MULTICRITICAL BEHAVIORS OF THE SPIN-3/2 BLUME-EMERYGRIFFITHS MODEL ON A CELLULAR AUTOMATON
N. SEFEROĞLU, G. SEZGİN
Gazi University, Institute of Science and Technology, Advanced Technologies Department, Ankara,
Turkey, E-mail: nurguls@gazi.edu.tr
Received September 6, 2011
A computational study based on the cellular automaton has been done to investigate the
phase transition of the spin-3/2 Blume-Emery-Griffiths (BEG) model. In the study, the
ferromagnetic spin-3/2 BEG model with repulsive biquadratic coupling is simulated by
using the cooling algorithm improved from the Creutz cellular automaton (CCA). The
temperature dependence of the order parameters and associated fluctuations are
calculated at various of the model parameters and the phase diagrams of the model are
constructed in the (D/J, kT/J) and (K/J, kT/J) plane in the absence and presence of the
external magnetic field. The phase diagrams of the model exhibit a rich variety of
behaviors. It is shown that different kinds of phase transitions take place between the
disordered, ferromagnetic, ferrimagnetic and antiquadrupolar phases for various of
model parameters at a magnetic field value in the interval –0.06 ≤h/J ≤ 0. In contrast,
there is only a transition between disordered and ferrimagnetic phases in the interval
– 0.2≤ h/J< –0.06. Results are compared with other approximate methods.
Key words: Spin-3/2 BEG model, cellular automaton, simple cubic lattice, external magnetic
field, phase diagram.
1. INTRODUCTION
The spin-3/2 Blume-Emery-Griffiths (BEG) model is a spin-3/2 Ising model
with bilinear (J) and biquadratic (K) exchange interactions and a single-ion
potential or crystal field interaction (D). The spin-3/2 BEG model with J and K
interactions has been initially introduced to give a qualitative description of phase
transition observed in the compound DyVO4 [1] and to study tricritical properties
in ternary mixtures [2]. The Hamiltonian of the model in an external magnetic field
is
(1)
where the spin variables Si located at site i on a discrete lattice take the values ±3/2,
±1/2 and the first two summations run over all nearest-neighboring pairs. h
describes the effect of an external magnetic field.
Rom. Journ. Phys., Vol. 57, Nos. 7–8, P. 1053–1064, Bucharest, 2012
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The spin-3/2 BEG model for K/J > 0 has been studied and the phase diagrams
have been presented by various methods, such as renormalization-group (RG)
methods [3], effective-field theory (EFT) [4], Monte Carlo (MC) and a densitymatrix RG method [5]. On the other hand, the model with K/J < 0 has also been
investigated within the MFA and MC [6], EFT [7], cluster variation method
(CVM) [8, 9], the recursion method [10], and cellular automaton (CA) [11]. In
these studies, some portion of the phase diagrams of the model are considered in
the absence of an external magnetic field. However, there are several studies to
obtain the critical behaviors and phase diagrams of the model in the presence of an
external magnetic field. For example, only one phase diagram in the (h/J, kT/J) for
K/J = –0.5 and D/J = 1.0 and the other one in the (K/J, kT/J) plane for h/J = 2.0 and
D/J = 0.5 on Bethe lattice with coordination number q = 3, 4, 6, and 8 using the
recursion method [12]. Recently, two different phase diagram topologies in the
(h/J, kT/J) plane in which only one is a new topology have been presented within
the lowest approximation of CVM [13].
The purpose of this work is to investigate the multicritical behaviors of the
ferromagnetic spin–3/2 BEG model in the absence and presence of an external
magnetic field. For this purpose, the model is simulated on the simple cubic lattice
by introducing two sublattices A and B using the cooling algorithm [11, 14, 15]
improved from Creutz cellular automaton (CCA) [16]. In this study, the
temperature dependence of the sublattice order parameters (MA,B, QA,B) and the
order parameter Qd and associated susceptibilities (χA,B and χd) are investigated and
the phase diagrams are obtained. The CCA algorithm is a microcanonical algorithm
interpolating between the canonical Monte Carlo and molecular dynamics
techniques on a cellular automaton and can be used as an alternative research tool
for Ising model investigations in computational physics. It was shown that the
improved CA algorithms from CCA have successfully produced the critical
behavior of the Ising model on the two, three and higher dimensions [17].
However, the CA algorithms have been used to study the critical behaviors of the
spin-1 Ising models [14, 18]. In contrast to spin-1 Ising models, the spin-3/2
models are not investigated extensively within this framework. The three
dimensional spin-3/2 Ising models have been studied by using the CA algorithms
[11, 15] and some portion of the phase diagrams of the model have been studied
only in the absence of an external magnetic field recently. The remainder of this
paper is organized as follows: The computational details are given in Section 2, the
results are discussed in Section 3 and a conclusion is given in Section 4.
2. COMPUTATIONAL DETAILS
The three-dimensional spin-3/2 BEG model is simulated on the simple cubic
lattice on a cellular automaton. The details of the simulation procedure are given
elsewhere [11, 15]. In the calculations, the cooling algorithm is used. The
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algorithm is divided into two basic parts, initialization procedure and the taking of
measurements. In the initialization procedure, firstly, all spins in the lattice sites
take an ordered structure according to selected (D/J, K/J, h/J) values and the
kinetic energy per site which is equal to the maximum change in the Ising spin
energy for the any spin flip is given to the lattice sites. This configuration is run
during the 10.000 cellular automaton time steps. At the end of the this step, the
configuration in the disordered structure at the high temperature is obtained. In the
next steps, the last configuration in the disordered structure has been chosen as a
starting configuration for the cooling run. Rather than resetting the starting
configuration at each energy, it is used the final configuration at a given energy as
the starting point for the next. During the cooling cycle, energy is subtracted at a
certain amount from the system after 1.000.000 cellular automaton steps.
The simulations are performed on simple cubic lattices with linear dimension
(L) from 12 to 30, containing N = L3 spins, and the periodic boundary conditions
are used to update the lattice configurations. The computed values of the quantities
are averages over the lattice and over the number of time steps (1.000.000) with
discard of the first 100.000 time steps during which the cellular automaton
develops. The calculations are done for K/J = – 1.5 and D/J = –1.5 in the –5 ≤ D/J ≤ 0
and –3.5 ≤ K/J ≤ 0 parameter regions, respectively, in the absence and presence of
an external magnetic field, i.e., h/J = 0 and h/J ≠ 0, and also for K/J = – 0.5 and
D/J = –3.0 in the interval –3 ≤ D/J ≤ 3 and –4 ≤ K/J ≤ 0, respectively, with an
external magnetic field. The magnetic field value is selected in the interval
–0.2 ≤ h/J ≤ 0 in the calculations.
3. RESULTS AND DISCUSSION
The physical quantities of use are the sublattice order parameters and
quadrupolar order parameters (Mα = <Si>α and Qα = <Si2>α, α = A or B) and Qd
order parameter (Qd = < Si2 > A – <Si2 >B). However, the susceptibilities associated
to the Mα and Qd (χα, χd) are studied in order to obtain the phase boundaries.
According to the behavior of the sublattice order parameters, four possible phases
of the ferromagnetic spin-3/2 Ising model are defined: (i) The disordered (d) phase
with MA = MB = 0, QA = QB ≠ 0, (ii) the ferromagnetic (f) phase with MA = MB ≠ 0,
QA = QB ≠ 0, (iii) the ferrimagnetic (fr) phase with MA ≠ MB ≠ 0, QA ≠ QB ≠ 0, and
(iv) the antiquadrupolar (a) phase with MA = MB = 0, QA ≠ QB ≠ 0.
3.1. THE BEHAVIORS OF THE ORDER PARAMETERS
In this section, we study the thermal variations of the sublattice order
parameters and Qd order parameter. According to our analysis, the obtained
behaviors of the order parameters depend on K/J, D/J and h/J values and are given
in the following:
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In Fig.1, the temperature variation of the order parameters are given at L = 18
for selected (K/J, D/J) values in the absence of an external magnetic field. For
K/J = 0 and D/J = – 1.5, the values of the sublattice order parameters (MA, MB) are
equal each other and MA=MB=0 at high temperature. Similarly, the quadrupolar
order parameters are also equal and QA = QB ≠ 0. This means that d phase occurs at
high temperature. While the temperature decreases, the sublattice order parameters
increase from zero continuously and tend to be ground state values, i.e., MA = MB = 1.5
and f phase occurs at low temperature.
Fig. 1 – The temperature dependence of the sublattice order parameters MA, MB, QA, QB at h/J = 0
for a) D/J = –1.5 and K/J = 0, D/J = –1.5 and K/J = –3, b) K/J = –1.5 and D/J = –1, c) D/J = –1.5
and K/J = –1.6, d) K/J = –1.5 and D/J = –4 on a lattice with L = 18. The variation
of the Qd parameter is seen for K/J = –1.5 and D/J = –4 in (d).
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However, the quadrupolar order parameters, QA and QB, change from the value
QA = QB ≠ 0 at high temperature to QA = QB = 2.25 at zero temperature. Similarly,
for K/J = –3 and D/J = –1.5, the order parameters increase from MA = MB = 0 and
QA = QB ≠ 0 to MA = MB = 0.5 and QA = QB = 0.25 with decreasing the temperature.
Hence, for these (K/J, D/J) values, the model exhibits the transition from the d to f
phase (d→f) at h/J = 0 (Fig. 1a).
For K/J = –1.5, D/J = –1 at h/J = 0, the sublattice order parameters increase
from zero (MA = MB = 0) continuously to MA = MB ≠ 0. Hence, there is a transition
from d phase to f phase as temperature decreases. If the temperature continuous to
decline, MA increases and MB decreases to the ground state values, i.e., MA = 1.5
and MB = 0.5 and fr phase occurs. Therefore, the system undergoes two second
order phase transition from d to f phase and from f to fr phase (d→f→fr), seen in
Fig. 1b. The d→f→fr transition is also seen for K/J = –1.6 and D/J= –1.5 at h/J = 0
(Fig. 1c). For this parameter set, the order parameters (Mα and Qα) are discontinuous
at transition temperature for f→fr transition and it is first order. Hence, the system
exhibits d→f→fr transitions in which the d→f transition is second order and f→fr
transition is first order.
For K/J = –1.5, D/J = –4 at h/J = 0, the sublattice order parameters and
quadrupolar order parameter are equal (MA = MB = 0 and QA = QB ≠ 0) at high
temperature. With decreasing temperature, the value of MA and MB remain at zero
value and QA and QB become unequal. This means, the transition from d to a phase
occurs. And then, if decreasing temperature continues, MA and MB increase
continuously to MA = 1.5 and MB = 0.5. Therefore, for this case, the second order
transition takes place from a to fr phase (d→a→fr), seen in Fig. 1d. In this figure,
the Qd order parameter is also seen. It exhibits a continuous behavior which
characterizes the transition from d to a phase.
The data of the sublattice susceptibilities for the d→f→fr and d→f→a
transitions are seen in Fig. 2 for the mentioned parameters in above. For K/J = –1.6
and D/J = –1.5, the susceptibilities show two peaks which belong to the second
order d→f and first order f→fr transitions. Similarly, the susceptibilities exhibit
two peaks for K/J = –1.5 and D/J = –1 which characterize the second order d→f
and f→fr transitions. However, the χd susceptibility shows a peak for f→fr
transitions at the same temperature value of the sublattice susceptibilities, is not
shown in here.
On the other hand, for K/J = –1.5 and D/J = –4, the sublattice susceptibilities
have no peak for d→a and only one peak which belongs the a→fr transition. For
the transition d→a, the value of MA and MB remain at zero and the sublattice
susceptibilities have no peak, as expected. Hence, the Qd order parameter and χd
susceptibility are useful to detect the transition d→a. As is seen inset the figure, the
χd has a peak belongs to d→a transition.
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Fig. 2 – The temperature dependence of the susceptibilities χA,B and χd at h/J = 0 for selected D/J
and K/J values on a lattice with L = 18.
In the following, the effect of the external magnetic field is given. In Fig. 3,
the order parameters are seen for selected (K/J, D/J) parameter sets in the presence
of an external magnetic field. It is obtained that the behavior of the order
parameters is not changing with the applied the magnetic field value in the interval
–0.06 ≤ h/J < 0.
For K/J = –1.8, D/J = –1.5 at h/J = –0.02 and h/J = –0.1, the sublattice order
parameters are shown in Fig. 3a and 3b. From the figures, the system undergoes
two phase transitions in which the first one is a second order from d to f phase and
the second one is a first order from f to fr phase at h/J = –0.02. These behavior are
also seen for h/J = 0. On the other hand, at h/J = –0.1, the transition d→f
disappeared and one transition from d to fr phase occurs. Similarly, while the
second order d→a→fr transition is seen for K/J = –1.5, D/J = –4 at h/J = –0.02
(Fig. 3c) which is the same behavior at h/J=0 (Fig. 1d), there is one transition
occurs from d to fr phase (d→fr) at h/J = –0.1 (Fig. 3d). As expected, while the χd
susceptibility and sublattice susceptibilities have a peak characterize the d→a and
a→fr transition, respectively, for h/J = –0.02, they have a peak belongs to the d→fr
at the same temperature for h/J = –0.1, is seen inset the Fig. 3c and d.
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3.1. PHASE DIAGRAMS
In this study, firstly, two phase diagrams are obtained for selected K/J = –1.5
and D/J = –1.5 values in the absence of an external magnetic field. Secondly, the
phase diagrams are constructed for K/J = –0.5 and –1.5, D/J = –1.5 and –3 in the
presence of an external magnetic field. To obtain the phase boundaries, the finitecritical temperatures are estimated from the maxima of the susceptibilities on the
lattice with L = 18. We calculate the quantities on different lattice size, from 12 to
30, and obtain that the finite lattice temperature does not change significantly after
L = 18. In Fig.4, the obtained phase diagrams are given. As far as we know, these
phase diagrams have gone unnoticed in the other approximations for simple cubic
lattice. The phase diagram for D/J = –1.5 in the (K/J, kT/J) plane contains d, f and
fr phases (Fig. 4a). From our calculations, there is only d→f transition for K/J ≤ –2
and K/J ≥ –0.45 values. The transition d→f→fr occurs in the interval –2< K/J <–0.45.
The phase boundary between d and f phases is a second order line and between f
and fr phases is also a second order line in the interval –2 < K/J ≤ –1.91 and
–0.5 ≤ K/J < –0.45. Moreover, for –1.91 < K/J < –0.52 parameter region, there is a
first order line between f and fr phases. However, the phase diagram also exhibits
two tricritical points (T) at which the transition changes from second order to first
order. This is a new phase diagram topology in this plane, which is either absent
from previous approaches or has gone unnoticed. The phase diagram for K/J = –1.5
in the (D/J, kT/J) plane is given in Fig. 4b. As far as we know, this diagram has
also unnoticed in the other approximation for simple cubic lattice. The boundary
between d and a phases is estimated from the maxima of the χd susceptibility
because of no peak in the sublattice susceptibilities mentioned in above text. It is
obtained that the diagram contains d, a, f and fr phases and there are the transitions
d→a→fr in the interval –5 ≤ D/J < –1.88, d→f→fr in the interval –1.89 < D/J < –0.5
and d→f for D/J ≥ –0.5. However, all boundaries are second order line except for
–1.89 < D/J < –1.18 parameter region. In this region, the d→f→fr transition takes
place and f→fr transition is first order. Moreover, the diagram contains two
tricritical points (T) and a multicritical point (A).
We have also presented the phase diagrams of the model in the presence of
an external magnetic field (Fig. 5–6). According to our calculations, the same
topology are obtained for a magnetic field value in the interval –0.06 ≤ h/J < 0 and
for h/J = 0. In Fig. 5a, (b) and (c), at h/J = –0.02, the phase diagrams for D/J =
–1.5, D/J = –3 and K/J = –0.5 are illustrated. As is seen in figures, the phase
diagram for D/J = –1.5 at h/J = –0.02 are not different from h/J = 0 given in
Fig. 3a. Similarly, for D/J = –3 and K/J = –0.5 at h/J = –0.02, the topology is the
same with those obtained at h/J = 0 given in Ref. 11.
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Fig. 3 – The temperature dependence of the sublattice order parameters MA, MB, QA, QB for D/J= –1.5
and K/J=1.8 at a) h/J = –0.02, b) h/J = –0.1, for K/J = –1.5 and D/J = –4 at c) h/J =–0.02, d) h/J = –0.1
on a lattice with L = 18. The variations of the susceptibilities χA and χd are seen for K/J= –1.5
and D/J = –4 at h/J = –0.02 and h/J = –0.1 in (c) and (d), respectively.
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Fig. 4 – The obtained phase diagram at h/J=0 a) in the (K/J, kT/J) plane for D/J = –1.5, b) in the (D/J,
kT/J) plane for K/J = –1.5. Dashed line with hollow triangles and solid line with solid triangles
represent first and second order phase transitions, respectively.
Finally, we have presented the phase diagrams at h/J = –0.1 in the (K/J, kT/J)
and (D/J, kT/J) planes (Fig. 6). For selected parameters, the similar phase diagram
topologies are obtained except of the first order line appearing for D/J = –1.5 and
K/J = –1.5. The phase diagram for D/J = –1.5 in the (K/J, kT/J) plane and for
K/J = –1.5 in the (D/J, kT/J) plane at h/J = –0.1 have the same topology and there
are the first order line and two tricritical points (Fig. 6a and b). These diagrams are
new phase topology which are either absent or have gone unnoticed. For K/J = –0.5
and D/J = –3, there is only the transition from d to fr phase and the phase boundary
is second order line (Fig. 6c and d). The phase diagrams exhibits no reentrant
behavior. It should be mentioned that the similar topology was presented for
various parameters in the literature [12, 13] except the occurrence of reentrancy.
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Fig. 5 – The obtained phase diagram at h/J = –0.02 a) in the (K/J, kT/J) plane for D/J = –1.5, b)
for D/J = –3, c) in the (D/J, kT/J) plane for K/J = –0.5. Dashed line with hollow triangles represents
first order phase transitions.
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Fig. 6 – The obtained phase diagram at h/J = –0.1 a) D/J = –1.5, b) K/J = –1.5, c) K/J = –0.5,
d) D/J = –3. Dashed lines with hollow triangles represent first order phase transitions.
4. CONCLUSION
In this paper, the ferromagnetic spin-3/2 BEG model is simulated using the
cooling algorithm of the CA on simple cubic lattice. The behaviors of the order
parameters and susceptibilities are studied for selected (K/J, D/J) parameter sets
and the phase diagrams of the model have been obtained in the (K/J, kT/J) and
(D/J, kT/J) planes in the absence and presence of an external magnetic field. The
temperature dependence of the sublattice order parameters (MA,B, QA,B) and the
order parameter Qd and associated susceptibilities (χA,B and χd) are investigated and
the phase diagrams are obtained in the absence and presence of the external
magnetic field. According to the behaviors of the sublattice order parameters, the
model exhibits the different phase transitions such as d→f, d→f→fr, d→a→fr at
h/J=0 and in the interval –0.06≤h/J<0, only d→fr for h/J< –0.06. Also, while the
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d→f and d→a transitions are second order, the f→fr and d→fr transitions can be
second order or first order depending on selected (K/J, D/J) values (Figs. 1–3). The
phase diagrams are obtained by investigating these behaviors of the sublattice order
parameters and the boundaries are estimated from the maxima of the
susceptibilities. From the results, two new topologies of the phase diagrams are
obtained in the absence of the magnetic field and also one new topology in the
presence of the magnetic field. The obtained diagrams in the (K/J, kT/J) plane for
D/J = –1.5 and in the (D/J, kT/J) plane for K/J = –1.5 at h/J = 0 (Fig. 4) have a new
topologies which are either absent or have gone unnoticed. In addition, the phase
diagrams in the (K/J, kT/J) and (D/J, kT/J) planes are constructed for the value h/J
in the interval –0.1 ≤ h/J < 0. It is obtained that the topology of the diagram change
for the value h/J < –0.06 (Figs. 5–6). However, the diagrams for K/J = –1.5 and
D/J = –1.5 at h/J = –0.1 have the similar topology with two tricritical points
(Fig. 6a and b). And also, the similar one has been obtained for K/J = –0.5 and
D/J = –3 and there are only second order line and no reentrant behaviors. The
similar topology of the diagrams for h/J = –0.1 have been presented in literature
[12,13] except the existence of two tricritical points.
Acknowledgements. This work is supported by a grant from Gazi University (BAP:18/2009-04)
REFERENCES
1. J. Sivardière, M. Blume, Phys. Rev. B 5, 1126 (1972).
2. S. Krinsky and D. Mukamel, Phys. Rev. B 11, 399 (1975).
3. A. Bakchich, A. Bassir, and A. Benyoussef, Physica A 195, 188 (1993).
4. T. Kaneyoshi and M. Jascur, Phys. Lett. A 177, 172 (1993).
5. N. Tsushima, Y. Honda, T. Horiguchi, J. Phys. Soc. Japan 66, 3053 (1997).
6. F.C. Sa´ Barreto and O.F. de Alcantara Bonfim, Physica A 172, 378 (1991).
7. A. Bakkali, M. Kerouad and M. Saber, Physica A 229, 563 (1996).
8. J.W. Tucker, J. Magn. Magn. Mater. 214, 121 (2000).
9. M. Keskin, O. Canko, J. Magn. and Magn. Mater. 320, 8 (2008).
10. C. Ekiz, E. Albayrak, M. Keskin, J. Magn. Magn. Mater. 256, 311 (2003).
11. N. Seferoğlu, Commun. Comput. Phys. 7, 779 (2010).
12. C. Ekiz, Phys. Stat. Sol. (b) 241, 1324 (2004).
13. M. Ali Pınar, M. Keskin, A. Erdinc, O. Canko, Ze.Naturforsch. 62a, 127 (2007).
14. B.Kutlu, A. Özkan, N. Seferoğlu, A. Solak, B. Binal, Int. J. Mod. Phys. C 16, 933 (2005).
15. N. Seferoğlu, Commun. Theor. Phys. 56, 168 (2011).
16. M. Creutz, Ann. Phys. Rev. 167, 62 (1986).
17. N. Aktekin, in: D. Stauffer (Ed.), Annual Reviews of Computational Physics, vol. VII, World
Scientific, Singapore, pp. 1–23 (2000); B. Kutlu, Physica A 234, 807 (1997); B. Kutlu, Int. J.
Mod. Phys. C 14, 1305 (2003).
18. N. Seferoğlu, B. Kutlu, J. Stat. Phys. 129, 453 (2007); N. Seferoglu, B. Kutlu, Chin. Phys. Lett.
24, 2040 (2007); N. Seferoğlu, B. Kutlu, Physica A 374, 165 (2007); N. Seferoglu, B. Kutlu,
Centr. Eurp. J. of Phys. 6, 230 (2008); A. Özkan, B. Kutlu, Int. J. of Mod. Phys. C 20, 1617
(2009).