Magnetic orderings and phase separations in the zero-bandwidth limit of the extended Hubbard model with intersite magnetic interactions
ABSTRACT A simple effective model for a description of magnetically ordered insulators
is analysed. The tight binding Hamiltonian consists of the effective on-site
interaction (U) and intersite magnetic exchange interactions (Jz, Jxy) between
nearest-neighbours. The phase diagrams 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
show that, depending on the values of interaction parameters and the electron
concentration, the system can exhibit not only homogeneous phases:
(anti-)ferromagnetic (F) and nonordered (NO), but also phase separated states
Magnetic orderings and phase separations in the zero-bandwidth limit
of the extended Hubbard model with intersite magnetic interactions
Waldemar Kłobus, Konrad Kapcia,∗and Stanisław Robaszkiewicz
Electron States of Solids Division, Faculty of Physics,
Adam Mickiewicz University, ul. Umultowska 85, 61-614 Poznań, POLAND
(Dated: July 5, 2010)
A simple effective model for a description of magnetically ordered insulators is analysed. The tight
binding Hamiltonian consists of the effective on-site interaction (U) and intersite magnetic exchange
interactions (Jz, Jxy) between nearest-neighbours. The phase diagrams 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 show that, depending on the
values of interaction parameters and the electron concentration, the system can exhibit not only
homogeneous phases: (anti-)ferromagnetic (Fα) and nonordered (NO), but also phase separated
states (PSα: Fα–NO).
PACS numbers: 71.10.Fd, 75.10.-b, 75.30.Gw, 64.75.Gh, 71.10.Hf
The extended Hubbard model with anisotropic spin
exchange interactions [1–5] is a conceptually simple phe-
nomenological model for studying correlations and for
a description of magnetism and other types of electron or-
derings in narrow band systems with easy-plane or easy-
axis magnetic anisotropy.
In this report we will focus on the zero-bandwidth limit
of the extended Hubbard model with magnetic interac-
tions for the case of arbitrary electron density 0 < n < 2.
We consider the U-JzHamiltonian of the following form:
where U is the on-site density interaction, Jzis z-
component of the intersite magnetic exchange interac-
bours. ˆ c+
with spin σ at the site i, ˆ ni=?
on the concentration of electrons is calculated from
ˆH = U
ˆ ni↑ˆ ni↓− 2Jz?
?i,j?restricts the summation to nearest neigh-
iσdenotes the creation operator of an electron
σˆ niσ, ˆ niσ= ˆ c+
2(ˆ ni↑− ˆ ni↓). The chemical potential µ depending
with 0 ≤ n ≤ 2 and N is the total number of lattice sites.
The model (1) can be treated as an effective model
of magnetically ordered insulators. The interactions U
and Jzwill be assumed to include all the possible contri-
butions and renormalizations like those coming from the
strong electron-phonon coupling or from the coupling be-
tween electrons and other electronic subsystems in solid
or chemical complexes. In such a general case arbitrary
values and signs of U are important to consider. We re-
strict ourselves to the case of positive Jz> 0, because of
∗Electronic address: email@example.com
the symmetry between ferromagnetic (Jz> 0) and anti-
ferromagnetic (Jz< 0) case for lattice consisting of two
interpenetrating sublattices such as for example sc or bcc
We have performed extensive study of the phase dia-
gram of the model (1) for arbitrary n and µ [6, 7]. In the
analysis we have adopted a variational approach (VA)
which treats the on-site interaction U exactly and the
intersite interaction Jzwithin the mean-field approxi-
mation (MFA). We restrict ourselves to the case of the
positive Jz, as it was mentioned above.
Let us point out that in the MFA, which does
not take into account collective excitations, one ob-
tains the same results for the U-Jzmodel and the
U-Jxymodel, where the term 2Jz?ˆ sz
tween xy-components of spins at neighbouring sites,
In both cases the self-consistent
equations have the same form, only the replacement
Jz→ Jxyis needed and a magnetization along the z-axis
becomes a magnetization in the xy-plane .
For the model (1) only the ground state phase dia-
gram as a function of µ  and special cases of half-filling
(n = 1)  and U → ∞  have been investigated till
Within the VA the intersite interactions are decou-
pled within the MFA, what let us find a free energy
per site f(n). The condition (2) for the electron con-
centration and a minimization of f(n) with respect to
the magnetic-order parameter leads to a set of two self-
consistent equations (for homogeneous phases), which are
solved numerically. The order parameter is defined as
average magnetization in a sublattice γ = A,B in the
α = z,xy direction (sxy
non-zero the ferromagnetic phase (Fα) is a solution, oth-
erwise the non-ordered phase (NO) occurs.
Phase separation (PS) is a state in which two domains
with different electron concentration exist in the system
(coexistence of two homogeneous phases). The free ener-
with Jxy?(ˆ s+
i= ˆ c+
j+ ˆ s+
i), describing interactions be-
i↑ˆ ci↓= (ˆ s−
B), where mα
i? is the
ihere). If mαis
arXiv:1007.1147v1 [cond-mat.str-el] 7 Jul 2010
gies 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 solutions and
sity n+. We find numerically the minimum of fPSwith
respect to n+and n−.
In the model considered only PSαstate (i. e. a coexis-
tence of Fαand NO phases) can occur.
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” transition.
During this transition a size of one domain in the PS state
decreases continuously to zero at the transition temper-
Second order transitions are denoted by solid lines on
phase diagrams, dotted curves denote first order transi-
tions and dashed lines correspond to the “third order”
transitions. We also introduce the following denotation:
Obtained phase diagrams are symmetric with respect
to half-filling because of the particle-hole symmetry of
the Hamiltonian (1), so the diagrams will be presented
only in the range 0 ≤ n ≤ 1.
n+−n−is a fraction of the system with a charge den-
0= z1Jαfor α = z,xy, where z1is the number of near-
II.RESULTS AND DISCUSSION
A.The ground state
In the ground state the energies of homogeneous
phases have the form:
EF= U(n − 1) − (1/2)Jα
paring the energies we obtain diagram shown in Fig. 1.
At U = −Jα
Fα–NO takes place in the system.
associated with a discontinuous disappearance of the
The firstderivative of
∂µ/∂n = ∂2E/∂n2for U/Jα
phases is negative what implies that homogeneous phases
are not stable (except n = 1).
for NO: ENO= (1/2)Un
0(2 − n)2
n ≤ 1
n ≥ 1.
0(1 − |n − 1|) the first order transition
This transition is
0> −1 in the lowest energy
Finite temperature phase diagrams taking into account
only homogeneous phases and plotted as a function of
cal point T1, which is connected with a change of tran-
sition order, for n = 1 is located at kBT/Jα
0for chosen n are shown in Fig. 2a. The tricriti-
0= 1/3 and
0= −2/3ln2 .
FIG. 1: Ground state phase diagrams as a function of n with-
out consideration of PS states. The dotted line denotes dis-
The range of the occurrence of Fα phase is reduced
with decreasing n.For n > 0.67 and any U/Jα
we observe only one transition Fα–NO with increas-
ing temperature. In the range 0.67 < n ≤ 1 the U/Jα
coordinate of the T1-point remains constant, so for
However, for n < 0.67 in some range of U/Jα
appear a sequence of two transitions: NO–Fα–NO.
0< −2/3ln2 the Fα–NO transition is discontinuous.
In Fig. 2b there are shown dependencies of the tran-
sition temperature Fα–NO as a function of n for chosen
values of U/Jα
decreasing of U/Jα
only one second order transition Fα–NO with increas-
ing temperature. There exist ranges of n and U/Jα
where the sequence of transitions: NO–Fα–NO is present.
0. The range of Fαstability is reduced with
0. For U/Jα
0> 0 and any n we observe
At sufficiently low temperatures homogeneous phases
are not states with the lowest free energy and there PS
state can occur. On the phase diagrams, where we con-
sidered the possibility of appearance of the PS states,
there is a second order line at high temperatures, sep-
arating Fα and NO phases. A “third order” transition
takes place at lower temperatures, leading to a PS into
Fαand NO phases. The critical point for the phase sep-
aration (denoted as T2, a tricritical point) lies on the
second order line Fα–NO. Phase diagrams for U/Jα
0= 10 are shown in Fig. 3.
In the ranges of PS stability the homogeneous
phases can be metastable (if ∂µ/∂n > 0) or unstable (if
∂µ/∂n < 0). We leave a deeper analyses of meta- and
unstable states to future publications.
-1.0 -0.8 -0.6 -0.4 -0.20.0
n = 0.67
n = 0.5
n = 0.33
n = 0.25
n = 0.125
n = 1
n = 0.75
0.0 0.20.40.60.8 1.0
FIG. 2: Phase diagrams (a) kBT/Jαvs. U/Jα
PS states. Dotted and solid lines denote first and second order transitions, respectively.
0 for fixed n and (b) kBT/Jα
0 vs. n for fixed U/Jα
0 without the consideration of
0 = 1
0 = 10
FIG. 3: Phase diagrams kBT/Jα
dashed lines indicate second order and “third order” boundaries, respectively.
0 vs. n with the consideration of PS states for: U/Jα
0 = 1 (a) and U/Jα
0 = 10 (b). Solid and
We considered a simple model for magnetically ordered
insulators. It was shown that at the sufficiently low tem-
peratures homogeneous phases do not exist and the states
with phase separation are states with the lowest free en-
ergy. On phase diagrams we also observe the tricritical
points, which are associated with a change of transition
order (T1-point, Fig. 2) or are located in the place where
the second order line connects with “third order” lines
(T2-point, Fig. 3).
Let us stress that the knowledge of the zero-bandwidth
limit can be used as starting point for a perturbation ex-
pansion in powers of the hopping and as an important
test for various approximate approaches (like dynami-
cal MFA) analyzing the corresponding finite bandwidth
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