- Email: [email protected]
2 Blume–Capel model
Journal Pre-proof Phase diagrams of the spin-5/2 Blume–Capel model M. Karimou, A.S. de Arruda, M. Godoy
PII: DOI: Reference:
S0378-4371(19)31746-7 https://doi.org/10.1016/j.physa.2019.123096 PHYSA 123096
To appear in:
Physica A
Received date : 10 June 2019 Revised date : 25 September 2019 Please cite this article as: M. Karimou, A.S.d. Arruda and M. Godoy, Phase diagrams of the spin-5/2 Blume–Capel model, Physica A (2019), doi: https://doi.org/10.1016/j.physa.2019.123096. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Published by Elsevier B.V.
Journal Pre-proof Highlights:
- Several phase diagrams were obtained for the spin-5/2 Blume-Capel model.
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- The influence of the random single-ion anisotropy and magnetic field was examined for the system. - First- and second-order phase transition lines were found.
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- The tricritical behavior was examined.
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Phase diagrams of the spin-5/2 Blume-Capel model M. Karimou Ecole Nationale Supérieure de Génie Energétique et
of
Procédés (ENSGEP), Université d’Abomey, Bénin and
Institute of Mathematics and Physical Sciences (IMSP),
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University of Abomey Calavi, Dangbo, Benin∗ A. S. de Arruda and M. Godoy†
Instituto de Física, Universidade Federal de Mato Grosso, 78060-900, Cuiabá, Mato Grosso, Brazil
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Abstract
We have used the Curie-Weiss mean-field approximation to study the effects of the random singleion anisotropy and random magnetic field in the phase diagram and thermodynamic properties of the spin-5/2 Blume-Capel model. The phase diagrams have been presented in the temperature T /J versus strength random single-ion anisotropy D/J and strength random magnetic field h/J planes. The magnetization dependence was plotted as a function of temperature and the strength
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single-ion anisotropy. These diagrams showed that the different phase transition types, first- or second-order between the ferromagnetic and paramagnetic phases, are dependent on the random
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parameters. Therefore, within these conditions, the model presents tricritical behavior.
∗
[email protected]
†
[email protected]
1
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I.
INTRODUCTION
The standard Ising model is undoubtedly the most studied model of magnetism in the literature [1]. Its importance lies in the fact that it has exact solutions in one [2] and two
of
[3] dimensions, which serve as a guide for the production of new analytical approaches. It is well known that the spin-1/2 Ising model represents very well idealized systems, so due
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to the need to understand the behavior of real systems, it becomes necessary to generalize the model, i.e., first taking into account spins greater than spin-1/2, second, introduce new interactions present in the real systems. Thus, several generalizations this model have been made such as the Blume-Capel (BC) model, the Blume-Emery-Griffiths (BEG) [6].
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The BC model was introduced independently by Blume [4] and by Capel [5]. They introduced spins-1 and also included a new anisotropy, know as single-ion anisotropy, to investigate the tricritical behavior. This can be the simplest lattice model which could exhibit a tricritical point (TCP). One of the most interesting features of the BC model is the occurrence of a TCP. A TCP is in the phase diagram and separates a smooth phase transition (known as the second-order phase transitions) of a sudden phase transition (known as the first-order phase transitions). In special, the version of the spin-3/2 BC model was
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introduced to understand the phase transitions in DyV O4 [7–10] and the critical properties of certain mixtures of ternary fluids [11]. Extensions of this model featuring interactions of
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higher-spins spin-3/2 were also considered in the literature [12], where Plascak et al used a mean-field theory via Bogoliubov inequality and studied the critical properties of the BC model for spins-S.
The need to consider large spins in magnetic models is increasing, since many atoms and magnetic alloys have large spins, also besides, has been characterized numerous molecules
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which in the fundamental state presents a very large value of spins [13, 14]. Until then, most studies about the BC model address the critical properties focused on versions with the spin1 and spin-3/2, however, our justification for studying the version of the spin-5/2 BC model is based on the fact that there are several physical realizations such as Rb2 M nF [15, 16] and K2 M nF4 and M n(HCOO)2 2H2 O [17] which, besides being quasi-two-dimensional, present M n atoms with spin-5/2. More recently, high spin materials such as BiM n2 P O6 [18] and BaM n2 O3 [19] were obtained. The material BaM n2 O3 is made up by atoms M n2 + (3d5 ) that have spin-5/2, and on the other hand, the BiM n2 P O6 is an AFM compound with a 3D 2
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topology of magnetic interactions, a significant 1D anisotropy, and a moderate frustration of interladder couplings for long-range AFM order. The critical and thermodynamic properties of the BC model have been investigated in
of
the last decades by a variety of analytical approaches. For example, in the mean-field approximations [12, 20, 21], effective-field theory [22–25], renormalization group [26–29], and Monte Carlo simulations [30–33]. Some articles were used extrapolations of series expansions
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of high and low temperatures [34–39]. Moreover, dynamic magnetic properties of BC model have been thoroughly studied with different dynamic methods [40–50]. Currently, the disordered systems became objects of many investigations, and thus several models were proposed to simulate their physical properties. These models take into account
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various ingredients, such as, random magnetic field [51–54], single-ion anisotropy [55–58], disorders in the couplings between spins [59], and systems formed by two sub-lattices with different spins and different interactions of field crystals [60, 61]. One other option to introduce disorder into the model is to consider transverse and random single-ion anisotropy [62–64] and a transverse and random magnetic field, which transforms the classical model into a quantum model.
Now we are interested in understanding an issue that until then has not been fully ex-
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plained in disordered systems, which refers to the dependence of the phase diagram on the random magnetic fields and a single-ion anisotropy. Therefore, in this work, we are inter-
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ested in to understand how the random single-ion anisotropy and random magnetic field influence the topology of phase diagrams and thermodynamic behavior of the spin-5/2 BC model, which were partially treated in Refs. [53, 54] for the case spin-3/2 BC model. The outline of this work is organized as follows: In Section II, we introduced the spin5/2 BC model and analytical expressions for Gibbs free energy, and the equation of state
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is obtained by used Curie-Weiss mean-field approximation. In Section III, we describe our theoretical results and discuss several phase diagrams. Finally, in Section IV we present our conclusions.
II.
FORMULATION
The goal is to study the effects of the random single-ion anisotropy and random magnetic field in the phase diagrams and the thermodynamic behavior of the spin-5/2 BC model. 3
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Therefore, we considered a system with N spins distributed in the sites of a hypercubic lattice. Each spin is connected with all the others spins by long-range ferromagnetic exchange interaction, and all they are submitted the actions of a random single-ion anisotropy and a random magnetic field. This system can be represented by a generalized BC model which
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is given by following Hamiltonian model:
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N N N X X J X 2 Si Sj + H=− Di Si − Hi Si , 2N i i hi,ji
(1)
where J > 0 is the long-range ferromagnetic exchange interaction which connects all spins Si of the system, Di and Hi are the random single-ion anisotropy and a random magnetic field, respectively. The first sum is performed over all spin pairs in the lattice. Thus, the
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interactions among all the different pairs of spins must be equal to J, and the factor 1/N is only to ensure the thermodynamic limit. The random single-ion anisotropy Di and random external magnetic field Hi are present in all lattice sites. Si are variables of spins that assume six states (Si = ±5/2, ±3/2 and ±1/2). When D < 0, it induces the spins to assume the states with higher spin component and, on the other hand, when D > 0 are favored those states with the lower spin component. The second and third sums are executed on all N
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points of the lattice.
In order to study the phase diagrams and to obtain the thermodynamic properties of the
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model is necessary to calculate the free energy and the magnetization. Instead of we use an approximate solution (mean-field) on a Bravais lattice, it is often interesting to introduce a simple modification on the physical model itself for become it a new problem which can be exactly solved, this approach is known as Curie-Weiss mean-field approximation. Therefore,
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the canonical partition function is given as follows: Z=
X
exp (−βH),
(2)
{Si }
where β = (kB T )−1 , kB is the Boltzmann constant and T is the absolute temperature. Using the Hamiltonian (Eq. (1)), in Z (Eq. (2)) results in Z=
X {Si }
N N N X X X βJ Si Sj − β Di Si2 + β Hi Si . exp 2N i i
hi,ji
4
(3)
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We can to use Si = Sj due to the coupling interaction between the spins to be long-range type, so Eq. (3) is rewritten as X
exp
{Si }
2N
Si
i
#2
−β
N X
Di Si2 + β
i
N X
Hi Si
i
(4)
.
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Z=
" N βJ 21 X
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In the above equation using the Gaussian identity, the quadratic term can be rewritten as exp where
1 =√ π
ˆ
−∞
a1 =
Thus, the Eq. (5) becomes,
2N
Si
i
dx exp −x2 + 2a1 x ,
#2
1 =√ π
βJ 2N
12 X N
ˆ
(5)
Si .
i
∞
−∞
"
dx exp −x2 + 2x
βJ 2N
12 X N i
#
Si .
(6)
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exp
" N βJ 21 X
∞
Pr e-
a21
p p Making x = ( βJN/2)m and dx = ( βJN/2)dm and substituting the new variables in " N βJ 21 X
2N
i
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exp
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Eq. (6), we obtain
Si
#2
βJN 2 dm exp − m × 2 −∞ " #r r 1 N βJN βJN βJ 2 X exp 2 Si m . 2N 2 2 i
1 =√ π
ˆ
∞
(7)
Therefore, the partition function can be written as
Z=
r
βJN 2π
ˆ
+∞
−∞
Y X βJN 2 2 dm exp − eβ (JmSi −Di Si +Hi Si ) . m 2 i
(8)
{Si }
Now, making Jm + Hi = Wi and substituting the values of Si = ±5/2, ±3/2 and ±1/2 5
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in Eq. (8), we obtain Z=
r
βJN 2π
ˆ
+∞
−∞
Jm2 25Di + −R , dm exp −βN 2 4
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where
h β(Jm + H ) i h 3β(Jm + H ) i 1 n i i ln 2 exp(6βDi ) cosh + 2 exp(4βDi ) cosh β 2 2 h 5β(Jm + H ) io i . + 2 cosh 2
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R=
as Z=
r
βJN 2π
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Let us now use the Curie-Weiss approach, therefore, the partition function can be written
ˆ
∞
dm exp[−βN g(T, D, m, H)],
(9)
−∞
where g is the Gibbs free energy per spin (g = G/N ) given by
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1 n 5 3 4βDi g = − ln 2 cosh β(Jm + Hi ) + 2e cosh β(Jm + Hi ) β 2 2 o 2 1 Jm 25 +2e6βDi cosh β(Jm + Hi ) + + Di . 2 2 4
(10)
We introduced the effects of randomness considering that the random single-ion anisotropy is governed by a bimodal probability distribution given by (11)
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P (Di ) = qδ(Di ) + (1 − q)δ(Di − D),
where the first term qδ(Di ) indicates that a amount q of spins are free of the influence of the single-ion anisotropy, and the second term (1 − q)δ(Di − D) indicates that (1 − q) spins
are under influence of the single-ion anisotropy, where D is the magnitude this anisotropy.
The average over the disorder in Gibbs free energy is calculated as: g1 =
ˆ
g P (Di )dDi , 6
(12)
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which results in
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5 3 q n g1 = − ln 2 cosh β(Jm + Hi ) + 2 cosh β(Jm + Hi ) β 2 2 o 1 +2 cosh β(Jm + Hi ) 2 (1 − q) n 5 3 2βD − ln 2 cosh β(Jm + Hi ) + 2e cosh β(Jm + Hi ) β 2 2 o 2 Jm 25 1 + + D(1 − q). +2e2βD cosh β(Jm + Hi ) 2 2 4
(13)
The effects of randomness due to the random external magnetic field is introduced considering that such field is governed by a trimodal probability distribution given by (1 − p) [δ(Hi + H) + δ(Hi − H)], 2
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P (Hi ) = pδ(Hi ) +
(14)
where the first term indicates that a portion p of spins are free of the influence this field, i.e, H = 0. The two other terms indicate that (1 − p)/2 of the spins are under the influence of the magnetic field along +H and −H directions.
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Therefore, the average over the disordered magnetic is given by
which results in
g1 P (Hi )dHi ,
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g2 =
ˆ
pq (1 − q)p (1 − p)q ln f (0, y) − ln f (x, y) − [ln f (0, y− ) + ln f (0, y+ )] β β 2β (1 − p)(1 − q) 1 25 − [ln f (x, y− ) + ln f (x, y+ )] + Jm2 + D(1 − q), 2β 2 4
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g2 = −
where we have defined the following recurrent functions, 1 3 5 4x f (x, y) = 2e cosh y + 2e cosh y + 2 cosh y , 2 2 2 1 3 5 6x 4x f (x, y− ) = 2e cosh βy− + 2e cosh βy− + 2 cosh βy− , 2 2 2 1 3 5 6x 4x f (x, y+ ) = 2e cosh βy+ + 2e cosh βy+ + 2 cosh βy+ , 2 2 2 6x
where x = βD, y = βJm, y− = (Jm − H) and y+ = (Jm + H). 7
(15)
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The Eq. (15) represents the complete free energy of the generalized BC model with random single-ion anisotropy and random magnetic field. The magnetization can be obtained by minimizing the free energy in relation to variable m, i.e., ∂g2 /∂m = 0, which results
q(1 − p) [G(0, y+ ) + G(0, y− )] , 2
where we also define the recurrent functions:
(16)
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+
(1 − p)(1 − q) [G(x, y+ ) + G(x, y− )] 2
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m = pqG(0, y) + p(1 − p)G(x, y) +
5 sinh(5y/2) + 3e4βx sinh(3y/2) + e6βx sinh(y/2) , 2 cosh(5y/2) + 2e4βx cosh(3y/2) + 2e6βx cosh(y/2) 5 sinh(5y+ ) + 3e4βx sinh(3y+ ) + e6βx sinh(y+ ) , G(x, y+ ) = 2 cosh(5y+ ) + 2e4βx cosh(3y+ ) + 2e6βx cosh(y+ ) 5 sinh(5y− ) + 3e4βx sinh(3y− ) + e6βx sinh(y− ) G(x, y− ) = , 2 cosh(y− ) + 2e4βx cosh(3y− ) + 2e6βx cosh(y− )
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G(x, y) =
where x = βD, y = Km, y− = (Km − h)/2, y+ = (Km + h)/2, h = H/kB T and K =
J/kB T . Thus, the statistical mechanics method together with the Curie-Weiss mean-field approximation provided the fundamental equation given by the Gibbs free energy (Eq. (15))
RESULTS AND DISCUSSIONS
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III.
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and the equation of state given by the magnetization (Eq. (16)).
In this section, we present the numerical solutions via Newton-Raphson method of Eqs. (15) and (16), which provide the phase diagrams in the D/J −T /J and h/J −T /J planes
in addition to the thermal properties of the spin-5/2 BC model with random anisotropy and
A.
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random magnetic field.
Ground-state
We used the mean-field theory (version Curie-Weiss) to study the ground-state phase diagram of a spin-5/2 BC model by varying the physical parameter random single-ion anisotropy and random magnetic field. For T /J = 0 (ground-state) the thermal fluctuations disappear, but the quantum fluctuations persist, which can lead the system to phase transitions. The ground-state diagram was determined from the Hamiltonian (Eq. (1)) comparing the energies 8
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(3 /
2,
(5 /2 ,3
(5/2, 5/2)
1/
2)
(5/2, 1/2)
2,
10
(3 /
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(5/2, - 1/2)
5
) /2
-3
(5/2, - 5/2)
(3 /2 ,
, /2
-1
(5
/2 )
h/|J|
3/ 2)
/2 )
15
(a)
p ro
(1/2, 1/2)
(3/2, - 3/2)
0 -1
0
1
(1/2, - 1/2)
2
3
4
6
8
D/|J|
/2 (3
,1
/2 )
(5
/2
(5/2, 5/2)
,3
,3 /2 )
/2
)
15
/2
(3
Pr e,-
1/ 2)
(5/2, - 1/2)
,/2
(5
(3
/2
h/|J|
(5/2, 1/2)
10
2) 3/
5 (5/2, - 5/2)
(b) 0 -2
(1/2, 1/2)
(1/2, - 1/2)
(3/2, - 3/2)
0
2
4
D/|J|
/2 ) /2 ,3
,-
/2
2)
/2 ,
1/ 2)
2)
(1/2, 1/2)
(1/2, - 1/2)
4
8
12
D/|J|
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( c ) (3/2, - 3/2) 0
3/
0
(3 /2 ,-
(5
(5/2, - 5/2)
5
1/
(5/2, 1/2)
(5/2, - 1/2)
(3
10
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h/|J|
(3
(5
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(5/2, 5/2)
/2
,3
/2 )
15
Figure 1: (Color online) Ground-state diagram of the spin-5/2 BC model. The black-solid lines represent first-order transitions for p = q = 0 and the red-solid lines represent first-order transition for p = 0: (a) q = 0.25, (b) q = 0.50, and (c) q = 0.75.
of the different possible states for a pairs of spins {Sa , Sb }, see Fig. 1. At zero temperature,
the structure of the ground-state for this system consists of nine ordered phases (ferromag9
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netic phase) defined as O2 ≡
5 3 ,− , 2 2
O6 ≡
5 3 , , 2 2
O3 ≡
3 3 , , 2 2
O7 ≡
O9 ≡
1 1 , , 2 2
5 1 , , 2 2
O4 ≡
3 1 , , 2 2
5 1 ,− , 2 2
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O5 ≡
5 5 , , 2 2
O8 ≡
3 1 ,− , 2 2
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O1 ≡
and three disordered phases (paramagnetic phase) defined as 5 5 ,− , 2 2
D2 ≡
3 3 ,− , 2 2
D3 ≡
1 1 ,− , 2 2
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D1 ≡
where the numbers ±5/2, ±3/2 and ±1/2 represent the different states. The ground-state energies are given by
D h E = Sa Sb + (1 − q)(Sa2 + Sb2 ) − (1 − p)(Sa + Sb ), J J J
(17)
where the values of Sa , Sb , q, p and D are used to obtain the energies for each state. All
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the phases are separated by first-order transition lines. The black-solid lines are for the case p = q = 0 and the red-solid lines for the cases p = 0: q = 0.25 (Fig. 1(a)), q = 0.50
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(Fig. 1(b)), and q = 0.75 (Fig. 1(c)). We can also observe that the phase boundaries are modified with the dilution (changing p and q) of anisotropy D using Eq. (11). Hence, our results, for p = 0 and different values of q (see Fig. 1), suggest an increase of some disordered (D1 , D2 ) and ordered (O1 , O2 , O3 , O4 , O5 ) phases and a decrease of other disordered (D3 ) and ordered (O6 , O7 , O8 , O9 ) phases, changing the topology of the phase diagram related
B.
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to the case p = q = 0.
Special cases
i - The first special case corresponds to the pure spin-5/2 Ising model which is obtained making p = 1.0 and q = 1.0 in Eq. (16), the magnetization is given by m=
[5 sinh(5m/2τ ) + 3 sinh(3m/2τ ) + sinh(m/2τ )] , [2 cosh(5m/2τ ) + 2 cosh(3m/2τ ) + 2 cosh(m/2τ )] 10
(18)
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5
h/J= 0.0 p = 1.0
h/J = 0.2 p = 0.1
4
T/J
T/J
4
3
0.8 0.7
2
0.6
2
p ro
0.5 0.4 0.3
1
q = 0.0
0 D/J
(a)
1
2
0 -1
Pr e-
-1
0.6 0.5 0.4 0.3
1
0.2 0.1
0 -2
1.0 0.9 0.8 0.7
of
3 1.0 0.9
0.2 0.1 q = 0.0
0
1 D/J
(b)
Figure 2: (Color online) Phase diagram in the D/J − T /J plane of the spin-5/2 BC model for several values of parameter q, as indicated in the figures. All the solid lines denote second-order phase transitions. (a) For h/J = 0 and p = 1.0, (b) for h/J = 0.2 and p = 0.1.
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where τ = kB T /J. In limit of m → 0 we obtained the critical temperature, Tc /J = 2.9166, axis.
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which is represented in the phase diagram of Fig. 2(a) by parallel line (q = 1) to the D/J
ii - Another special case corresponds to the pure spin-5/2 BC model simply by making p = 1.0 and q = 0 in Eq. (15) and Eq. (16), which corresponds to the case h = 0 and the single-ion anisotropy D is present in all points of the lattice. In this case the free energy
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and the magnetization are given by 1 1 3 5 6βD 4βD g2 = − ln 2e cosh βJm + 2e cosh βJm + 2 cosh βJm β 2 2 2 1 25 + Jm2 − D, 2 4
(19)
and 6βD e sinh 21 βJm + 3e4βD sinh 32 βJm + 5 sinh 25 βJm . m = 6βD 2e cosh 21 βJm + 2e4βD cosh 32 βJm + 2 cosh 25 βJm 11
(20)
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When m → 0, the critical temperature is given by
kB Tc 1 e6βD + 9e4βD + 25 = . J 4 e6βD + e4βD + 1
(21)
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The numerical solution of Eqs. (21) provide a continuous line indicating a second-order phase transition from ferromagnetic to paramagnetic phase. This line is represented in the phase diagram in Fig. 2(a) corresponding to line q = 0, which is the same phase diagram
C.
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obtained by Plascak et al [12].
Phase diagrams in the D/J − T /J plane
Pr e-
Now, we have considered the case p = 1.0 which indicates absence of random magnetic field for all points of the lattice, and q 6= 0 indicates that when q increase, an amount of spins are free of influence of the random single-ion anisotropy. For this case, the magnetization is given by
[5 sinh(5m/2τ ) + 3 sinh(3m/2τ ) + sinh(m/2τ )] [2 cosh(5m/2τ ) + 2 cosh(3m/2τ ) + 2 cosh(m/2τ )] 6βD e sinh 21 βJm + 3e4βD sinh 32 βJm + 5 sinh 52 βJm . + (1 − q) 6βD 2e cosh 21 βJm + 2e4βD cosh 23 βJm + 2 cosh 52 βJm
(22)
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m=q
When T tends to Tc (T → Tc ) the magnetization tends to zero (m → 0), therefore,
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expanding around Tc we obtain the following expression for the critical temperature: kB Tc 35q 1 (1 − q) e6βD + 9e4βD + 25 = + . J 12 4 e6βD + e4βD + 1
(23)
The numerical solution of Eqs. (23) provides the second-order phase transition lines that
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are displayed in Fig. 2(a). This phase diagram displays only second-order phase transition lines between the ferromagnetic (below of the curves) and paramagnetic phases (above of the curves). The line q = 0 indicates that the system corresponds to the pure spin-5/2 BC model, i.e, all the spins on the lattice are under the action of the anisotropy D/J. The case p = 1.0 and q = 1.0 is the line parallel to the axis D/J, and represents the pure spin-5/2 Ising model. Here, all others lines start in high temperature from the negative side of D/J and decrease until they cross at D/J = 0 and the critical temperature of the pure model, Tc /J = 2.9166. Then the curves decay until reaching a constant temperature in the region 12
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of positive large values of D/J. The increase of q indicates that a number q of spins are free from the influences of the anisotropy D/J, which allows the system to reach a constant critical temperature at a higher anisotropy value. These results for q 6= 0 have not been
published and it represents our first contribution to the study of the spin-5/2 BC model with
of
random single-ion anisotropy. Fig. 2(b) (phase diagram in the D/J − T /J plane) shows the case p = 0.1, where only a small number (ten percent) of spins are free of the magnetic field.
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The remaining portion (ninety percent) of spins are under the action of the magnetic field +h and −h with magnitude h/J = 0.2 and with the same probability (1 − p). In this case, the system presents the same behavior shown in Fig. 2(a), but now the random magnetic
field is responsible by lowering the critical temperatures, as shown in D/J = 0 for all q.
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Thus, a small number of free spins of the effects of magnetic field with small magnitude is not enough to dramatically change the critical behavior of the system, i.e, the system still does not present tricritical behavior. 3
3
q = 0.0
p = 0.1
2,5
T/J
T/J
2,5
2
p = 0.0 q = 0.0
h/J = 0.2 h/J = 0.6
2
h/J = 1.0
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h/J = 0.8
1,5
1,5 h/J = 1.0
h/J = 0.4
0,5
0
0
0,1
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1
0,2
0,3
0,4
1
0,5 h/J = 0.2 h/J = 0.3
0,5
0
0
0,1
0,2
0,3
0,4
0,5
0,6
D/J
D/J
(a)
(b)
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Figure 3: Phase diagram in the D/J − T /J plane of the spin-5/2 BC model for several values of the magnetic field h/J, as indicated in the figures. The solid and dotted lines denote second- and first-order phase transitions, respectively. The full black dots are the TCPs. (a) For q = 0 and p = 0.1, and (b) for p = q = 0. In Fig. 3(a), we have shown some detail of Fig. 2(b), for instance, the line q = 0 and p = 0.1 is shown for different values of the magnetic field. The line for h/J = 0.2 is reproduced in the two-phase diagrams (Fig. 3(a) and Fig. 2(b)). When the magnitude of the magnetic field is increased the critical behavior changes drastically, i.e., the first-order 13
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phase transition lines (dotted lines) appear for values of h/J > 0.3. The magnitude of the random magnetic field is enough to impose the tricritical behavior, i.e., the random magnetic field tends to disturb the system faster, so close to Tc the magnetization goes to zero discontinuously. In summary, at high temperatures,the phase transition is second-order
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(continuous lines) for all magnitudes of the random magnetic field, but at low temperatures, the phase transition is first-order for h/J > 0.3, and a TCP (full black dot) separates the
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second- and first-order phase transition.
Now, in Fig. 3(b) we have presented the phase diagram in the D/J − T /J plane for p = 0
(which indicates that all spins of the lattice suffer the action of the random magnetic field with magnitude +h or −h, with the same probability (1 − p)/2) and q = 0, i.e., all spins in
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the sites of the lattice are influenced by random single-ion anisotropy with magnitude given by D/J. Here, the tricritical behavior appears in the range of 0.2 < h/J < 1.0. Therefore, in the region of high temperatures the phase transitions between the ferromagnetic and paramagnetic phases are of second-order for values of the random magnetic field smaller than the one (h/J < 1.0), and in this range it is of first-order in regions of low temperatures for (h/J > 0.2). The TCPs separate these phase transitions. When h/J = 1.0 the phase transition is only of first-order in all regions (low and high temperatures). This fact indicates critical temperature.
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that the random magnetic field with large magnitude causes the system to reach quickly the
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To conclude, it can be noted that the phase diagrams in the D/J − T /J plane show
tricritical behavior only at low temperatures and for an applied magnetic field with magnitude h/J > 0.2. On the other hand, in the high temperatures regions, for any magnetic
field h/J < 1.0, the phase diagrams show that the phase transitions from the ferromagnetic
D.
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phase to the paramagnetic phase are only of second-order.
Phase diagram in the h/J − T /J plane
The phase diagrams, in the h/J − T /J plane, are obtained considering the numerical
solutions of the general expression for the critical temperature given by Eqs. (16). Fig. 4(a)
presents the case q = 1.0 (or D/J = 0) and p 6= 0 indicating that there are a random magnetic field in all sites of the lattice, therefore, this case refers to the spin-5/2 Ising model
in random magnetic field. Here, all phase transition lines start in h/J = 0 at the critical 14
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3
3 D/J = 0.0 q = 1.0
p = 0.0
D/J = 0.0
2,5
q = 0.0
2,5
0.1
p = 0.6
2
1,5
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T/J
T/J
0.2
2
1,5 p = 0.3
1
p = 0.0
0,5
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1 p = 0.2
0,5
0.7
0
0
0,5
1
1,5
2
2,5
3
h/J
0
1
0,5
1,5
h/J
(b)
Pr e-
(a)
0
Figure 4: Phase diagram in the h/J − T /J plane of the spin-5/2 BC model. (a) For several values of the parameter p as indicated in the figure, D/J = 0, and q = 1.0. (b) For several values of the anisotropy D/J as indicated in the figure and p = q = 0. The solid and dotted lines denote second- and first-order phase transitions, respectively. The black dot points are the TCPs. temperature (Tc /J = 2.9166) of the pure spin-5/2 Ising model and decreases when h/J
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increase. The lines p = 0 and p = 0.2 exhibit second-order transition (solid lines) in the region of high temperature and lower magnetic field. For lower temperatures and higher
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magnetic field the phase transitions are of first-order (dotted lines). When the magnetic field increase, it induces a first-order phase transition in the low-temperature region. This increase causes a rapid decrease at the critical temperature and it will goes to zero at hc /J = 1.55 and hc /J = 1.65. This result is expected because the random magnetic field tends to favor the disorder in the system and, therefore, in this way, the system reaches
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the disorder more quickly in lower temperatures. The two full-black dots represent TCPs whose coordinates are (Tc /J = 2.141, h/J = 0.929) and (Tc /J = 1.753, h/J = 1.199 ) for p = 0 and p = 0.2, respectively. These points separate the two-phase transition types, the second-order phase transition from the first-order phase transition. The lines with p > 0.2 show only second-order phase transition, i.e., the tricritical behavior disappears. All the lines start in h/J = 0 and Tc /J = 2.9166 and tend to a constant and very large value of h/J. In Fig. 4(b), we exhibited the phase diagram for the case p = q = 0 and three different 15
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values of anisotropy D/J. This case, we have a spin-5/2 BC model in a random magnetic field typical. Here all the spins are under the action of a magnetic field ±h/J with the same probability and under the action of the single-ion anisotropy D/J. The curves with
D/J = 0, 0.1 and 0.2 show that the system change from ferromagnetic to paramagnetic
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phase at a second-order phase transition (solid lines) in the region of high temperatures, and at first-order phase transition in low-temperature regions which are separated by TCPs.
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For D/J = 0.7, we can observe only the first-order phase transition in the region of low temperatures. In summary, the increase of the magnetic field makes the system becomes disordered because the randomness favors the disorder. These curves also show that the single-ion anisotropy D also contributes to the disorder of the system, leading to the phase
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transitions at lower temperatures. Therefore, the model presents a tricritical behavior for D/J < 0.7.
Thermodynamic properties
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E.
(b)
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(a)
Figure 5: Magnetization m as a function of temperature T /J. (a) For different values of q as indicated in the figure, D/J = 0.1, h/J = 0, and p = 1.0. (b) For different values of D/J as indicated in the figure, h/J = 0, p = 1.0, and q = 0. It is well known that the Mermin-Wagner’s theorem states that a two-dimensional isotropic system does not maintain long-range order. Therefore, this fact guarantees the importance of the anisotropies, in special, the single-ion anisotropy that plays a very important role in stabilizing of the long-range order in systems with low dimensionality [65, 66]. Here, 16
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our goal is to study the behavior of the magnetization in the first- and second-order phase transition regions, to verify the phase diagrams obtained in the previous subsection. In this model, the exchange interaction is responsible for favoring the magnetic order. On the contrary, the temperature, the random single-ion anisotropy and the random magnetic field
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favor disordering in the system. Therefore, we have restricted our study to the behavior of the magnetization in relation the temperature only for some select values of the parameters
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D/J, h/J, q and p.
The temperature T /J dependence of magnetization m is given by numerical solutions of Eq. (16). Fig. 5(a) shows the variation of magnetization m as a function of temperature T /J for the case p = 1.0, h/J = 0, where the region was chosen with D/J = 0.1. In this
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region, the magnetization presents a standard behavior for all values of q, i.e., q = 0 to q = 1.0. By standard behavior, it is understood that the magnetization is maximum at T /J = 0 and continuously decreases when the temperature increases and reaches the value zero in a critical temperature (Tc ). This behavior confirms the phase diagram of Fig. 2(a), since observing the line q = 0 and the single-ion anisotropy with the magnitude D/J = 0.1, it can be noted that the phase transition is of second-order for all q. In Fig. 5(b), we have shown the variation of magnetization m as a function of temperature
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T /J for the case p = 1.0, h/J = 0, q = 0 and for different values of D/J. This plot also presents the standard behavior of the magnetization which is characteristic of the region
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with a second-order phase transition. The line D/J = 0 corresponds to the case of the pure spin-5/2 BC model, where m is maximum in T /J = 0 and go to zero in Tc /J = 2.9166. The plot shows that the increase of D/J makes the magnetization goes faster to zero, i.e., indicating that the system goes faster to disordered state at a lower critical temperature. This case corresponds to line q = 0 of Fig. 2(a) for D/J values ranging from zero to 0.7
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indicating only second-order phase transition. The plot presented in Fig. 6(a) shows the standard behavior of the magnetization varying with the temperature in a special case with p = 0, q = 1.0 and D/J = 0, but now for several values of the magnetic field h/J. The line h/J = 0 corresponds to the case of the pure spin-5/2 BC model. Increasing the magnitude of the random magnetic field leads the magnetization to zero at a lower critical temperature. This case corresponds to high temperature region (with h/J < 1.0) and the line p = 0 in Fig. 4(a), which shows only second-order phase transition. On the other hand, the plot exhibited in Fig. 6(b) presents 17
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q = 1.0
p = 0.0 q = 1.0
2,5
2,5
h/J = 0.5 D/J = 0.0
D/J = 0.0
2
1,5
1,5
0.0
1
0.5 0.7
0,5
p ro
1
0
0
1
2
3
(a)
0
Pr e-
T/J
1.0 0.9 0.6 0.3
0,5
0.9 h/J = 0.95
0
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m
m
2
p = 0.0
1
2
3
T/J
(b)
Figure 6: Magnetization m as a function of temperature T /J. (a) For different values of h/J as indicated in the figure, D/J = 0, p = 0, and q = 1.0. (b) For different values of p as indicated in the figure, h/J = 0.5, D/J = 0, and q = 1.0.
the standard behavior of the magnetization in the region with second-order phase transition (case q = 1.0, h/J = 0.5 and D/J = 0). The line p = 1.0 corresponds to the case of
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the pure spin-5/2 BC model. The increase in the strength random magnetic field leads the magnetization quickly to zero. This plot can be seen as a measure of magnetization in a
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fixed magnetic field h/J = 0.5 and by varying the parameter p, which indicates the degree of randomness. Thus, this plot also confirms the information shown in Fig. 4(a), i.e., taking a line parallel to the T /J axis by cutting the h/J axis in h/J = 0.5, which crosses several lines corresponding to all values of p. In Fig. 7, we have displayed the variation of magnetization m versus single-ion anisotropy
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D/J for two cases: i) q = 0, h/J = 0.3 for p = 0 with temperature T /J = 0.03, and ii) for p = 0.4 with T /J = 0.1. It is observed that at low temperatures, the magnetization goes to well-defined values (in the first case, for T /J = 0.03 and p = 0 represented by solid line), i.e., the magnetization m starts at 2.5 and goes to 2.0, from 2.0 to 1.5, 1.5 to 1.0 and 1.0 to 0.5. In this case, all spins are under the action of the single-ion anisotropy and random magnetic field. Thus, this result is expected because at low temperatures the anisotropy easily favors the spins to assume the states ±5/2, ±3/2 and ±1/2, consequently generating
well defined values of magnetization. On the other hand, the random magnetic field, in this 18
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2.5
q = 0.0
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h/J = 0.3
m
2
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1.5 p= 0.0 ; T= 0.03
1 p= 0.4 ; T= 0.1
0 0
0.2
Pr e-
0.5
0.4
0.6
0.8
1.0
D/J
Figure 7: Magnetization m as a function of the anisotropy D/J for two different values of p and T , as indicated in the figure. Here, h/J = 0.3 and q = 0.
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case, induces magnetizations m = 2.0 and m = 1.0. Then, when the temperature increases, the possible number of values for magnetization also increases from m = 0 to m = 5/2.
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However, in the second case (p = 0.4 with T /J = 0.1) there is still forty percent of the spins under the action of the random magnetic field, but it is possible to observe that the values of magnetization no longer have their values very well defined.
CONCLUSIONS
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IV.
The main purpose of this paper was to study the phase diagram and the thermodynamic properties of the spin-5/2 BC model on the random magnetic field and a random single-ion anisotropy. The free energy and the equation of state were calculated using the mean-field approximation in the version Curie-Weiss. Therefore, as first results we rescue the case pure spin-5/2 BC model that was studied by Plascak et al [12], where the phase diagram (D/J − T /J plane) obtained by them presents only second-order phase transition and that corresponds to our line q = 0 in Fig. 2(a). Then, we can conclude that the phase diagram 19
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of this model does not present TCPs, i.e., there is no tricritical behavior. This behaviour persists in two situations: the first is even when the randomness is introduced (increase q) as shown in the phase diagram of Fig. 2(a). The increase of q indicates that a larger amount of spins are free of the influence of anisotropy D/J, thus the line p = 1 corresponds
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to the pure spin-5/2 Ising model whose critical temperature is Tc /J = 2.9166. The second situation is when all spins are under the action of the random anisotropy and ninety percent
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they are with a small fixed random magnetic field h ≤ 0.3. Here also the system does not
present tricritical behavior as shown in Fig. 2(b) and Fig. 3(a). The tricritical behavior appears for values of h/J > 0.3, as shown in Fig. 3(a). In Fig. 3(b), with all spins under the actions of the anisotropy and the random magnetic field, the tricritical behavior appears
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with h/J > 0.2, and for the field h/J ≥ 1.0 the phase transition is only of first-order.
Now, in the h/J − T /J plane, that is keeping the fixed magnitude of the single-ion
anisotropy and varying the random magnetic field. In this case, for D/J = 0 the tricritical behavior appears only when more than seventy percent of spins in the lattice are on the influence of the random magnetic field, as is shown in Fig. 4(a). This model would be a
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kind of a spin-5/2 Ising model in random magnetic field [52]. Our results show that when anisotropy and random magnetic field are present at all points of the lattice, the tricritical behavior is present in the region where the magnitude of the single-ion anisotropy is smaller
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than D/J < 0.7. Above D/J ≥ 0.7 the tricritical behavior disappears and the phase transition is only of first-order. These results are shown in Fig. 4(b).
Figs. 5 and 6 show only the standard behavior of magnetization in a second-order phase
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transition region, which means that at zero temperature the magnetization is maximum and decreases with increasing temperature and goes to zero at a critical temperature. In very low temperature is expected that the anisotropic parameter favors the spins to assume the states ±5/2, ±3/2 and ±1/2, but in this case, the random magnetic field induced magnetizations
m = 2 and m = 1, as shown in Fig. 7. In future, this problem can be treated classically by using the Monte Carlo simulation, and from the viewpoint more realistic using the quantum model version introducing a transverse single-ion anisotropy [25, 63, 64] and/or transverse magnetic field [62]. 20
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ACKNOWLEDGMENTS
Financial support from the Brazilian Agencies CAPES and FAPEMAT is gratefully ac-
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knowledged.
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PII: DOI: Reference:
S0378-4371(19)31746-7 https://doi.org/10.1016/j.physa.2019.123096 PHYSA 123096
To appear in:
Physica A
Received date : 10 June 2019 Revised date : 25 September 2019 Please cite this article as: M. Karimou, A.S.d. Arruda and M. Godoy, Phase diagrams of the spin-5/2 Blume–Capel model, Physica A (2019), doi: https://doi.org/10.1016/j.physa.2019.123096. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Journal Pre-proof Highlights:
- Several phase diagrams were obtained for the spin-5/2 Blume-Capel model.
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- The influence of the random single-ion anisotropy and magnetic field was examined for the system. - First- and second-order phase transition lines were found.
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- The tricritical behavior was examined.
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Phase diagrams of the spin-5/2 Blume-Capel model M. Karimou Ecole Nationale Supérieure de Génie Energétique et
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Procédés (ENSGEP), Université d’Abomey, Bénin and
Institute of Mathematics and Physical Sciences (IMSP),
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University of Abomey Calavi, Dangbo, Benin∗ A. S. de Arruda and M. Godoy†
Instituto de Física, Universidade Federal de Mato Grosso, 78060-900, Cuiabá, Mato Grosso, Brazil
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Abstract
We have used the Curie-Weiss mean-field approximation to study the effects of the random singleion anisotropy and random magnetic field in the phase diagram and thermodynamic properties of the spin-5/2 Blume-Capel model. The phase diagrams have been presented in the temperature T /J versus strength random single-ion anisotropy D/J and strength random magnetic field h/J planes. The magnetization dependence was plotted as a function of temperature and the strength
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single-ion anisotropy. These diagrams showed that the different phase transition types, first- or second-order between the ferromagnetic and paramagnetic phases, are dependent on the random
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parameters. Therefore, within these conditions, the model presents tricritical behavior.
∗
[email protected]
†
[email protected]
1
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I.
INTRODUCTION
The standard Ising model is undoubtedly the most studied model of magnetism in the literature [1]. Its importance lies in the fact that it has exact solutions in one [2] and two
of
[3] dimensions, which serve as a guide for the production of new analytical approaches. It is well known that the spin-1/2 Ising model represents very well idealized systems, so due
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to the need to understand the behavior of real systems, it becomes necessary to generalize the model, i.e., first taking into account spins greater than spin-1/2, second, introduce new interactions present in the real systems. Thus, several generalizations this model have been made such as the Blume-Capel (BC) model, the Blume-Emery-Griffiths (BEG) [6].
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The BC model was introduced independently by Blume [4] and by Capel [5]. They introduced spins-1 and also included a new anisotropy, know as single-ion anisotropy, to investigate the tricritical behavior. This can be the simplest lattice model which could exhibit a tricritical point (TCP). One of the most interesting features of the BC model is the occurrence of a TCP. A TCP is in the phase diagram and separates a smooth phase transition (known as the second-order phase transitions) of a sudden phase transition (known as the first-order phase transitions). In special, the version of the spin-3/2 BC model was
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introduced to understand the phase transitions in DyV O4 [7–10] and the critical properties of certain mixtures of ternary fluids [11]. Extensions of this model featuring interactions of
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higher-spins spin-3/2 were also considered in the literature [12], where Plascak et al used a mean-field theory via Bogoliubov inequality and studied the critical properties of the BC model for spins-S.
The need to consider large spins in magnetic models is increasing, since many atoms and magnetic alloys have large spins, also besides, has been characterized numerous molecules
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which in the fundamental state presents a very large value of spins [13, 14]. Until then, most studies about the BC model address the critical properties focused on versions with the spin1 and spin-3/2, however, our justification for studying the version of the spin-5/2 BC model is based on the fact that there are several physical realizations such as Rb2 M nF [15, 16] and K2 M nF4 and M n(HCOO)2 2H2 O [17] which, besides being quasi-two-dimensional, present M n atoms with spin-5/2. More recently, high spin materials such as BiM n2 P O6 [18] and BaM n2 O3 [19] were obtained. The material BaM n2 O3 is made up by atoms M n2 + (3d5 ) that have spin-5/2, and on the other hand, the BiM n2 P O6 is an AFM compound with a 3D 2
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topology of magnetic interactions, a significant 1D anisotropy, and a moderate frustration of interladder couplings for long-range AFM order. The critical and thermodynamic properties of the BC model have been investigated in
of
the last decades by a variety of analytical approaches. For example, in the mean-field approximations [12, 20, 21], effective-field theory [22–25], renormalization group [26–29], and Monte Carlo simulations [30–33]. Some articles were used extrapolations of series expansions
p ro
of high and low temperatures [34–39]. Moreover, dynamic magnetic properties of BC model have been thoroughly studied with different dynamic methods [40–50]. Currently, the disordered systems became objects of many investigations, and thus several models were proposed to simulate their physical properties. These models take into account
Pr e-
various ingredients, such as, random magnetic field [51–54], single-ion anisotropy [55–58], disorders in the couplings between spins [59], and systems formed by two sub-lattices with different spins and different interactions of field crystals [60, 61]. One other option to introduce disorder into the model is to consider transverse and random single-ion anisotropy [62–64] and a transverse and random magnetic field, which transforms the classical model into a quantum model.
Now we are interested in understanding an issue that until then has not been fully ex-
al
plained in disordered systems, which refers to the dependence of the phase diagram on the random magnetic fields and a single-ion anisotropy. Therefore, in this work, we are inter-
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ested in to understand how the random single-ion anisotropy and random magnetic field influence the topology of phase diagrams and thermodynamic behavior of the spin-5/2 BC model, which were partially treated in Refs. [53, 54] for the case spin-3/2 BC model. The outline of this work is organized as follows: In Section II, we introduced the spin5/2 BC model and analytical expressions for Gibbs free energy, and the equation of state
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is obtained by used Curie-Weiss mean-field approximation. In Section III, we describe our theoretical results and discuss several phase diagrams. Finally, in Section IV we present our conclusions.
II.
FORMULATION
The goal is to study the effects of the random single-ion anisotropy and random magnetic field in the phase diagrams and the thermodynamic behavior of the spin-5/2 BC model. 3
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Therefore, we considered a system with N spins distributed in the sites of a hypercubic lattice. Each spin is connected with all the others spins by long-range ferromagnetic exchange interaction, and all they are submitted the actions of a random single-ion anisotropy and a random magnetic field. This system can be represented by a generalized BC model which
of
is given by following Hamiltonian model:
p ro
N N N X X J X 2 Si Sj + H=− Di Si − Hi Si , 2N i i hi,ji
(1)
where J > 0 is the long-range ferromagnetic exchange interaction which connects all spins Si of the system, Di and Hi are the random single-ion anisotropy and a random magnetic field, respectively. The first sum is performed over all spin pairs in the lattice. Thus, the
Pr e-
interactions among all the different pairs of spins must be equal to J, and the factor 1/N is only to ensure the thermodynamic limit. The random single-ion anisotropy Di and random external magnetic field Hi are present in all lattice sites. Si are variables of spins that assume six states (Si = ±5/2, ±3/2 and ±1/2). When D < 0, it induces the spins to assume the states with higher spin component and, on the other hand, when D > 0 are favored those states with the lower spin component. The second and third sums are executed on all N
al
points of the lattice.
In order to study the phase diagrams and to obtain the thermodynamic properties of the
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model is necessary to calculate the free energy and the magnetization. Instead of we use an approximate solution (mean-field) on a Bravais lattice, it is often interesting to introduce a simple modification on the physical model itself for become it a new problem which can be exactly solved, this approach is known as Curie-Weiss mean-field approximation. Therefore,
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the canonical partition function is given as follows: Z=
X
exp (−βH),
(2)
{Si }
where β = (kB T )−1 , kB is the Boltzmann constant and T is the absolute temperature. Using the Hamiltonian (Eq. (1)), in Z (Eq. (2)) results in Z=
X {Si }
N N N X X X βJ Si Sj − β Di Si2 + β Hi Si . exp 2N i i
hi,ji
4
(3)
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We can to use Si = Sj due to the coupling interaction between the spins to be long-range type, so Eq. (3) is rewritten as X
exp
{Si }
2N
Si
i
#2
−β
N X
Di Si2 + β
i
N X
Hi Si
i
(4)
.
of
Z=
" N βJ 21 X
p ro
In the above equation using the Gaussian identity, the quadratic term can be rewritten as exp where
1 =√ π
ˆ
−∞
a1 =
Thus, the Eq. (5) becomes,
2N
Si
i
dx exp −x2 + 2a1 x ,
#2
1 =√ π
βJ 2N
12 X N
ˆ
(5)
Si .
i
∞
−∞
"
dx exp −x2 + 2x
βJ 2N
12 X N i
#
Si .
(6)
al
exp
" N βJ 21 X
∞
Pr e-
a21
p p Making x = ( βJN/2)m and dx = ( βJN/2)dm and substituting the new variables in " N βJ 21 X
2N
i
Jo
exp
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Eq. (6), we obtain
Si
#2
βJN 2 dm exp − m × 2 −∞ " #r r 1 N βJN βJN βJ 2 X exp 2 Si m . 2N 2 2 i
1 =√ π
ˆ
∞
(7)
Therefore, the partition function can be written as
Z=
r
βJN 2π
ˆ
+∞
−∞
Y X βJN 2 2 dm exp − eβ (JmSi −Di Si +Hi Si ) . m 2 i
(8)
{Si }
Now, making Jm + Hi = Wi and substituting the values of Si = ±5/2, ±3/2 and ±1/2 5
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in Eq. (8), we obtain Z=
r
βJN 2π
ˆ
+∞
−∞
Jm2 25Di + −R , dm exp −βN 2 4
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where
h β(Jm + H ) i h 3β(Jm + H ) i 1 n i i ln 2 exp(6βDi ) cosh + 2 exp(4βDi ) cosh β 2 2 h 5β(Jm + H ) io i . + 2 cosh 2
p ro
R=
as Z=
r
βJN 2π
Pr e-
Let us now use the Curie-Weiss approach, therefore, the partition function can be written
ˆ
∞
dm exp[−βN g(T, D, m, H)],
(9)
−∞
where g is the Gibbs free energy per spin (g = G/N ) given by
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al
1 n 5 3 4βDi g = − ln 2 cosh β(Jm + Hi ) + 2e cosh β(Jm + Hi ) β 2 2 o 2 1 Jm 25 +2e6βDi cosh β(Jm + Hi ) + + Di . 2 2 4
(10)
We introduced the effects of randomness considering that the random single-ion anisotropy is governed by a bimodal probability distribution given by (11)
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P (Di ) = qδ(Di ) + (1 − q)δ(Di − D),
where the first term qδ(Di ) indicates that a amount q of spins are free of the influence of the single-ion anisotropy, and the second term (1 − q)δ(Di − D) indicates that (1 − q) spins
are under influence of the single-ion anisotropy, where D is the magnitude this anisotropy.
The average over the disorder in Gibbs free energy is calculated as: g1 =
ˆ
g P (Di )dDi , 6
(12)
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which results in
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of
5 3 q n g1 = − ln 2 cosh β(Jm + Hi ) + 2 cosh β(Jm + Hi ) β 2 2 o 1 +2 cosh β(Jm + Hi ) 2 (1 − q) n 5 3 2βD − ln 2 cosh β(Jm + Hi ) + 2e cosh β(Jm + Hi ) β 2 2 o 2 Jm 25 1 + + D(1 − q). +2e2βD cosh β(Jm + Hi ) 2 2 4
(13)
The effects of randomness due to the random external magnetic field is introduced considering that such field is governed by a trimodal probability distribution given by (1 − p) [δ(Hi + H) + δ(Hi − H)], 2
Pr e-
P (Hi ) = pδ(Hi ) +
(14)
where the first term indicates that a portion p of spins are free of the influence this field, i.e, H = 0. The two other terms indicate that (1 − p)/2 of the spins are under the influence of the magnetic field along +H and −H directions.
al
Therefore, the average over the disordered magnetic is given by
which results in
g1 P (Hi )dHi ,
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g2 =
ˆ
pq (1 − q)p (1 − p)q ln f (0, y) − ln f (x, y) − [ln f (0, y− ) + ln f (0, y+ )] β β 2β (1 − p)(1 − q) 1 25 − [ln f (x, y− ) + ln f (x, y+ )] + Jm2 + D(1 − q), 2β 2 4
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g2 = −
where we have defined the following recurrent functions, 1 3 5 4x f (x, y) = 2e cosh y + 2e cosh y + 2 cosh y , 2 2 2 1 3 5 6x 4x f (x, y− ) = 2e cosh βy− + 2e cosh βy− + 2 cosh βy− , 2 2 2 1 3 5 6x 4x f (x, y+ ) = 2e cosh βy+ + 2e cosh βy+ + 2 cosh βy+ , 2 2 2 6x
where x = βD, y = βJm, y− = (Jm − H) and y+ = (Jm + H). 7
(15)
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The Eq. (15) represents the complete free energy of the generalized BC model with random single-ion anisotropy and random magnetic field. The magnetization can be obtained by minimizing the free energy in relation to variable m, i.e., ∂g2 /∂m = 0, which results
q(1 − p) [G(0, y+ ) + G(0, y− )] , 2
where we also define the recurrent functions:
(16)
p ro
+
(1 − p)(1 − q) [G(x, y+ ) + G(x, y− )] 2
of
m = pqG(0, y) + p(1 − p)G(x, y) +
5 sinh(5y/2) + 3e4βx sinh(3y/2) + e6βx sinh(y/2) , 2 cosh(5y/2) + 2e4βx cosh(3y/2) + 2e6βx cosh(y/2) 5 sinh(5y+ ) + 3e4βx sinh(3y+ ) + e6βx sinh(y+ ) , G(x, y+ ) = 2 cosh(5y+ ) + 2e4βx cosh(3y+ ) + 2e6βx cosh(y+ ) 5 sinh(5y− ) + 3e4βx sinh(3y− ) + e6βx sinh(y− ) G(x, y− ) = , 2 cosh(y− ) + 2e4βx cosh(3y− ) + 2e6βx cosh(y− )
Pr e-
G(x, y) =
where x = βD, y = Km, y− = (Km − h)/2, y+ = (Km + h)/2, h = H/kB T and K =
J/kB T . Thus, the statistical mechanics method together with the Curie-Weiss mean-field approximation provided the fundamental equation given by the Gibbs free energy (Eq. (15))
RESULTS AND DISCUSSIONS
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III.
al
and the equation of state given by the magnetization (Eq. (16)).
In this section, we present the numerical solutions via Newton-Raphson method of Eqs. (15) and (16), which provide the phase diagrams in the D/J −T /J and h/J −T /J planes
in addition to the thermal properties of the spin-5/2 BC model with random anisotropy and
A.
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random magnetic field.
Ground-state
We used the mean-field theory (version Curie-Weiss) to study the ground-state phase diagram of a spin-5/2 BC model by varying the physical parameter random single-ion anisotropy and random magnetic field. For T /J = 0 (ground-state) the thermal fluctuations disappear, but the quantum fluctuations persist, which can lead the system to phase transitions. The ground-state diagram was determined from the Hamiltonian (Eq. (1)) comparing the energies 8
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(3 /
2,
(5 /2 ,3
(5/2, 5/2)
1/
2)
(5/2, 1/2)
2,
10
(3 /
of
(5/2, - 1/2)
5
) /2
-3
(5/2, - 5/2)
(3 /2 ,
, /2
-1
(5
/2 )
h/|J|
3/ 2)
/2 )
15
(a)
p ro
(1/2, 1/2)
(3/2, - 3/2)
0 -1
0
1
(1/2, - 1/2)
2
3
4
6
8
D/|J|
/2 (3
,1
/2 )
(5
/2
(5/2, 5/2)
,3
,3 /2 )
/2
)
15
/2
(3
Pr e,-
1/ 2)
(5/2, - 1/2)
,/2
(5
(3
/2
h/|J|
(5/2, 1/2)
10
2) 3/
5 (5/2, - 5/2)
(b) 0 -2
(1/2, 1/2)
(1/2, - 1/2)
(3/2, - 3/2)
0
2
4
D/|J|
/2 ) /2 ,3
,-
/2
2)
/2 ,
1/ 2)
2)
(1/2, 1/2)
(1/2, - 1/2)
4
8
12
D/|J|
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( c ) (3/2, - 3/2) 0
3/
0
(3 /2 ,-
(5
(5/2, - 5/2)
5
1/
(5/2, 1/2)
(5/2, - 1/2)
(3
10
urn
h/|J|
(3
(5
al
(5/2, 5/2)
/2
,3
/2 )
15
Figure 1: (Color online) Ground-state diagram of the spin-5/2 BC model. The black-solid lines represent first-order transitions for p = q = 0 and the red-solid lines represent first-order transition for p = 0: (a) q = 0.25, (b) q = 0.50, and (c) q = 0.75.
of the different possible states for a pairs of spins {Sa , Sb }, see Fig. 1. At zero temperature,
the structure of the ground-state for this system consists of nine ordered phases (ferromag9
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netic phase) defined as O2 ≡
5 3 ,− , 2 2
O6 ≡
5 3 , , 2 2
O3 ≡
3 3 , , 2 2
O7 ≡
O9 ≡
1 1 , , 2 2
5 1 , , 2 2
O4 ≡
3 1 , , 2 2
5 1 ,− , 2 2
of
O5 ≡
5 5 , , 2 2
O8 ≡
3 1 ,− , 2 2
p ro
O1 ≡
and three disordered phases (paramagnetic phase) defined as 5 5 ,− , 2 2
D2 ≡
3 3 ,− , 2 2
D3 ≡
1 1 ,− , 2 2
Pr e-
D1 ≡
where the numbers ±5/2, ±3/2 and ±1/2 represent the different states. The ground-state energies are given by
D h E = Sa Sb + (1 − q)(Sa2 + Sb2 ) − (1 − p)(Sa + Sb ), J J J
(17)
where the values of Sa , Sb , q, p and D are used to obtain the energies for each state. All
al
the phases are separated by first-order transition lines. The black-solid lines are for the case p = q = 0 and the red-solid lines for the cases p = 0: q = 0.25 (Fig. 1(a)), q = 0.50
urn
(Fig. 1(b)), and q = 0.75 (Fig. 1(c)). We can also observe that the phase boundaries are modified with the dilution (changing p and q) of anisotropy D using Eq. (11). Hence, our results, for p = 0 and different values of q (see Fig. 1), suggest an increase of some disordered (D1 , D2 ) and ordered (O1 , O2 , O3 , O4 , O5 ) phases and a decrease of other disordered (D3 ) and ordered (O6 , O7 , O8 , O9 ) phases, changing the topology of the phase diagram related
B.
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to the case p = q = 0.
Special cases
i - The first special case corresponds to the pure spin-5/2 Ising model which is obtained making p = 1.0 and q = 1.0 in Eq. (16), the magnetization is given by m=
[5 sinh(5m/2τ ) + 3 sinh(3m/2τ ) + sinh(m/2τ )] , [2 cosh(5m/2τ ) + 2 cosh(3m/2τ ) + 2 cosh(m/2τ )] 10
(18)
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5
h/J= 0.0 p = 1.0
h/J = 0.2 p = 0.1
4
T/J
T/J
4
3
0.8 0.7
2
0.6
2
p ro
0.5 0.4 0.3
1
q = 0.0
0 D/J
(a)
1
2
0 -1
Pr e-
-1
0.6 0.5 0.4 0.3
1
0.2 0.1
0 -2
1.0 0.9 0.8 0.7
of
3 1.0 0.9
0.2 0.1 q = 0.0
0
1 D/J
(b)
Figure 2: (Color online) Phase diagram in the D/J − T /J plane of the spin-5/2 BC model for several values of parameter q, as indicated in the figures. All the solid lines denote second-order phase transitions. (a) For h/J = 0 and p = 1.0, (b) for h/J = 0.2 and p = 0.1.
al
where τ = kB T /J. In limit of m → 0 we obtained the critical temperature, Tc /J = 2.9166, axis.
urn
which is represented in the phase diagram of Fig. 2(a) by parallel line (q = 1) to the D/J
ii - Another special case corresponds to the pure spin-5/2 BC model simply by making p = 1.0 and q = 0 in Eq. (15) and Eq. (16), which corresponds to the case h = 0 and the single-ion anisotropy D is present in all points of the lattice. In this case the free energy
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and the magnetization are given by 1 1 3 5 6βD 4βD g2 = − ln 2e cosh βJm + 2e cosh βJm + 2 cosh βJm β 2 2 2 1 25 + Jm2 − D, 2 4
(19)
and 6βD e sinh 21 βJm + 3e4βD sinh 32 βJm + 5 sinh 25 βJm . m = 6βD 2e cosh 21 βJm + 2e4βD cosh 32 βJm + 2 cosh 25 βJm 11
(20)
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When m → 0, the critical temperature is given by
kB Tc 1 e6βD + 9e4βD + 25 = . J 4 e6βD + e4βD + 1
(21)
of
The numerical solution of Eqs. (21) provide a continuous line indicating a second-order phase transition from ferromagnetic to paramagnetic phase. This line is represented in the phase diagram in Fig. 2(a) corresponding to line q = 0, which is the same phase diagram
C.
p ro
obtained by Plascak et al [12].
Phase diagrams in the D/J − T /J plane
Pr e-
Now, we have considered the case p = 1.0 which indicates absence of random magnetic field for all points of the lattice, and q 6= 0 indicates that when q increase, an amount of spins are free of influence of the random single-ion anisotropy. For this case, the magnetization is given by
[5 sinh(5m/2τ ) + 3 sinh(3m/2τ ) + sinh(m/2τ )] [2 cosh(5m/2τ ) + 2 cosh(3m/2τ ) + 2 cosh(m/2τ )] 6βD e sinh 21 βJm + 3e4βD sinh 32 βJm + 5 sinh 52 βJm . + (1 − q) 6βD 2e cosh 21 βJm + 2e4βD cosh 23 βJm + 2 cosh 52 βJm
(22)
al
m=q
When T tends to Tc (T → Tc ) the magnetization tends to zero (m → 0), therefore,
urn
expanding around Tc we obtain the following expression for the critical temperature: kB Tc 35q 1 (1 − q) e6βD + 9e4βD + 25 = + . J 12 4 e6βD + e4βD + 1
(23)
The numerical solution of Eqs. (23) provides the second-order phase transition lines that
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are displayed in Fig. 2(a). This phase diagram displays only second-order phase transition lines between the ferromagnetic (below of the curves) and paramagnetic phases (above of the curves). The line q = 0 indicates that the system corresponds to the pure spin-5/2 BC model, i.e, all the spins on the lattice are under the action of the anisotropy D/J. The case p = 1.0 and q = 1.0 is the line parallel to the axis D/J, and represents the pure spin-5/2 Ising model. Here, all others lines start in high temperature from the negative side of D/J and decrease until they cross at D/J = 0 and the critical temperature of the pure model, Tc /J = 2.9166. Then the curves decay until reaching a constant temperature in the region 12
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of positive large values of D/J. The increase of q indicates that a number q of spins are free from the influences of the anisotropy D/J, which allows the system to reach a constant critical temperature at a higher anisotropy value. These results for q 6= 0 have not been
published and it represents our first contribution to the study of the spin-5/2 BC model with
of
random single-ion anisotropy. Fig. 2(b) (phase diagram in the D/J − T /J plane) shows the case p = 0.1, where only a small number (ten percent) of spins are free of the magnetic field.
p ro
The remaining portion (ninety percent) of spins are under the action of the magnetic field +h and −h with magnitude h/J = 0.2 and with the same probability (1 − p). In this case, the system presents the same behavior shown in Fig. 2(a), but now the random magnetic
field is responsible by lowering the critical temperatures, as shown in D/J = 0 for all q.
Pr e-
Thus, a small number of free spins of the effects of magnetic field with small magnitude is not enough to dramatically change the critical behavior of the system, i.e, the system still does not present tricritical behavior. 3
3
q = 0.0
p = 0.1
2,5
T/J
T/J
2,5
2
p = 0.0 q = 0.0
h/J = 0.2 h/J = 0.6
2
h/J = 1.0
al
h/J = 0.8
1,5
1,5 h/J = 1.0
h/J = 0.4
0,5
0
0
0,1
urn
1
0,2
0,3
0,4
1
0,5 h/J = 0.2 h/J = 0.3
0,5
0
0
0,1
0,2
0,3
0,4
0,5
0,6
D/J
D/J
(a)
(b)
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Figure 3: Phase diagram in the D/J − T /J plane of the spin-5/2 BC model for several values of the magnetic field h/J, as indicated in the figures. The solid and dotted lines denote second- and first-order phase transitions, respectively. The full black dots are the TCPs. (a) For q = 0 and p = 0.1, and (b) for p = q = 0. In Fig. 3(a), we have shown some detail of Fig. 2(b), for instance, the line q = 0 and p = 0.1 is shown for different values of the magnetic field. The line for h/J = 0.2 is reproduced in the two-phase diagrams (Fig. 3(a) and Fig. 2(b)). When the magnitude of the magnetic field is increased the critical behavior changes drastically, i.e., the first-order 13
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phase transition lines (dotted lines) appear for values of h/J > 0.3. The magnitude of the random magnetic field is enough to impose the tricritical behavior, i.e., the random magnetic field tends to disturb the system faster, so close to Tc the magnetization goes to zero discontinuously. In summary, at high temperatures,the phase transition is second-order
of
(continuous lines) for all magnitudes of the random magnetic field, but at low temperatures, the phase transition is first-order for h/J > 0.3, and a TCP (full black dot) separates the
p ro
second- and first-order phase transition.
Now, in Fig. 3(b) we have presented the phase diagram in the D/J − T /J plane for p = 0
(which indicates that all spins of the lattice suffer the action of the random magnetic field with magnitude +h or −h, with the same probability (1 − p)/2) and q = 0, i.e., all spins in
Pr e-
the sites of the lattice are influenced by random single-ion anisotropy with magnitude given by D/J. Here, the tricritical behavior appears in the range of 0.2 < h/J < 1.0. Therefore, in the region of high temperatures the phase transitions between the ferromagnetic and paramagnetic phases are of second-order for values of the random magnetic field smaller than the one (h/J < 1.0), and in this range it is of first-order in regions of low temperatures for (h/J > 0.2). The TCPs separate these phase transitions. When h/J = 1.0 the phase transition is only of first-order in all regions (low and high temperatures). This fact indicates critical temperature.
al
that the random magnetic field with large magnitude causes the system to reach quickly the
urn
To conclude, it can be noted that the phase diagrams in the D/J − T /J plane show
tricritical behavior only at low temperatures and for an applied magnetic field with magnitude h/J > 0.2. On the other hand, in the high temperatures regions, for any magnetic
field h/J < 1.0, the phase diagrams show that the phase transitions from the ferromagnetic
D.
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phase to the paramagnetic phase are only of second-order.
Phase diagram in the h/J − T /J plane
The phase diagrams, in the h/J − T /J plane, are obtained considering the numerical
solutions of the general expression for the critical temperature given by Eqs. (16). Fig. 4(a)
presents the case q = 1.0 (or D/J = 0) and p 6= 0 indicating that there are a random magnetic field in all sites of the lattice, therefore, this case refers to the spin-5/2 Ising model
in random magnetic field. Here, all phase transition lines start in h/J = 0 at the critical 14
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3
3 D/J = 0.0 q = 1.0
p = 0.0
D/J = 0.0
2,5
q = 0.0
2,5
0.1
p = 0.6
2
1,5
of
T/J
T/J
0.2
2
1,5 p = 0.3
1
p = 0.0
0,5
p ro
1 p = 0.2
0,5
0.7
0
0
0,5
1
1,5
2
2,5
3
h/J
0
1
0,5
1,5
h/J
(b)
Pr e-
(a)
0
Figure 4: Phase diagram in the h/J − T /J plane of the spin-5/2 BC model. (a) For several values of the parameter p as indicated in the figure, D/J = 0, and q = 1.0. (b) For several values of the anisotropy D/J as indicated in the figure and p = q = 0. The solid and dotted lines denote second- and first-order phase transitions, respectively. The black dot points are the TCPs. temperature (Tc /J = 2.9166) of the pure spin-5/2 Ising model and decreases when h/J
al
increase. The lines p = 0 and p = 0.2 exhibit second-order transition (solid lines) in the region of high temperature and lower magnetic field. For lower temperatures and higher
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magnetic field the phase transitions are of first-order (dotted lines). When the magnetic field increase, it induces a first-order phase transition in the low-temperature region. This increase causes a rapid decrease at the critical temperature and it will goes to zero at hc /J = 1.55 and hc /J = 1.65. This result is expected because the random magnetic field tends to favor the disorder in the system and, therefore, in this way, the system reaches
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the disorder more quickly in lower temperatures. The two full-black dots represent TCPs whose coordinates are (Tc /J = 2.141, h/J = 0.929) and (Tc /J = 1.753, h/J = 1.199 ) for p = 0 and p = 0.2, respectively. These points separate the two-phase transition types, the second-order phase transition from the first-order phase transition. The lines with p > 0.2 show only second-order phase transition, i.e., the tricritical behavior disappears. All the lines start in h/J = 0 and Tc /J = 2.9166 and tend to a constant and very large value of h/J. In Fig. 4(b), we exhibited the phase diagram for the case p = q = 0 and three different 15
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values of anisotropy D/J. This case, we have a spin-5/2 BC model in a random magnetic field typical. Here all the spins are under the action of a magnetic field ±h/J with the same probability and under the action of the single-ion anisotropy D/J. The curves with
D/J = 0, 0.1 and 0.2 show that the system change from ferromagnetic to paramagnetic
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phase at a second-order phase transition (solid lines) in the region of high temperatures, and at first-order phase transition in low-temperature regions which are separated by TCPs.
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For D/J = 0.7, we can observe only the first-order phase transition in the region of low temperatures. In summary, the increase of the magnetic field makes the system becomes disordered because the randomness favors the disorder. These curves also show that the single-ion anisotropy D also contributes to the disorder of the system, leading to the phase
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transitions at lower temperatures. Therefore, the model presents a tricritical behavior for D/J < 0.7.
Thermodynamic properties
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E.
(b)
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(a)
Figure 5: Magnetization m as a function of temperature T /J. (a) For different values of q as indicated in the figure, D/J = 0.1, h/J = 0, and p = 1.0. (b) For different values of D/J as indicated in the figure, h/J = 0, p = 1.0, and q = 0. It is well known that the Mermin-Wagner’s theorem states that a two-dimensional isotropic system does not maintain long-range order. Therefore, this fact guarantees the importance of the anisotropies, in special, the single-ion anisotropy that plays a very important role in stabilizing of the long-range order in systems with low dimensionality [65, 66]. Here, 16
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our goal is to study the behavior of the magnetization in the first- and second-order phase transition regions, to verify the phase diagrams obtained in the previous subsection. In this model, the exchange interaction is responsible for favoring the magnetic order. On the contrary, the temperature, the random single-ion anisotropy and the random magnetic field
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favor disordering in the system. Therefore, we have restricted our study to the behavior of the magnetization in relation the temperature only for some select values of the parameters
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D/J, h/J, q and p.
The temperature T /J dependence of magnetization m is given by numerical solutions of Eq. (16). Fig. 5(a) shows the variation of magnetization m as a function of temperature T /J for the case p = 1.0, h/J = 0, where the region was chosen with D/J = 0.1. In this
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region, the magnetization presents a standard behavior for all values of q, i.e., q = 0 to q = 1.0. By standard behavior, it is understood that the magnetization is maximum at T /J = 0 and continuously decreases when the temperature increases and reaches the value zero in a critical temperature (Tc ). This behavior confirms the phase diagram of Fig. 2(a), since observing the line q = 0 and the single-ion anisotropy with the magnitude D/J = 0.1, it can be noted that the phase transition is of second-order for all q. In Fig. 5(b), we have shown the variation of magnetization m as a function of temperature
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T /J for the case p = 1.0, h/J = 0, q = 0 and for different values of D/J. This plot also presents the standard behavior of the magnetization which is characteristic of the region
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with a second-order phase transition. The line D/J = 0 corresponds to the case of the pure spin-5/2 BC model, where m is maximum in T /J = 0 and go to zero in Tc /J = 2.9166. The plot shows that the increase of D/J makes the magnetization goes faster to zero, i.e., indicating that the system goes faster to disordered state at a lower critical temperature. This case corresponds to line q = 0 of Fig. 2(a) for D/J values ranging from zero to 0.7
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indicating only second-order phase transition. The plot presented in Fig. 6(a) shows the standard behavior of the magnetization varying with the temperature in a special case with p = 0, q = 1.0 and D/J = 0, but now for several values of the magnetic field h/J. The line h/J = 0 corresponds to the case of the pure spin-5/2 BC model. Increasing the magnitude of the random magnetic field leads the magnetization to zero at a lower critical temperature. This case corresponds to high temperature region (with h/J < 1.0) and the line p = 0 in Fig. 4(a), which shows only second-order phase transition. On the other hand, the plot exhibited in Fig. 6(b) presents 17
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q = 1.0
p = 0.0 q = 1.0
2,5
2,5
h/J = 0.5 D/J = 0.0
D/J = 0.0
2
1,5
1,5
0.0
1
0.5 0.7
0,5
p ro
1
0
0
1
2
3
(a)
0
Pr e-
T/J
1.0 0.9 0.6 0.3
0,5
0.9 h/J = 0.95
0
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m
m
2
p = 0.0
1
2
3
T/J
(b)
Figure 6: Magnetization m as a function of temperature T /J. (a) For different values of h/J as indicated in the figure, D/J = 0, p = 0, and q = 1.0. (b) For different values of p as indicated in the figure, h/J = 0.5, D/J = 0, and q = 1.0.
the standard behavior of the magnetization in the region with second-order phase transition (case q = 1.0, h/J = 0.5 and D/J = 0). The line p = 1.0 corresponds to the case of
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the pure spin-5/2 BC model. The increase in the strength random magnetic field leads the magnetization quickly to zero. This plot can be seen as a measure of magnetization in a
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fixed magnetic field h/J = 0.5 and by varying the parameter p, which indicates the degree of randomness. Thus, this plot also confirms the information shown in Fig. 4(a), i.e., taking a line parallel to the T /J axis by cutting the h/J axis in h/J = 0.5, which crosses several lines corresponding to all values of p. In Fig. 7, we have displayed the variation of magnetization m versus single-ion anisotropy
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D/J for two cases: i) q = 0, h/J = 0.3 for p = 0 with temperature T /J = 0.03, and ii) for p = 0.4 with T /J = 0.1. It is observed that at low temperatures, the magnetization goes to well-defined values (in the first case, for T /J = 0.03 and p = 0 represented by solid line), i.e., the magnetization m starts at 2.5 and goes to 2.0, from 2.0 to 1.5, 1.5 to 1.0 and 1.0 to 0.5. In this case, all spins are under the action of the single-ion anisotropy and random magnetic field. Thus, this result is expected because at low temperatures the anisotropy easily favors the spins to assume the states ±5/2, ±3/2 and ±1/2, consequently generating
well defined values of magnetization. On the other hand, the random magnetic field, in this 18
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2.5
q = 0.0
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h/J = 0.3
m
2
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1.5 p= 0.0 ; T= 0.03
1 p= 0.4 ; T= 0.1
0 0
0.2
Pr e-
0.5
0.4
0.6
0.8
1.0
D/J
Figure 7: Magnetization m as a function of the anisotropy D/J for two different values of p and T , as indicated in the figure. Here, h/J = 0.3 and q = 0.
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case, induces magnetizations m = 2.0 and m = 1.0. Then, when the temperature increases, the possible number of values for magnetization also increases from m = 0 to m = 5/2.
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However, in the second case (p = 0.4 with T /J = 0.1) there is still forty percent of the spins under the action of the random magnetic field, but it is possible to observe that the values of magnetization no longer have their values very well defined.
CONCLUSIONS
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IV.
The main purpose of this paper was to study the phase diagram and the thermodynamic properties of the spin-5/2 BC model on the random magnetic field and a random single-ion anisotropy. The free energy and the equation of state were calculated using the mean-field approximation in the version Curie-Weiss. Therefore, as first results we rescue the case pure spin-5/2 BC model that was studied by Plascak et al [12], where the phase diagram (D/J − T /J plane) obtained by them presents only second-order phase transition and that corresponds to our line q = 0 in Fig. 2(a). Then, we can conclude that the phase diagram 19
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of this model does not present TCPs, i.e., there is no tricritical behavior. This behaviour persists in two situations: the first is even when the randomness is introduced (increase q) as shown in the phase diagram of Fig. 2(a). The increase of q indicates that a larger amount of spins are free of the influence of anisotropy D/J, thus the line p = 1 corresponds
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to the pure spin-5/2 Ising model whose critical temperature is Tc /J = 2.9166. The second situation is when all spins are under the action of the random anisotropy and ninety percent
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they are with a small fixed random magnetic field h ≤ 0.3. Here also the system does not
present tricritical behavior as shown in Fig. 2(b) and Fig. 3(a). The tricritical behavior appears for values of h/J > 0.3, as shown in Fig. 3(a). In Fig. 3(b), with all spins under the actions of the anisotropy and the random magnetic field, the tricritical behavior appears
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with h/J > 0.2, and for the field h/J ≥ 1.0 the phase transition is only of first-order.
Now, in the h/J − T /J plane, that is keeping the fixed magnitude of the single-ion
anisotropy and varying the random magnetic field. In this case, for D/J = 0 the tricritical behavior appears only when more than seventy percent of spins in the lattice are on the influence of the random magnetic field, as is shown in Fig. 4(a). This model would be a
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kind of a spin-5/2 Ising model in random magnetic field [52]. Our results show that when anisotropy and random magnetic field are present at all points of the lattice, the tricritical behavior is present in the region where the magnitude of the single-ion anisotropy is smaller
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than D/J < 0.7. Above D/J ≥ 0.7 the tricritical behavior disappears and the phase transition is only of first-order. These results are shown in Fig. 4(b).
Figs. 5 and 6 show only the standard behavior of magnetization in a second-order phase
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transition region, which means that at zero temperature the magnetization is maximum and decreases with increasing temperature and goes to zero at a critical temperature. In very low temperature is expected that the anisotropic parameter favors the spins to assume the states ±5/2, ±3/2 and ±1/2, but in this case, the random magnetic field induced magnetizations
m = 2 and m = 1, as shown in Fig. 7. In future, this problem can be treated classically by using the Monte Carlo simulation, and from the viewpoint more realistic using the quantum model version introducing a transverse single-ion anisotropy [25, 63, 64] and/or transverse magnetic field [62]. 20
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ACKNOWLEDGMENTS
Financial support from the Brazilian Agencies CAPES and FAPEMAT is gratefully ac-
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knowledged.
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