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 .
The other result of the interplay between magnetism
and superconductivity could be appearance of triplet
pairing . Such a solution could appear together with
ferromagnetic spin ordering. However, in the model (1)
which assumes t
= 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
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
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
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
model, i. e. model (1),
and the U-I-J
model, where the term 2J
placed with J
), describing interactions
between xy-components of spins at neighboring sites,
. In both cases the self-consistent
equations have the same form, only a magnetization
along the z-axis becomes a magnetization in the xy-
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-
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