The two dimensional mixed spin system consisting of spin-1/2 and spin-1 with four-spin interactions exhibits tricritical behaviour. The influence of a transverse field on this behaviour is studied with the finite cluster approximation based on a single-site cluster theory. The state equations are derived for the square lattice. It has been shown that the system keeps its tricritical behaviour, but when the strength of the transverse field approaches a critical value, all transitions become of second-order for any value of four-spin interaction

Benayad, N.

Ghliyem, M.

English

Revues Scientifiques Marocaines

Phase diagram of the mixed spin transverse Ising model with four-spin     Interactions

M. J. CONDENSED MATTER                      VOLUME 13, NUMBER 2                                   OCTOBER 2011  MJCM, VOLUME 13, NUMBER 2                                            © 2011 The Moroccan Statistical Physical and Condensed Matter Society      jjxSiixzzzzzz)lk,j,i,(lSjSkσiσ4JijjSiσ2JH    Phase diagram of the mixed spin transverse Ising model with four-spin     Interactions  N.Benayad*, M.Ghliyem  Groupe de Mécanique Statistique, Laboratoire de physique théorique et appliquée, Faculté des  sciences Aïn-Chock, Université Hassan II-Casablanca, 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: The two dimensional mixed spin system consisting of spin-1/2 and spin-1 with four-spin interactions exhibits tricritical behaviour. The influence of a transverse field on this behaviour is studied with the finite cluster approximation based on a single-site cluster theory. The state equations are derived for the square lattice. It has been shown that the system keeps its tricritical behaviour, but when the strength of the transverse field approaches a critical value, all transitions become of second-order for any value of four-spin interaction. Keywords: mixed spin, four-spin interactions, transverse field.  I. Introduction  Over recent years, a particular interest has been devoted to the theoretical and experimental study of Ising models with multispin interactions. The increasing interest in investigating models with higher-order interactions arises from the fact that, on the one hand, they may exhibit rich phase diagrams and can describe phase transitions in some physical systems. On the other hand, they show physical behaviours not observed in the usual spin systems. For instance, they display a nonuniversal critical phenomena [1, 2]. Recently, the influence of multispin interactions and the crystal field on the mixed spin Ising models has been theoretically studied within different methods. Besides, finite cluster approximation [3], Monte Carlo simulation [4], and exact calculations [5]. Very recently, we have investigated [6], using the finite cluster approximation the influence of the fluctuation of the crystal field interaction on the mixed spin-1/2 and spin-1 Ising model with four-spin interaction. The transverse Ising model was originally introduced by Blinc [7] and de Gennes [8] as a valuable model for the tunneling of the proton in hydrogen-bonded ferroelectrics [9] such as the KH2PO4 type. Since, then, it has been successfully applied to several physical systems, such as cooperative Jahn-teller systems [10] (like DyVO4, and TbVO4), ordering in rare earth compounds with a singlet crystal- field ground state [11] and also to some real magnetic materiels with strong uniaxial anisotropy in a transverse field [12]. From the theoretical point of view, monoatomic transverse Ising model with multispin interactions have been studied within the finite cluster approximation for the honeycomb and square lattice [13]. The mixed spin transverse Ising model [14] has been investigated by the finite cluster approximation [15], mean field renormalization group technique [16], and exact calculations [17, 18].     The purpose of this paper is to study the influence of the transverse field on the phase diagram of the mixed spin-1/2 and spin-1 Ising model with two and four-spin interactions on the square lattice. This system can be described by the following Hamiltonian:                                                                                                                                                                                                                                                                                    (1) where σα and Sα (α=x,z) are components of spin-1/2 and spin-1 operators. The first summation is carried out only over nearest-neighbour pair of spins. The second term represents the four-spin interactions, where the summation is over all alternate squares, shaded in Fig.1. The last term describes the uniform transverse field. To this end, we use the finite cluster approximation [19,20] within the framework of a single-site cluster theory.            Fig .1: Part of the square lattice.  and × correspond to σ and s-sublattice sites, respectively.  The outline of this work is as follows: In section 2, we give a description of the theoretical framework. In section 3, the phase diagrams of the system are investigated and discussed. We conclude this work in section 4.  13 52 N.Benayad, M.Ghliyem  MJCM, VOLUME 13, NUMBER 2                                            © 2011 The Moroccan Statistical Physical and Condensed Matter Society  oxSBozSAs 22oH  lzSzSlkkzzSj,j,σ4J41jj2J1Α  jzSkzkjiiziiz ,,σ4J412J2Α  oσσoσoσc βHexpβHexpσTrTrσooo   osSosnoSc βHexpβHexpTrSTrSooo  oσσoσoσco βHexpβHexpσσμTrTroo  sSsSSSmoooco βHexpβHexpTrTroo  sSsSSSqoo2c2βHexpβHexpTrTroooo  2121212112tanh2BABAAz  2222222222222cosh21sinh2BABABAAmzoxBozA σ1σ1oσH I. Theoretical framework The theoretical framework to be used in the study of the system described by the Hamiltonian (1) is the finite cluster approximation (FCA), based on a single-site cluster theory. In this approach, attention is focused on a cluster comprising just a single selected spin σo(So) and its nearest neighbour spins {σ1,σ2,S1,S2,S3,S4}({S1,S2,σ1,σ2,σ3,σ4}) with which it directly interacts (see Fig .2(a,b)). We split the total Hamiltonian (1) into two parts, H=Ho+H’, where Ho includes all parts of H associated with the lattice site o. In the present system, Ho takes the form   (2)       (3) where                                                                        ,        B1=-Ω                                                                                                                                                                          ,   B2=-Ω          (4)  whether the lattice site o belongs to σ or S- sublattice, respectively.            (a)                                                  (b)  Fig.2: (a) Neighbours of spin σo with which it directly interacts.             (b) Neighbours of spin So with which it directly interacts.  First, the problem consists in evaluating the sublattice longitudinal and transverse components of the  magnetizations and its quadrupolar moments. In order to calculate them, we choose a representation in which σoz and Soz are diagonal and denote by <σoα>c (<(Soα)n>c , n=1,2) the mean value of σoα ((Soα)n) for a given configuration c, of all other spins (i.e.when all other spin σi  and Sj (i, j≠0) are kept fixed). Neglecting the fact that Ho and H’ do not commute,  <σoα>c and <(Soα)n>c are given by:                                                                                               (5)  and                                                                                               (6)   where Trσo (or Trso ) means the trace performed over  σo (or So) only. As usual ß=1/T where T is the absolute temperature. Therefore, the magnetizations µα, mα (α=x,z) and the quadrupolar moment qα are given by                                                                            ,                  (7)                                                                              ,                 (8)  and                                                                          ,                   (9)  which can be considered as the starting point of the single-site cluster approximation. <…..> denotes the average over all spin configurations. Eqs (7) to (9) are not exact. Nevertheless they have been accepted as a reasonable   starting point [21] for transverse ising systems and have been successfully applied to a number of interesting transverse Ising systems [22,23-25].We have to emphasize that the Ising limit (Ω=0),Hamiltonian (1) contains only σiz  and Sjz.Then, relations(7)-(9) become exact identities. To calculate <σoα>c and <(Soα)n>c one has first to diagonalize the single site  Hamiltonians Hoσ and Hos , respectively. To do this, it is convenient to use the following rotation transformations: oxozoz 'sin'cos                                                    (10) oxozox 'cos'sin                                                   (11) oxozoz SSS 'sin'cos                                                      (12) oxozox SSS 'cos'sin                                                      (13)  where                                                                                 (14)                        (15)  The coordinate rotations (10) to (13) turn the Hamiltonians Hoσ and Hos into diagonal forms. Then, evaluating the inner traces in (7) to (9) over the states of the selected spins σo(So), we obtain:                                                                                                                                                                                         (16)                                                                                              (17) 22222sinBAB22222cosBAA21211cosBAA21211sinBAB13 53 Phase diagram of the mixed spin transverse Ising model with four-spin Interactions  MJCM, VOLUME 13, NUMBER 2                                            © 2011 The Moroccan Statistical Physical and Condensed Matter Society    2222222222222cosh21sinh2BABABAAqzSσfσSfSfSfσff),(26524321 Sf    zzzzzzzzzzzzzzzzzzzzzzz qqA4mμA2mμ]qA4A[4m]qA2qA4A[2mμ]qA[4μm]qA4qA4qA4A[4m]A2qA4qA[2μμ1532144131232111092382372654433221][4mμ]qB2B[2mμ]B[4mμ]qB2B[2mμ]qB4qB8B[4μ]qB2B[2m]qB4qB8B[4μm142313124112109228763542321Bzzzzzzzzzzzzzzzzzzz][Cμm]C[6mμ]qC4C[4μm]qCqC2[Cμ]qC4C[4mμ][Cm]qC6qC12C[6μ]qCqC2[Cq1642152214133212111049872265422321zzzzzzzzzzzzzzzzzzzzba1μ2 z                                                                                             (18)   It is a formidable task to calculate the average on the right-hand side of Eqs.(16) to (18) over all spin configurations. We can easily observe that any function such as ƒ(σ,S) of  σ and S can be written as the linear superposition                                                                                                                                                                                                                   (19) with appropriate coefficients fi ( i=1,….,6) . After applying this to all spins σi and Sj in expressions between brackets in Eqs. (16) to (18), we average over all spin configurations. In this paper, we use the simplest approximation in which we treat all spin self-correlations exactly while still neglecting correlations between quantities pertaining to different sites. This leads to the following coupled equations:                                                                                                   (20)                                                                                                (21)                                                                                                 (22) where the coefficients are functions of K, α, and Ω. After some algebraic manipulations of Eqs (20) to (22), we obtain an equation for μz of the form μz = a μz + b μz 3 +……                                                      (23) As usual the condition    a(K, α, Ω)=1                                                                  (24) determines the second-order transition line. The magnetization μz  in the vicinity of the second-order transition is given by                                                                                                                                                                                                                                    (25)  The right hand side of (25) must be positive. If this is not the case, the transition is of the first-order, and the point at which     1α,K,a     and     0α,K,b                                  (26)  characterizes the tricritical point.  II.Results and discussion  In this section, we illustrate some results for the system under investigation. Indeed, we study the effect of the transverse field Ω, as well as the four-spin interactions J4 on the phase diagram of the mixed spin Ising model. First of all, we determine the phase diagram of the mixed spin transverse Ising system in the absence of the four-spin interactions (i.e.α=J4/J2=0). As clearly seen in figure 3, the critical temperature decreases from its maximum value in the mixed Ising model Tc(Ω=0) to vanish at a critical value Ωc/J2 =2.117.              Fig.3: The phase diagram in T-Ω plane of the mixed spin-1/2 and spin-1 transverse Ising model on the square lattice when J4/J2=0.  In order to have an idea on the effects of the transverse field Ω on the mixed spin Ising model with four-spin interactions, we have plotted in figure4, the phase diagram in the (T/J2,J4/J2) plane, for selected values of the transverse field. When this latter belongs to the range 0≤Ωc/J2≤1.826 the critical temperature Tc increases with increasing values of J4 and passes by a smooth maximum and then reduces to a tricritical point. The remaining part of the phase diagram (1.826<Ωc/J2≤2.117) shows a qualitatively different behaviour from the previous range of Ω. Thus, the tricritical behaviour disappears and all transitions are always of second-order. We have to mention that the transverse field reduces the longitudinal ferromagnetic order and therefore acts against the order.               Fig.4: The phase diagram in T-J4 plane. The number accompanying each curve denotes the value of Ω/J2. The black point denotes the tricritical point. 13 54 N.Benayad, M.Ghliyem  MJCM, VOLUME 13, NUMBER 2                                            © 2011 The Moroccan Statistical Physical and Condensed Matter Society  In figure 5, we plot various transition lines in the (T/J2, Ω /J2) plane for fixed values of the J4/J2. So, the figure illustrates the influence of the four-spin interaction on the phase diagram of the mixed spin transverse Ising model. It shows that, for any given value of J4/J2 the critical temperature line starts from its J4/J2 dependent value at Ω=0 and decreases gradually with increasing values of the transverse field. In this case, we note that the system exhibits second-order transition for large range of J4/J2 but in a narrow range of the four-spin interaction, the critical lines end in a tricritical point as is shown in Fig.5.b. We also notice that the Ω-component of the tricritical point increases with increasing the strength of the four-spin interaction.                                                           (a)                                                           (b)  Fig.5: The phase diagram in T- Ω plane, when all transition lines are of second order. The number accompanying each curve denotes the value of J4/J2. The black point denotes the tricritical point.      III.Conclusion  In this work, we have studied the mixed spin transverse Ising model, consisting of spin-1/2 and spin-1, with two and four-spin interactions J4 on the square lattice. We have used the finite cluster approximation within the framework of a single-site cluster theory. 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Phase diagram of the mixed spin transverse Ising model with four-spin Interactions

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