The exact phase diagrams of spin-1 Ising model on a two-layer Bethe lattice

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Abstract

The two-layer spin-1 Ising model on the Bethe lattice is studied in terms of the intralayer coupling constants J1 and J2 of the two layers, interlayer coupling constant J3 between the layers and the external magnetic fields, which are coupled to the two layers and assumed to be different for each layer, for given values of the coordination number q by using the recursion relation scheme. The ground-state configurations of the system are obtained on the (J2/|J1|,J3/q|J1|) planes depending on J1<0 or J1>0. Then, the phase diagram of the system is obtained on the (kT/J1,J3/J1) plane for given values of α=J2/J1 and q in zero external magnetic fields. It was found that the system presents both first- and second-order phase transitions for all values of q. Besides, we also present the thermal change of the total and staggered magnetizations of the two layers and also the spin–spin correlation function between the nearest-neighbor spins of the adjacent layers.

Introduction

The study of magnetic thin films consisting of various magnetic layered structures or superlattices has been receiving intense attention in the recent years for both theoretical and experimental point of views [1]. Since these materials is made up with multiple layers of different magnetic substances, there is a high potential for technological advances in information storage and retrieval and in synthesis of new magnets for a variety of applications [2]. These materials present some interesting novel magnetic properties such as giant magnetoresistance [3], surface magnetic anisotropy [4], enhanced surface magnetic moment [5] and surface magnetoelastic coupling [6].

The multilayered structures or Ising films containing spin-12 are studied theoretically within several different frameworks such as mean-field approximations [7], effective-field theories [8], Monte Carlo methods [9], spin-fluctuation theory [10], renormalization-group techniques [11], two-site cluster approximations [12], by means of the modified spin wave method [13], by using a bond-operator mean-field method [14] and the iteration technique on the Bethe lattice [15]. Then these methods were expanded to the magnetic layered structures containing higher spin values. The most studied spin system after the spin-12 is the spin-1 Ising system which was originally introduced for studying the superfluidity and phase separation in helium mixtures [16] and afterwards this model was extended for the systems characterized with three states such as solid–liquid–gas systems, multicomponent fluid and liquid crystal mixtures [17], microemulsions [18], semiconductor alloys [19], ternary mixtures [20], electron conduction models [21], martensitic transformations [22] and two-dimensional Blume–Emery–Griffiths–Potts model [23].

Therefore, the multilayered systems consisting of spin-1 are also of interest: the effects of surface single-ion anisotropy and surface dilution on surface phase transitions of a semi-infinite Ising model with diluted spins on the surface were investigated by the use of an effective-field theory [24]; ellipsometric and calorimetric studies of argon and krypton adsorbed on triangular lattice graphite substrates have found a novel sequence of apparent reentrant layering transitions between integer-plus-one-half coverages [25], which motivated the study of preroughening and layering transitions on triangular lattice substrates by the use of the solid–solid models [26]; the magnetic phase diagram of a thin film is determined at T=0 by including the exchange coupling, the magnetic dipole coupling, as well as second- and fourth-order lattice anisotropies [27], the complete global phase diagram for a spin-1 bilayer Blume–Emery–Griffiths (BEG) model is studied by the use of cluster variational theory in the pair approximation [28]; motivated by the experiments, the effect of the interlayer exchange interaction on the magnetic properties of coupled Co/Cu/Ni trilayer was studied theoretically and the magnetization and susceptibility of the coupled ferromagnetic trilayers calculated with a Green's function type theory [29], respectively; the order–disorder layering transitions are studied in the presence of a variable crystal field by using the Monte Carlo simulations and mean-field theory [30], respectively; the effect of the transverse field on bulk melting and layering sublimation transitions of the BEG model was studied by using the mean-field theory [31]; the phase transitions of a transverse spin-1 Ising L-layer film of simple cubic symmetry with nearest-neighbor (NN) exchange interactions was examined within the framework of the effective-field theory and again using the effective-field theory with a probability distribution technique that accounts for the self-spin-correlation functions, the layer longitudinal magnetizations and quadrupolar moments of a spin-1 Ising film and their averages are examined [32], respectively.

As a result we study the bilayer spin-1 Ising system on the Bethe lattice by using the recursion relations obtained by the pairwise approach [15]. A Bethe lattice is an infinitely Cayley or regular tree, i.e. a connected graph without circuits, and historically gets its name from the fact that its partition function is exactly that of an Ising model in the Bethe approximation [33]. In this study, the original one-layer lattice is replaced with the two-layer Bethe lattice with the same coordination number q. It should be mentioned that the one-layer Bethe lattice provides exact solutions and the results of which are qualitatively better approximations for the regular lattices than solutions obtained by the conventional mean-field theories [34]. We should also point out that the cluster variation method in the pair approximation studies on regular lattices yield results that are exact for the same model on the Bethe lattice [35]. Thus two-layer spin-1 Ising model on the Bethe lattice is studied in terms of the intralayer coupling constants of the two layers and interlayer coupling constant between the layers including the external magnetic fields for given values of the coordination number q by using the recursion relation scheme in a pairwise approach.

The rest of the paper is organized as follows. In Section 2, bilayer Ising model is introduced and then the ground-state phase diagram is obtained and discussed. Section 3 is devoted to obtaining of the order-parameters and the free energy of the system in terms of the recursion relations exactly. In Section 4 we have presented the thermal change of the total and staggered magnetizations of the two layers and also the spin–spin correlation function between the nearest-neighbor spins of the adjacent layers, therefore the phase diagram of the system is obtained on the (kT/J1,J3/J1) plane for given values of α=J2/J1 and q in zero external magnetic fields. Finally, in the last section we give a brief summary, discussion and concluding remarks.

Section snippets

Introduction to two-layer Bethe lattice and its ground states

Two-layer Bethe lattice is an extension of its one-layer version [36]. The one-layer version consists of a central spin which maybe called the first generation spin. This central spin has q NNs, i.e. coordination number, which forms the second-generation spins. Each spin in the second-generation is joined to (q-1) NNs. Therefore, in total the second generation has q(q-1) NNs which form the third generation and so on to infinity. In correspondence with, its two-layer version contains two

The calculations of the order-parameters and the free energy of the bilayer system

In order to obtain the order-parameters and the free energy in terms of the recursion relations on the bilayer Bethe lattice, first we have to calculate the partition function of the system by using the Ising Hamiltonian given in Eq. (1). Therefore, it is assumed that adjacent NN spins of G1 and G2 are considered as pairs. The first pair deep inside the bilayer lattice is called the central pair which forms the first-generation spins. This central pair of spins is connected by q NN spin pairs,

The thermal change of the order-parameters and the phase diagrams

After having obtained the necessary equations to calculate the order-parameters, i.e. the total and staggered magnetizations, and the spin–spin correlation functions, we are ready to present their thermal behaviors. These are required for the determination of the second- and first-order phase transition temperatures and as well as for the classification of the different phase regions in the phase diagrams.

The first figures for the thermal variations of the order-parameters, Figs. 3, explain how

Conclusions

In this work, we have studied the two-layer spin-1 Ising model on the Bethe lattice in detail in terms of the intralayer coupling constants J1 and J2 of the two layers, interlayer coupling constant J3 between the layers for given values of the coordination number q by using the recursion relation scheme. The ground-state configurations of the system are obtained on the (J2/|J1|,J3/q|J1|) planes depending on J1<0 or J1>0. Then, the phase diagram of the system is obtained on the (kT/J1,J3/J1)

Appendix

The explicit definitions of fi(Xn-1(i))(i=1,,8), functions are given as:f1(Xn(i))=[eβ(J1+J2+J3+h1+h2)(Xn1)q-1+eβ(J1+h1)(Xn2)q-1+eβ(J1-J2-J3+h1-h2)(Xn3)q-1+eβ(J2+h2)(Xn4)q-1+eβ(-J2-h2)(Xn5)q-1+eβ(-J1+J2-J3-h1+h2)(Xn6)q-1+eβ(-J1-h1)(Xn7)q-1+eβ(-J1-J2+J3-h1-h2)(Xn8)q-1+1]/D1,f2(Xn(i))=[eβ(J1+J3+h1+h2)(Xn1)q-1+eβ(J1+h1)(Xn2)q-1+eβ(J1-J3+h1-h2)(Xn3)q-1+eβ(h2)(Xn4)q-1+eβ(-h2)(Xn5)q-1+eβ(-J1-J3-h1+h2)(Xn6)q-1+eβ(-J1-h1)(Xn7)q-1+eβ(-J1+J3-h1-h2)(Xn8)q-1+1]/D1,f3(Xn(i))=[eβ(J1-J2+J3+h1+h2)(Xn1)q-1+eβ(J1

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