Dynamic phase transitions and dynamic phase diagrams of the Ising model on the Shastry-Sutherland lattice

Dynamic phase transitions and dynamic phase diagrams of the Ising model on the Shastry-Sutherland lattice

Journal of Magnetism and Magnetic Materials 402 (2016) 94–100 Contents lists available at ScienceDirect Journal of Magnetism and Magnetic Materials ...

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Journal of Magnetism and Magnetic Materials 402 (2016) 94–100

Contents lists available at ScienceDirect

Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm

Current Perspectives

Dynamic phase transitions and dynamic phase diagrams of the Ising model on the Shastry-Sutherland lattice Şeyma Akkaya Deviren a,n, Bayram Deviren b a b

Department of Science Education, Education Faculty, Nevsehir Hacı Bektaş Veli University, 50300 Nevşehir, Turkey Department of Physics, Nevsehir Hacı Bektaş Veli University, 50300 Nevsehir, Turkey

art ic l e i nf o

a b s t r a c t

Article history: Received 6 July 2015 Received in revised form 11 November 2015 Accepted 13 November 2015

The dynamic phase transitions and dynamic phase diagrams are studied, within a mean-field approach, in the kinetic Ising model on the Shastry-Sutherland lattice under the presence of a time varying (sinusoidal) magnetic field by using the Glauber-type stochastic dynamics. The time-dependence behavior of order parameters and the behavior of average order parameters in a period, which is also called the dynamic order parameters, as a function of temperature, are investigated. Temperature dependence of the dynamic magnetizations, hysteresis loop areas and correlations are investigated in order to characterize the nature (first- or second-order) of the dynamic phase transitions as well as to obtain the dynamic phase transition temperatures. We present the dynamic phase diagrams in the magnetic field amplitude and temperature plane. The phase diagrams exhibit a dynamic tricritical point and reentrant phenomena. The phase diagrams also contain paramagnetic (P), Néel (N), Collinear (C) phases, two coexistence or mixed regions, (Nþ C) and (N þP), which strongly depend on interaction parameters. & 2015 Elsevier B.V. All rights reserved.

Pacs: 03.65.Vf 05.50. þq 64.60.Ht 75.10.Hk Keywords: Shastry-Sutherland lattice Dynamic phase transition Dynamic phase diagram Glauber-type stochastic dynamics

1. Introduction Geometric frustration is an important research area in condensed matter and statistical physics, and has exhibit very rich magnetic properties so it has drawn much attention during the last several decades [1]. The Shastry-Sutherland lattice (SSL), one of the simplest systems with geometric frustration, which was first introduced by Shastry and Sutherland in 1981 [2]. This frustrated lattice can be described as a square lattice with antiferromagnetic (AFM) coupling (J’) between the nearest neighbors and additional AFM coupling (J) between next-nearest neighbors in every second square. The SSL and its experimental realization in the material SrCu2BO3 [3], have much attention due to its fascinating behavior in magnetic field [4,5]. Recently similar magnetization behavior has been also observed in a few rare-earh tetraborides RB4 (R¼Tb, Dy, Ho, Tm etc) with the magnetic ions of these compounds located on a lattice which is topologically equivalent to the SSL [6– 11]. Unlike the SrCu2(BO3)2, TmB4 has a large total magnetic mo+3 is ∼6.0 mB) and can be conment (the magnetic moment of Tm sidered as a classical spin system. Moreover, due to the strong n

Corresponding author. E-mail address: [email protected] (Ş.A. Deviren).

http://dx.doi.org/10.1016/j.jmmm.2015.11.045 0304-8853/& 2015 Elsevier B.V. All rights reserved.

crystal field effects of TmB4, the effective spin model for TmB4 has been suggested to be described by the spin-1/2 Shastry Sutherland model under a strong Ising (or easy axis) anisotropy [10]. Based on this fact, theoretically, the magnetization process has been studied in great detail by using various approaches on the SSL. Chang and Yang have investigated the plateau in the magnetization process for the classical AFM Ising model on the SSL by using the tensor renormalization group approach [12]. Slavin and Krivchikov have numerically studied two dimensional SSL in the framework of classical Heisenberg model with Monte Carlo (MC) simulations [13]. Fangzhou and Subir have examined the phase diagrams of the SS antiferromagnet within the magnetic field in the limit of strong anisotropy using the MC simulation [14]. The magnetic properties of the spin-1/2 Ising like XXZ model have studied by using quantum MC method [15–16]. Moreover, a model that is combination of electron and spin subsystems has examined to understand the magnetization process in RB4 [17]. The magnetic behaviors of the classical Ising model have been studied by means of MC simulation with the long range RKKY (Ruderman-Kittel-Kasuya-Yosida) interaction on the Archimedian lattice that is topologically equivalent to SSL one [18]. More recently, Verkholyak et. al. have obtained the exact ground states of a spin-1/2 Ising-Heisenberg model on the SSL in a magnetic field [19], and Deviren has investigated the magnetization properties of a two-dimensional

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spin-1/2 Ising model on the SSL by using the effective-field theory (EFT) with correlations [20]. On the other hand, the nonequilibrium or dynamic magnetization process may play important role to describe the magnetization behaviors and dynamic phase transitions due to the influence of sinusoidally oscillating magnetic field. Although dynamic magnetic properties of the Ising systems have attracted much attention for over two decades, the mechanism behind the dynamic phase transition (DPT) of these systems is not that well known. The DPTs originate due to a competition between time scales of the relaxation time of the system and oscillating period of the external applied magnetic field. In particular, the kinetic Ising models on the lattice are very often used to describe the time evolution and the corresponding steady states of a great variety of interacting particle systems, such as catalysis, contact process, domain growth, phase separation, and transport phenomena. After Tomé and Oliveira [21] first studied this type of investigation regarding the DPT properties of kinetic spin-1/2 Ising model in the presence of a time-dependent oscillating field by using the meanfield theory (MFT) based on Glauber dynamics [22], a great deal of studies concerning the DPTs as well as dynamics phase diagrams of different types of magnetic systems have been investigated by a variety of techniques such as MFT [23–27], MC simulations [28– 32], EFT [33–36], real space renormalization group technique [37] and using recursion relations on the Bethe Lattice [38]. Not only is it an interesting system from a purely theoretical point of view, but also the kinetic Ising model can be used to describe experimental evidence in highly anisotropic (Ising-like) and ultra thin Co/Cu (001) ferromagnetic films [39], a [Co/Pt]3 magnetic multilayer system with strong perpendicular anisotropy [40], amorphous YBaCuO films [41], nanocomposites [42]. Aside from a few approximate analytical methods were used to investigate these systems, computational methods have been the main tool to study nonequilibrium steady states. It is also mentioning that we have found a few works about the dynamic magnetization behavior of the classical Ising model on the SSL such as: Huang et al. have studied classical Shastry Sutherland Ising model by means of the Glauber dynamics with long range interactions [43] and magnetization dynamics in frustrated triangular spin chain compound Ca3Co2O6 investigated [44–46]. Thus, the main motivation of this paper is to understand the dynamic behaviors of the classical Ising model on the SSL with the Glauber dynamics. Thermal variations of the dynamic magnetizations, the hysteresis loop areas and the dynamic correlations are investigated in order to characterize the nature (first- or secondorder) of the dynamic transitions as well as to obtain the DPT temperatures and the dynamic phase diagrams. The phase diagrams exhibit a dynamic tricritical point and reentrant phenomena. Moreover, phase diagrams contain paramagnetic (P), Néel (N), Collinear (C) states, two coexistence or mixed regions, (N þC) and (N þP), which strongly depend on interaction parameters. The organization of the remaining part of this paper is as follows. In Section 2, the model and formulation are given. Section 3 contains the numerical results and discussions. Finally, the summary and conclusions are given in Section 4.

Fig. 1. Schematic representation of the Shastry-Sutderland lattice. The anti-ferromagnetic exchange interaction (J) between all nearest neighbor bonds and additional anti-ferromagnetic exchange interaction (J′) between next-nearest neighbor bonds in every second square.

H = − J′

∑ ( σi σj + Si Sj ) − J ∑ ( σi Si + σi Sj ) ⟨ij⟩

− h (t )

∑ ( σi

⟨⟨ij⟩⟩

+ σj + Si + Sj ) ,

i

The SSL can be described as a square lattice with AFM exchange interaction (J′) between all nearest neighbor bonds and additional AFM exchange interaction (J) between next-nearest neighbor bonds in every second square, as sketched in Fig. 1. The Hamiltonian of the spin-1/2 Ising model on the SSL is given by

(1)

where si, sj, Si and Sj,¼ 71 are the spin variables; o ij4 indicates a summation over all the diagonal bonds; ij runs over all the square bonds. h(t) is a time-dependent external oscillating magnetic field and is given by

h (t ) = h0 sin (wt),

(2)

where h0 and w¼2πν are the amplitude and the angular frequency of the oscillating field, respectively. The system is in contact with an isothermal heat bath at absolute temperature TA. Now, we apply the Glauber-type stochastic dynamics to obtain the set of the mean-field dynamic equations. Thus, the system evolves according to a Glauber-type stochastic process at a rate of 1/τ transitions per unit time. Since the derivation of the meanfield dynamic equations was described in detail for spin-1/2 system [21] and different spin systems [25,47,48], in here, we shall only give a brief summary. The spins are assumed to interact not only with the neighbors and oscillating external magnetic field but also with heat bath, based on the Glauber-type stochastic dynamics. The master equation for the first s-spins can be written as

d P ( σ1, σ 2, ... , σ N; t) = − dt +

2. Model and formulations

95



∑ i

Wi ( −σi ) P ( σ1, σ 2, ... , σ i, ... σ N; t)

i

Wi ( σi ) P ( σ1, σ 2, ... , − σ i, ... σ N; t).

(3)

where Wi (σi ) is the probability per unit time that the ith s spin changes from si to – si ( while the spins on other sublattice momentarily fixed). Since the system is in contact with a heat bath at absolute temperature TA, each spin s can flip with the probability per unit time;

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Wi ( σi ) =

1 exp ( −β ΔE ( σi ) ) , τ ∑σ exp ( −β ΔE ( σi ) )

(4)

i

where β = 1/kB TA,kB is the Boltzmann factor,∑σ is the sum over the i

two possible values of σi , ±1, and

⎛ ΔE ( σi ) = 2σi ⎜⎜ H + z1 J′ ⎝



∑ σj + z2 J ∑ ( Si + Sj ) ⎟⎟, j

ij



(5)

gives the change in the energy of the system when the si-spin changes. The probabilities satisfy the detailed balance condition and by using these equations with mean-field approach, the first mean-field dynamic equation can be written in the presence of a time-varying field as Ω

d m1 = − m1 + tanh ⎡⎣ β ( z1 J′ m2 + z2 J ( m3 + m4 ) + h 0 sin (ξ ) ) ⎤⎦ , dξ

(6)

where m1 = σi , m2 = σj , m3 = Si , m4 = Sj , ξ = w t , and Ω ¼ τ w . Moreover z1 ¼ 1 and z2 ¼2 correspond to the number of nearestand next nearest-neighboor pairs of spins, respectively. We can also obtain the other mean-field dynamical equations for the other spins on the SSL by using the similar calculations as follows; Ω

d m2 = − m2 + tanh ⎡⎣ β ( z1 J′ m1 + z2 J ( m3 + m4 ) + h 0 sin (ξ ) ) ⎤⎦ , dξ

(7)

Ω

d m3 = − m3 + tanh ⎡⎣ β ( z1 J′ m4 + z2 J ( m1 + m2 ) + h 0 sin (ξ ) ) ⎤⎦ , dξ

(8)

Ω

d m4 = − m4 + tanh ⎡⎣ β ( z1 J′ m3 + z2 J ( m1 + m2 ) + h 0 sin (ξ ) ) ⎤⎦ , dξ

(9)

Thus, the set of the mean-field dynamical equations for the average magnetizations are obtained, namely Eqs. (6–9). We fixed Ω ¼2π. The dynamic order parameters or dynamic magnetizations as the time-averaged magnetization over a period of the oscillating magnetic field are given as

Mα =

w 2π

∮ mα (t ) dt ,

(10)

where α ¼1, 2, 3, and 4 which correspond to the dynamic magnetizations on the SSL. On the other hand, the hysteresis loop areas are defined by Acharyya [28] as

Aα = −

∮ mα (t ) dh =

− h0 w

∮ mα (t ) cos (wt ) dt,

(11)

which correspond to the energy loss due to the hysteresis. The dynamic correlations are calculated as

Cα =

w 2π

∮ mα (t ) h (t ) dt = w2hπ0 ∮ mα (t ) sin (wt ) dt.

(12)

We should also mention that in the numerical calculations, the hysteresis loop areas Aα and the dynamic correlations Cα are also measure in units J and J′. In the next section, we will give the numerical results of these equations.

3. Numerical results and discussions

coupled dynamical mean-field equations, given in Eqs. (6–9), when the parameters J, J′, T and h are varied. The stationary solutions of these equations will be a periodic function of ξ with period 2π; that is, m2 (ξ + 2π ) = m2 (ξ ), m1 (ξ + 2π ) = m1 (ξ ), m3 (ξ + 2π ) = m3 (ξ ) and m4 (ξ + 2π ) = m4 (ξ ). Moreover, they can be one of three types according to whether they have or do not have the properties m1 (ξ + π ) = − m1 (ξ ), m2 (ξ + π ) = − m2 (ξ ), m3 (ξ + π ) = − m3 (ξ ), and m4 (ξ + π ) = − m4 (ξ )

(13)

The first type of solution satisfies Eq. (13) is called a symmetric solution which corresponds to a paramagnetic (P) solution. In this solution, the magnetizations m1, m2, m3 and m4 are equal to each other (m1 = m2 = m3 = m4 ) and they oscillate around zero and are delayed with respect to the external magnetic field. The second type of solution, which does not satisfy Eq. (13), is called a nonsymmetric solution that corresponds to a Néel (N) solution. In this solution, the m1 (ξ ) and m2 (ξ ) are equal to each other (m1 (ξ ) = m2 (ξ ) = − 1.0), and they do not follow the external magnetic field any more, and instead of oscillating around zero, they oscillate around nonzero values, namely  1.0; but the m3 (ξ ) and m4 (ξ ) are equal to each other and oscillate around þ 1.0 (m3 (ξ ) = m4 (ξ ) = + 1.0); hence, we have the Néel (N) state. The third type of solution, which does also not satisfy Eq. (13), is called a nonsymmetric solution but this solution corresponds to a Collinear (C) solution; because the (m1 (ξ ) = m3 (ξ ) = − 1.0) and (m2 (ξ ) = m4 (ξ ) = + 1.0). These facts are seen explicitly by solving Eqs. (6–9) using the Adams–Moulton predictor-corrector method for a given set of parameters and initial values, and obtained results are presented in Fig. 2. From Fig. 2, one can see following five different solutions or phases, namely the P, N and C fundamental solutions, and two coexistence state or solutions, namely the C þN in which C and N solutions coexist; the C þP in which C and P solutions coexist, have been found. In Fig. 2(a) only the symmetric solution is always obtained, in this case (m1 (ξ ) = m2 (ξ ) = m3 (ξ ) = m4 (ξ )) oscillate around zero value (m1 (ξ ) = m2 (ξ ) = m3 (ξ ) = m4 (ξ ) = 0.0). Hence, we have a P solution. On the other hand in Fig. 2(b) and (c) only the nonsymmetric solutions are found; therefore, we have the N and C solutions, respectively. In Fig. 2(b), m1 (ξ ) and m2 (ξ ) oscillate around  1 and m3 (ξ ) and m4 (ξ ) oscillate around þ1; hence we have the Néel (N) state (m1 (ξ ) = m2 (ξ ) = − 1.0, m3 (ξ ) = m4 (ξ ) = + 1.0). In Fig. 2(c), m1 (ξ ) and m3 (ξ ) oscillate around  1 and m2 (ξ ) and m4 (ξ ) oscillate around þ1, this solution corresponds to the Collinear (C) state (m1 (ξ ) = m3 (ξ ) = − 1.0, m2 (ξ ) = m4 (ξ ) = + 1.0). In Fig. 2 (d), m1 (ξ ) and m3 (ξ ) oscillate around either 1 and m2 (ξ ) and m4 (ξ ) oscillate around either þ 1, that corresponds to the C state, or m1 (ξ ) and m2 (ξ ) oscillate around either  1 and m3 (ξ ) and m4 (ξ ) oscillate around either þ1 which corresponds to the N state; hence we have the coexistence solution (C þN), as explained above. In Fig. 2 (e), m1 (ξ ) and m3 (ξ ) oscillate around either  1 and m2 (ξ ) and m4 (ξ ) oscillate around either þ1, that corresponds to the C state, or oscillate around zero value which corresponds to the P phase; hence we have the coexistence solution (C þP). A symmetric solution does not depend on the initial values, but the other solutions depend on the initial values. 3.2. Thermal behavior of the dynamic magnetizations, hysteresis loop areas and correlations

3.1. Time variations of the average order parameters In this section, first we study the time variations of the average magnetizations to find the phases in the system. In order to investigate the behaviors of time variations of the average magnetizations, we have to study the stationary solutions of the set of

In this section, we investigate the behavior of the dynamic magnetizations (Mα), hysteresis loop areas (Aα) and correlations (Cα) as a function of the temperature on the SSL for several values of J and J’′in the presence of the external magnetic field. In order to investigate the thermal behavior of the Mα, Aα and Cα, we solve

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Fig. 2. Time variations of the average magnetizations: (a) Exhibiting a paramagnetic (P) phase: J¼  0.1, J’¼  1.0, h ¼1.5 and T ¼1.0. (b) Exhibiting a Néel (N) phase: J¼  0.1, J’¼  1.0, h ¼ 0.5 and T ¼ 0.4. (c) Exhibiting a Collinear (C) phase: J¼  2.0, J’¼  1.0, h ¼ 0.25 and T ¼ 0.3. (d) Exhibiting a coexistence region (Nþ C): J¼  1.0, J’¼  1.0, h ¼0.2 and T ¼ 0.1. (e) Exhibiting a coexistence region (N þ P): J¼  0.5, J’¼  1.0, h ¼ 1.0 and T ¼0.5.

Eqs. (10–12) by combining the numerical methods of Adams– Moulton predictor corrector method with the Romberg integration. This study leads us to characterize the nature (continuous or discontinuous) of DPTs, to check the stability of each dynamical phase, as well as to obtain the DPT points. A few explanatory and

interesting results are plotted in Fig. 3(a–e) to illustrate the calculation of the DPT points. Fig. 3(a) illustrates the thermal variation of the Mα, Aα and Cα for J¼  0.1, J′¼  1.0 and h¼1.0. In this figure, the dynamic magnetizations M1 ¼M2 ¼  M3 ¼  M4 ¼  1.0 at zero temperature, and they go to zero continuously as the

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Fig. 3. Thermal variations of the dynamic magnetizations for the various values of J′and h. TN/|J| and Tt/|J| are the second- and first-order phase transition temperatures, respectively. (a) Exhibiting a second-order phase transition from the Néel (N) phase to the paramagnetic (P) phase for for J’¼  1.0, J¼  0.1 and h ¼ 1.0; TN/|J|¼ 0.85 is found. (b) Exhibiting a first-order phase transition from the N phase to the P phase for J’ ¼  1.0, J¼  1.0 and h ¼ 1.25; Tt/|J| ¼0.175 is found. (c) Exhibiting two successive phase transition at two different phase transition temperatures for J’¼  1.0, J ¼  0.1, h/|J|¼ 1.5, and 0.16 and 0.485 are found Tt/|J| and TN/|J|, respectively. (d) Exhibiting a dynamic second-order phase transition from the N phase to the P phase for J’¼  1.0, J¼  1.0 and h ¼0.2 and the initial values of M1 ¼ M2 ¼  M3 ¼  M4 ¼  1.0; 0.745 is found TN1/|J|. e) Exhibiting a dynamic second-order phase transition from the C phase to the P phase for J’¼  1.0, J¼  1.0 and h ¼ 0.2 and the initial values of M1 ¼  M2 ¼M3 ¼  M4 ¼  1.0; 0.245 is found TN2/|J|.

temperature increases; therefore, a second-order phase transition occurs at TN/|J| ¼ 0.85 and the DPT is from the Néel (N) phase to the paramagnetic (P) phase. We have checked the stability of DPT points between the phases of the system by investigating the Aα and Cα. The Aα and Cα become a maximum and a minimum (negative) at the second-order phase transition temperature TN/|J|, respectively. This has also been tested by our calculations, because we have found exactly the same critical temperature (TN/|J|) for these calculations. Fig. 3(b) is plotted for J ¼  1.0, J′¼  1.0 and h¼1.25. In Fig. 3(b), M1 ¼M2 ¼ M3 ¼ M4 ¼  1.0 at the zero temperature and they go to zero discontinuously as the temperature increases; hence, the system undergoes a first-order phase transition from the N phase to the P phase at Tt/|J| ¼0.175. Therefore, Tt/|J| is the first-order phase transition temperature where the discontinuity or jump occurs. We also checked this dynamic

discontinuous transition by investigating the thermal behavior of the dynamic hysteresis loop areas Aα and dynamic correlations Cα, seen in Fig. 3(b). From the figure, one can see that if the temperature increases from zero, the Aα and Cα increase and decrease from zero to a certain positive non zero values, and Aα and Cα suddenly jump to the higher positive and lower negative values, respectively; hence, the first order phase transition occur at Tt/|J| ¼ 0.175, which is exactly the first-order phase transition that is found by investigation of the thermal behavior of the Mα, Aα and Cα. Fig. 3(c) is obtained for the values of J ¼  0.1, J′¼  1.0 and h¼1.5. In this case, the system undergoes two successive phase transitions: The first one is first-order phase transition occurred at Tt/|J| ¼0.175 from the P phase to the N phase that is M1 ¼ M2 ¼  M3 ¼ M4 ¼ 1.0. The second is a second-order phase transition from the N phase to the P phase at TN/|J| ¼0.485. Two successive phase

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Fig. 4. The phase diagrams in the (T/|J|, h/|J|) plane for J ¼  1.0. Dashed and solid lines represent the first- and second-order phase transition temperature, respectively. The tricritical point are indicated with filled triangle. (a) J’¼  0.1, (b) J’¼  0.5, (c) J’¼  1.0, and (d) J’¼  2.0.

transitions were also experimentally found in TbB4 at TN1 ¼24 K and TN2 ¼44 K in the magnetic susceptibility measurements [49]. Moreover, two successive transitions have also been theoretically found on the magnetic properties of the spin-1/2 Ising-like XXZ model on the SSL with long-range interactions, using the quantum MC method [50]. On the other hand, in order to see the Collinear þNéel mixed state the temperature dependence of Mα, Aα and Cα are plotted for J ¼  1.0, J ¼  1.0, h¼0.2 and various initial values, seen in Fig. 3(d) and (e). The behavior of Fig. 3(d) is similar to Fig. 3(a); hence the system undergoes a second-order phase transition from the N phase to the P phase at TN1/|J| ¼0.745. In Fig. 3 (e), the system undergoes a second order phase transition at TN2/| J| ¼0.245, that is from the Collinear (C) phase (M1 ¼  M2 ¼M3 ¼  M4 ¼  1.0) to the P phase. If one compares Fig. 3(d) and (e) with Fig. 4(c), the system exhibits the Collinear þNéel (C þN) mixed phase until TN2/|J| ¼0.245, the N phase between TN2/|J| ¼ 0.245 and TN1/|J| ¼0.745, the P phase after TN1/|J| ¼ 0.745. Therefore, the coexistence region i.e, the C þ N mixed state exists in the system between TN2/|J| ¼0.245 and TN1/|J| ¼0.745. By investigating the Aα and Cα, the stability of DPT between the phases in the system have been checked, because the same critical temperature for these calculations have exactly been found. 3.3. The dynamic phase diagrams Since we have characterized the nature of DPTs and obtained the DPT points in subsection 3.2, we can now present the dynamic phase diagrams of the system. The calculated phase diagrams in the (T/|J|, h/|J|) plane are presented in Fig. 4 for J¼  1.0 and various values of J′ In Fig. 4, the solid and dashed lines represent the dynamic second- and first-order phase transition lines, respectively,

and the dynamic tricritical point is denoted by a filled triangle. From these phase diagrams following interesting phenomena have been observed. (i) Fig. 4(a) is obtained for J′¼0.1, and in the phase diagram, at high temperature (T) and high external magnetic field (h), the solutions are paramagnetic (P); and at low values of T and h, are Néel (N) phase. The dynamic phase boundaries between these regions, N-P, are the second- and first-order phase transition lines at low and high values of the temperature, respectively. Fig. 4(a) also shows that the second- and first-order lines combine at the tricritical points and this combination separates the N phase from P phase. We have also found similar phase diagram to the one seen in works [28,33,38], except the ferromagnetic phase occurs instead of the N phase. (ii) Fig. 4(b) is illustrated for J′¼  0.5. In addition to the combination of the second- and first order lines combined at the tricritical points as in Fig. 4(a), there is a phase region with the coexistence Nþ C and Nþ P phases separated from each other with only second-order lines at lower temperature and magnetic fields. This figure is similar with Fig. 8 of [28], with 3 (b) of Ref. [38] and also with Fig. 7(b) of Ref. [51]. The phase diagram also exhibits a reentrant behavior, i.e., for the 1.28 r h/| J| o 1.38 as the temperature increases, the system passes from the P phase to the N þP phase, and passes to the P phase again. In spin systems, reentrant behavior can be understood as follows. At high temperatures, the entropy is the most important factor and uncorrelated fluctuations determine the thermodynamics. The system is then in the P phase bias due to the applied field. As the temperature is lowered, the energy and entropy are both important and the correlated fluctuations affect the dominance of either phase significantly. The system enters the ordered phase. At low temperatures, the energy is important, not the entropy, and the system reenters the P phase again [52]. (iii) Fig. 4(c) is

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illustrated for J′¼  1.0. This phase diagram is similar to Fig. 4(b), except that the reentrant phenomenon disappears. Moreover, the second-order phase transition between the N þC and N þP phases is increased, but other phase transition line between the Nþ P and P states is decreases due to the impact of bilinear interaction parameter. (iv) For J′¼  2.0, the phase diagram is presented in Fig. 4(d). While this phase diagram has the same phase topology as the diagram in Fig. 4(a), but only differs from Fig. 4(a) in which the C phase is settled instead of the N phase. A similar phase diagram is found to the one seen in works [28,33,8], except the ferromagnetic phase occurs instead of the C phase.

4. Summary and conclusion In this study, we have studied the dynamical aspects of a SSL in the presence of a time-dependent oscillating external magnetic field within the mean-field approach and the Glauber-type stochastic dynamics. The mean-field dynamic equations for the magnetizations are derived by employing the Glauber transition rates. First, we have studied time variations of the magnetizations in order to find the phases in the system. Second, the dynamic magnetizations, dynamic hysteresis loop areas, and the dynamic correlations have been calculated and studied as functions of the temperature. These studies lead us to characterize the nature (continuous and discontinuous) of dynamic phase transitions as well as to obtain the dynamic phase transition (DPT) points. The dynamic phase diagrams are presented in the (T/|J|, h/|J|) plane. We found that the behavior of the system strongly depends on the values of interaction parameter and four different phase diagram topologies are found. The phase diagrams exhibit the dynamic tricritical point where the first-order phase transition line signals the change from the first- to the second-order phase transition. According to values of Hamiltonian parameters, besides the paramagnetic (P), Néel (F) and Collinear (C) fundamental phases, two mixed phase regions, i.e., N þC, N þP, exist in the system. The obtained results are in good agreement with some experimental and theoretical results. We expect that our detailed theoretical investigation may stimulate further researches to study the nonequilibrium or dynamic theoretical and experimental researches on the magnetic properties of the frustrated model as well as to research on its magnetism. Moreover this work may shed some light on future experimental researches for the further study of all the SSL magnets.

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