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Tricritical behavior of a Heisenberg model with Dzyaloshinski-Moriya interaction
___ ll!!!
15 August 1994
2s
m
PHYSICS
WV
LETTERS
A
4._
ELSEVIER
Physics Letters A 191 (1994) 275-278
Tricritical behavior of a Heisenberg model with Dzyaloshinski-Moriya interaction * J. Ricardo de Sousa, Douglas F. de Albuquerque, I.P. Fittipaldi
’
Departamentode Fisica, UniversidadeFederal de Pernambuco,50732-910 Recife - PE, Brazil Received 18 May 1993; revised manuscript received 28 February 1994;accepted for publication 26 May 1994
Communicated by A.R. Bishop
Abstract
The phase diagram of a spin-t anisotropic Heisenberg model with Dzyaloshinski-Moriya (DM ) interaction is studied in the mean-field approximation. It is shown that for the easy direction (D= Dz; where D is the DM vector coupling) the model exhibits tricritical point (TCP ). The TCP is obtained explicitly and the tricritieal temperature, T,, is independent of the exchange anisotropy parameter A (A= 0 and A= 1, corresponds to the isotropic Heisenberg and Ising models, respectively), while the trieritical parameter, 4, is dependent on A. We obtain Tt= f (z- 1) an d Dt( A= 0)-zDt( A= 1 ), where z is the lattice coordinate number.
There has been increasing interest in the study of the effect of anisotropies on the critical behavior of magnetic systems. The interest in most of these studies are due to the fact that in real magnetic materials, anisotropies of different origins are always present resulting in deviations from the ideal behavior. In particular, an important type of anisotropy which has attracted much attention in recent years (see, e.g., Ref. [ 1] ) is the Dzyaloshinski-Moriya (DM ) interaction [ 21, which plays an important role to describe certain class of insulators (see, e.g. Ref. [ 3 ] ) as well as in the study of spin glasses (see, e.g., the recent work of Yi et al. [ 41). More recently, the DM interaction has also found to be essential for the emergence of weak ferromagnetism observed in the lowtemperature orthorhombic phase of all lame&u copper oxide superconductors [ 5 1. * Work supported in part by CNPq, CAPESand FINEP (Brazilian research agencies). ’ Temporary address: Diretoria do Centro de Ciencias Exatas e da Naturexxa-CCEN, Universidade Federal de Fkmambuco, Cidade Univenitia, 50732-9 10 Recife-PE, Brazil. Elsevier Science B.V. SSDI 0375-9601(94)00426-P
On the other hand, due to anisotropic interactions, many spin model Hamiltonians exhibit a tricritical point (TCP ) in the phase diagram, at which the phase transition changes from second to first order. Such typical systems are the Blume-Cape1 and the BlumeEmery-Grifftths [ 6,7] models for spin-l Ising, the spin-4 Ising model with a random tield [ 81 and the spin-f Ising model with four-spin interactions (see, e.g. Ref. [ 91). Examples for Heisenberg models exhibiting tricritical behavior are the spin-l biquadratic Heisenberg ferromagnet [ lo], and the spin-4 Heisenberg model, in which each spin is subject to a randomly oriented quenched local magnetic field
[ill. In all earlier investigations in which tricritical point has been predicted the admittedly very simplified mean-field approximations (MFA) have successfully been used, and a subsequent corroboration is then made by using more sophisticated treatments. In this paper, by the use of the MFA method we investigate an anisotropic Heisenbcrg model with DM interaction. Within this framework we discuss the
276
J.R. de Sousa et al. /Physics Letters A 191 (1994) 275-278
second-order ferromagnetic-phase stability limit for any values of both the DM coupling parameter D, and the exchange anisotropy parameter A, as well as the first-order phase transition which may occur in the system. It is shown that the present mean-field approximative procedure, although analytically simple, leads to the prediction of a TCP in the anisotropic Heisenberg model with DM interaction. It turns out, as far as we know, that such a prediction for the model under consideration is here, for the first time, reported in the literature. The Heisenberg model with DM interaction for a system consisting of N ions in a uniform magnetic field h = 0, has the Hamiltonian H=-J
c [(l-d)(s;s~+s~s~)+s;s:]
+e-2Kcosh[2K~~]},
(4)
where K=PJ (/3= (k,T)-‘) and D,=D/J. The magnetization m is then obtained by minimizing the free energy (3)) which is given by the solution of the transcendental equation m=sinh[2K(z-
l)m]{cosh[2K(z-
l)m]
+e-2Kcosh[2K,/m]}-1
.
(5)
At this point it is worth mentioning that Eq. ( 5 ) with D,,=O reduces to the same mean-field type of equation of state previously obtained in Ref. [ 12 1.
Let us now address our attention to the study of the critical ferromagnetic frontiers. Thus, by expanding the right-hand side (RHS) of (3) around m=O, one obtains the following sort of Landau order-parameter serie expansion for the reduced free energy f (m, T) = F/ JN
where the coupling constant J is restricted to the nearest-neighbors pairs of spins, Sy (V= X, y, z) are components of a spin-j operator at site i, A and D, are the exchange anisotropy and the DM interaction parameters, respectively. We shall assume Dii= Dt, i.e. D, parallel to the easy direction which really is a special choice of the DM interaction term in ( 1). Here, we shall also restrict ourselves to the case where the exchange coupling constant Jis ferromagnetic (i.e., J> 0 and 0 G A< 1). According to the mean-field approximative procedure, the pair-spin Hamiltonian, HzFA, associated to the two-spin cluster, obtained from ( 1) is given by
f(m,T)-f(O, T) + (z-1) (1 -a)m2-bm4-cm6...
,
(6)
where the coefficients a, b and c are given by 2(z- 1 )Kc a= 1 +@,(A, D) ’ b= (~-l)~Kc 3 C=
(z- 1 )3K,z 90
[k(A,
(7) D) -21a2,
(8)
[~34(A,D)-13~K,(A,D)+161a3, (9)
with -D(s;g-S~Sf)-J(x-l)m(Sf+S~),
(2)
where z is the coordination number and m is the averaged magnetic moment per spin along a fixed direction 2 in space. Hence, within the present two-spin cluster mean-field calculation the free energy as a function of the temperature T and of the magnetization m is given by w=(z-l)m2)-K--IlnZ(m,T), with Z(m, T)=2eK{cosh[2K(z-l)m]
(3)
o&f,
D) =e- 2”cosh[2&J], (10)
in which K; l = (j&T) -’ is the reduced critical temperature. In the vicinity of the second-order phase transition line the magnetization m is then given by m2=(1-a)/b.
(11)
As the r.h.s. of (11) must be positive, the secondorder phase transition line is then determined by a = 1, with b c 0. When b > 0 an unstable solution appears. Hence, the point at which a = 1 and b= 0 (with c < 0), separating the stable (b< 0) and the unstable (b> 0) solutions, is the TCP at which the system
277
J. R. de Sousa et al. / PhysicsLettersA I91 (1994) 2 75-2 78
undergoes a first-order phase transition. By solving the critical condition of a = 1 and b = 0 (with c
(12)
and D,(d) = (DO&P = JfT:{arccosh[2exp(2/T,)
]}2- (1 -4)2
.
(13) For Do> D,(A) in the m-T plane the magnetization suffers a discontinuous jump from a finite value mF( D, A) to zero, at a temperature T:(D, A), which characterizes a first-order phase transition. This firstorder phase transition temperature TF(D, A) is obtained from Eqs. (3), (4) and (5) in the region Do> Dt( A), by equaling the free energy of the ferromagnetic phase (i.e., m # 0, with m given in (5) ) to the corresponding free energy of the paramagnetic phase (i.e., m=O). This procedure leads to a set of two coupled transcendental equations from which TF( D, A) and mF( D, A) are obtained numerically. We have also solved numerically the relation II= 1, for b < 0. The resulting critical frontiers separating the ferromagnetic phase from the paramagnetic one, are represented in Fig. 1 for z= 6, in which the variation of the reduced transition temperature k,T,/J is plotted as a function of a convenient reduced parameter X, defined by X= In the figure, solid and dashed lines denote, respectively, the second- and the first-order phase transitions, and the black point refers to the TCP given by Eqs. ( 12) and ( 13 ) . As is seen from Fig. 1, when X increases from zero, the critical second-order phase transition line falls from its value in the absence of the DM interaction, reaching the TCP at ( T,, 0,). According to ( 12 ) and ( 13 ), the tricritical point, T,, is found to be independent of the exchange anisotropy parameter A, with the constant value of T,= y, while the tricritical parameter, Dt, as a function of A, is them given by D,=$O.75(1 -A)2, leading to the result D,(A=O)
,/m
.
0.0
I 0.0
I
I
2.0
4.0 X
6.0
Fig. 1. Nature of variation of the critical temperature ( kBTC/J) with respect to the reduced parameterX= ,/( 1-A)* +Da for the case of z= 6. The continuous and broken-regions of the curves refer to the second- and first-order phase transitions, respectively. The black point denotes the TCP and the corresponding constant value of the tricritical temperature r, is shown by the arrow.
X,= 3.5. Therefore, the phase diagram presented in Fig. 1 is exact within the context of the present twospin cluster MFA approach. This critical behavior of the system may be interpreted as resulting from a competition between the exchange interaction that tries to align the spins in the same direction and the effect of the DM interaction, which has the tendency to destroy this alignment. At this point, it is worth mentioning that the reduced transition temperature in the absence of the DM interaction (i.e., K;‘(A, D=O)), obtained within the present two-spin cluster MFA procedure, for the limit cases of A= 1 (Ising) and A=0 (isotropic Heisenberg) are given by the distinct values: K;‘(A=1)=5.847 and K;‘(A=0)=5.719. We recall that the series expansions results are, respectively, 4.5 11 and 3.360 [ 13 1. Hence, the present MFA scheme correctly distinguishes between this two different models, while the usual one-spin cluster MFA provides for both model the unique value of 6.0. In conclusion, the applicability of the two-spin cluster MFA scheme for the anisotropic Heisenberg model with DM interactions has been demonstrated and used to investigate the nature of variation of the critical temperature in the T-X plane. It is shown that this MFA treatment leads to the conclusion that the tricritical point at which the phase transition changes
278
J.R. deSousa et al. /Physics Letters A 191(1994) 275-278
from second to first order may exist in the system for all values of de [0,11, although previous works (see Ref. [ 141, and references therein) have not obtained this tricritical behavior. Certainly, the reason for that is due to the fact that in all of these above mentioned works [ 4,14 ] the analysis has been devoted only to some qualitative aspects of the phase diagrams and it was restricted to some region of the parameter space in which there was no possibility of finding a TCP. Nevertheless, it is also clear that the present MFA results have limited qualitative and quantitative confidence. As is well-known, within such a kind of meanfield framework, the strict criticality of the system is lost (in particular, the critical exponents are going to be the classical ones), and the real dimensionality of the system is only partially incorporated through the coordination number of the lattice. For instance, for bidimensional lattices (i.e., 2=4) our two-spin cluster MFA treatment still predict the unphysical results of T,# 0 [15 1. However, for tridimensional lattices (as is the case of z = 6 ) , to which our calculation was devoted, we believe that the present results are, to a certain extent, of qualitative relevance. Thus, in order to examine such aspects as well as to elucidate the present MFA predictions, it might be worth and necessary to extend our calculation by employing more accurate treatments. These aspects of the problem are now under investigation and will be reported on in the near future.
References [ 11 G.G. Kenning, D. Chu and R. Orbach, Phys. Rev. Lett. 66 (1991) 2923. [ 21 I.E. Dzyaloshinski, J. Phys. Chem. Solids 4 ( 1958) 241; T. Moriya, Phys. Rev. Lett. 4 ( 1960) 228. [3] F. Keffer, in: Encyclopedia of physics, Vol. 18/2, eds. S. Fhigge and H.P.J. Wijn (Springer, Berlin, 1966) p. 1. [ 41 L. Yi, G. Biittner and K.D. Usadel, Phys. Rev. B 47 ( 1993) 254, and references therein. [ 5 ] W. Koshibae, Y. Ohta and S. Maekawa, Phys. Rev. Lett. 7 1 ( 1993) 467, and references therein. [6] M. Blume, Phys. Rev. 141 (1966) 517; H.W. Capel, Physica 32 ( 1966) 966. [ 71 M. Blume, V.J. Emergy and R.B. Grifliths, Phys. Rev. A 4 (1971) 1071. [8] J. Mizumo, J. Phys. C 7 (1974) 3755; A. Aharony, Phys. Rev. B 18 ( 1978) 33 18. [9] C.L. Wang, Z.K. QinandD.L. Lin, J. Magn. Magn. Mat. 88 (1990) 87. [lO]H. ChenandP.M.Levy,Phys.Rev.B 7 (1973) 4267. [ 111 V.K. Saxena, J. Phys. C 14 ( 198 1) L745. [ 121 J.S. Smart, Effective field theories of magnetism (Saunders, London, 1966). [ 13 ] C. Domb, in: Phase transitions and critical phenomena, Vol. 3, eds. C. Domb and M.S. Green (Academic Press, New York, 1974). [ 141 C.E. Cordeiro, E.V. de Mello and M.A. Continentino, Z. Phys. B Condensed Matter 85 ( 1991) 307. [ 151 N.D. Mermin and H. Wagner, Phys. Rev. Lett. 17 ( 1966) 1133.
15 August 1994
2s
m
PHYSICS
WV
LETTERS
A
4._
ELSEVIER
Physics Letters A 191 (1994) 275-278
Tricritical behavior of a Heisenberg model with Dzyaloshinski-Moriya interaction * J. Ricardo de Sousa, Douglas F. de Albuquerque, I.P. Fittipaldi
’
Departamentode Fisica, UniversidadeFederal de Pernambuco,50732-910 Recife - PE, Brazil Received 18 May 1993; revised manuscript received 28 February 1994;accepted for publication 26 May 1994
Communicated by A.R. Bishop
Abstract
The phase diagram of a spin-t anisotropic Heisenberg model with Dzyaloshinski-Moriya (DM ) interaction is studied in the mean-field approximation. It is shown that for the easy direction (D= Dz; where D is the DM vector coupling) the model exhibits tricritical point (TCP ). The TCP is obtained explicitly and the tricritieal temperature, T,, is independent of the exchange anisotropy parameter A (A= 0 and A= 1, corresponds to the isotropic Heisenberg and Ising models, respectively), while the trieritical parameter, 4, is dependent on A. We obtain Tt= f (z- 1) an d Dt( A= 0)-zDt( A= 1 ), where z is the lattice coordinate number.
There has been increasing interest in the study of the effect of anisotropies on the critical behavior of magnetic systems. The interest in most of these studies are due to the fact that in real magnetic materials, anisotropies of different origins are always present resulting in deviations from the ideal behavior. In particular, an important type of anisotropy which has attracted much attention in recent years (see, e.g., Ref. [ 1] ) is the Dzyaloshinski-Moriya (DM ) interaction [ 21, which plays an important role to describe certain class of insulators (see, e.g. Ref. [ 3 ] ) as well as in the study of spin glasses (see, e.g., the recent work of Yi et al. [ 41). More recently, the DM interaction has also found to be essential for the emergence of weak ferromagnetism observed in the lowtemperature orthorhombic phase of all lame&u copper oxide superconductors [ 5 1. * Work supported in part by CNPq, CAPESand FINEP (Brazilian research agencies). ’ Temporary address: Diretoria do Centro de Ciencias Exatas e da Naturexxa-CCEN, Universidade Federal de Fkmambuco, Cidade Univenitia, 50732-9 10 Recife-PE, Brazil. Elsevier Science B.V. SSDI 0375-9601(94)00426-P
On the other hand, due to anisotropic interactions, many spin model Hamiltonians exhibit a tricritical point (TCP ) in the phase diagram, at which the phase transition changes from second to first order. Such typical systems are the Blume-Cape1 and the BlumeEmery-Grifftths [ 6,7] models for spin-l Ising, the spin-4 Ising model with a random tield [ 81 and the spin-f Ising model with four-spin interactions (see, e.g. Ref. [ 91). Examples for Heisenberg models exhibiting tricritical behavior are the spin-l biquadratic Heisenberg ferromagnet [ lo], and the spin-4 Heisenberg model, in which each spin is subject to a randomly oriented quenched local magnetic field
[ill. In all earlier investigations in which tricritical point has been predicted the admittedly very simplified mean-field approximations (MFA) have successfully been used, and a subsequent corroboration is then made by using more sophisticated treatments. In this paper, by the use of the MFA method we investigate an anisotropic Heisenbcrg model with DM interaction. Within this framework we discuss the
276
J.R. de Sousa et al. /Physics Letters A 191 (1994) 275-278
second-order ferromagnetic-phase stability limit for any values of both the DM coupling parameter D, and the exchange anisotropy parameter A, as well as the first-order phase transition which may occur in the system. It is shown that the present mean-field approximative procedure, although analytically simple, leads to the prediction of a TCP in the anisotropic Heisenberg model with DM interaction. It turns out, as far as we know, that such a prediction for the model under consideration is here, for the first time, reported in the literature. The Heisenberg model with DM interaction for a system consisting of N ions in a uniform magnetic field h = 0, has the Hamiltonian H=-J
c [(l-d)(s;s~+s~s~)+s;s:]
+e-2Kcosh[2K~~]},
(4)
where K=PJ (/3= (k,T)-‘) and D,=D/J. The magnetization m is then obtained by minimizing the free energy (3)) which is given by the solution of the transcendental equation m=sinh[2K(z-
l)m]{cosh[2K(z-
l)m]
+e-2Kcosh[2K,/m]}-1
.
(5)
At this point it is worth mentioning that Eq. ( 5 ) with D,,=O reduces to the same mean-field type of equation of state previously obtained in Ref. [ 12 1.
Let us now address our attention to the study of the critical ferromagnetic frontiers. Thus, by expanding the right-hand side (RHS) of (3) around m=O, one obtains the following sort of Landau order-parameter serie expansion for the reduced free energy f (m, T) = F/ JN
where the coupling constant J is restricted to the nearest-neighbors pairs of spins, Sy (V= X, y, z) are components of a spin-j operator at site i, A and D, are the exchange anisotropy and the DM interaction parameters, respectively. We shall assume Dii= Dt, i.e. D, parallel to the easy direction which really is a special choice of the DM interaction term in ( 1). Here, we shall also restrict ourselves to the case where the exchange coupling constant Jis ferromagnetic (i.e., J> 0 and 0 G A< 1). According to the mean-field approximative procedure, the pair-spin Hamiltonian, HzFA, associated to the two-spin cluster, obtained from ( 1) is given by
f(m,T)-f(O, T) + (z-1) (1 -a)m2-bm4-cm6...
,
(6)
where the coefficients a, b and c are given by 2(z- 1 )Kc a= 1 +@,(A, D) ’ b= (~-l)~Kc 3 C=
(z- 1 )3K,z 90
[k(A,
(7) D) -21a2,
(8)
[~34(A,D)-13~K,(A,D)+161a3, (9)
with -D(s;g-S~Sf)-J(x-l)m(Sf+S~),
(2)
where z is the coordination number and m is the averaged magnetic moment per spin along a fixed direction 2 in space. Hence, within the present two-spin cluster mean-field calculation the free energy as a function of the temperature T and of the magnetization m is given by w=(z-l)m2)-K--IlnZ(m,T), with Z(m, T)=2eK{cosh[2K(z-l)m]
(3)
o&f,
D) =e- 2”cosh[2&J], (10)
in which K; l = (j&T) -’ is the reduced critical temperature. In the vicinity of the second-order phase transition line the magnetization m is then given by m2=(1-a)/b.
(11)
As the r.h.s. of (11) must be positive, the secondorder phase transition line is then determined by a = 1, with b c 0. When b > 0 an unstable solution appears. Hence, the point at which a = 1 and b= 0 (with c < 0), separating the stable (b< 0) and the unstable (b> 0) solutions, is the TCP at which the system
277
J. R. de Sousa et al. / PhysicsLettersA I91 (1994) 2 75-2 78
undergoes a first-order phase transition. By solving the critical condition of a = 1 and b = 0 (with c
(12)
and D,(d) = (DO&P = JfT:{arccosh[2exp(2/T,)
]}2- (1 -4)2
.
(13) For Do> D,(A) in the m-T plane the magnetization suffers a discontinuous jump from a finite value mF( D, A) to zero, at a temperature T:(D, A), which characterizes a first-order phase transition. This firstorder phase transition temperature TF(D, A) is obtained from Eqs. (3), (4) and (5) in the region Do> Dt( A), by equaling the free energy of the ferromagnetic phase (i.e., m # 0, with m given in (5) ) to the corresponding free energy of the paramagnetic phase (i.e., m=O). This procedure leads to a set of two coupled transcendental equations from which TF( D, A) and mF( D, A) are obtained numerically. We have also solved numerically the relation II= 1, for b < 0. The resulting critical frontiers separating the ferromagnetic phase from the paramagnetic one, are represented in Fig. 1 for z= 6, in which the variation of the reduced transition temperature k,T,/J is plotted as a function of a convenient reduced parameter X, defined by X= In the figure, solid and dashed lines denote, respectively, the second- and the first-order phase transitions, and the black point refers to the TCP given by Eqs. ( 12) and ( 13 ) . As is seen from Fig. 1, when X increases from zero, the critical second-order phase transition line falls from its value in the absence of the DM interaction, reaching the TCP at ( T,, 0,). According to ( 12 ) and ( 13 ), the tricritical point, T,, is found to be independent of the exchange anisotropy parameter A, with the constant value of T,= y, while the tricritical parameter, Dt, as a function of A, is them given by D,=$O.75(1 -A)2, leading to the result D,(A=O)
,/m
.
0.0
I 0.0
I
I
2.0
4.0 X
6.0
Fig. 1. Nature of variation of the critical temperature ( kBTC/J) with respect to the reduced parameterX= ,/( 1-A)* +Da for the case of z= 6. The continuous and broken-regions of the curves refer to the second- and first-order phase transitions, respectively. The black point denotes the TCP and the corresponding constant value of the tricritical temperature r, is shown by the arrow.
X,= 3.5. Therefore, the phase diagram presented in Fig. 1 is exact within the context of the present twospin cluster MFA approach. This critical behavior of the system may be interpreted as resulting from a competition between the exchange interaction that tries to align the spins in the same direction and the effect of the DM interaction, which has the tendency to destroy this alignment. At this point, it is worth mentioning that the reduced transition temperature in the absence of the DM interaction (i.e., K;‘(A, D=O)), obtained within the present two-spin cluster MFA procedure, for the limit cases of A= 1 (Ising) and A=0 (isotropic Heisenberg) are given by the distinct values: K;‘(A=1)=5.847 and K;‘(A=0)=5.719. We recall that the series expansions results are, respectively, 4.5 11 and 3.360 [ 13 1. Hence, the present MFA scheme correctly distinguishes between this two different models, while the usual one-spin cluster MFA provides for both model the unique value of 6.0. In conclusion, the applicability of the two-spin cluster MFA scheme for the anisotropic Heisenberg model with DM interactions has been demonstrated and used to investigate the nature of variation of the critical temperature in the T-X plane. It is shown that this MFA treatment leads to the conclusion that the tricritical point at which the phase transition changes
278
J.R. deSousa et al. /Physics Letters A 191(1994) 275-278
from second to first order may exist in the system for all values of de [0,11, although previous works (see Ref. [ 141, and references therein) have not obtained this tricritical behavior. Certainly, the reason for that is due to the fact that in all of these above mentioned works [ 4,14 ] the analysis has been devoted only to some qualitative aspects of the phase diagrams and it was restricted to some region of the parameter space in which there was no possibility of finding a TCP. Nevertheless, it is also clear that the present MFA results have limited qualitative and quantitative confidence. As is well-known, within such a kind of meanfield framework, the strict criticality of the system is lost (in particular, the critical exponents are going to be the classical ones), and the real dimensionality of the system is only partially incorporated through the coordination number of the lattice. For instance, for bidimensional lattices (i.e., 2=4) our two-spin cluster MFA treatment still predict the unphysical results of T,# 0 [15 1. However, for tridimensional lattices (as is the case of z = 6 ) , to which our calculation was devoted, we believe that the present results are, to a certain extent, of qualitative relevance. Thus, in order to examine such aspects as well as to elucidate the present MFA predictions, it might be worth and necessary to extend our calculation by employing more accurate treatments. These aspects of the problem are now under investigation and will be reported on in the near future.
References [ 11 G.G. Kenning, D. Chu and R. Orbach, Phys. Rev. Lett. 66 (1991) 2923. [ 21 I.E. Dzyaloshinski, J. Phys. Chem. Solids 4 ( 1958) 241; T. Moriya, Phys. Rev. Lett. 4 ( 1960) 228. [3] F. Keffer, in: Encyclopedia of physics, Vol. 18/2, eds. S. Fhigge and H.P.J. Wijn (Springer, Berlin, 1966) p. 1. [ 41 L. Yi, G. Biittner and K.D. Usadel, Phys. Rev. B 47 ( 1993) 254, and references therein. [ 5 ] W. Koshibae, Y. Ohta and S. Maekawa, Phys. Rev. Lett. 7 1 ( 1993) 467, and references therein. [6] M. Blume, Phys. Rev. 141 (1966) 517; H.W. Capel, Physica 32 ( 1966) 966. [ 71 M. Blume, V.J. Emergy and R.B. Grifliths, Phys. Rev. A 4 (1971) 1071. [8] J. Mizumo, J. Phys. C 7 (1974) 3755; A. Aharony, Phys. Rev. B 18 ( 1978) 33 18. [9] C.L. Wang, Z.K. QinandD.L. Lin, J. Magn. Magn. Mat. 88 (1990) 87. [lO]H. ChenandP.M.Levy,Phys.Rev.B 7 (1973) 4267. [ 111 V.K. Saxena, J. Phys. C 14 ( 198 1) L745. [ 121 J.S. Smart, Effective field theories of magnetism (Saunders, London, 1966). [ 13 ] C. Domb, in: Phase transitions and critical phenomena, Vol. 3, eds. C. Domb and M.S. Green (Academic Press, New York, 1974). [ 141 C.E. Cordeiro, E.V. de Mello and M.A. Continentino, Z. Phys. B Condensed Matter 85 ( 1991) 307. [ 151 N.D. Mermin and H. Wagner, Phys. Rev. Lett. 17 ( 1966) 1133.