Condensed
Zeitschrift Matter
for Physik B
9 Springer-Verlag 1990
Real-space renormalization group investigation of pure
and disordered m i x e d spin Ising models on d-dimensional lattices*
N. Benayad**
Institut fiir Theoretische Physik, Universit/it zu K61n, D-5000 K61n 41, Federal Republic of Germany Received March 6, 1990
The mixed spin Ising model (spins a= 89 and S = 1) on d-dimensional hypercubic lattices with nearest-neigh- bout exchange interactions is studied via a renormaliza- tion group transformation in position space. The phase diagrams in (L, K) space, i.e. in dependence of the bilinear (K) and the biquadratic (L) interaction coefficients, are qualitatively different for d = 2 and d > 2. For any dimen- sion d however it is found that all transitions are of second order. At zero-temperature ( K = 0% L = oo), the ferromagnetic order disappears at ,--(K)o = 2, which does not depend on d. Using an extension of this real-space renormalization group analysis we study the two-dimen- sional random disordered version of the above model.
L is kept homogeneous and the bilinear interactions Kij are assumed to be independent random variables with distribution P (Kq) = p 3 ( K q - K) + (1 - p) 6 (K~-- a K);
where K > 0 . The phase diagrams for different values of p are obtained. At zero temperature, it is found that
/ x
in the bond diluted model (0~=0)the value ( ~ ) depends
o
continuously on p, whereas in the random _ K interac-
/ x
tions ( ~ = - - 1 ) ( ~ - / is unique and does not de- pend on p. \ ~ / o
I. Introduction
The biquadratic exchange interaction [1-5] and the sin- gle ion anisotropy [4-7] has been pointed out to give significant effects on the Curie temperature and other magnetic properties of ferromagnets. In all these investi- gations, the spins had the same value on each lattice site. Therefore it is interesting now to extend the investi-
* Supported by the agreement of cooperation between the D F G -
W. Germany and the CNR-Maroc
** On leave from Facult6 des Sciences I, Univ6rsit6 Hassan II, Casablanca, Morocco; and Laboratoire de Magn6tisme, Universit6 de Rabat, Morocco
gation to pure and random mixed spin-89 and spin-1 Ising models defined on d-dimensional hypercubic lattices and described by the following reduced Hamiltonian:
fill=- Z Kij ~i Sj+ Z L,~(cq Sj) 2. (1)
<i j ) <i j )
The underlying lattice is composed of two interpenetrat- ing sublattices, one occupied by spins with spin moment a-- _+ 89 and the other one is occupied by spins with spin moment S=0, _+ 1. The summation in (1) extends over all pairs of nearest-neighbour sites of the lattice. Kij and L o. denote the reduced bilinear and biquadratic exchange interactions, respectively. For isotropic L~i(Lij= Lj) one should note that the Hamiltonian (1) is equivalent to a mixed spin Ising system with crystal field interaction
d 2
- X o, sj + 2 rj sj (2)
(i j> j
where d is the dimensionality of the hypercubic lattice.
In the simplest form of (1), namely Kij=-K and L~j
= 0, the above model has been studied by renormaliza- tion group techniques [8] (in two dimensions) and high- temperature series expansions [9]. It has been shown that the critical exponents are in good agreement with those suggested by universality ideas. Recently, attention has been directed to the pure model (K~j=K, Lq=_L)
described by the Hamiltonian (1). The effects of the bi- quadratic exchange interaction on the Curie temperature has been discussed [10-12] by the use of the Bethe- Peierls method. It was shown that this extra-term signifi- cantly changes the value of the transition temperature but it was not clarifed whether the system exhibits a tricritical behaviour or not. Using an effective-field ap- proximation [13] and the finite-cluster approximation [14], it was shown that a tricritical point exists if the coordination number z is larger than 3.
The first purpose of this paper is to study the pure
d-dimensional mixed spin Ising model described by the
Hamiltonian (1). This investigation is performed using
the Migdal-Kadanoff renormalization group method
[15-16]. In addition, we study the existence of tricritical behaviour for any dimension d of the system. We remark that the method we employ can not be restricted to a mixed spin Ising model with only bilinear term (L=0), since in the two-dimensional parameter space (L, K), the subspace L = 0 is not invariant.
Several authors used real-space renormalization group techniques to study the magnetic properties of Ising models on monoatomic lattices with quenched bond dilution [17-20]. The same model with a biquad- ratic term (or with crystal field) has also been investigat- ed by various methods [6, 7]. It was shown that this extra term modifies significantly the phase diagram.
The second purpose of this work is to present an extension of this study to randomly disordered mixed spin -1 and spin-1 Ising models. We investigate this model using an extension of the Migdal-Kadanoff transforma- tion to disordered systems. The reduced bilinear interac- tions K~j are assumed to be independent random vari- ables with distribution
P (K,j) = p 6 ( K , j - K) + (1 - p) 6 ( K , j - aK), (3) where K > 0. The biquadratic exchange interactions L~j are kept constant and homogeneous
Q (L, 9 = 6 (L~ s - L). (4)
In Sect. 2 phase diagrams of the d-dimensional pure model are determined within the framework of a real- space renormalization group (RSRG) technique. Using an extension of this method in Sect. 3, we determine the phase diagrams for both kinds of disorder. In particular we investigate the influence of dilution and the influence of random _+ K interactions on the critical value ( ~ )
at zero-temperature, o-
where the sums are over all spin configurations and Z is the partition function. Sine the 0-- and S-sublattices are translationally invariant, m and q do not depend on the site i. According to the values of m and q, three different phases can be distinguished as follows:
Paramagnetic(P_): m - 0 , q<v- Paramagnetic (P+): m--0, q > l Ferromagnetic (F): m :~ 0, q ~ 89
In terms of the renormalization group theory, these phases and all transitions are determined by the topolo- gy of the R S R G flow linking the various fixed points.
In this section we determine the phase diagram in (L, K) space for d-dimensional hypercubic lattices using the Migdal-Kadanoff renormalization group method. This renormalization technique is applicable for all space di- mensionalities. We give the recursion relations for any d after briefly describing the method.
Because of the symmetry of the model, we have to restrict ourselves to an odd scale factor b. In the present study we choose b = 3 and consider a one-dimensional chain with spins 0-~, St, 0-2, $2. All these spins are cou- pled by the same bilinear and biquadratic exchange in- teraction K and L, respectively. The corresponding re- duced Hamiltonian reads
- g (0-1 81 q- S1 0-2 q- 0-2 S2) q- g(0-12 $2 $2 0-2
+ (6)
Performing the trace over $2 and 0- 2 we obtain the trans- formed reduced Hamiltonian
fl/4= - / ~ a ~ $2 + L a ~ S 2. (7) The renormalized one-dimensional /~ and L are given as functions of K and L by
e k _
3 L 3 K K L / ~
e - T ( e T + 3 e - 7 ) + 2 e -~ cosh
3L 89 K L (2-)'
e - T ( e 3 e ~-) + 2e -~ cosh
(8a)
LV 3L 3K .~ L ( ~ ) ] [ 3L 3~ K L ( K ) ]
[ e - T ( e ~ - + 3 e-~) + 2e-~ cosh e ~ (e + 3eY)+2e-~ cosh
L
L
[2 e - ~ (1 + cosh (K)) + 2] 2
(8b)
II. Pure system
The different phases of the model described by Hamil- tonian (1) can be characterized by two parameters: the magnetization ms (or m~) and the quadrupole parameter
m ( K , L ) =- ( S i ) - = Z - ~ ~ Si e x p ( - fill), (5a)
o',S
q(K, L) - (S z) = Z -1 ~ S 2 exp( - fill), (5 b)
o',S
The Migdal-Kadanoff recursion relations for the d-di- mensional hypercubic lattice are obtained using the usual moving procedure, as illustrated in Fig. 1. They are given by
K' = 3d-1/~(L, K), (9a)
E = 3 d- t L(L, K). (9b)
These relations have been obtained by first performing
the trace and then moving the bonds. We could have
first moved the bonds and then performed the trace.
C
)
"x
k_
f
Fig. 1. Bond moving in the case d=2. Open circles remain after decimation has been performed
Table 1. Coordinates and classification of the fixed points of trans- formation (11) for d = 2
Fixed Type (L*, K*) coordinates Domaine
points in the
(L, K) space F (m#:0, c7~89 phase (-oo, oo) surface F P+ (m = 0, q > 89 phase ( - 0% 0) surface P+
P_ (m=0, q<89 phase (0% 0) surface P_
I second order ( - 0% 0.80) line NBI
l k
N s e c o n d o r d e r ((~, K ~ ~ - / p o i n t N
\ --/
O separation (0, 0) line NO
between R and P+
This last procedure would have given a recursion rela- tion of the following form:
K ' = / ~ ( 3 a-1 L, 3 a
- 1K), (10a)
E = L ( 3 a - ~ L , 3a- 1K), (10b)
w h e r e / ( a n d L are the functions defined by (8).
In general, the first procedure underestimates the crit- ical temperature, while the second procedure overesti- mates them. To obtain m o r e precise values, we use a symmetrized procedure which gives the following recur- sion relations:
K ' = 89 [3a-1/~(L, K ) + / ( ( 3 a - 1 L, 3 a - ' K)], ( l l a ) E = 1 [3 a-~ L(L, K) + L(3 a-1 L, 3 a-~ K)]. (11 b) The various fixed points of this t r a n f o r m a t i o n have been determined for d = 2, d = 3, and d = 4. Their coordinates and the phases as well as transitions they characterize are given in Tables 1 a n d 2.
F o r d = 2 the phase d i a g r a m is shown in Fig. 2. The ferromagnetic phase F is separated from the p a r a m a g - netie phase P+ by the second order transition line NBI.
The two p a r a m a g n e t i c phases P_ and P+ are separated by the line NO. These two phases, however, have no different s y m m e t r y a n d one can continuously pass from one to the other. The a b o v e lines meet at the point N
Table 2. Coordinates and classification of the fixed points of trans- formation (11) for d=3, and d=4
Fixed Type (L*, K*) coordinates Domaine
points in the
(L, K) space F (m4=0, q~89 phase ( - 0% oo) surface F P+ (m = 0, q > 89 phase ( - 0% 0) surface P+
P (m=0, q<89 phase (~, 0) / surface P_
N second order (or, K* = ~-) point N
\
I second order ( - co, 0.120) for d= 3 line B1 I (--oo, 0.021) for d=4
B~ critical (0.058, 0.152) for d=3 point B1 (0.005, 0.026) for d = 4
B 2
second order (10.80, 5.78) for d = 3 line
N B 2 B 1(30.38, 17.18) for d=4
O separation (0, 0) line B10
between P_ and P+
K N
F
P+
-2. -1. 0 1. 2.
L
Fig. 2. Phase diagram in (L, K) space obtained by the Migdal-Ka- danoff renormalization group treatment for d= 2. Arrows indicate the relative stability and connectivity of fixed points
( L)
which is situated at L = o% K = 0% ~ = 2 . As shown in Table 2, we have (for d = 3 and d = 4), two other non- trivial fixed points in addition to those obtained for d = 2.
One of them is completely unstable whereas the second
describes the separation between the ferromagnetic and
p a r a m a g n e t i c P_ phases. The phase diagrams for d = 3
and d = 4 are very similar. A typical one showing the
location of the various fixed points, is represented in
Fig. 3 in arbitrary units. We r e m a r k that at zero-temper-
ature ( K = o% L = oo) the ferromagnetic order is des-
troyed for 0 = 2 . We can show, from (11), that this
point does not depend on d. Using the Nienhuis-Nauen-
berg criterion [21], we point o u t that there is no fixed
point characterizing a first order transition. Therefore
all transitions are of second order for any dimension
d.
P+
K N
//
P_
L
Fig. 3. A typical phase diagram in (L, K) space obtained by the Migdal-Kadanoff renormalization-group treatment for d= 3 and d = 4. Arrows indicate the relative stability and connectivity of fixed points
d = 2
-2. -1. 0 1, 2.
L / K
Fig. 4. Variation of the Curie temperature K~- ~ with biquadratic exchange interaction ~ for d = 2 and d = 3 L
The Curie temperature K 7 ~ has been calculated for both cases of positive and negative values of L. F o r d = 2 and d = 3, the results are shown in Fig. 4. It turns out that the effect of the biquadratic exchange interaction
(
L or the single-ion anisotropy on the transition temperature is considerably stronger for L > 0 than in the region L < 0 . This means that for L > 0 there exists competition between the bilinear and biquadratic ex- change interaction and as a consequence there is a de- crease in the transitions temperature.
One should note that for L ~ - 1 , the spin configura- tions is completely d o m i n a t e d by S = __ 1 on each site of the S-sublattice. When we restrict the sums to these configurations, the second term in (1) becomes just an additive constant which does not affect further ordering and the Hamittonian reduces to
Kla KI~ Kx~
Kzl K2z K2s
K31 K32 K3*
K"
9 9
Fig. 5. Interactions K~j that enter Migdal's recursion relations in the case of a disordered two-dimensional mixed spin Ising model.
A similar diagram can be drawn for the biquadratic exchange inter- actions Lij and the renormalised E
-flH(K,{a, S})=K ~2 a~ St, (12)
<it>
a = +_ 89 and S = _ 1, which describes the d-dimensional spin-89 Ising model, exactly solved by Onsager [22, 23]
in the case d = 2. Therefore the K component of the fixed point I should give the critical temperature of the spin-89 Ising model. We found Kc = 0.80 in good agreement with the exact value Kc = 0.88.
III. The disordered system
In the disordered system under investigation, the reduced nearest-neighbour interactions K,t are assumed to be in- dependent r a n d o m variables according to (3). The bi- quadratic exchange interaction is kept fixed.
F o r such a disordered system the renormalized inter- actions are given by an extension of relation (9). As illus- trated in Fig. 5, K' and E are functions of the 3 d original nearest-neighbour K~j and L i t ( i = 1, 2, . . . , 3 d- 1, j - - 1, 2, 3). F o r d = 2 and b = 3, we have
K' = / ( 1 (Kll, KI2, K13) +/~z(K21, K22, K23)
+ / ( ( K a l , K32, K33),
E - - L l ( L l l , L12, L1 a)+ L2(LE1, L22, Lza)
+ L ( L 3 , , L32, L33). (13)
The functions/s and L~ are determined by the following relations
l n [ a i b i ]
L,i=--2 \ c~ ]" (14)
where
ai = e - 88 + L,2 + L,3) { exp [3 ( K i 1 + K i z + K~ 3)]
+ exp [89 (Ki 1 - K i 2 - - Ki3)]
+ exp [3 ( - K i a - K i2 + K~ 3)] + exp [3 (-- Ki x + K i z
-K,2]}
bl--e ~e p[~[--lkil--l~i2-- Ki3)]
+ exp [3(Kil + Ki 2 -- K~3)]
(15a)
+ exp [89 - Ki2 + Ki3)] + exp [ 89 Kil + Ki2 + K J ] }
+ 2 exp ( - ~ -3) 9 cosh ( ~ ) . (15b)
Ci=4e-88 cosh (~-2) + 2. (15C,
A straightforward but tedious analysis of all the possible values for the interactions K~j and Lij taking into ac- count their respective probabilities (3) and (4), gives an expression for the renormalized probability distributions P'(K;j) and Q'(Eij ) which are not of the same form as the initial ones. Although it is possible in principle to
where
3L 3K K L [ ~ \
e ~-(e -z- + 3 e - ~ ) + 2 e - g c o s h
fl(L,K)=ln 3L
Le - T ( e 3 ~ + 3 e ~ ) + 2 e - 4 cosh
f2 (p) = _6p3 {q6 +6pq5 + 15p2q4+20p3q3 + 15p4q 2
+6pSq+p6}' L 3K K / \ K
q=l--p, a = e - 4 , b = e 2 - + 3e - T , c = cosh ~ - ) ,
3K K
d = e 2 +3eT, e = l + c o s h ( K ) .
b) Random + K interactions (~ = - 1) 1 --p' _ 3q 9 + 9 p q s + 27p2 q 7 + 57p 3 q6 + 81 p4q5 + 99p5 q4+ 81 p6q3 + 27pTq2
p' 3p9+3pSq+27p7q2+57p6q3+81pSq4+99p4qS+81p3q6+27pZq 7' (18a) p'K'=fl(L,K).{3p9+9paq+27p7qe+57p6q3+81pSq4+99p4qS+81p3q6+27pZqT}, (18b)
e - - Z - ( e ~ - + 3 e - 7 ) + 2 e - a c o s h e 4 (e ~- + 3 J ) + 2 e - * c o s h
E = - 6 in __L (18c)
[2e 2(1 +cosh(K))+2] 2
follow the evolution of the probability distributions under successive renormalizations and to determine the various fixed distribution, it is much simpler to approxi- mate the transformed distributions by ones having the form of the initial ones but with renormalized parame- ters, K', E, and p'.
P~'ppro,,(K~j)=p'3(K'ij-K')+(1-p')6(K~j-~xK'), (16a)
Q'approx (Eij) --- • (/~j --/-~)- (16 b)
Equating the first moments of P'(K' 0 and P,~prox(K'ij) on the one hand and of Q'(E,j ) and Q'appro,,(Eij) on the other hand we obtain recursion relations for the vari- ables p, K, and L.
In the case of a two-dimensional model the recursion relations for the variables p, K, and L read
a) Bond dilution (a = 0)
1-p'=q9 +9pqS + 36pZqT +81p3 q 6
+ 108p4q5 + 81p5 q4 + 27p6 q3, (17a) p'K' =fl(L, K). {3p3q 6 + 18p4q 5 +45p 5 q4
+ 60p6q 3 +45p7q 2 + 18pSq + 3p9}, (17b)
fln (a3 b + 2ac) (a3 d + 2ac) ' , . , [ l - p \ E =f2(p). (2a2e+2) ~ -t- z ~ - )
[ln 2 a a e + 2 a 2a3e+2ac]
4aZc2+ 2 r ~a~a2c~ ]
m
1--p 2 2aac+ac 2 a 3 c +
+ 2 ( i p p ) 3 , 2 a a + a ]
m 2a--U j , (17c)
One should note that p = 1 is a subspace which corre- sponds to the pure system and (17) and (18) reduce, in this case, to (9).
The recursion relations (17) were used to determined the phase diagram of the diluted model. Phase bound- aries were determined, as usual, as the separation be- tween the regions of attraction of the trivial fixed points.
For L = 0 the model reduces to the mixed spin Ising model. Figure 6 shows the transition temperature of this model as a function of dilution. This line is very similar in shape to the critical curves of the diluted spin-89 [18]
and spin-1 [20] Ising models. At zero-temperature, this model exhibits a transition at the percolation threshold p * = 0.682 which may be compared with the exact value of 0.5 [24]. Phase diagrams in the ( K , K - 1 ) p l a n e for different values of p are shown in Fig 7. At zero-tempera- ture the value of the biquadratic exchange interaction, at which the ferromagnetic order disappears, depends continuously on p.
For the model with random + K interactions (a = 0),
the recursion relations (18) were used to determined the
phase diagrams. For the zero-biquadratic exchange in-
teraction, the long range ferromagnetic order is des-
troyed if the concentration of the + K bonds is lower
than p =0.893 (which is not a fixed point) as is shown
in Fig. 8. The corresponding phase diagrams of this mod-
el for different values of p are represented in Fig. 9. At
zero-temperature the ferromagnetic order is destroyed
beyond ,--(K)o = 2. It is important to note that this value
of the biquadratic exchange interaction is unique and
does not depend on p.
i
0
1. .9 .8 .7 ,6
P
Fig. 6. Critical curve for the bond diluted (c~=0) mixed spin Ising model (L = 0) determined by Migdal's method
.61
(a) K I
.4 (e)
\
\
.5 1. 1.5 2.
L/K
Fig. 7. The phase diagrams of the bond diluted mixed spin Ising model in the .T[___L K - 1~ plane. Transition lines are shown for differ- \ K ' ] ent values of the concentration p. (a): p = 1; (b): p=0.9; (c): p=0.8;
(d): p = 0.7
.4
.2
0 1,
.6 (~)
(r
K 1 (d)
(e)
.4 (f)
.2 K - I
.6
.95 .9
P
Fig. 8. The variation of the critical temperature with the concentra- tion of ferromagnetic bonds for the random +_ K interaction model,
= - 1 and L = 0
.5 1. 1.5 2.
