any J/I, with increas-
ing U/I
0
the SS–NO transition temperature decreases
monotonically from k
B
T/I
0
= 1 at U −∞, whereas
the F–NO transition temperature is an increasing func-
tion of U/I
0
(to its maximum k
B
T/J
0
= 1 at U +).
Let us concentrate now on temperature dependencies
of the order parameters and thermodynamic properties
of the system at the sequence of transitions: SSFNO
for J/I = 0.3, U/I
0
= 0.69 and n = 1.
Page 3
4
B. The order parameters
The temperature dependencies of the order parame-
ters: and m for J/I = 0.3, U/I
0
= 0.69 and n = 1 are
presented in Fig. 2. It is clearly seen that at the SS–F
transition (at k
B
T
c1
/I
0
= 0.16) the both order parame-
ters: superconducting order parameter and magneti-
zation m change discontinuously. In the SS phase 6= 0
and m = 0 whereas in the F phase m 6= 0 and ∆ = 0. The
F–NO transition (at k
B
T
c2
/I
0
= 0.24) is connected with
a continuous decay of m at the transition temperature.
C. The thermodynamic properties
Calculating the free energy per site f one can obtain
thermodynamic characteristics of the system for arbi-
trary temperature. The double occupancy per site D
is defined as:
D =
1
N
X
i
hˆn
i
ˆn
i
i =
f
U
T
(4)
and it is related with the local magnetic moment γ by
the following formula:
γ =
1
N
X
i
hˆs
z
i
i =
1
2N
X
i
h|ˆn
i
ˆn
i
|i (5)
=
1
2
n
X
i
hˆn
i
ˆn
i
i =
1
2
n
f
U
T
=
1
2
n D,
because |ˆn
i
ˆn
i
| = (ˆn
i
ˆn
i
)
2
= ˆn
i
+ ˆn
i
2ˆn
i
ˆn
i
,
ˆn
2
= ˆn
= 0, 1 and |ˆn
i
ˆn
i
| = 0, 1.
The entropy s and the specific heat c can be derived
as:
s =
f
T
, c = T
2
f
T
2
. (6)
The temperature dependencies of the thermodynamic
parameters for J/I = 0.3, U/I
0
= 0.69 and n = 1 are
shown in Fig. 3.
The concentration of paired electrons n
p
= 2D (nor-
malized to the total electron concentration n) as a func-
tion of temperature is presented in Fig. 3a. At the SS–F
transition large amount of electron pairs is destroyed.
Thus n
p
has a sharp break at the SS–F transition tem-
perature T
c1
and a substantial fraction of single particles
exists above T
c1
. As temperature is lowered, the conden-
sate growths both from a condensation of pre-existing
pairs and from binding and condensation of single parti-
cles. At the F–NO transition (at T
c2
) n
p
is continuous.
In the NO phase it increases to n
p
0.5 at T +
(two electrons at the site is one of four equal probable
configurations at the site and n = hˆn
i
i = 1).
The temperature dependencies of the entropy s and the
specific heat c are shown in Figs. 3b and 3c, respectively.
s increases monotonically with increasing T . At T
c1
the
entropy s is discontinuous whereas it is continuous at T
c2
.
FIG. 3. (Color online) Thermodynamic parameters (a) the
concentration of paired electrons n
p
/n = 2D/n, (b) the en-
tropy s and (c) the specific heat c as a function of k
B
T/I
0
for
J/I = 0.3, U/I
0
= 0.69 and n = 1.
One can notice that in the high-temperature limit the en-
tropy s/(k
B
I
0
) ln(4) 1.386 (there are four possible
configurations at each site). The peak in c(T ) is asso-
ciated with the first order transition (at T
c1
), while the
λ-point behavior is typical for the second order transition
(at T
c2
).
Page 4
5
III. FINAL REMARKS
We have studied a simple model of a magnetic super-
conductor with very short coherence length (i. e. with the
pair size being of the order of the radius of an effective
lattice site) and considered the situation where the single
particle mobility is much smaller than the pair mobility
and can be neglected.
One has found that the system considered for n = 1 ex-
hibits various multicritical behaviors (determined by the
ratio J/I) including tricritical, critical-end and bicritical
points. It has been shown that, depending on the values
of interaction parameters, three homogeneous phases: su-
perconducting, (anti-)ferromagnetic and nonordered oc-
cur on the phase diagrams of the model (1) at half-filling.
The transitions between ordered phases (SS, F) and the
NO phase can be first order as well as second order ones,
whereas the SS–F transition is first order one. For n 6= 1
several types of phase separated states could be also sta-
ble in definite ranges of model parameters [6].
The other result of the interplay between magnetism
and superconductivity could be appearance of triplet
pairing [17]. Such a solution could appear together with
ferromagnetic spin ordering. However, in the model (1)
which assumes t
ij
= 0 such a state cannot be found. To
investigate the possibility of occurrence of a supercon-
ducting state with triplet pairing, the model should be
extended to the case of finite bandwidth (t
ij
6= 0) and
be analyzed taking into account intersite pairing (in par-
ticular triplet pairing), e. g. using Hartree-Fock broken
symmetry framework [17–19].
The mean-field approximation used to the intersite
term is best justified if the I
ij
interactions are long-
ranged or if the number of nearest neighbors is relatively
large. The derived VA results are exact in the limit of
infinite dimensions d +, where the MFA treatment
of the intersite interactions I and J terms becomes the
rigorous one.
Let us point out that in the MFA, which does not
take into account collective excitations, one obtains the
same results for the U-I-J
z
model, i. e. model (1),
and the U-I-J
xy
model, where the term 2J
P
ˆs
z
i
ˆs
z
j
is re-
placed with J
P
(ˆs
+
i
ˆs
j
+ ˆs
+
j
ˆs
i
), describing interactions
between xy-components of spins at neighboring sites,
ˆs
+
i
= ˆc
+
i
ˆc
i
= (ˆs
i
)
. In both cases the self-consistent
equations have the same form, only a magnetization
along the z-axis becomes a magnetization in the xy-
plane [12].
ACKNOWLEDGMENTS
The author is indebted to Professor Stanisław
Robaszkiewicz for very fruitful discussions during this
work and careful reading of the manuscript. The work
has been financed by National Science Center (NCN)
as a research project in years 2011-2013, under grant
No. DEC-2011/01/N/ST3/00413. We would also like to
thank the European Commission and Ministry of Science
and Higher Education (Poland) for the partial financial
support from European Social Fund Operational Pro-
gramme “Human Capital” POKL.04.01.01-00-133/09-
00 Proinnowacyjne kształcenie, kompetentna kadra,
absolwenci przyszłości”.
[1] R. Micnas, J. Ranninger, S. Robaszkiewicz, Rev. Mod.
Phys. 62, 113 (1990).
[2] D. C. Johnston, Adv. in Physics 59, 803 (2010);
P. M. Aswathy, J. B. Anooja, P. M. Sarun,
U. Syamaprasad, Supercond. Sci. Technol. 23, 073001
(2010).
[3] J. T. Park, D. S. Inosov, Ch. Niedermayer, G. L. Sun,
D. Haug, N. B. Christensen, R. Dinnebier, A. V. Boris,
A. J. Drew, L. Schulz, T. Shapoval, U. Wolff, V. Neu,
Xiaoping Yang, C. T. Lin, B. Keimer, V. Hinkov, Phys.
Rev. Lett. 102, 117006 (2009).
[4] A. Ricci, N. Poccia, G. Campi, B. Joseph, G. Arrighetti,
L. Barba, M. Reynolds, M. Burghammer, H. Takeya,
Y. Mizuguchi, Y. Takano, M. Colapietro, N. L. Saini,
A. Bianconi, Phys. Rev. B 84, 060511(R) (2011).
[5] J. Xu, S. Tan, L. Pi, Y. Zhang, J. Appl. Phys. 104,
063914 (2008).
[6] K. Kapcia, S. Robaszkiewicz, in preparation.
[7] R. A. Bari, Phys. Rev. B 7, 2128 (1973).
[8] W.-C. Ho, J. H. Barry, Phys. Rev. B 16, 3172 (1977).
[9] S. Robaszkiewicz, G. Pawłowski, Physica C 210, 61
(1993).
[10] S. Robaszkiewicz, Acta. Phys. Pol. A 85, 117 (1994).
[11] K. Kapcia, S. Robaszkiewicz, R. Micnas, in preparation.
[12] W. Kłobus, K. Kapcia, S. Robaszkiewicz, Acta. Phys.
Pol. A 118, 353 (2010).
[13] S. Murawski, K. Kapcia, G. Pawłowski,
S. Robaszkiewicz, Acta. Phys. Pol. A 121 (2012),
in press; arXiv:1109.2620.
[14] U. Brandt, J. Stolze, Z. Phys. B 62 433 (1986).
[15] K. Bernardet, G. G. Batrouni, J.-L. Meunier, G. Schmid,
M. Troyer, A. Dorneich Phys. Rev. B 65, 104519 (2002).
[16] R. Micnas, S. Robaszkiewicz, T. Kostyrko, Phys. Rev. B
52, 6863 (1995).
[17] J. F. Annett, B. L. Gyrffy, G. Litak, K. I. Wysokiski,
Phys. Rev. B 78, 054511 (2008).
[18] W. R. Czart, S. Robaszkiewicz, Acta. Phys. Pol. A 109,
577 (2006).
[19] R. Micnas, J. Ranninger, S. Robaszkiewicz, S. Tabor,
Phys. Rev. B 37, 9410 (1988); R. Micnas, J. Ranninger,
S. Robaszkiewicz, Phys. Rev. B 39, 11653 (1989).
Page 5
  • Source
    [Show abstract] [Hide abstract] ABSTRACT: We have studied the extended Hubbard model with pair hopping in the atomic limit for arbitrary electron density and chemical potential. The Hamiltonian considered consists of (i) the effective on-site interaction U and (ii) the intersite charge exchange interactions I, determining the hopping of electron pairs between nearest-neighbour sites. The model can be treated as a simple effective model of a superconductor with very short coherence length in which electrons are localized and only electron pairs have a possibility of transferring. The phase diagrams and thermodynamic properties of this model have been determined within the variational approach, which treats the on-site interaction term exactly and the intersite interactions within the mean-field approximation. We have also obtained rigorous results for a linear chain (d = 1) in the ground state. Moreover, at T = 0 some results derived within the random phase approximation (and the spin-wave approximation) for d = 2 and 3 lattices and within the low-density expansions for d = 3 lattices are presented. Our investigation of the general case (as a function of the electron concentration n and as a function of the chemical potential μ) shows that, depending on the values of interaction parameters, the system can exhibit not only the homogeneous phases, superconducting (SS) and nonordered (NO), but also the phase separated states (PS: SS-NO). The system considered exhibits interesting multicritical behaviour including tricritical points.
    Full-text · Article · Apr 2012 · Journal of Physics Condensed Matter
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    [Show abstract] [Hide abstract] ABSTRACT: We have studied the extended Hubbard model with pair hopping in the atomic limit for arbitrary electron density and chemical potential and focus on paramagnetic effects of the external magnetic field. The Hamiltonian considered consists of (i) the effective on-site interaction U and (ii) the intersite charge exchange interactions I, determining the hopping of electron pairs between nearest-neighbour sites. The phase diagrams and thermodynamic properties of this model have been determined within the variational approach (VA), which treats the on-site interaction term exactly and the intersite interactions within the mean-field approximation. Our investigation of the general case shows that the system can exhibit not only the homogeneous phases-superconducting (SS) and non-ordered (NO)-but also the phase separated states (PS: SS-NO). Depending on the values of interaction parameters, the PS state can occur in higher fields than the SS phase (field induced PS). Some ground state results beyond the VA are also presented.
    Full-text · Article · Jan 2013 · Journal of Physics Condensed Matter
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    [Show abstract] [Hide abstract] ABSTRACT: We present studies of an effective model which is a simple generalization of the standard model of a local pair superconductor with on-site pairing (i.e., the model of hard core bosons on a lattice) to the case of finite pair binding energy. The tight binding Hamiltonian consists of (i) the effective on-site interaction U, (ii) the intersite density-density interactions W between nearest-neighbours, and (iii) the intersite charge exchange term I, determining the hopping of electron pairs between nearest-neighbour sites. In the analysis of the phase diagrams and thermodynamic properties of this model we treat the intersite interactions within the mean-field approximation. Our investigations of the U<0 and W>0 case show that, depending on the values of interaction parameters, the system can exhibit three homogeneous phases: superconducting (SS), charge-ordered (CO) and nonordered (NO) as well as the phase separated SS-CO state.
    Full-text · Article · Mar 2013 · Journal of Superconductivity and Novel Magnetism
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