Critical behavior and phase diagrams of a spin-1 Blume-Capel model withrandom crystal field interactions: An effective field theory analysis

Abstract

A spin-1 Blume-Capel model with dilute and random crystal fields is examined
for honeycomb and square lattices by introducing an effective-field
approximation that takes into account the correlations between different spins
that emerge when expanding the identities. For dilute crystal fields, we have
given a detailed exploration of the global phase diagrams of the system in
$k_{B}T_{c}/J-D/J$ plane with the second and first order transitions, as well
as tricritical points. We have also investigated the effect of the random
crystal field distribution characterized by two crystal field parameters $D/J$
and $\triangle/J$ on the phase diagrams of the system. The system exhibits
clear distinctions in qualitative manner with coordination number $q$ for
random crystal fields with $\triangle/J,D/J\neq0$. We have also found that,
under certain conditions, the system may exhibit a number of interesting and
unusual phenomena, such as reentrant behavior of first and second order, as
well as a double reentrance with three successive phase transitions.

Full text

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arXiv:1107.4194v1 [cond-mat.stat-mech] 21 Jul 2011
Critical behavior and phase diagrams of a spin-1 Blume-Capel model with random
crystal field interactions: An effective field theory analysis
Yusuf Y¨ uksel,∗ ¨Umit Akıncı, and Hamza Polat†
Department of Physics, Dokuz Eyl¨ ul University, TR-35160 Izmir, Turkey
(Dated: July 22, 2011)
A spin-1 Blume-Capel model with dilute and random crystal fields is examined for honeycomb and
square lattices by introducing an effective-field approximation that takes into account the correla-
tions between different spins that emerge when expanding the identities. For dilute crystal fields, we
have given a detailed exploration of the global phase diagrams of the system in kBTc/J −D/J plane
with the second and first order transitions, as well as tricritical points. We have also investigated
the effect of the random crystal field distribution characterized by two crystal field parameters D/J
and △/J on the phase diagrams of the system. The system exhibits clear distinctions in qualitative
manner with coordination number q for random crystal fields with △/J,D/J ?= 0. We have also
found that, under certain conditions, the system may exhibit a number of interesting and unusual
phenomena, such as reentrant behavior of first and second order, as well as a double reentrance with
three successive phase transitions.
PACS numbers 75.10.Dg, 75.10.Hk, 75.30.Kz
Contents
I. Introduction
1
II. Formulation
2
III. Results and Discussion
A. Phase diagrams of the system with dilute crystal field
B. Phase diagrams of the system with random crystal field
5
5
7
IV. Conclusions
9
Acknowledgements
9
A. Fundamental correlation functions of the system for a square lattice
10
B. The complete set of twenty one linear equations of a honeycomb lattice
11
References
13
I. INTRODUCTION
Spin-1 Blume-Capel (BC) model1,2is one of the most extensively studied models in statistical mechanics and con-
densed matter physics. The model exhibits a variety of multicritical phenomena such as a phase diagram with ordered
ferromagnetic and disordered paramagnetic phases separated by a transition line that changes from a continuous phase
transition to a first-order transition at a tricritical point. On the other hand, as an extension of the model, BC model
with a random crystal field represents the critical behavior of3He −4He mixtures in a random media, i.e., aerogel
where S = 0 and S = ±1 states represent3He and4He atoms, respectively3,4. From the theoretical point of view,
BC model with a random crystal field (RCF) has been studied by a variety of techniques such as cluster variational
method (CVM)4, Bethe lattice approximation (BLA)5, effective field theory (EFT)6–10, finite cluster approximation
(FCA)11,12, mean field theory (MFT)13–18, Monte Carlo (MC) simulations19, pair approximation (PA)20, and renor-
malization group (RG) method21. Among these studies, EFT and MFT have been widely used to investigate the
thermal and magnetic properties of BC model with a RCF distribution. For example, Kaneyoshi and Mielnicki8inves-
tigated the phase diagram of the system for a honeycomb lattice by using EFT with correlations and they found some
important differences from the results obtained by the standard MFT. Similarly, in a recent paper, Yan and Deng10
considered the same model within the framework of EFT, and they derived the expressions of magnetizations for

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honeycomb and square lattices. On the other hand, in several studies based on MFT,13,15the authors paid attention
to the effects of crystal field dilution on the phase diagrams of the system and they observed that the system may
exhibit a reentrant behavior, as well as first order phase transitions.
However, EFT and MFT studies mentioned above have some unsatisfactory results. Namely, the results obtained
by EFT are limited to second-order phase transitions and tricritical points, and a detailed description of first-order
transitions has not been reported. Other than this, it is well known that magnetic systems with dilute crystal fields
exhibit qualitatively similar characteristics when compared to site dilution problem of magnetic atoms. From this
point of view, for a BC model with diluted crystal fields, MFT predicts that the phase transition temperature of the
system will remain at a finite value until zero concentration is reached. In this context, we believe that BC model
with RCF still deserves particular attention for investigating the proper phase diagrams, especially the first-order
transition lines that include reentrant phase transition regions. Conventional EFT approximations include spin-spin
correlations resulting from the usage of the Van der Waerden identities, and provide results that are superior to those
obtained within the traditional MFT. However, these conventional EFT approximations are not sufficient enough to
improve the results, due to the usage of a decoupling approximation (DA) that neglects the correlations between
different spins that emerge when expanding the identities. Therefore, taking these correlations into consideration will
improve the results of conventional EFT approximations. In order to overcome this point, recently we proposed an
approximation that takes into account the correlations between different spins in the cluster of a considered lattice22.
Namely, an advantage of the approximation method proposed by this study is that no decoupling procedure is used
for the higher-order correlation functions.
In this paper, we intent to investigate the effects of RCF distributions on the phase diagrams of spin-1 BC model
on 2D lattices, namely honeycomb (q = 3) and square (q = 4) lattices. For this purpose, the paper is organized as
follows: In Sec. II we briefly present the formulations. The results and discussions are presented in Sec. III, and
finally Sec. IV contains our conclusions.
II.FORMULATION
In this section, we give the formulation of the present study for a 2D lattice which has N identical spins arranged.
We define a cluster on the lattice which consists of a central spin labeled S0, and q perimeter spins being the nearest
neighbors of the central spin. The cluster consists of (q + 1) spins being independent from the spin operatorˆS. The
nearest-neighbor spins are in an effective field produced by the outer spins, which can be determined by the condition
that the thermal average of the central spin is equal to that of its nearest-neighbor spins. The Hamiltonian describing
our model is
H = −J
?
?i,j?
Sz
iSz
j−
?
i
Di(Sz
i)2, (1)
where the first term is a summation over the nearest-neighbor spins with Sz
summation represents a random crystal field, distributed according to a given probability distribution. In this paper,
we primarily deal with two kinds of probability distributions, namely, a quenched diluted crystal field distribution
and a double peaked delta distribution which are given by Eqs. (2) and (3), respectively as follows
i= ±1,0 and the term Dion the second
P(Di) = pδ(Di− D) + (1 − p)δ(Di),
1
2{δ[Di− (D − △)] + δ[Di− (D + △)]}.
(2)
P(Di) =
(3)
where p denotes the concentration of the spins on the lattice which are influenced by a crystal field D.
We can construct the mathematical background of our model by using the approximated spin correlation identities23
by taking into account random configurational averages
??{fi}Sz
i??r=
??
{fi}Tri(Sz
i)exp(−βHi)
Triexp(−βHi)
??
r
,(4)
??{fi}(Sz
i)2??r=
??
{fi}Tri(Sz
i)2exp(−βHi)
Triexp(−βHi)
??
r
,(5)
where β = 1/kBT, {fi} is an arbitrary function which is independent of the spin variable Siand the inner ?...? and
the outer ?...?rbrackets represents the thermal and random configurational averages, respectively.

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In order to apply the differential operator technique24,25, we should separate the Hamiltonian (1) into two parts as
H = Hi+ H
other part H
′. Here, the effective Hamiltonian Hiincludes all the contributions associated with the site i, and the
′does not depend on the site i.
− Hi= ESz
i+ Di(Sz
i)2,(6)
where E = J?
jSz
jis the local field on the site i. If we use the matrix representations of the operators Sz
for the spin-1 system then we can obtain the matrix form of Eq. (6)
iand (Sz
i)2
− Hi=


E + D 0
0
0
0
00
0 −E + D

. (7)
Hereafter, we apply the differential operator technique in Eqs. (4) and (5) with {fi} = 1. From Eq. (4) we obtain
the following spin identity with thermal and configurational averages of a central spin for a lattice with a coordination
number q as
??Sz
0??r=
??
q?
j=1
?1 + Sz
jsinh(J∇) + (Sz
j)2{cosh(J∇) − 1}?
??
r
F(x)|x=0.(8)
The function F(x) in Eq. (8) is defined by
F(x) =
?
dDiP(Di)f(x,Di),(9)
where
f(x,Di) =
1
?3
2cosh(βx) + e−βDi.
n=1exp(βλn)
2sinh(βx)
3
?
n=1
?ϕn|Sz
i|ϕn?exp(βλn),(10)
=
In Eq. (10), λndenotes the eigenvalues of −Himatrix in Eq. (7), and ϕnrepresents the eigenvectors corresponding
to the eigenvalues λnof −Himatrix. With the help of Eq. (10), and by using the distribution functions defined in
Eqs. (2) and (3), the function F(x) in Eq. (9) can be easily calculated by numerical integration. Hereafter, we will
focus our attention on the construction of the correlation functions, as well as magnetization and quadrupole moment
identities of a honeycomb lattice with q = 3. A brief formulation of the fundamental spin identities for a square lattice
with q = 4 can be found in Appendix A.
By expanding the right-hand side of Eq. (8) for a honeycomb lattice with q = 3, we get the longitudinal magneti-
zation as
mz= ??Sz
0??r = l0+ 3k1??S1??r+ 3(l1− l0)??S2
+6(k2− k1)??S1S2
+k3??S1S2S3??r+ 3(l4− l2)??S1S2S2
+3(k1− 2k2+ k4)??S1S2
+(−l0+ 3l1− 3l3+ l5)??S2
1??r+ 3l2??S1S2??r
2??r+ 3(l0− 2l1+ l3)??S2
1S2
2??r
3??r
2S2
3??r
1S2
2S2
3??r,(11)
We note that, for the sake of simplicity, the superscript z is omitted from the correlation functions on the right-hand
side of Eq. (11). The coefficients in Eq. (11) are defined as follows:
l0= F(0),
l1= cosh(J∇)F(x)|x=0,
l2= sinh2(J∇)F(x)|x=0,
l3= cosh2(J∇)F(x)|x=0,
l4= cosh(J∇)sinh2(J∇)F(x)|x=0,
l5= cosh3(J∇)F(x)|x=0,
k1= sinh(J∇)F(x)|x=0,
k2= cosh(J∇)sinh(J∇)F(x)|x=0,
k3= sinh3(J∇)F(x)|x=0,
k4= cosh2(J∇)sinh(J∇)F(x)|x=0,
(12)

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Next, the average value of the perimeter spin in the system can be written as follows, and it is found as
m1= ??Sz
δ??r= ??1 + Sz
0sinh(J∇) + (Sz
0)2{cosh(J∇) − 1}??rF(x + γ)|x=0, (13)
??S1??r= a1
?1 − ??(Sz
0)2??r
?+ a2??Sz
0??r+ a3??(Sz
0)2??r, (14)
with the coefficients
a1 = F(γ),
a2 = sinh(J∇)F(x + γ)|x=0,
a3 = cosh(J∇)F(x + γ)|x=0,(15)
where γ = (q − 1)A is the effective field produced by the (q − 1) spins outside the system and A is an unknown
parameter to be determined self-consistently. In the effective-field approximation, the number of independent spin
variables describes the considered system. This number is given by the relation ν = ??(Sz
the spin-1 system, 2S = 2 which means that we have to introduce the additional parameters ??(Sz
resulting from the usage of the Van der Waerden identity for the spin-1 Ising system. With the help of Eq. (5),
quadrupolar moment of the central spin can be obtained as follows
i)2S??r. As an example for
0)2??rand ??(Sz
δ)2??r
??(Sz
0)2??r=
??
q?
j=1
?1 + Sz
jsinh(J∇) + (Sz
j)2{cosh(J∇) − 1}???
r
G(x)|x=0, (16)
where the function G(x) is defined as
G(x) =
?
dDiP(Di)g(x,Di). (17)
Definition of the function g(x,Di) in Eq. (17) is given as follows and the expression in Eq. (18) can be evaluated by
using the eigenvalues and corresponding eigenvectors of the effective Hamiltonian matrix in Eq. (7).
g(x,Di) =
1
?3
2cosh(βx) + e−βDi.
n=1exp(βλn)
2cosh(βx)
3
?
n=1
?ϕn|(Sz
i)2|ϕn?exp(βλn),(18)
=
Hence, we get the quadrupolar moment by expanding the right-hand side of Eq. (16)
??(Sz
0)2??r = r0+ 3n1??S1??r+ 3(r1− r0)??S2
+6(n2− n1)??S1S2
+3(r4− r2)??S1S2S2
+(−r0+ 3r1− 3r3+ r5)??S2
1??r+ 3r2??S1S2??r
2??r+ 3(r0− 2r1+ r3)??S2
3??r+ 3(n1− 2n2+ n4)??S1S2
1S2
1S2
2??r+ n3??S1S2S3??r
2S2
3??r
2S2
3??r,(19)
with
r0= G(0),
r1= cosh(J∇)G(x)|x=0,
r2= sinh2(J∇)G(x)|x=0,
r3= cosh2(J∇)G(x)|x=0,
r4= cosh(J∇)sinh2(J∇)G(x)|x=0,
r5= cosh3(J∇)G(x)|x=0,
n1= sinh(J∇)G(x)|x=0,
n2= cosh(J∇)sinh(J∇)G(x)|x=0,
n3= sinh3(J∇)G(x)|x=0,
n4= cosh2(J∇)sinh(J∇)G(x)|x=0,
(20)
Corresponding to Eq. (13),
??(Sz
δ)2??r= ??1 + Sz
0sinh(J∇) + (Sz
0)2{cosh(J∇) − 1}??rG(x + γ),(21)
??S2
1??r= b1
?1 − ??(Sz
0)2??r
?+ b2??Sz
0??r+ b3??(Sz
0)2??r.(22)

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where
b1 = G(γ),
b2 = sinh(J∇)G(x + γ)|x=0,
b3 = cosh(J∇)?G(x + γ)|x=0. (23)
Eqs. (11), (14), (19) and (22) are the fundamental spin identities of the system. When the right-hand sides of Eqs.
(8) and (16) are expanded, the multispin correlation functions appear. The simplest approximation, and one of the
most frequently adopted is to decouple these correlations according to
??Sz
i(Sz
j)2...Sz
l
??
r∼= ??Sz
i??r
??(Sz
j)2??
r...??Sz
l??r,(24)
for i ?= j ?= ... ?= l26. The main difference of the method used in this study from the other approximations in
the literature emerges in comparison with any decoupling approximation (DA) when expanding the right-hand sides
of Eqs. (8) and (16). In other words, one advantage of the approximation method used in this study is that no
uncontrolled decoupling procedure is used for the higher-order correlation functions.
For spin-1 Ising system with q = 3, taking Eqs. (11), (14), (19) and (22) as a basis, we derive a set of linear equations
of the spin identities. At this point, we assume that (i) the correlations depend only on the distance between the
spins, (ii) the average values of a central spin and its nearest-neighbor spin (it is labeled as the perimeter spin) are
equal to each other with the fact that, in the matrix representations of spin operatorˆS, the spin-1 system has the
properties (Sz
Ising system with q = 3 reduces to twenty one, and the complete set is given in Appendix B.
If Eq. (B1) is written in the form of a 21×21 matrix and solved in terms of the variables xi[(i = 1,2,...,21)(e.g.,x1=
??Sz
0??r,x2= ??S1S0??r,...)] of the linear equations, all of the spin correlation functions, as well as magnetizations and
quadrupolar moments can be easily determined as functions of the temperature and Hamiltonian parameters. Since
the thermal and configurational average of the central spin is equal to that of its nearest-neighbor spins within the
present method then the unknown parameter A can be numerically determined by the relation
δ)3= Sz
δand (Sz
δ)4= (Sz
δ)2. Thus, the number of the set of linear equations obtained for the spin-1
??Sz
0??r= ??S1??r
orx1= x4.(25)
By solving Eq. (25) numerically at a given fixed set of Hamiltonian parameters we obtain the parameter A. Then
we use the numerical values of A to obtain the spin correlation functions which can be found from Eq. (B1). Note
that A = 0 is always the root of Eq. (25) corresponding to the disordered state of the system. The nonzero root of A
in Eq. (25) corresponds to the long-range ordered state of the system. Once the spin identities have been evaluated
then we can give the numerical results for the thermal and magnetic properties of the system. Since the effective
field γ is very small in the vicinity of kBTc/J, we can obtain the critical temperature for the fixed set of Hamiltonian
parameters by solving Eq. (25) in the limit of γ → 0 then we can construct the whole phase diagrams of the system.
Depending on the values of Hamiltonian and crystal field distribution parameters, there may be two solutions [i.e.,
two critical temperature values which satisfy Eq. (25)] corresponding to the first (or second) and second-order phase-
transition points, respectively. We determine the type of the transition by looking at the temperature dependence of
magnetization for selected values of system parameters.
III.RESULTS AND DISCUSSION
In this section, we will discuss the effect of the crystal field distributions defined in Eqs. (2) and (3) on the global
phase diagrams of the system where the second and first order transitions are shown by solid and dashed curves,
respectively with tricritical points (shown by hollow circles) for honeycomb (q = 3) and square (q = 4) lattices. Also,
in order to clarify the type of the transitions in the system, we will give the temperature dependence of the order
parameter.
A. Phase diagrams of the system with dilute crystal field
In this section, we illustrate the phase diagrams and magnetization curves of the system with a dilute crystal field
distribution defined in Eq.(2) where crystal field D is turned on, or turned off with probabilities p and (1−p) on the
lattice sites, respectively. In Figs. (1a) and (1c), we plot the phase diagrams of the system in (kBTc/J − D/J) plane
for honeycomb and square lattices with coordination numbers q = 3 and q = 4, respectively. As seen in Figs. (1a)

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and (1c), phase diagrams of the system can be divided into three parts with different concentration values p. For
the curves in the first group with p < p∗, the system always exhibits a second order phase transition with a finite
critical temperature kBTc/J which extent to D/J → −∞. If the concentration p reaches its critical value p∗then the
critical temperature depresses to zero. Physical reason underlying this behavior can be explained as follows; when we
select sufficiently large negative crystal field values (i.e.D/J → −∞), all of the spins in the system tend to align in
S = 0 state. As p increases starting from zero, the ratio of spins which aligned in S = 0 state increases, and therefore,
magnetization weakens, and accordingly, critical temperature of the system decreases. According to our numerical
results, the critical concentration value is obtained as p∗= 0.3795 for q = 3 and p∗= 0.5875 for q = 4. In the second
group of the phase diagrams in Figs. (1a) and (1c), the system exhibits a reentrant behavior of second order, whereas
the curves in the third group, exhibit a reentrant behavior of first order with a tricritical point at which a first order
transition line turns into a second order transition line. Besides, the curves which exhibit a reentrant behavior of
first (or second) order, depress to zero at three successive values of crystal field D/J = −3.0,−2.0,−1.0. Moreover,
in D/J → ∞ limit, the system behaves like spin-1/2 for p = 1.0. In the case of p ?= 0, the ratio of spins that behave
like S = ±1 increases as p increases. Therefore, for 0 ≤ p ≤ 1.0, all transition lines have finite critical temperatures
which increase with increasing p values for D/J → ∞. At this point, we also note that if we select D/J = 0 in Eq.
(2), all lattice sites expose to a crystal field Di/J = 0 independent from p. Hence, all transition lines intersect each
other on the point (D/J,kBTc/J) = (0,1.3022) for q = 3, and (D/J,kBTc/J) = (0,1.9643) for q = 4. Meanwhile,
previous studies based on EFT are not capable of obtaining first order transition lines of the system. From this point
of view, we see that our method improves the results of the other EFT works and we take the conventional EFT
method one step forward by investigating the global phase diagrams, especially the first-order transition lines that
include reentrant phase transition regions.
TABLE I: Critical concentration p∗obtained by several methods
and the present work for honeycomb (q = 3) and square (q = 4)
lattices.
Lattice EFT-I8,9
EFT-II10
MFT13,15
PA20
Present Work
q = 30.4840.492 1.00.5 0.3795
q = 40.6040.6101.00.6670.5875
On the other hand, Figs. (1b) and (1d) shows the phase boundary in (kBTc/J − p) plane which separates the
ferromagnetic and paramagnetic phases with D/J → −∞. According to this figure, critical temperature kBTc/J of
system decreases gradually, and ferromagnetic region gets narrower as p increases, and kBTc/J value depresses to
zero at p = p∗. Such a behavior is an expected fact in dilution problems. Numerical value of critical concentration
p∗for honeycomb (q = 3) and square (q = 4) lattices is given in Table I, and compared with the other works in the
literature. As seen in Table I, numerical values of pcfor q = 3 and q = 4 are new results in literature. Furthermore,
MFT13,15predicts that the system always has a finite critical temperature and exists in a ferromagnetic state at lower
temperatures in D/J → −∞ limit, except that p = 1.0. This artificial result can be regarded as a failure of the MFT.
In Fig.(2), we plot the temperature dependencies of magnetization curves corresponding to the phase diagrams
depicted in Fig. (1) for q = 3. As seen in Fig. (2), as p increases then critical temperature kBTc/J values decrease for
D/J < 0, except the reentrant phase transition temperatures which occur at low temperatures. On the other hand,
effect of increasing p values on the shape of magnetization curves depends on value of D/J. Namely, in Figs. (2a) and
(2b) we see that ground state saturation values of magnetization curves decreases as p increases for D/J = −10.0 and
−3.1. Moreover, for D/J = −3.1, magnetization curves of the system exhibit a broad maximum at low temperatures
for p = 0.37, and a reentrant behavior of second order for p = 0.4. If we select D/J = −2.5 as in Fig. (2c), saturation
values of magnetization curves remain unchanged for p = 0,0.2,0.3 and tend to decrease for p > 0.3. If p increases
further, a reentrant behavior of second order appears for p = 0.53, and we see a broad maximum at low temperatures
for p = 0.517 and 0.5 which tends to depress as p decreases. This broad maximum behavior of magnetization curves
originates from the increase in the number of spins directed parallel to the z-direction with increasing temperature, due
to the thermal agitation. For D/J = −2.0 in Fig. (2d), magnetization curves saturates at m = 1 at the ground state
and reentrant behavior disappears. If we increase D/J further, for example for D/J = −1.5 (Fig. (2e)), another type
of reentrant behavior occurs in the system in which a first order transition is followed by a second order transition.
Finally, for sufficiently large positive values of crystal field, magnetization curves always saturate at m = 1 and the
system always undergoes a second order phase transition from a ferromagnetic phase to a paramagnetic phase with
increasing temperature, which can be seen in Fig. (2f) with D/J = 10.0. As a common property of the curves in
Fig. (2), we see that effect of p on the saturation values, as well as temperature dependence of magnetization curves
strictly depend on the strength of D/J. Hence, according to us, the presence of dilute crystal fields on the system

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should produce a competition effect on the phase diagrams of the system. We also note that, although it has not been
shown in the present work, magnetization curves for q = 4 corresponding to the phase diagrams depicted in Fig. (1c)
exhibit qualitatively similar behavior with those of Fig. (2) with q = 3.
As seen in Fig. (1), for a dilute crystal field distribution defined in Eq. (2), the global phase diagrams which are
plotted in (kBTc/J−D/J) plane, as well as the phase boundaries in (kBTc/J−p) plane for q = 3 exhibit qualitatively
similar characteristics when compared with those for q = 4. Hence, in order to examine the phase diagrams which are
plotted in (kBTc/J − D/J) plane in Figs. (1a) and (1c) in detail, we plot the evolution of the global phase diagrams
in Fig. (3) only for q = 4. From this point of view, Fig. (3a) shows how the phase diagrams in Fig. (1c) evolve
when the concentration p changes from 0.5 to 0.6. As seen in Fig. (3a), we observe a second order phase transition
line with a finite critical temperature kBTc/J which extent to D/J → −∞ for p = 0.575. If p increases, namely
for p = 0.580 and 0.583, we see that a low temperature transition line arises between −4.0 < D/J < −3.0, as well
as a high temperature phase boundary which extents to D/J → −∞. If p increases further, such as for p = 0.584,
0.585 and 0.587, high temperature phase boundary is gradually connected to the transition line which arises between
−4.0 < D/J < −3.0, and the phase diagrams exhibit a bulge on the right hand side of (kBTc/J−D/J) plane, whereas
another transition line emerges within the range of −∞ < D/J < −4.0, which disappears for p > 0.587. Similarly,
evolution of the phase diagrams in Fig. (1c) when the concentration p changes from 0.6 to 0.7 can be seen in Fig.
(3b). As seen in this figure, the curves for p = 0.60, 0.62, 0.64, 0.66 exhibit a reentrant behavior of second order,
while for p = 0.68 reentrance disappears and for p = 0.70 and 0.71 double reentrance with three successive second
order phase transitions occurs in a very narrow region of D/J. On the other hand, increasing values of p generates
first order phase transitions with tricritical points, as well as reentrant behavior of first order. This phenomena is
illustrated in Fig. (3c). From Fig. (3c), we see that, the second order transition temperatures decrease as absolute
value of D/J increases, and turn into first order transition lines at tricritical points. Evidently, the phase diagrams
change abruptly for p ≥ 0.7164. Hence, the behavior of the p = 0.7163 curve is completely different from that of
p = 0.7164. Namely, first order transition temperatures of the system for p ≤ 0.7163 and p ≥ 0.7164 depress to zero
at D/J = −3.0 and D/J = −2.0, respectively. In order to investigate the phase transition features of the system
further, we should continue increasing the value of p. In Fig. (3d), we see that the curves for p = 0.72, 0.74, 0.76
and 0.78 exhibit a reentrant behavior of first order, whereas the curves with p = 0.80, 0.82, and 0.84 exhibit double
reentrance with two first order and a second order transition temperature.
B.Phase diagrams of the system with random crystal field
Next, in order to investigate the effect of the random crystal fields defined in Eq.(3) on the thermal and magnetic
properties of the system, we represent the phase diagrams and corresponding magnetization curves for honeycomb
(q = 3) and square lattices (q = 4) throughout Figs. (4) and (8).
We note that random crystal field distribution given in Eq. (3) with D/J = 0 corresponds to a bimodal distribution
function, while for △/J = 0, we obtain pure BC model with homogenous crystal field D/J. In Fig. (4), phase diagrams
of the system corresponding to the bimodal distribution function are shown in (kBTc/J −△/J) plane. For a bimodal
distribution, the phase diagrams have symmetric shape with respect to △/J which comes from the fact that p = 1/2,
and as seen in Fig. (4), transition temperatures are second order, and it is clear that the system exhibit different
characteristic features depending on the coordination number q. Namely, for q = 3, transition temperature decreases
with increasing △/J and exhibits double reentrance with three second order phase transition temperatures, then
falls to zero at △/J = 3.0 (left panel in Fig. (4)). On the other hand, as seen on the right panel in Fig. (4), as
△/J increases then the transition temperature of the system for q = 4 decreases and remains at a finite value for
△/J → ∞ which means that ferromagnetic exchange interactions for q = 3 are insufficient for the system to keep its
ferromagnetic order for △/J > 3.0, while for q = 4 these interactions are dominant in the system, and the presence
of a disorder in the crystal fields cannot destruct the ferromagnetic order.
At the same time, in order to see the effect of the random crystal fields with △/J,D/J ?= 0 on the phase diagrams
and magnetization curves of the system for q = 3 and 4, we plot the phase diagrams in (kBTc/J − D/J) plane in
Fig. (5) and variation of the corresponding magnetization curves with temperature in Figs. (6) and (7), respectively.
At first sight, it is obvious that the phase diagrams in Fig.(5) represent evident differences in qualitative manner
with coordination number q. That is, as seen in Fig. (5a), the curve corresponding to △/J = 0 represents the
phase diagram of pure BC model for a honeycomb lattice which exhibits a reentrant behavior of first order with first
and second order transition lines, as well as a tricritical point. From Fig. (5a), we see that as △/J increases then
the tricritical point and first order transitions disappear, and the first order reentrance turns into double reentrance
with three transition temperatures of second order, and phase transition lines shift to positive crystal field direction
without changing their shapes. On the other hand, the situation is very different for a square lattice. Namely, as
seen in Figs. (5a) and (5b), △/J = 0 curves for q = 3 and q = 4 are qualitatively identical to each other. However,

Page 8

8
as seen in Fig. (5b), for △/J ?= 0, first order transition lines and tricritical points do not disappear from the system
for q = 4, but shift to negative crystal field values. Besides, the system does not exhibit double reentrance for q = 4.
Furthermore, for △/J ≥ 1.5 in Fig. (5a), and △/J ≥ 0 in Fig. (5b), all phase diagrams exhibit similar behavior as
D/J varies. Namely, critical temperature kBTc/J in Fig. (5a) reduces to zero at D/J = △/J − 3.0. On the other
hand, first order transition temperatures in Fig. (5b), reduce to zero at D/J = −△/J.
It is important to note that these observations are consistent with the results shown in Figs. (1a) and (1c). In
other words, the distribution function given in Eq. (2) with p = 0.5 and D/J = 2D0/J is identical to Eq. (3) for
△/J = ±D0/J and D/J = D0/J. For example, according to Eq. (2), if we select D0/J = 4.0 with p = 0.5, it means
that half of the spins on the lattice sites expose to a crystal field D/J = 0, while a crystal field given by D/J = 8.0
acts on the other half of the spins. On the other hand, if we select △/J = ±D0/J and D/J = D0/J by using Eq. (3),
we generate the same distribution again. Hence, we expect to get the same results in Figs. (1) and (5) for these system
parameters. For instance, for D0/J = 4.0 in Fig. (1a), we get D/J = 8.0, and the system exhibits a ferromagnetic
order in the ground state, which can also be seen in Fig. (5a) with a critical temperature kBTc/J = 1.4395. These
conditions are also valid for q = 4, and for the whole temperature region on the phase diagrams. Therefore, the state
(para-or ferro), as well as thermal and magnetic properties of a selected (kBT/J,D/J) point with respect to p = 0.5
curves in Figs. (1a) and (1c) is identical to the state of a point (kBT/J,D/2J) in Figs. (5a) and (5b) with respect to
the curve △/J = ±D/J, respectively. Moreover, the qualitative differences between Figs. (5a) and (5b) mentioned
above are strongly related to the percolation threshold value of the lattice. Namely, distribution function Eq.(3) is
valid only for p = 0.5. However, as seen in Table I, we obtain pc< 0.5 for q = 3, and pc> 0.5 for q = 4.
In Fig. (6), we examine the temperature dependence of magnetization curves for q = 3, corresponding to the phase
diagrams shown in Fig. (5a) with D/J = −1.0. Fig. (6a), shows how the temperature dependence of magnetization
curves evolve when △/J changes. According to Fig. (6a), magnetization curves saturate at a partially ordered state
at low temperatures. Besides, for △/J = 1.68,1.74 and 1.80, the system undergoes three successive phase transitions
of second order, which confirms the existence of double reentrance. Similarly, Fig. (6b) shows how the shape of the
magnetization curves change as D/J changes for constant △/J = 6.0. As seen in Fig. (6b), magnetization curves
exhibit a second order phase transition from a ferromagnetic (fully ordered) to a paramagnetic phase at certain values
of crystal field, namely at D/J = 4.0,4.1 and 4.4, whereas for D/J = 3.3,3.5,3.7 and 3.9 the system can only achieve
a partially ordered phase. In addition, the curves corresponding to D/J = 3.5,3.7 and 3.9 exhibit a broad maximum
at low temperatures, and then decrease as the temperature increases, whereas for D/J = 3.3, we observe double
reentrance. Fig. (7) shows the magnetization curves for q = 4, corresponding to the phase diagrams shown in Fig.
(5b). In Fig. (7a), we see that magnetization curves exhibit a second order phase transition from a paramagnetic
phase to a fully ordered ferromagnetic phase for △/J = 0 and 5.0. On the other hand, the curves corresponding to
D/J = 5.5, 5.8 and 6.0, saturate at a partially ordered state at low temperatures, and exhibit a broad maximum with
increasing temperature which depresses gradually as △/J increases, then fall rapidly at a second order phase transition
temperature. The broad maximum behavior observed in these curves disappears for △/J = 8.0. Additionally, Fig.
(7b) represents the magnetization versus temperature curves for q = 4 with with D/J = −4.0. In Fig. (7b), it
is clearly evident that, at low temperatures, the system saturates at a partially ordered phase for △/J = 4.0, 5.0,
6.0 and 7.0, while for △/J = 3.7 and 3.8, a reentrant behavior of first order occurs. Again we see that, there is a
competition between ferromagnetic exchange interactions and disorder effects in crystal fields which determines the
saturation values and temperature dependence of magnetization curves of the system.
Finally, dependence of magnetization of the system on the crystal field △/J for fixed temperature values kBT/J =
0.01,0.05,0.1 and 0.2 with D/J = 2.0 is shown in Fig. (8) for q = 3 and 4, respectively. We see that at sufficiently
low temperatures such as kBT/J = 0.01, magnetization curves exhibit three phases for q = 3. A first order transition
is characterized by a gap in this figure. On the left panel in Fig. (8), which is plotted for q = 3, we observe two
successive first order phase transitions for kBT/J = 0.01. The first one is from the fully ordered ferromagnetic phase
(m = 1.0) to the partly ordered phase (m = 0.47), and the other is from partly ordered phase to disordered phase
(m = 0.0). On the other hand, according to the right panel in Fig. (8), the system can not reach a paramagnetic
phase at the ground state for q = 4.Hence, we observe two phases. Namely, for kBT/J = 0.01, a first order phase
transition from a fully ordered phase (m = 1.0) to a partly ordered phase (m = 0.59). Then, saturation magnetization
of the partly ordered phase reduces continuously to (m = 0.527). Moreover, the first order transitions disappear with
increasing temperatures, both for q = 3 and 4. Since in Figs. (5a) and (5b), all phase diagrams exhibit similar
behavior as D/J varies, behavior of magnetization curves in Fig. (8) should be the same for different D/J values.
Namely, the left panel of Fig. (8) can be regarded as the variation of magnetization with △/J within the range
D/J + 1.0 < △/J < D/J + 4.0 for q = 3. for a selected value of D/J. It is possible to observe the similar behavior
on the right panel in Fig. (8). This general behavior is an expected result, since the behavior of the system depends
on the relation between △/J and D/J parameters for a given temperature, not their values independently.

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IV.CONCLUSIONS
In this work, we have studied the phase diagrams of a spin-1 Blume-Capel model with diluted and random crystal
field interactions on two dimensional lattices. We have introduced an effective-field approximation that takes into
account the correlations between different spins in the cluster of a considered lattice and examined the phase diagrams
as well as magnetization curves of the system for different types of crystal field distributions, namely, dilute crystal
fields and a double peaked delta distribution, given by Eqs. (2) and (3), respectively.
For dilute crystal fields, we have given a detailed exploration of the global phase diagrams of the system in kBTc/J−
D/J plane with the second and first order transitions, as well as tricritical points. We have also shown that the system
with dilute crystal fields exhibits a percolation threshold value pcwhich can not be predicted by standard MFA. In
addition, we have observed multi-reentrant phase transitions for specific set of system parameters.
On the other hand, we have investigated the effect of the random crystal field distribution characterized by two
crystal field parameters D/J and △/J on the phase diagrams of the system.
focused on a bimodal distribution with D/J = 0. Particulary, we have reported the following observations for a
bimodal distribution: It has been found that the phase diagrams have symmetric shape with respect to △/J which
comes from the fact that p = 1/2. The transition temperatures are of second order, and the system exhibit different
characteristic features depending on the coordination number q. Besides, we have realized that the system may exhibit
clear distinctions in qualitative manner with coordination number q for random crystal fields with △/J,D/J ?= 0.
Moreover, we have discussed a competition effect which arises from the presence of dilution, as well as random crystal
fields, and we have observed that saturation values of the magnetization curves are strongly related to these effects.
As a result, we can conclude that all of the points mentioned above show that our method improves the conventional
EFT methods based on decoupling approximation. Therefore, we hope that the results obtained in this work may be
beneficial from both theoretical and experimental points of view.
As a limited case, we have also
Acknowledgements
One of the authors (Y.Y.) would like to thank the Scientific and Technological Research Council of Turkey
(T¨UB˙ITAK) for partial financial support.This work has been completed at Dokuz Eyl¨ ul University, Graduate
School of Natural and Applied Sciences. Partial financial support from SRF (Scientific Research Fund) of Dokuz
Eyl¨ ul University (2009.KB.FEN.077) (H.P.) is also acknowledged.

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Appendix A: Fundamental correlation functions of the system for a square lattice
Magnetization of the central spin for a square lattice is given as follows
??Sz
0?? = µ0+ 4c1??S1??r+ 4(µ2− µ0)??S2
+6µ1??S1S2??r+ 12(c2− c1)??S1S2
+6(µ0− 2µ2+ µ3)??S2
+12(µ4− µ1)??S1S2S2
+4(µ5− 3µ3+ 3µ2− µ0)??S2
+4(c5− c3)??S1S2S3S2
+4(c6− 3c4+ 3c2− c1)??S1S2
+(µ0− 4µ2+ 6µ3− 4µ5+ µ7)??S2
1??r
2??r
1S2
3??r+ 12(c4− 2c2+ c1)??S1S2
1S2
4??r+ 6(µ1− 2µ4+ µ6)??S1S2S2
2S2
1S2
2??r+ 4c3??S1S2S3??r
2S2
3??r
2S2
3??r+ µ8??S1S2S3S4??r
3S2
4??r
3S2
4??r
2S2
3S2
4??r,(A1)
where the coefficients are given by
µ0= F(0),
µ1= sinh2(J∇)F(x)|x=0,
µ2= cosh(J∇)F(x)|x=0,
µ3= cosh2(J∇)F(x)|x=0,
µ4= sinh2(J∇)cosh(J∇)F(x)|x=0,
µ5= cosh3(J∇)F(x)|x=0,
µ6= sinh2(J∇)cosh2(J∇)F(x)x=0,
µ7= cosh4(J∇)F(x)|x=0,
µ8= sinh4(J∇)F(x)x=0.
c1= sinh(J∇)F(x)x=0,
c2= sinh(J∇)cosh(J∇)F(x)x=0,
c3= sinh3(J∇)F(x)x=0,
c4= cosh2(J∇)sinh(J∇)F(x)x=0,
c5= sinh3(J∇)cosh(J∇)F(x)x=0,
c6= cosh3(J∇)sinh(J∇)F(x)x=0,
Quadrupolar moment corresponding to equation (B1) defined as
??(Sz
0)2?? = ρ0+ 4η1??S1??r+ 4(ρ2− ρ0)??S2
+6ρ1??S1S2??r+ 12(η2− η1)??S1S2
+6(ρ0− 2ρ2+ ρ3)??S2
+12(ρ4− ρ1)??S1S2S2
+4(ρ5− 3ρ3+ 3ρ2− ρ0)??S2
+4(η5− η3)??S1S2S3S2
+4(η6− 3η4+ 3η2− η1)??S1S2
+(ρ0− 4ρ2+ 6ρ3− 4ρ5+ ρ7)??S2
1??r
2??r
1S2
3??r+ 12(η4− 2η2+ η1)??S1S2
1S2
4??r+ 6(ρ1− 2ρ4+ ρ6)??S1S2S2
2S2
1S2
2??r+ 4η3??S1S2S3??r
2S2
3??r
2S2
3??r+ ρ8??S1S2S3S4??r
3S2
4??r
3S2
4??r
2S2
3S2
4??r, (A2)
where
ρ0= G(0),
ρ1= sinh2(J∇)G(x)|x=0,
ρ2= cosh(J∇)G(x)|x=0,
ρ3= cosh2(J∇)G(x)|x=0,
ρ4= sinh2(J∇)cosh(J∇)G(x)|x=0,
ρ5= cosh3(J∇)G(x)|x=0,
ρ6= sinh2(J∇)cosh2(J∇)G(x)x=0,
ρ7= cosh4(J∇)G(x)|x=0,
ρ8= sinh4(J∇)G(x)x=0.
η1= sinh(J∇)G(x)x=0,
η2= sinh(J∇)cosh(J∇)G(x)x=0,
η3= sinh3(J∇)G(x)x=0,
η4= cosh2(J∇)sinh(J∇)G(x)x=0,
η5= sinh3(J∇)cosh(J∇)G(x)x=0,
η6= cosh3(J∇)sinh(J∇)G(x)x=0,
Finally, perimeter spin identities are as follows
??S1??r = α1(1 − ??(S0)2??r) + α2??S0??r+ α3??(S0)2??r,
??S2
(A3)
(A4)
1??r = ω1+ ω2??S0??r+ (ω3− ω1)??S2
0??r

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with the coefficients
α1= F(γ)
α2= sinh(J∇)F(x + γ)
α3= cosh(J∇)F(x + γ)
ω1= G(γ)|x=0
ω2= sinh(J∇)G(x + γ)|x=0
ω3= cosh(J∇)G(x + γ)|x=0
where γ = (q − 1)A with q = 4, and the functions F(x) and G(x) are defined in equations (9) and (17).
Appendix B: The complete set of twenty one linear equations of a honeycomb lattice
??Sz
0??r = l0+ 3k1??S1??r+ 3(l1− l0)??S2
+6(k2− k1)??S1S2
+k3??S1S2S3??r+ 3(l4− l2)??S1S2S2
+3(k1− 2k2+ k4)??S1S2
+(−l0+ 3l1− 3l3+ l5)??S2
??S1S0??r = (3l1− 2l0)??S1??r+ 3k1??S2
+6(k2− k1)??S2
+(−l0+ 3l1− 3l2− 3l3+ 3l4+ l5)??S1S2
+3(k1− 2k2+ k4)??S2
??S1S2S0??r = (l0− 3l1+ 3l2+ 3l3)??S1S2??r+ (6k2− 3k1)??S1S2
+(−l0+ 3l1− 3l2− 3l3+ 3l4+ l5)??S1S2S2
+(3k1− 6k2+ k3+ 3k4)??S1S2
??S1??r = a1(1 − ??(Sz
??S1S2??r = a1??S1??r+ a2??S0S1??r+ (a3− a1)??S1S2
??S1S2S3??r = a1??S1S2??r+ a2??S0S1S2??r+ (a3− a1)??S1S2S2
??S2
??S1S2
??S2
??S0S2
0??r
??S0S1S2
??S0S2
??S1S2S2
??S1S2
??S2
??S2
+6(n2− n1)??S1S2
+3(r4− r2)??S1S2S2
+(−r0+ 3r1− 3r3+ r5)??S2
??S1S2
+6(n2− n1)??S2
+(−r0+ 3r1− 3r2− 3r3+ 3r4+ r5)??S1S2
+3(n1− 2n2+ n4)??S2
??S2
+6(n2− n1)??S1S2
+(−r0+ 3r1− 3r2− 3r3+ 3r4+ r5)??S2
??S1S2S2
1??r+ 3l2??S1S2??r
2??r+ 3(l0− 2l1+ l3)??S2
1S2
2??r
3??r
2S2
3??r
1S2
1??r+ 3(l0− 2l1+ l2+ l3)??S1S2
2??r+ k3??S1S2S2
2S2
3??r
2??r
1S2
3??r
2S2
3??r
1S2
2S2
3??r
2??r
3??r
2S2
3??r
0)2??r) + a2??Sz
0??r+ a3??(Sz
0)2??r
0??r
0??r
1??r = b1(1 − ??(Sz
2??r = b1??S1??r+ b2??S0S1??r+ (b3− b1)??S1S2
1S2
1??r = b3??S0??r+ b2??S2
2??r = b3??S0S1??r+ b2??S1S2
1S2
3??r = b1??S1S2??r+ b2??S0S1S2??r+ (b3− b1)??S1S2S2
2S2
1S2
0??r = r0+ 3n1??S1??r+ 3(r1− r0)??S2
0)2??r) + b2??Sz
0??r+ b3??(Sz
0)2??r
0??r
1S2
2??r = b1??S2
1??r+ b2??S0S2
1??r+ (b3− b1)??S2
0??r
0??r
0??r
2??r = b3??S0S2
1??r+ b2??S2
1S2
0??r
0??r
2S2
3??r = b1??S1S2
2S2
2??r+ b2??S0S1S2
1S2
2??r+ (b3− b1)??S1S2
1S2
1??r+ 3r2??S1S2??r
2??r+ 3(r0− 2r1+ r3)??S2
3??r+ 3(n1− 2n2+ n4)??S1S2
1S2
1??r+ (3r2+ 3r0− 6r1+ 3r3)??S1S2
1S2
2S2
3??r = b1??S2
2??r+ b2??S0S2
2??r+ (b3− b1)??S2
1S2
0??r
1S2
2??r+ n3??S1S2S3??r
2S2
3??r
2S2
3??r
0??r = (3r1− 2r0)??S1??r+ 3n1??S2
2??r
2??r+ n3??S1S2S2
3??r
2S2
3??r
1S2
2S2
3??r
1S2
0??r = (3r1− 2r0)??S2
1??r+ 3n1??S1??r+ (3r2+ 3r0− 6r1+ 3r3)??S2
2??r+ (3n1− 6n2+ n3+ 3n4)??S1S2
1S2
2??r
2S2
3??r
1S2
2S2
3??r
0??r = (r0− 3r1+ 3r2+ 3r3)??S1S2??r+ (−3n1+ 6n2)??S1S2
2??r

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12
(−r0+ 3r1− 3r2− 3r3+ 3r4+ r5)??S1S2S2
+(3n1− 6n2+ n3+ 3n4)??S1S2
0??r = (r0− 3r1+ 3r2+ 3r3)??S1S2
(−r0+ 3r1− 3r2− 3r3+ 3r4+ r5)??S1S2
+(3n1− 6n2+ n3+ 3n4)??S1S2S2
2S2
+(3n1− 6n2+ n3+ 3n4)??S1S2
+(−r0+ 3r1− 3r2− 3r3+ 3r4+ r5)??S2
3??r
2S2
3??r
??S1S2
2S2
2??r+ (−3n1+ 6n2)??S1S2??r
2S2
3??r
1S2
2S2
1S2
3??r
??S2
1S2
0??r = (r0− 3r1+ 3r2+ 3r3)??S2
2??r+ (−3n1+ 6n2)??S1S2
3??r
2??r
2S2
3??r.
(B1)

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13
∗Also at Dokuz Eyl¨ ul University, Graduate School of Natural and Applied Sciences, Turkey
†Electronic address: hamza.polat@deu.edu.tr
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?
?
?
?
?
??????
??????
?
?
?
?
?
?
?
?????
?
??????
?
?
FIG. 1: (a) Phase diagrams of the system for q = 3 in a (kBTc/J −D/J) plane corresponding to dilute crystal field distribution
defined in Eq. (2). The solid and dashed lines correspond to second- and first-order phase transitions, respectively. The open
circles denote the tricritical points, and the numbers on each curve represent the value of concentration p. (b) Phase diagrams
of the system for q = 3 in a (kBTc/J − p) plane with a selected value of the crystal field D/J = −10.0. (c) Phase diagrams of
the system for q = 4 in a (kBTc/J −D/J) plane corresponding to dilute crystal field distribution defined in Eq. (2). The solid
and dashed lines correspond to second- and first-order phase transition, respectively. The open circles refer to the tricritical
points, and the numbers on each curve represent the value of concentration p. (d) Phase diagrams of the system for q = 4 in a
(kBTc/J − p) plane with a selected value of the crystal field D/J = −10.0.

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FIG. 2: (Color online) Temperature dependence of magnetization corresponding to Fig. (1a) with some selected values of
crystal field. (a) D/J = −10.0, (b) D/J = −3.1, (c) D/J = −2.5, (d) D/J = −2.0, (e) D/J = −1.5, and (f) D/J = 10.0. The
numbers on each curve denote the value of concentration p. The solid and dashed lines correspond to second- and first-order
phase transitions, respectively.