Phase transitions in the spin-32 Blume-Emery-Griffiths model

..ysica North-Holland

Phase transitions in the spin- 3 Blume-Emery-Griffiths model A . B a k c h i c h a'b, A . B a s s i r b a n d A . B e n y o u s s e f b aGroupe de Physique de la Mati~re Condens~e, D~partement de Physique, Facult~ des Sciences, B.P. 20, El Jadida, Morocco bLaboratoire de Magn~tisme, D~partement de Physique, Facult~ des Sciences, B.P. 1014, Rabat, Morocco

Received 31 July 1992 The spin-~ Ising model on the square lattice with nearest-neighbor ferromagnetic exchange interactions (both bilinear (J) and biquadratic (K)) and crystal-fieldinteraction (A) is studied using a renormalization-group transformation in position-space based on the MigdalKadanoff recursion relations. The global phase diagram in (J, K, A) space (with J, K/> 0) is found to have two surfaces of critical phase transitions and two surfaces of first-order phase transitions. These surfaces are variously bounded by an ordinary tricritical line, an isolated critical line of end points, and a line of multicritical points. The global connectivity and local exponents of the thirteen separate fixed points underlying this quite complicated structure are determined.

1. Introduction T h e B l u m e - E m e r y - G r i f f i t h s ( B E G ) model is a spin-1 Ising model, which presents a rich variety of critical and multicritical p h e n o m e n a . The B E G model was initially introduced in connection with phase separation and superfluid ordering in H e 3 - H e 4 mixtures [1]. Subsequently, the model was reinterpreted to describe critical p h e n o m e n a in magnetic systems and simple [2] and multic o m p o n e n t fluids [2,3]. A n extention of the B E G model is the possibility of inclusion of higher spin values. The spin-3 B E G model with dipolar and quadrupolar interactions was introduced to explain phase transitions in DyWO 4 and its phase diagram was obtained within the mean-field approximation ( M F A ) [4]. A n o t h e r spin -3 m o d e l was later introduced to study tricritical properties in ternary fluid mixtures [5], which was also solved in the MFA. Recently, the phase transitions in the spin-3 B E G model with nearest-neighbor interactions, both bilinear and biquadratic, and with a crystal-field interaction has been studied within 0378-4371/93/$06.00 (~ 1993- Elsevier Science Publishers B.V. All rights reserved

A. Bakchich et al. / Phase transitions in the spin-~ BEG model

189

MFA and Monte Carlo simulations [6]. Besides a second-order transition line occurring at high temperatures, they locate two low temperature lines. By contrast, in the research reported here we use a position-space renormalization-group (PSRG) method based on the Migdal-Kadanoff (MK) [7,8] recursion relations to study this model with a very rich phase diagram, which exhibits a wide variety of transitions of first and higher order. We obtain the unified global phase diagram in (J, K, A) space, with two supplementary constant K cross sections necessary for comparison with the results obtained from other theories [6], and other local features which have not been reported previously. We find a total of 13 different fixed points, yielding first-order phase boundaries, critical, tricritical and multicritical points. All of these arise from a very simple set of recursion relations. The global phase diagram is determined by the topology of the PSRG flows linking the various fixed points. A local analysis of the recursion relations near the fixed points gives all the exponents. We study the spin -3 BEG model on the square lattice (d = 2) described by the following Hamiltonian: -H(J,K,A)

= JEgigj+K (ij)

~ (ij)

S i2 S2 j + A E S

2,

Si =-+3,-+ ½ ,

(1)

i

where (ij) indicates a summation over nearest-neighbor pairs (we restrict this study to J,K>~O). The usual (KnT) -~ factor has been absorbed into this Hamiltonian. The terms on the right-hand side are, respectively, the bilinear exchange, the biquadratic exchange and crystal-field interactions. In order to obtain self-consistent recursion relations, the adding of a new coupling

c ~, (S,S~ + S3Sj)+ f ~, S3S~ (ij)

(2)

(ij)

to the Hamiltonian must also be considered. Here, we will not be concerned about the physical origin of these couplings and we will treat them as parameters in the calculations. The plan of our presentation is as follows: in section 2 we develop our PSRG transformation which enables us to exhibit the global phase diagram in (J, K, z~) space. This phase diagram is composed of four surfaces: two of first-order transitions and two of critical (second-order) transitions. These four surfaces are variously bounded by an ordinary tricritical line which separates one of the critical surfaces and one of the first-order surfaces, an isolated line of critical end points bounding the remaining first-order surface, and a line of multicritical points. We locate the 13 separate fixed points underlying this quite complicated structure and study the connectivity of the renormalization-group flows linking them. In section 3 we draw our final conclusion.

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A. Bakchich et al. / Phase transitions in the spin-~ BEG model

2. Renormalization group transformation 2.1. General considerations

Given a Hamiltonian HK(S ) which depends on spin variables S and a set of coupling constants K, the basic idea of the real-space renormalization-group approach [9] is to remove progressively degrees of freedom, usually by performing a trace over a certain fraction of the spins, and to look at the effective interactions K ' between the remaining spins. Performing the trace is sometimes a complicated task, so to make calculations tractable we use the approximative recursion relations of Migdal [7]. This approximation, which has been reinterpreted and extended by Kadanoff [8], has the advantage of its simplicity and has been found to give a reasonably quantitative description of the critical behavior. Let us briefly describe Migdal's method. Choose a scale factor b and consider a one-dimensional chain of b + 1 spins coupled by nearest-neighbour interactions K. Perform the trace over all spins on the chain except those at the end. The end spins are now coupled b y effective interactions /~ which are a function of K. The relationship between /~ and K is a renormalization-group transformation for a one-dimensional (1D) lattice within the framework of what is called the cluster approximation. In a d-dimensional cubic lattice Migdal argues that the renormalization-group transformation which gives the new coupling constants K ' as a function of K is simply K'= bd-'k(K).

(3)

Fig. 1 illustrates how a d-dimensional renormalization interaction can be considered to be the sum of b d-I 1D renormalized interactions.

Fig. 1. Original lattice (solid lines) and renormalized lattice (broken lines) in the case b = 2 and d = 2. The 2D renormalized interaction is the sum of two 1D renormalized interactions.

A. Bakchich et al. / Phase transitions in the spin-a2 BEG model

191

2.2. Renormalization-group analysis

As mentioned earlier, the renormalization-group phase diagram is derived from the global study of flows in a Hamiltonian space, which are governed by fixed point (points invariant under the transformation). As such a fixed point, the correlation length ff of the system is either zero or infinite [10]. In the latter case only, the entire domain of attraction (the subspace which eventually flows into the fixed point in question) shares the ~ = ~ property. The two cases usually are distinguished quite easily by examination of the fixed point Hamiltonian. For example, ~: = 0 is ruled out when the fixed point Hamiltonian contains finite couplings. Accordingly, fixed points and their domains can be classified following the notation established by Berker and Wortis [11]: (a) higher-order fixed point (if* = 0% where the asterisk denotes a fixed point): the domain is the locus of higher-order phase transition; (b) first-order fixed point (~* = 0): the domain is the locus of first-order phase transitions; (c) trivial fixed point (~'*= 0): the domain is either (i) an entire thermodynamic phase, or (ii) the smooth continuation of one thermodynamic phase into another. We call these "phase sinks" or "continuation fixed points", respectively. Critical (higher-order) exponents are obtained by linearizing the recursion relations at the higher-order fixed point whose domain is the locus of the transitions in question. In our case, the recursion relations yield A= by ,

(4)

where the length rescaling factor b is 2 in our case. Unlike the eigenvalues A, the "eigenvalue exponents" y are transformation independent. The relevant eigenvalue exponents y > 0 give higher-order (critical and tricritical) exponents. Renormalization-group trajectories flow away from the fixed point along the eigendirections associated with these relevant y. The irrelevant eigenvalues y < 0 give corrections to scaling exponents [12]. Trajectories flow into the fixed point (from its domain) along the associated eigendirections.

2.3. Overall description

The different phases of the spin-3 BEG model can be characterized by two parameters: the magnetization m = (Sj) and the quadrupole parameter q = ( S 2 ). Since the lattice is translationally invariant, m and q do not depend upon the site i. According to the values of m and q, four different phases can be distinguished as follows:

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A. Bakchich et al. / Phase transitions in the spin-~2 B E G model

Paramagnetic, P a r a ( - ) : Paramagnetic, Para( + ): Ferromagnetic, Ferro( ½): Ferromagnetic, Ferro( 3 ):

m m m m

= = ~ ~

0, 0, 0, 0,

q<5/4, q>5/4, q<5/4, q >5/4.

The (J, K, za) phase diagram, as obtained in the PSRG treatment, is presented in fig. 2. The volume (J, K/> 0) under study is divided by transition surfaces into four regions. Two of these are occupied by paramagnetic (m = 0) phases: one (labelled Para(+)) has large quadrupole order parameter q, the other ( P a r a ( - ) ) has small q. These two paramagnetic phases are separated by a first-order transition surface (characterized by the fixed point F~) bounded by a multicritical line of L*, but merge at an isolated line of end points (critical end line) provided by G*. In the remaining volumes, two ferromagnetic (m ~ 0) phases, referred to as Ferro(3) and Ferro(½), are separated by the first-order transition surface represented by F*. In fact, in each of these ordered regions, two ferromagnetic (m X 0) phases coexist. Thus, these regions are actually a locus of first-order transitions between up and down magnetizations. The ferromagnetic phase Ferro(23-) is separated from the paramagnetic phases ,J / / /

FerroO/~)

\

,

/"

/

/ ./~r'r'(~(vO

/ . . . . . . . . . . . . ~-C.L~_-Z . . . . l - Z - i - - ' /

o

17 .......... ,,

-.-/........................i- ..................

,

3

,

/

.......................J

Fig. 2. BEG phase diagram obtained in the position-space renormalization group treatment.

Critical and first-order transitions are, respectively, drawn with full and dotted lines. The notched line denotes the ordinary tricritical line and the dash-dotted (. . . . . . . ) line represents the multicritical line. Wavy lines denote smooth continuation of surfaces.

A. Bakchich et al. / Phase transitions in the spin-~2 BEG model

193

P a r a ( - ) and P a r a ( + ) by a critical transition surface represented respectively by C* and C~, which is bounded by an ordinary tricritical line of T*, whereas F e r r o ( 1 ) and P a r a ( + ) are separated by the critical surface of E*. Thirteen fixed points underlie the phase diagram in fig. 2. Their classification and locations are given in table I. Four fixed points in table I provide the first-order transitions in our phase diagram. These fulfill the Neinhuis and Nauenberg conditions [13] for seeing first-order transitions in PSRG. Specifically, the largest eigenvalue exponent y whose eigenfield couples to the discontinuous order parameter must equal the lattice dimensionality of the system (d = 2 in our case). This is exactly fulfilled for the four fixed points in question. The eigenvalue exponents associated with the higher-order fixed points are given in table II. Representative constant-K cross sections are presented in fig. 3. Let us discuss the portion of the phase diagram which corresponds to K / J = O, T / J versus A/J (fig. 3a). The phase diagram is qualitatively similar to that obtained within M F A and Monte Carlo simulations [6]. It shows a paramagnetic phase and two ordered phases Ferro( 3 ) and Ferro(½ ). A second-order transition line separates the paramagnetic phase from the two ordered phases, and a firstTable I Classification and locations of the fixed points underlying the phase diagram of the spin-~2BEG model in two dimensions. Fixed point

Type

(J*, K*, A*, c*, f*) coordinates

Domainin the (J, K, A, c, f) space

(0.168; 0; +~; 0; 0.083) (0; 0.609; -3.046; 0; 0) (1.819; 0.013; -3.954; 0.018; 0.014) (0; +~; -~; 0; +~) (2.646; 0.775; -3.880; -0.663; 0.118) (2.646; 0.077; -3.880; -0.663; 0.118)

surface line line line surface surface

(+~; -~o; +~; -~; +oo) (+~; -~; -~; -~; +~) (0; +oo;-c~; 0; 0) (+~; +~; -~; -~; +~)

volume Ferro( ~) volume Ferro( ½) surface surface

(0; 0; -~; 0; 0)

volume Para(+ )

(0; 0; +~; 0; 0)

volume Para(-)

(0; 0; 0; 0; 0)

surface

Higher-order fixed points

C~' G* T* L* C~ E*

critical critical ordinary tricritical multicritical critical critical

First-order fixed points

F~* F* F~ F*

discontinuous m discontinuous m discontinuous q discontinuous q

Trivial fixed points

P* P_* S*

sink for (m = 0, large q) phase sink for (m = 0, small q) phase smooth continuation between preceding two phases

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A. Bakchich et al. / Phase transitions in the spin-~ BEG model (a) 3

2

_T J Ferro(a/2)

i

-o.s

-,',o

-1~s

i

Ferro~h)

-zo

LX J

(b}

K=I 0.6

0,4

F~ro~)

J

0.2

;~ro&)

i

-3

-5

A

Fig. 3. (a) Critical temperature (T/J) as function of the single-ion anisotropy (A/J) in the PSRG for the biquadratic interaction K/J = 0. A first-order phase boundary (dotted line) separates the two ordered phases designated by Ferro( ~) and Ferro( ½). (b) Phase diagram in the (J, A) plane for K = 1. A first-order phase boundary separates the two paramagnetic phases, Para(+) and Para(-). o r d e r transition line separates the two o r d e r e d phases. In fig. 3b we present the phase d i a g r a m in the ( T , A ) plane for K = 1. Besides two s e c o n d - o r d e r transition lines separating the o r d e r e d phase Ferro(-~) f r o m the two param a g n e t i c phases Para(+-), the phase diagram exhibits a first-order transition

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A . Bakchich et al. / Phase transitions in the spin-~ B E G model

Table II Critical exponents of the higher-orderfixed points. Fixed points c* c~ E* T* G*

Eigenvalue exponents Yl

Y2

Y3

Y4

Y5

0.635 0.921 0.527 0.512 1.841

1.273 1.311 1.565 1.365 6.826

-0.843 2.803 1.368 2.795 -0.075

-0.843 -5.765 - 1.765 -2.566 -0.075

-5.765 - 1.558 -2.566 -2.299

line separating Para(+) from P a r a ( - ) , similar to that obtained for the spin-1 B E G model [11].

3. Conclusion We have dealt with the phase transitions of a spin -3 BEG model with a bilinear interaction (J), a single-ion anisotropy (A) and a biquadratic interaction (K). By means of a position-space renormalization-group method we obtain the global phase diagram in (J, K, A) space (with J, K < 0), which exhibits four different phases (two paramagnetic and two ferromagnetic phases). Besides a first-order transition surface separating the two ferromagnetic phases a t low temperature, the phase diagram exhibits a supplementary high temperature first-order transition surface separating the two paramagnetic phases, which merge at an isolated line of critical end points. Two critical transition surfaces separate the ferromagnetic from the paramagnetic phases, with an ordinary tricritical line and a line of multicritical points. We locate the 13 separate fixed points with their global connectivity and local exponents.

Acknowledgements This work has been done in the framework of the agreement of cooperation between the CNR (Morocco) and the DFG (Germany), for which we wish to thank both organisations. One of the authors (A. Benyoussef) would like to thank Prof. J. Zittartz for the hospitality of the Institut fiir Theoretische Physik of the University of K61n.

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A . Bakchich et al. / Phase transitions in the spin- ~ B E G model

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