Monte Carlo investigation of the mixed spin Ising model with four-spin interaction and next-nearest neighbor couplings

Accepted Manuscript

Monte Carlo investigation of the mixed spin Ising model with four-spin inter- action and next-nearest neighbor couplings

M. Azhari, N. Benayad, M. Mouhib

PII: S0749-6036(14)00483-2

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Reference: YSPMI 3532

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Received Date: 20 March 2014

Revised Date: 5 December 2014 Accepted Date: 8 December 2014

Please cite this article as: M. Azhari, N. Benayad, M. Mouhib, Monte Carlo investigation of the mixed spin Ising model with four-spin interaction and next-nearest neighbor couplings,

(2014), doi: http://dx.doi.org/10.1016/j.spmi.2014.12.015

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1

Monte Carlo investigation of the mixed spin Ising model with four-spin interaction and next-nearest neighbor couplings

M. Azhari, N. Benayad_*, M. Mouhib

Laboratory of High Energy Physics and Scientific Computing, Hassan II University–Casablanca, Faculty of Sciences-Aïn Chock,

B.P : 5366 Maarif, Casablanca 20100, Morocco.

*Corresponding author: E-mail: n.benayad@fsac.ac.ma; noureddine_benayad@yahoo.fr Phone: + 212 5 22 23 06 84, Fax: + 212 5 22 23 06 74

Abstract

A Monte Carlo method is used to investigate the phase diagram of the ferromagnetic two-dimensional mixed spin-1 and spin- 1/2 Ising model with four-spin interaction J4 and next-nearest couplings J’. In the absence of this latter interaction, the system does not present any evidence for the tricritical point, whereas such behavior was suggested by finite cluster approximation.

In the presence of the next-nearest neighbor interaction, the phase diagram is qualitatively and quantitatively different from that obtained with J’=0. It undergoes two kinds of behaviors according to the negative and positive value of J’. In particular, for J4/J₂=-4 the system keeps the coexistence of the two ground states up to a J’-dependent finite temperature. In addition to the phase diagrams, the thermal dependence of the magnetizations have also been examined. Some characteristic features have been found which depend on the strengths of the interactions J₄ and J’.

Keywords: mixed spin Ising model, four-spin interactions, next-nearest neighbor, phase diagram, magnetizations, Monte Carlo simulation.

1. Introduction

The Ising model continues to be one of the most frequently studied models in statistical mechanics, because of its simplicity and wide applicability. Recently, there has been considerable interest in experimental and theoretical researches of Ising model with multispin interactions. These models are interesting because they found their theoretical explanation in the theories of super exchange interaction, the magnetoelastic effect and the spin-phonon coupling [1]. Moreover, it was pointed out that the models with the higher-order exchange interactions may exhibit rich phase diagrams and can describe phase transition in some physical systems.

Additionally, they show physical behavior not detected in the usual spin systems. For example, the deviation from ^/ Block law at low temperature [2] and the non universal critical phenomena [3,4]. For instance, the violation of universality is observed in the two-dimensional square lattice Ising model with nearest-neighbor, next-nearest neighbor, and four spin interactions [5,6].

From the theoretical point of view, the monoatomic Ising models with multispin interactions have been investigated in detail within different methods, such as mean field approximation [7,8], effective field theory [9,10], some more accurate treatments such as series expansion [11,12], renormalization group methods [13], Monte Carlo simulations [14], and also exact calculations [15,16]. Experimentally, an interesting fact for the models with multispin interactions has been reported. Indeed, it can be used to describe various physical systems such as classical fluid [17], solid ³He [18], lipid bilayers [19], and rare gases [20]. Moreover for some materials it has been shown that the multispin interactions play a significant role; and they are comparable or even much important than the bilinear ones. The models with pair and quartet interaction have been applied successfully to study and explain the existence of first order phase transition in squaric acid crystal H2C2O4[21]. Such models have been also used to describe thermodynamical properties of hydrogen-bonded ferroelectric PbHPO4, PbDPO4[22], some copolymers [23] and optical conductivity [24] observed in cuprate ladder LaxCa14-XCu24O41. On the other hand, some experimental studies on La6Ca8Cu24O41[25,26] and La4Sr10Cu24O41[27] reveal that they

2

could be explained by the introduction of the four-spin interaction. It is worthy to note here that this later plays an important role in the two dimensional antiferromagnet La2CuO2[28], the parent material of high-TC

superconductors.

The inclusion of further-neighbor interactions would allow for a better modeling of real magnetic systems [29] and of course of all other systems that can be mapped onto the Ising models such as models of microemulsions [30]. Concerning the Ising model with next-nearest neighbor interactions (NNN), it is of interest, not only because of the existing theoretical questions (universality), but also because experimental work [31] has shown that real pseudo-two-dimensional systems exist and the model (Ising model with NNN) may have physical significance. Thus, it can model interesting physical system as notably gases adsorbed on a crystalline surface, or on layered crystals. Indeed, adsorbed monolayers at 50% coverage are also good physical analogs of NNN Ising square lattice [32].

Intense interest has been directed to study the magnetic properties of two-sublattices mixed spin Ising system. They have less translational symmetry than their single spin counterparts, and are well adapted to study a certain type of ferrimagnetism [33]. Experimentally, it has been shown that theMnNi(EDTA)-6H2O complex[34]

is a good example of a mixed system. The mixed Ising model consisting of spin-1/2 and spin-1 with only two- bilinear interaction has been studied by the renormalization group techniques [35,36], by high temperature series expansion [37], by free fermions approximation [38] and by finite cluster approximation [39]. The introduction of the next nearest neighbor (N.N.N.) coupling has been studied using numerical transfer matrix techniques [40]

and Monte Carlo simulation [41]. Attention has been devoted to study the ground-state and the influence of the NNN on the transition temperature.

In a very recent work [42], one of us (N.B) has studied the thermodynamical properties of the mixed spin Ising model with four-spin interactions on square lattice, using finite cluster approximation (FCA). For instance, it has been shown that the phase diagram displays a second-order transition line which ends in a tricritical point which depends on the strength of the four-spin interaction.

The first purpose of this paper is to investigate the phase diagrams and the magnetic properties of the ferromagnetic mixed spin Ising model with four-spin interaction on square lattice using Monte Carlo simulation and; in particular, compare our phase diagram with that obtained very recently by the finite cluster approximation. The second goal of this work is to examine the effects of the next-nearest neighbor interactions (N.N.N) on the obtained magnetic properties. Since the NNN and the four spin interactions have important effects on 2D Ising real systems, this makes the suggested model extremely interesting from theoretical and simulation viewpoints. Our system can be described by the following Hamiltonian:

,,,

′

1 The underlying lattice is composed of two interpenetrating sublattices. One occupied by spins with spin moment σ=±1/2 and the other one is occupied by spins with moment S= 0,±1. The first summation carried out only over nearest-neighbor pair of spins. The second and the third summations represent the four-spin and NNN interactions, respectively, where the summations concern all alternate squares shaded in Fig.1.

The ground states of the Hamiltonian (1) can be easily found by comparing the energies of corresponding configurations. The resulting ground state phase diagram is shown in Fig.2. In the present work, our study is focused on the ferromagnetic state I.

The presentation of this paper is as follows. In section 2, we outline the formalism of Monte Carlo approach we used in the simulation. The results of our simulation and discussions are presented in section 3. Finally, our concluding remarks are summarized in section 4.

2. Monte Carlo simulation

To simulate the system described by the Hamiltonian (1), we have used the Monte Carlo technique. This latter is an accurate tool for the investigation of various statistical mechanical problems. In this approach the statistical mechanical expectation value <A> of an operator A is estimated over a relatively small sample of the total collection of states of the system. The configurations used for this estimation are chosen so that the probability of each state appearing is related to the importance of its contribution to this expectation value.

Indeed, our two-dimension square lattice contains LxL spins. We use the metropolis algorithm with single-spin flips with periodic boundary conditions. In this approach, the probability of a successful spin flip is given by

3

exp 2 where δE is the energy involved in the change of the state of the spin. Since spin S has tree states, the possible

“New” state is chosen randomly before each spin-flip trial, whereas for the spin σ we first choose its corresponding opposite state. The physical quantity described by the operator A is estimated by

1

/ 3 where the summation runs over all configurations obtained by using the Metropolis algorithm to update the lattice over one sweep of N spins of the lattice (one Monte Carlo step (MCS)) counted after the thermal equilibrium . N is the number of MCS. The thermal equilibrium of the system can be tested by monitoring the energy as a function of MCS. Most of data were obtained with L=100 but we also performed some simulations with L=10, 50 and 100 in order, for instance, to examine the finite size effects as shown in Fig.3. In this investigation, the data were generated with 5.10⁴ MCS per spin after discarding the first 10⁴steps.

Let us denote by and the σ- and S–sublattice magnetizations, respectively. They are defined as

1

4

1

5 where (C) and (C) are the magnetizations obtained from one MCS, defined as

2

# $ #

6

2

# $ #

7 The total magnetization is defined by

'

2 8 The second order phase transition is determined by the observation of a peak in the susceptibility χ (Fig.4(a)),

) /(

)

( Δ m

k

TL

χ =

where the fluctuation of the magnetization m is given by

2 2 >-< >

<

= m m

Δm (10) In order to accurate its numerical value, we use the Binder cumulant [43]

) 1 /3 11 Thus, the critical temperature is the fixed point where the cumulant (11) for different system sizes must intersect as is shown in Fig.4(b).

4 3. Simulation results and discussions

Let us first consider the system in the absence of the four-spin interaction (J4=0) and NNN couplings (J’=0).We recover the usual two dimensional Ising model on the square lattice. The critical temperature is 0.980, which is determined from the remarkably pronounced peak of magnetic susceptibility obtained for L=100, and confirmed by the intersection of the Binder cumulant for different sizes, as seen from Fig.4.This value, which improves the FCA result [42], is very close to 0.974 obtained from high-temperature series [44] and can be compared with the result (1.15) obtained by the renormalization group[35] and a previous Monte Carlo simulation (0.976) [45]. In the absence of the next-nearest neighbor interaction (J’=0),the system reduces to two- sublattice mixed spin-1/2 and spin-1 Ising model with four-spin interaction. In the present work, we study essentially the ferromagnetic state. The variation of the order-disorder transition temperature with strength of the four-spin interaction is represented in Fig.5. As is expected, the transition temperature is an increasing function of J4. Indeed, a positive value of the four-spin term enhances the tendency for long range order; whereas for J4<0, this latter, disturbs the ferromagnetic state and therefore the transition temperature decreases from the value TC (J4=0) to vanish at J4=-4J2. We note that the system exhibits second order transition for any value of the four-spin interaction J4/J2>-4. It’s worthy of notice that the obtained MC phase diagram is qualitatively similar to that calculated by high-temperature series expansion for monoatomic spin-1/2 Ising model with four-spin interaction [46]. We have to mention that our MC phase diagram (Fig.5) agrees in part (at low temperature) with that obtained very recently [42] by finite cluster approximation, since this effective method predicted a tricritical behavior for a particular value of the four-spin interaction. This discordance at high temperature is due to fact that the finite cluster approximation has been designed to treat all spin self-correlation exactly while still neglecting correlations between different spins.

Secondly, we study the influence of the next-nearest neighbor coupling on the phase diagram of the ferromagnetic mixed spin-1/2 and spin-1 Ising model with four-spin interactions, plotted in Fig.5. The resulting phase diagram is depicted in Fig.6. It presents the dependence of the critical temperature as a function of the four-spin interaction J4 for both negative and positive values of the NNN interaction J’. These curves give the sections of the critical surface TC(J4,J’) with planes of fixed values of NNN coupling. We note that TC increases with increasing values of J4 and J’, which is physically reasonable. As seen from this figure, the critical line J’=0 separates two different qualitatively behaviors corresponding to J’<0 and J’>0. Indeed, when the value of the NNN coupling belongs to the range -4<J’/J2<0 it acts against the order and therefore reduces the ferromagnetic domain. Thus, and as can be expected, the system undergoes, at the ground state, a phase transition at a finite critical value (J4/J2)C. This latter belongs to the transition line J4/J2+J’/J2=-4 which is obtained from the exact calculation of the ground state of the system. As shown in Fig.2, this line separates the states I and III. On the other hand, a positive value of J’ strengthens the order at low temperature. The long-range ferromagnetic order domain becomes wider with increasing values of NNN couplings which is clearly shown in Fig.6. Furthermore, at zero-temperature, the line (J’/J2>-4,J4/J2=-4) separates the two ground states I and II shown in Fig.2, which correspond to J4>-4J2 and J4<-4J2, respectively. This means that at each point of this line, the phase I and phase II coexist. As seen in Fig.6,our MC investigation shows that this coexistence, at J4/J2= -4, is preserved at low temperatures. Thus, for any given J’/J2>0, when J4/J2 approaches -4, the transition temperature decreases and depresses to a J’-dependent finite value T*/J2 (J’) at J4= -4J2. In Fig.7, we plot the variation of T*/J2 with J’/J2, which delimits the stability domain of the above coexistence for any positive strength of the NNN coupling.

Since the ferromagnetic version of the system under study shows qualitatively interesting features, it is worthy to investigate the thermal behaviors of the sublattice magnetizations m_σand mS as well as the total magnetization mT defined by mT=(m_σ+mS)/2. Let us first examine these quantities in the absence of the next- nearest neighbor interaction (J’=0), and how the four-spin interaction J4 influences the magnetic properties of the mixed spin -1/2 and spin-1 Ising model. In Fig. 8, we plot the temperature dependence of m_σ, mS and mT, selecting various values of J4/J2. For J4/J2>-4, the system is ferromagnetic. Therefore, at T=0K, the total magnetization is mT=0.75, since the sublattice magnetizations are given by m_σ=1/2 and mS=1. With the increase of the temperature, the m_σ, mS and mT curves decrease continuously from their saturation values to vanish at a J4- depended critical value TC(J4) which can be determined using the procedure described in the previous section.

We note that the effect of the four-spin interaction J4 is to reinforce the ordering of m_σ, mS and mT. So, as seen from Fig.8, the long range ferromagnetic order domain becomes wider with increasing values of the four-spin interaction. As shown in this section, the next-nearest neighbor interaction has significant effects on the phase

5

diagram of the system as depicted in Fig.6. This latter is qualitatively and quantitatively different from that obtained for J’=0. Therefore, it is interesting to examine the influence of the NNN interaction on the sublattice and total magnetizations (m_σ,mS and mT). In particular, the system has two kinds of behaviors according to the positive or negative value of J’; especially when the (J4/J2, J’/J2) belongs to the borders of the state I (see Fig.2).

In Fig.9, we plot the temperature dependences of magnetizations for 6positive and negative values of J’/J2

(J’/J2=-1, 2) with various values of J4/J2 (J4/J2=-1, 1). At the ground state, the magnetizations m_σ, mS and mT take their saturation values 0.5, 1 and 0.75, respectively. For higher temperature, they decease continuously with the increase of T to vanish at (J’, J4)-dependent critical value. In the same figure, we have added the magnetization curves for (J’/J2=3.8, J4/J2=-3.8) and ((J’/J2=-2, J4/J2=-1.8)) which are located, respectively, close to the borders with the states II and III. In this case, the mS (m_σ) temperature dependence shows downward curvature. This behavior is expected in this area of the ground state phase diagram, since at low temperature, there is a competition between ferromagnetic state I and the state II (III).

As we have mentioned, when J4/J2 and J’/J2 belong to the common borders of i) state I and state II (J’/J2>0, J4/J2=-4) and ii) state I and state III (-4<J’/J2<0, J’+J4=-4J2), the system exhibits two kinds of behaviors. In the first case, the phase I and II coexist. The system keeps this coexistence up to J’-dependent temperature T*/J2. By averaging the previous coexisting phases, the sublattice and the total magnetizations take the values m_σ=1/2, mS=1/2 and mT=1/2. In Fig.10 (a) we plot the temperature dependence of magnetizations for J4/J2=-4 with different positive values of J’/J2(J’/J2=0.01, 3.8). At the ground state, all magnetization curves start from the above unique value 1/2. As shown in this figure, for low values of the NNN interactions and when the temperature is increased the S-sublattice as well as the total magnetizations increase and pass through maximums, then decrease and vanish at the transition temperature. This initial increase of mS and mT results from the fact that the phase I conflicts with the phase II. In the second case, the phases I and III coexist at the ground state and the system undergoes a transition at J’-dependent value which satisfies the relation J’+J4=-4J2. Fig.10 (b) shows the temperature dependences of the magnetizations. At T=0K, the sublattice and total magnetizations take the values m_σ=0.25, mS=1 and mT=0.625. These values are obtained by averaging the magnetizations of the phases I and III. As shown in Fig. 10 (b), with the increase of the temperature, m_σ, mS and mT curves decrease continuously from the above values to vanish at a (J’, J4)-dependent value. Finally, we have to mention that the investigation of the frustrated phases II, III, and IV is expected to show many characteristic features; in particular the existence of multitude compensation point. This is precisely our goal in future works.

4- Conclusion

In this work, we have study the magnetic properties of the two-dimensional mixed spin-1 and spin-1/2 with four-spin interaction J4 and next-nearest neighbor coupling J’. The phase diagrams obtained in the framework of the Monte Carlo simulation show very interesting features. Let us summarize by stating the main results of this investigation. In the absence of the next-nearest neighbor interactions, the system does not present a tricritical behavior suggested very recently by finite cluster approximation. We note that our Monte Carlo simulation results are qualitatively similar to those obtained for the monoatomic spin-1/2 model with four-spin interactions using high-temperature series expansions.

The influence of the next-nearest neighbor interaction J’ on the obtained phase diagram showed some characteristic behaviors, which depend on the positive and negative value of J’. In particular, for J’<0, the system exhibits, at zero-temperature, a phase transition at a finite critical value (J4/J2)C; while for J’>0 and J4/J2=-4 the system keeps the coexistence of two ground states up to a J’-dependent finite temperature.

Furthermore, the thermal dependence of the sublattice and the total magnetizations has also been examined when the ground state of the system is ferromagnetic. Some characteristic behaviors have been found; in particular when the hamiltonian parameters belong to the frontiers between ferromagnetic phase and neighboring phases in the ground state phase diagram.

Finally, we can note that in the absence of the four spin interactions, the critical behavior of the transition, from the ferromagnetic phase to the high temperature paramagnetic phase, is the same as in the square lattice ferromagnetic Ising model with only nearest-neighbor interaction. But in the presence of J4, a breakdown of universality can be expected and critical exponents will continuously vary depending on the four spin interaction strength. This situation could be similar to that shown by one of us (N.B) for the ferromagnetic Ashkin Teller

6

model [47], where the transition from the Baxter phase to the paramagnetic phase is characterized by continuously varying critical exponent of the correction length with the four spin interaction. On the other hand, it is worth to mention that the monoatomic two-dimensional square lattice Ising model with NN, NNN, and four spin interactions shows non-universal critical behavior [5,6]. Concerning the calculation of the critical exponents from the ordered states II, III, or IV to paramagnetic phase, we can expect that in these cases, the models possesses “anomalous” exponents due to special ground-state degeneracy.”

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9 Figure captions

Fig.1: Part of the square lattice. and × correspond to σ and S-sublattice sites, respectively.

Fig.2: The Ground state phase diagram.

Fig.3: The MC temperature dependence of the sublattice and the total magnetizations (mS, m_σ& mT) for different sizes.

Fig.4: The temperature dependence of (a) magnetic susceptibility, and (b) Binder parameter, for different system sizes, withJ4=0 and J’=0.

Fig.5: The MC phase diagram of the mixed spin-1/2 and spin-1 Ising model with four-spin interactions. L=100.

Fig.6: The phase diagrams in the (TC/J2, J4/J2) plane for various values of the NNN coupling. L=100.

Fig.7: The dependence of the transition temperature T*/J2 with the next-nearest neighbor coupling (J’> 0) whenJ4/J2=-4. L=100.

Fig.8: The temperature dependence of the sublattice and the total magnetizations in the absence of the NNN interaction, with various values of J4/J2. L=100.

Fig.9: The temperature dependence of the sublattice and the total magnetizations for (a) positive and (b) negative J’, with various values of J4/J2. L=100.

Fig.10: The temperature dependence of the sublattice and the total magnetizations when the interactions J4 and J’ belong to the frontier between (a) phase I-phase II and (b) phase I-phase III, at T=0K, (see Fig.2).

L=100.

10

111

122

13

14

155

166

17

18

199

20

Highlights

●We investigate a two-dimensional mixed spin Ising model with four-spin interaction J4 and next-nearest neighbor coupling J’

●In the absence of the next-nearest neighbor interactions, the system does not present a tricritical behavior.

●The influence of the next-nearest neighbor interaction depends on the positive and negative value of J’

●For J’<0, the system exhibits, at zero-temperature, a phase transition at a finite critical value (J4)C.

●For J’>0 and J4 = -4 the system keeps the coexistence of two ground states up to a J’-dependent finite temperature.