Condensed
Zeitschrift Matter
fi3r Physik B
9 Springer-Verlag 1990
Real-space renormalization group investigation
of the three-dimensional semi-infinite mixed spin Ising model*
N. Benayad** and J. Zittartz
Institut fiir Theoretische Physik, Universitfit zu K61n, D-5000 K61n 41, Federal Republic of Germany Received May 10, 1990
Within a real-space renormalization group framework we study the three-dimensional semi-infinite mixed spin Ising model (spins a = 8 9 and S = 1). The bitinear (Ks) and the biquadratic (Ls) interactions on the surface might be different from the bulk ones, K~ and L~. The parameter space is four dimensional. We find 26 fixed points describing a large variety of critical behaviour.
The effect of LB and Ls on the surface transition is inves- tigated.
I. Introduction
Surface magnetism is an interesting problem which has been the subject of numerous theoretical and experimen- tal studies. Detailed review articles containing an exten- sive list of references have been published by Binder [1]
and Diehl [2]. One of the simplest three-dimensional models is the spin-89 Ising model on the semi-infinite simple cubic lattice. It has been extensively studied using a variety of approximations, in particular renormaliza- tion group methods [3-13], Monte Carlo technique [14], and series expansions [15]. The reduced coupling constant Ks(Ks > 0) on the free surface is not necessarily equal to the bulk one KB(K8 > 0). One finds four different types of phase transitions associated with the surface.
The accepted terminology [-3, 4] is the following: If the ratio R =Kss is larger than a critical value Rc the system KB
orders at the bulk ferromagnetic transition temperature.
This is the ordinary phase transition. If R is less than R~, the system exhibits two successive transitions. First the surface orders at a temperature higher than the bulk.
Secondly, as the temperature is lowered, the bulk orders
* 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
at the bulk transition temperature. These two phase tran- sitions are called the surface and the extraordinary transi- tion, respectively. If R -- R c the system orders at the bulk transition temperature, but in this case the critical expo- nents differ from those of the ordinary transition. This is the special phase transition.
Using real-space renormalization group methods, similar transitions have been found for the three-dimen- sional semi-infinite spin-1 ferromagnetic Ising model with crystal field interaction /-16]. As a function of the ratio R of bulk and surface interactions and the ratio D of the bulk and surface crystal fields, it has been shown that the ordinary transition can be either first order or second order. According to the value of the surface crys- tal field, the surface and extraordinary transitions are always of second order.
In the above systems, the spins had the same value on each lattice site. Therefore it is interesting to extend the investigation to mixed spin Ising models. More pre- cisely we shall consider a semi-infinite mixed spin-89 and spin-1 Ising model defined on the simple cubic lattice and described by the following reduced Hamiltonian:
flH = - K s Z ai S j+ L s Z (ai S j) a - KB Z ak Sl
(i j) <i j> <kl>
s f . (1)
<kt>
The unterlying lattice is composed of two interpenetrat-
ing sublattices, one occupied by spins with spin moment
o-= _+ 89 and the other one occupied by spins with spin
moment S = 0 , +1. The first and second summations
run over all pairs of neighboring spins located on the
two-dimensional surface of the system, the third and
fourth summations run over all pairs of remaining neigh-
boring spins. Ks( m and Ls(B) denote the reduced bilinear
and biquadratic exchange interactions, respectively. The
subscripts S and B refer to the surface and the bulk,
respectively. One should note that the Hamiltonian (1)
is equivalent to a semi-infinite mixed spin Ising model
with crystal field interactions:
, H : - K s 2 o, Sj+AsES -I.:. Z (2)
( i j> j ( k l ) l
where A s = L s and An=~LB. Recently the Hamiltonian (2) has been studied [17] using the effective-field theory with correlation and by molecular-field theory. The effect of bulk and surface crystal fields has been investigated only when the bulk exhibits a second order transition [18].
The purpose of this paper is to investigate the semi- infinite mixed spin Ising model described by the Hamil- tonian (1) using a real-space renormalization group tech- nique. In particular, we shall determine explicitly the effect of any bulk and surface biquadratic interactions on the surface transition.
We remark that the method we employ can not be restricted to a semi-infinite mixed spin Ising model with only bilinear terms (Ls = Ls--0), since in the four-dimen- sional parameter space (LB, KB, Ls, Ks) the subspace L~j = Ls = 0 is not invariant.
One should note that the different phases (on the surface or in the bulk) of the model described by the Hamiltonian (1) can be characterized by two parameters:
the magnetization m = ( S ) (or ( a ) ) and the quadrupole parameter q = ($2). According to the values of m and q, three different phases can be distinguished:
Paramagnetic (P_): m = 0, q < 89 Paramagnetic (P§ m = 0, q > 89 Ferromagnetic (F): m =t = 0, q X 89
II. Real-space renormalization group approach
In this section we investigate the three-dimensional mixed spin ferromagnetic Ising model described by the Hamiltonian (1). This study will be performed using an extension of the Migdal-Kadanoff renormalization group method [19, 20]. This technique is applicable for all space dimensionalities. In what follows, we shall give a brief description of the method.
Because of the symmetry of the model, we m a y re- strict ourselves to an odd scale factor b. In the present study we choose b = 3 and consider a one-dimensional chain with spins al, S~, 0-2, $2. All these spins are cou- pled by the same bilinear and biquadratic exchange in- teraction K and L, respectively. We perform the trace over all spins on the chain except those at the end. The end spins are then coupled by an effective interaction /~ and L which are functions of K and L [217 given by:
_~\ _L_ I K \ e 3~ e_3~+3e 2 } + 2 e 4 c o s h | ~ - }
e R =
/ \ z /
e (e
L IK\q r 3L [ 3K L
2e- cosh (e- 3e )+2e
[ 3 L [ 3 K
[e-TqeT+ge- )+
e - - _ L
[2e 2 (1 + c o s h ( K ) ) + 2 ] ~
Table 1. Coordinates and classification of the fixed points of trans- formation (4)
Fixed Type (L*, K*) Domain in
points coordinates the (L, K) space
X
F (m =# 0, q ~ 89 phase ( - co, co) surface F P+ (m = 0, q > 89 phase ( - co, 0) surface P+
P_ (m = 0, q < 89 phase (0% 0) surface P_
/
N second order (co, K* = 2 ) point N
\
I second order ( - co, 0.708) line B 1 I B 1 critical (0.438, 0.903) point B1
B 2 second order (18.05, 9.685) line N B 2 B 1
0 separation (0, 0) line B10
between P_ and P+
In an infinite d-dimensional cubic lattice Migdal argues that the new couplings K ' and E as functions of K and L are simply
K'=bd-~ ~2(L,K), (4a)
E -= b d- 1L(L, K). (4b)
In the previous work [21], we have calculated the var- ious fixed points of similar recursion relations (4). We found that all transitions are of second order for any dimension (d = 2, 3, 4). In particular for d = 3, these rela- tions have 8 fixed points which are given in Table 1.
The corresponding phase diagram is shown in Fig. 1.
We note that for d = 2 we have only 6 fixed points (F, P+, P_, I, O, N) which have the same meaning as in d = 3 .
In the case of a semi-infinite three-dimensional cubic (d= 3) Ising model described by the Hamiltonian (1) it is straightforward to extend Migdal's approximate recur- sion relations (4) [12, 16]. We obtain
EB = b ~- ~ L(LB, K.), (5 a)
K~ = b d- 1/~ (Ls, KB), (5b)
Es =ba-2~,(Ls, Ks)+ abd-2(b - 1) L(LB, KB), (da) K's =ba-2F2(Ls, Ks)+ lbd-2(b - 1)/~(L B, Ks), (db) where/~ and L are the functions defined by (3). As usual for semi-infinite systems, the renormalized bulk interac- tions given by (5) depend only on the initial bulk interac- tions LB and KR. Therefore, in order to determine the four coordinates (L*, K*, L*, K~) of each fixed point, we shall first determine the (L*, K*) coordinates from (5) which problem was already solved in [21]. Subse-
(3a)
cosh( )]
(3b)
KB
F 10.
1,
P+ P_
I I i / / p
-2. - 1 . 0 .5 18.
N
LB
Fig. 1. Phase diagram in the (L~, KB) space obtained for the infinite three- dimensional mixed spin Ising model
quently we determine the remaining (L*, K~) coordinates from (6) after having replaced L B and K B by the fixed point values of L* and K~. This means that in the four- dimensional parameter space there are 8 two-dimension- al invariant subspaces in which the fixed points are deter- mined by the recursion relations (6).
To denote the various fixed points in the four-dimen- sional parameter space, we shall use the symbols defined in Table 1. More precisely, each fixed point will be denot- ed [16] by a pair (X, Y) where the symbol X refers to the coordinates (L*, K*) and Y to the coordinates (L*, K~) of the fixed points. This procedure therefore implies that the symbol X characterizes the two-dimen- sional invariant subspacc in which the fixed point (X, Y) is located.
a. The determination of the fixed points located in the invariant subspace X = 0 is very simple. Their (L*, K*) coordinates are equal to zero and their (L}, K*) coordi- nates are those of the two-dimensional infinite system given in Table 1 of [21] only with a new numerical value for the K-component (K~ = 1.443) of the fixed point I.
In this case, the bulk exhibits a smooth continuation between P§ and P_ while the surface can be in any of the three phases F, P+, and P_ or it can undergo one of the 3 different phase transitions of the infinite system [21]. Altogether we have 6 fixed points.
b. In the invariant subspace X = F there is only one fixed point: Y= F. This means that if the bulk is ferromagnetic, the surface is necessarily also ferromagnetic.
c. In the invariant subspace X = P§ there are three fixed points: Y= F, P§ I. The first two are trivial and charac- terize the ferromagnetic F and the paramagnetic P§
phase of the surface, respectively. The last point (L}-- - 0% K~ = 1.443) is nontrivial. It characterizes the three- dimensional hypersurface of second order transitions in the four-dimensional (LB, KB, Ls, Ks) phase diagram between the phases F and P§ on the surface, while the
bulk is in the paramagnetic P+ phase. This phase transi- tion is similar to the surface transition in the three-di- mensional semi-infinite Ising model.
d. In the invariant subspace X = P_ there are two fixed points: Y= P_, N. The first is trivial and characterizes the paramagnetic phase P_ of the surface. The second one is nontrivial but is located at infinite coupling. In this case the surface does not exhibit a phase transition at finite temperature.
e. In the invariant subspace X = I there are three fixed points: Y= F, I, I s. The first point characterizes a three- dimensional hypersurface in the phase diagram which corresponds to the transition in the bulk between phases F and P+, the surface being in the ferromagnetic phase.
This phase transition is similar to the extraordinary phase transition of the three-dimensional semi-infinite Ising models. The coordinates of the point (I, I) are L~
= - 0% K~=0.708, L * = - o% K~ =0.247. It character- izes a three-dimensional transition hypersurface. At any point of this hypersurface the bulk and the surface of the system exhibit a transition between the phases F and P§ Such a phase transition is similar to the ordinary phase transition in the three-dimensional semi-infinite Ising model.
The coordinates of the point (I, Is) are L * = - 0%
K* =0.708, L* = - 0% K* = 1.229. It characterizes a two- dimensional transition surface. At a point of this surface the bulk and the surface of the system exhibit a transition between the phases F and P§ However, in this case the critical exponents are different from the previous case. Such a phase transition is similar to the special phase transition.
f In the invariant subspace X = N , there is only one fixed point Y= N. At this point the bulk and the surface exhibit an ordinary transition between the phases F and P_.
g. In the invariant subspace X - = B 1 there are 6 fixed
Table 2. Coordinates and classification of the fixed points located in the subspace X=B~. Dim. is the dimensionality of the domain in the (L~, KB, Ls, Ks) space
Y Type (L*, K*) coordinates Dim.
F (re:l:0, q<>89 phase ( - ~ , o0) 2
I second order (-0% 0.326) 2
I~ special second order ( - ~ , 1.155) 1
P second order (0% 0.300) 2
P~ special second order (0.002, 0.317) 1
N second order 0% K* = 0
Table 3. Coordinates and classification of the fixed points located in the subspace X = B2. Dim. is the dimensionality of the domain in the (Ln, K~, Ls, Ks) space
Y Type ( s, K~) coordinates L* Dim.
F (m~:0, q<>89 phase ( - 0% ~) 3
P second order (0% 3.228) 3
P~ special second order (11.69, 6.580) 2
/ ~.
\ 21
( s, K*) coordinates are points. Their classification and L*
listed in Table 2. F o r the bulk undergoing a critical phase transition we only give the information concerning the surface. The fixed points (B~, 1) and (B 1, Is) characterize similar phase transition. The first one is said to be ordi- nary and the second one special. They have different critical exponents. If we denote by dim(X, Y) the dimen- sionality of the hypersurface characterized by the fixed point (X, Y), we have d i m ( B , Is)=dim(B1, /)-1. This is also valid for the fixed points (B~, P) and (B~, P~).
The nomenclature adopted here is in analogy with what occurs for the usual semi-infinite Ising model.
h. In the subspace X = B2 there are 4 fixed points. Their classification and L* ( s, K*) coordinates are listed in Ta- ble 3. For the bulk undergoing a phase transition be- tween the phases F and P_, we only give the information concerning the surface. Here again we find the special transition.
III. D i s c u s s i o n a n d c o n c l u s i o n s
In the previous section we have investigated the three- dimensional semi-infinite mixed spin Ising model with both bilinear and biquadratic exchange interactions (or crystal field). In the four-dimensional parameter space we found 26 fixed points. Referring to the 8 fixed points of the infinite mixed spin Ising model, the fixed points of the semi-infinite system have been classified and den- oted by a symbol (X, Y), where the first symbol refers to the bulk and the second to the surface.
Our new results have been obtained for all values of the biquadratic exchange interactions L~ and Ls. We
LB 4
found that for - ~ > 0 . 85 there is only an ordinary tran- sition between the phases F and P_. Therefore there is
K s L~
no surface transition for any ratio R = ~ . For K 8
<0.485, according to the value of Ls K~' a surface transi- tion exists for any ratio R lower than a critical value Its variation in the --L(~, R)" plane is shown in Fig. 2 Re.
for some fixed values of ~ . LB We remark that for each
R C
(a)
(b)
(c)
. 5
. 4
I - 1 o
(,): ~ = 04
( b ) : ~
KB
= 0.(C) : ~ ----2.
( d ) : ~ : - o o
