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Physica A
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Thermodynamical properties of the mixed spin transverse Ising model with four-spin interactions
Q1
N. Benayad ∗ , M. Ghliyem, M. Azhari
Laboratory of High Energy Physics and Scientific Computing, HASSAN II University—Casablanca, Faculty of Sciences-Aïn Chock, B.P.: 5366 Maarif, Casablanca 20100, Morocco
h i g h l i g h t s
• We derive the state equations for a mixed spin transverse Ising model with four-spin interactions.
• We investigate the magnetic properties.
• The system exhibits tricritical behaviour in certain ranges of the four-spin interaction and transverse field strengths.
• We examine the effects of the four-spin interaction and transverse field on the internal energy and the specific heat.
a r t i c l e i n f o
Article history:
Received 30 July 2013
Received in revised form 6 December 2013 Available online xxxx
Keywords:
Mixed spin Four-spin interactions Transverse field
Thermodynamical properties
a b s t r a c t
The thermodynamical properties of the mixed spin transverse Ising system consisting of spin-1/2 and spin-1 with four-spin interactions are studied within the frame work of the finite cluster approximation based on single-site cluster theory. In this approach, the state equations are derived for the two-dimensional square lattice. In addition to the phase diagrams which show qualitatively interesting features (variety of transitions and tricritical behaviour), we find some characteristic behaviours of the longitudinal and transverse sublattice magnetizations. In particular, the gap of the longitudinal magnetization at first order transition decreases with increasing values of the transverse field. Moreover, the effects of the transverse field on the internal energy and the specific heat are also investigated.
©2014 Published by Elsevier B.V.
1. Introduction 1
In recent years, a particular interest has been devoted to the theoretical and experimental study of Ising models with
2multispin interactions. The increasing interest in investigating models with higher-order interactions arises from the fact
3that, on the one hand, they may exhibit rich phase diagrams and can describe phase transitions in some physical systems.
4On the other hand, they show physical behaviours not observed in the usual spin systems. For instance, they display the non
5universal critical phenomena [1,2].
6From the theoretical point of view, monoatomic Ising models with multispin interactions have been studied
7within different methods, such as mean field approximation [3,4], effective field theory [5–7], series expansions [8,9],
8renormalization group methods [10], Monte Carlo simulation [11], and exact calculations [12]. Experimentally, the models
9with multispin interactions can be used to describe different physical systems such as classical fluids [13], solid
3He [14],
10lipid bilayers [15], metamagnets [16], and rare gases [17]. It is worthy to note here that the four-spin interaction plays an
11important role in the two dimensional antiferromagnet La
2CuO
4[18], the parent material of high-T
csuperconductors.
12∗Corresponding author. Tel.: +212 5 22 23 06 84; fax: +212 5 22 23 06 74.
E-mail addresses:n.benayad@fsac.ac.ma,noureddine_benayad@yahoo.fr(N. Benayad).
0378-4371/$ – see front matter©2014 Published by Elsevier B.V.
http://dx.doi.org/10.1016/j.physa.2014.01.023
Fig. 1. Part of the square lattice.•and×correspond toσands-sublattice sites, respectively.
Recently, attention has been directed to the study of the magnetic properties of a two-sublattice mixed spin Ising
1
systems. They are of interest for the following main reasons. They have less translational symmetry than their single spin
2
counterparts, and are well adopted to study a certain type of ferrimagnetism [19]. The influence of multispin interactions
3
on the mixed spin Ising models, in the presence of the crystal field, has been theoretically studied within finite cluster
4
approximation [20], Monte Carlo Simulation [21] and exact calculations [22]. Very recently, we have investigated [23], using
5
the finite cluster approximation, the influence of the fluctuation of the crystal field interaction on the mixed spin-1 / 2 and
6
spin-1 Ising model with four-spin interaction.
7
The transverse Ising model was originally introduced by Blinc [24] and De Gennes [25] as a valuable model for the
8
tunnelling of the proton in hydrogen-bonded ferroelectrics [26] such as potassium dihydrogen phosphate (KH
2PO
4). Since
9
then, it has been successfully applied to several physical systems, such as cooperative John-Teller systems [27] (like DyVo
410
and TbVo
4), ordering in rare earth compounds with a singlet crystal field ground state [28], and also to some real magnetic
11
materials with strong uniaxial anisotropy in a transverse field [29]. We mention that the mixed spin transverse Ising
12
model has been investigated by the finite cluster approximation [30], effective field methods [31], renormalization group
13
technique [32], and exact calculations [33,34]. It is worthy to note here that the diluted versions [35] of this model as well
14
as its presence in random fields [36] has also been studied.
15
The first purpose of this paper is to study the influence of the transverse field on the magnetic properties of the two-
16
dimensional mixed spin-1 / 2 and spin-1 Ising system with four-spin interaction. Such a system may be described by the
17
following Hamiltonian
18
H = − J
2
⟨ij⟩
σ
izS
jz− J
4
{ijkl}
σ
izσ
kzS
jzS
lz− Ω
i
σ
ix+
j
S
jx
(1)
19
where σ
αand S
α(α = x , z ) are components of spin-1 / 2 and spin-1 operators. The first summation is carried out only
20
over nearest neighbour pair of spins. The second term represents the four-spin interactions, where the summation is
21
over all alternate squares, shaded in Fig. 1. Ω denotes the uniform transverse field. To this end, we use the finite cluster
22
approximation within the framework of a single-site cluster theory. The second purpose is to examine the internal energy,
23
and the specific heat of the system.
24
The outline of this paper is as follows: In Section 2, we describe the theoretical framework and derive the state equations.
25
In Section 3, we investigate and discuss the magnetic properties of the system. In Section 4 other relevant thermodynamical
26
quantities are presented and analysed. Finally, our concluding remarks are given in Section 5.
27
2. Theoretical framework
28
The method we adopt in the study of the system described by the Hamiltonian (1) is the finite cluster approximation
29
(FCA), based on a single-site cluster theory. In this approach, attention is focused on a cluster comprising just a single selected
30
spin σ
o( S
o) , and its neighbour spins { σ
1, σ
2, S
1, S
2, S
3, S
4} ( { S
1, S
2, σ
1, σ
2, σ
3, σ
4} ) with which it directly interacts (see Fig. 2).
31
a b
Fig. 2. (a) Neighbours of spinσowith which it directly interacts. (b) Neighbours of spinSowith which it directly interacts.
We split the total Hamiltonian (1) into two parts, H = H
o+ H
′, where H
oincludes all parts of H associated with the lattice
1site o. In the present system, H
otakes the form.
2H
oσ= λ
1σ
oz− Ω σ
ox(2)
3H
os= λ
2S
oz− Ω S
ox(3)
4where
5λ
1= − J
24
j=1
S
jz− J
4
{j,k,l}
σ
kzS
jzS
lz 6λ
2= − J
24
i=1
σ
iz− J
4
{i,l,k}
σ
izσ
kzS
lz(4)
7whether the lattice site o belongs to σ or S-sublattice, respectively.
8First, the problem consists in evaluating the sublattice longitudinal and transverse components of the magnetizations
9and its quadrupolar moments. In order to calculate them, we choose a representation in which σ
ozand S
ozare diagonal and
10denote by ⟨ σ
oα⟩
c( ⟨ ( S
oα)
n⟩
c, n = 1 , 2 ) the mean value of σ
oα(( S
oα)
n) for a given configuration c of all other spins (i.e. when
11all other spins σ
iand S
j( i , j ̸= 0 ) are kept fixed). Neglecting the fact that H
oand H
′do not commute, ⟨ σ
oα⟩
cand ⟨ ( S
oα)
n⟩
c 12are given by
13⟨ σ
oα⟩
c= Tr
σoσ
oαexp ( − β H
oσ)
Tr
σoexp ( − β H
oσ) (5)
14and
15 S
onα
= cTr
SoS
onαexp ( − β H
os)
Tr
Soexp ( − β H
os) . (6)
16Tr
σo(or Tr
So) means the trace performed over σ
o(or S
o) only. As usual ß = 1 / T , where T is the absolute temperature.
17Therefore, the magnetizations µ
α, m
α(α = x , z ) and the quadrupolar moments q
αare given by
18µ
α≡ ⟨⟨ σ
oα⟩
c⟩ =
Tr
σoσ
oαexp ( − β H
oσ) Tr
σoexp ( − β H
oσ)
, (7)
19m
α≡ ⟨⟨ S
oα⟩
c⟩ =
Tr
s0S
oαexp ( − β H
os) Tr
s0exp ( − β H
os)
, (8)
20q
α≡
S
o2α
c
=
Tr
s0S
o2αexp ( − β H
os) Tr
s0exp ( − β H
os)
, (9)
21which can be considered as the starting point of the single-site cluster approximation. ⟨· · ·⟩ denotes the average over all
22spin configurations. Eqs. (7)–(9) are not exact. Nevertheless, they have been accepted as a reasonable starting point [37]
23for transverse Ising systems and have been successfully applied to a number of interesting transverse Ising systems [30,35,
2438–40]. We have to emphasize that in the Ising limit ( Ω = 0 ) , Hamiltonian (1) contains only σ
izand S
jz. Then, relations
25(7)–(9) become exact identities.
26To calculate ⟨ σ
oα⟩
cand ⟨ ( S
oα)
n⟩
cone has first to diagonalize the single-site Hamiltonians H
oσand H
os, respectively. To do
1
this, it is convenient to use the following rotation transformations.
2
σ
oz= cos ϕσ
oz′− sin ϕσ
ox′(10)
3
σ
ox= sin ϕσ
oz′+ cos ϕσ
ox′(11)
4
S
oz= cos φ S
oz′− sin φ S
ox′(12)
5
S
ox= sin φ S
oz′+ cos φ S
ox′(13)
6
where
7
cos ϕ = − λ
1 λ
21+ Ω
2, sin ϕ = Ω
λ
21+ Ω
2(14)
8
cos φ = − λ
2 λ
22+ Ω
2, sin φ = Ω
λ
22+ Ω
2. (15)
9
The coordinate rotations (10)–(13) turn the Hamiltonians H
oσand H
osinto diagonal forms. Then, evaluating the inner traces
10
in (7)–(9) over the states of the selected spins σ
oand S
o, we obtain
11
µ
z=
− λ
12
λ
21+ Ω
2tanh
β 2
λ
21+ Ω
2
(16)
12
m
z=
− λ
2 λ
22+ Ω
22 sinh
β
λ
22+ Ω
2
1 + 2 cosh
β
λ
22+ Ω
2
(17)
13
q
z=
Ω
2+
2 λ
22+ Ω
2 cosh
β
λ
22+ Ω
2
λ
22+ Ω
2
1 + 2 cosh
β
λ
22+ Ω
2
(18)
14
µ
x=
Ω
2
λ
21+ Ω
2tanh
β 2
λ
21+ Ω
2
(19)
15
m
x=
Ω
λ
22+ Ω
22 sinh
β
λ
22+ Ω
2
1 + 2 cosh
β
λ
22+ Ω
2
(20)
16
q
x=
λ
22+
2 Ω
2+ λ
22 cosh
β
λ
22+ Ω
2
λ
22+ Ω
2
1 + 2 cosh
β
λ
22+ Ω
2
. (21)
17
To calculate the average on the right-hand sides of Eqs. (16)–(21) over all spin configurations, we use the procedure
18
described in our previous work [23]. This leads to the following coupled equations:
19
µ
z= µ
z[ 2A
1q
2z+ 4A
2q
3z+ 2A
3q
4z] + m
z[ 4A
4+ 4A
5q
z+ 4A
6q
2z+ 4A
7q
3z] + m
zµ
2z[ 4A
8q
3z]
+ µ
zm
2z[ 2A
9+ 4A
10q
z+ 2A
11q
2z] + m
3z[ 4A
12+ 4A
13q
z] + µ
zm
4z[2A
14] + µ
2zm
3z[4A
15q
z] (22)
20
m
z= µ
z[ 4B
1+ 8B
2q
z+ 4B
3q
2z] + m
z[ 2B
4+ 2B
5q
z] + µ
3z[ 4B
6+ 8B
7q
z+ 4B
8q
2z]
+ m
zµ
2z[ 2B
9+ 2B
10q
z] + µ
zm
2z[ 4B
11] + m
zµ
4z[ 2B
12+ 2B
13q
z] + m
2zµ
3z[ 4B
14] (23)
21
q
z= [ C
1+ 2C
2q
z+ C
3q
2z] + µ
2z[ 6C
4+ 12C
5q
z+ 6C
6q
2z] + m
2z[ C
7] + µ
zm
z[ 4C
8+ 4C
9q
z]
+ µ
4z[ C
10+ 2C
11q
z+ C
12q
2z] + m
zµ
3z[ 4C
13+ 4C
14q
z] + µ
2zm
2z[ 6C
15] + m
2zµ
4z[ C
16] (24)
22
µ
x= D
o+ D
1µ
2z+ D
2q
z+ D
3m
2z+ D
4µ
zm
z+ D
5q
zµ
2z+ D
6µ
2zm
2z+ D
7µ
zm
3z+ D
8µ
zm
zq
z+ D
9q
2z+ D
10q
zm
2z+ D
11m
4z+ D
12µ
2zq
2z+ D
13µ
2zm
2zq
z+ D
14µ
2zm
4z+ D
15µ
zq
zm
3z+ D
16µ
zm
zq
2z+ D
17m
2zq
2z+ D
18q
3z+ D
19q
4z+ D
20µ
zm
zq
3z+ D
21µ
2zq
3z+ D
22µ
2zm
2zq
2z+ D
23µ
2zq
4z(25)
1m
x= E
0+ E
1q
z+ E
2q
2z+ E
3µ
2z+ E
4µ
2zq
z+ E
5µ
2zq
2z+ E
6m
2z+ E
7µ
zm
z+ E
8µ
zm
zq
z+ E
9µ
4z+ E
10µ
4zq
z+ E
11µ
4zq
2z+ E
12µ
3zm
z+ E
13µ
3zm
zq
z+ E
14µ
2zm
2z+ E
15µ
4zm
2z(26)
2q
x= F
0+ F
1q
z+ F
2q
2z+ F
3µ
2z+ F
4µ
2zq
z+ F
5µ
2zq
2z+ F
6m
2z+ F
7µ
zm
z+ F
8µ
zm
zq
z+ F
9µ
4z+ F
10µ
4zq
z+ F
11µ
4zq
2z+ F
12µ
3zm
z+ F
13µ
3zm
zq
z+ F
14µ
2zm
2z+ F
15µ
4zm
2z. (27)
3Expressions for non-zero coefficients quoted in Eqs. (22)–(24) can be taken from Appendix in Ref. [23], where only the
4definition of the functions f ( x ), g ( x ) and h ( x ) must be changed, respectively, to:
5f ( x ) = − x 2
x
2+
Ω J2
2tanh
K 2
x
2+
Ω J
2
2
, g ( x ) = − x
x
2+
Ω J2
22 sinh
K
x
2+
Ω J2
2
1 + 2 cosh
K
x
2+
Ω J2
2 ,
6h ( x ) =
Ω J2
2+
2x
2+
Ω J2
2 cosh
K
x
2+
Ω J2
2
x
2+
Ω J2
2
1 + 2 cosh
K
x
2+
Ω J2
2
7with K = β J
2and x depends on α = J
4/ J
2.
8While the coefficients mentioned in (25)–(27) are listed in Appendix A.
9If we replace m
zand q
zin (22) by their expressions taken from (23) and (24), we obtain an equation for µ
zof the form:
10µ
z= a µ
z+ b µ
3z+ · · · . (28)
11In order to obtain the second-order transition temperature, we neglect higher-order terms in the magnetizations in
12(22)–(24). Therefore, the critical temperature is analytically obtained through a determinantal equation, i.e.
131 = 2A
1q
20+ 4A
2q
30+ 2A
3q
4o+
4B
1+ 8B
2q
0+ 4B
3q
20
4A
4+ 4A
5q
0+ 4A
6q
20+ 4A
7q
30
1 − ( 2B
4+ 2B
5q
0) (29)
14where q
ois the solution of:
15q
o= C
1+ 2C
2q
o+ C
3q
2o. (30)
16Eq. (29) means that its right-hand side corresponds to the coefficient a in (28).
17In the vicinity of the second-order transition, the sublattice magnetization µ
zis given by
18µ
2z= 1 − a
b . (31)
19The right hand side of (31) is positive since we are in the long-ranged ordered regime. This means that the signs of 1 − a and
20b are the same. When b changes sign (1 − a keeping its sign), µ
2zbecomes negative. It means that we are not in the vicinity
21of a second-order transition line. So, the transition is of the first order. Therefore the point at which
22a ( K , α, Ω ) = 1 and b ( K , α, Ω ) = 0 (32)
23characterizes the tricritical point.
24To obtain the expression for b, one has to solve (22)–(24) for small µ
zand m
z. The solution is of the form
25q
z= q
0+ q
1µ
2z+ q
2m
2z+ q
3µ
zm
z(33)
26where q
1, q
2and q
3are given by
27q
1= 6C
4+ 12C
5q
0+ 6C
6q
201 − 2C
2− 2C
3q
0, q
2= C
71 − 2C
2− 2C
3q
0, q
3= 4C
8+ 4C
9q
01 − 2C
2− 2C
3q
0.
28Fig. 3. The phase diagram inT−J4plane. The number accompanying each curve denotes the value ofΩ/J2. The black point denotes the tricritical point.
After some algebraic manipulations, (23) and (33) can be written in the following forms.
1
m
z= A B µ
z+
8B
2q
4B + 8B
3q
0q
4B + C
B + AD
B
2+ 2AB
5q
4B
2+ 4B
11A
2
B
3
µ
3z+ · · · (34)
2
q
z= q
0+ q
4µ
2z(35)
3
where
4
A = 4B
1+ 8B
2q
0+ 4B
3q
20B = 1 − ( 2B
4+ 2B
5q
0) C = 4B
6+ 8B
7q
0+ 4B
8q
205
D = 2B
9+ 2B
10q
0q
4= q
1+
A B
2q
2+
A B
q
3.
6
By substituting m
zand q
zin Eq. (22), with their expressions taken from (34) and (35), we obtain Eq. (28), where b is given
7
by
8
b = 4A
1q
0q
4+ 12A
2q
4q
20+ 8A
3q
4q
3o+ A
4A
5q
4+ 8A
6q
4q
0+ 12A
7q
4q
20
B + 4A
8q
3 0
A B + EA
2
B
2+ FA
3
B
39
+
4A
4+ 4A
5q
0+ 4A
6q
20+ 4A
7q
30
×
8B
2q
4+ 8B
3q
0q
4+ C
B + AD + 2AB
5q
4B
2+ 4B
11A
2
B
3
(36)
10
with
11
E = 2A
9+ 4A
10q
0+ 2A
11q
20, F = 4A
12+ 4A
13q
0.
12
3. Magnetic properties
13
Eqs. (22)–(27) constitute a set of relations according to which we can study the system described by the Hamiltonian
14
(1). In particular, in this section, we illustrate the effect of the transverse field Ω , as well as the four-spin interactions J
4on
15
the magnetic properties. At high temperature the longitudinal magnetization moments µ
zand m
zare both equal to zero.
16
Below a transition temperature T
c, we have spontaneous ordering ( µ
z̸= 0 and m
z̸= 0), while the corresponding transverse
17
magnetizations µ
xand m
xare expected to be non-zero at all temperatures.
18
Let us first summarize the main results for the phase diagrams of the system under investigation. They are reported in
19
Figs. 3 and 4. In order to have an idea on the effects of the transverse field Ω on the mixed spin Ising model with four-spin
20
interaction, we have plotted in Fig. 3, the phase diagram in the ( T
c/ J
2, J
4/ J
2) plane, for selected values of the transverse field.
21
We note here that in the absence of the transverse field ( Ω = 0 ) , the system reduces to the two-sublattice mixed spin-1 / 2
22
and spin-1 Ising model with four-spin interaction. The phase diagram of this model was studied very recently by us [23]
23
and which corresponds to the curve Ω = 0 in Fig. 3. When the transverse field belongs to the range 0 ≤ Ω / J
2≤ 1 . 826,
24
the critical temperature T
cincreases with increasing values of J
4and passes by a smooth maximum and then reduces to a
25
tricritical point. The remaining part of the phase diagram ( 1 . 826 < Ω / J
2< 2 . 117 ) shows a qualitatively different behaviour
26
Fig. 4. The phase diagram inT−Ωplane, when (a) all transition lines are of second order, (b) the system exhibits a tricritical behaviour. The number accompanying each curve denotes the value ofJ4/J2. The black point denotes the tricritical point.
from the previous range of Ω . Thus, the tricritical behaviour disappears and all transitions are always of second-order. We
1have to mention that the transverse field reduces the longitudinal ferromagnetic order and therefore acts against the order.
2To explore more precisely the transitions undergone by the system, we investigate the influence of the four-spin interaction
3on the phase diagram of the mixed spin transverse Ising model. The results are summarized in Fig. 4(a, b). These give the
4sections of the critical surface T
c( J
4, Ω ) with planes of fixed values of the four-spin interactions. In the absence of the four-
5spin interaction the curve, corresponding to J
4= 0 in Fig. 4(a), shows that the critical temperature decreases gradually from
6its value in the mixed Ising model T
c( Ω = 0 ) to vanish at a critical value Ω
c/ J
2= 2 . 117 of the transverse field strength.
7Fig. 4(a) represents the phase diagrams when the system exhibits second-order transition for any value of the transverse field
8Ω , namely when J
4/ J
2belongs to the range − 4 < J
4/ J
2< 1 . 114. Indeed, as seen in this figure, the critical temperature line
9starts from its J
4/ J
2-dependent value at Ω = 0 and decreases with increasing values of the transverse field. So, the system
10undergoes at the ground state a phase transition at a finite critical value Ω
cwhich depends on the ratio J
4/ J
2. As indicated
11in Fig. 4(b), the phase diagram is qualitatively different for J
4belonging to the narrow range of the four-spin interaction
12( 1 . 114 ≤ J
4/ J
2< 1 . 565 ) . In fact the system shows a tricritical behaviour. The critical lines end in a tricritical point and we
13note that the Ω -component of this latter increases with the strength of the four-spin interaction.
14Secondly, since the system under study exhibits interesting phase diagram, it is worthy to investigating the temperature
15dependence of the longitudinal and transverse components of the sublattice magnetizations µ
αand m
α(α = z , x ) as
16a
b
Fig. 5. The temperature dependence of the longitudinal and transverse components of the magnetizations(µα,mα,Mα;α=x,z)when the transition is of second-order, for selected values ofΩ, with (a)J4/J2=1 and (b)J4/J2= −1.
well as their corresponding total magnetizations defined by M
α= (µ
α+ m
α)/ 2 by solving numerically the coupled
1
Eqs. (22)–(27). In Fig. 5, we plot the longitudinal and transverse magnetizations µ
α, m
αand M
αin the case of an applied
2
transverse field ( Ω = 0 . 5J
2, 1J
2, 1 . 5J
2) for fixed values of the four-spin interactions J
4/ J
2= 1 , J
4/ J
2= − 1, which
3
correspond to a second-order transition. As is shown in the figure, the longitudinal sublattice (µ
z, m
z) and the total
4
( M
z) magnetizations decrease continuously from their saturation values to vanish at a Ω -dependent critical temperature
5
T
c( Ω ) . We note that the role of the transverse field Ω is to inhibit the ordering of µ
z, m
zand M
z. So the long range
6
ferromagnetic order domain becomes less and less wide with increasing values of the transverse field Ω . The behaviours
7
of the corresponding transverse components µ
x, m
xand M
xare also shown in Fig. 5. As is clearly seen in this figure, they
8
are ordered at all temperatures, and pass through a cusp at the second-order transition. As can be expected, the sublattice
9
and the total transverse magnetizations increase with the value of Ω . On the other hand, the temperature dependences
10
of the magnetizations µ
α, m
αand M
α(α = z , x ) , when the system exhibits a first order transition (for instance ( J
4/ J
2=
11
1 . 5 , Ω / J
2= 0 . 5 ), ( J
4/ J
2= 1 . 5 , Ω / J
2= 1 . 5 ) ), are depicted in Fig. 6. This kind of transition is characterized at the transition
12
temperature, by a remarkable gap of the longitudinal magnetizations µ
z, m
zand M
zand weak jump of the transverse
13
components.
14
a
b
Fig. 6. The temperature dependence of the longitudinal and transverse components of the magnetizations(µα,mα,Mα;α=x,z)when the transition is of first-order, for (a)(J4/J2=1.5, Ω/J2=0.5)and (b)(J4/J2=1.5, Ω/J2=1.5).
4. Other thermodynamical properties 1
