CHINESE JOURNAL OF PHYSICS VOL.22,N0.3 AUTUMN 1984
Renormalization-Group Study of a Simple Cubic
Ising Model with Four-Spin Interactions
Chin-Kun Hu and Hong-Yuh L EE
Institute of Physics, Academic Sinica
lVankang, Taipei, Taiwa’n Il.5
Republic of China
(Received 29 September 1984)
The modified Kadanoff variational method has been used
to study a simple cubic Ising model with the nearest-neighbor
two-spin interactions of strength J2 and the four-spin interactions
of strength Jq. The free energy f. spontaneous magnetization M,
magnetic susceptibility x, internal energy U, and the specific
heat Ch are calculated and plotted. The results indicate that for
0 < J4/Jz < 4, the phase transitions are second-order.
II N thecr =simple Ising model, each site (NN) pairs of G withhavesites is occupiedstrengthspin-i Since
spin ?I and the nearest-neighbor
of the lattice
Onsager’ solved exactly the spin-i- ample Isin, model in 1944, there have been many extensions of the
simple Ising model. One important extension which will be the subject of this paper is spin-7 Ising
models with multispin interactions. The study’of such models could help us to understand more about
the factors which influce the behavior of phase transitions, such as the order of phase transitions and the
univerrsality of critical exponents.
The Ising models with multispin interactions which have been studied included the Askin-Teller
modei2. Baxter mode13, Baxter-Wu mode14. and many other modelsS-9 studied in recent years. Based
on the modified Kadanoff variational method (MKVM)“. in I981 Hu and Kleban” formulated a syste-
matic method to calculate the free energy and its derivatives for Ising models in one to three dimensions
with a wide class of interactions. This method is particularly useful for studying Ising ‘m odels
with multispin interactions. As a first step toward such study. in this paper we calculate the free energy
f, spontaneous magnetization M, magnetic susceptibility x. internal energy U and the specibic heat Ch
for a simple cubic (SC) Ising model with the nearest-neighbor (NN) interactions of strength J2 and four-
spin interactions of strength Jq. It seems that this model has not been extensively studied before. The
results of our calculations are plotted and discussed. In this paper, we emphasize on the global properties.
The analysis of the critical properties, including the correction to scaling, from the calculated data, will
be published in another paper,
The Hamiltonian of the simple cubic Ising model with two-spin and four-spin interactions is given
H = -J’N~~i~j - J4 ScQOiajOkUp - hi Ui ) (1)
12 RENORMALIZATIONGROUP STUDY OF A SIMPLE CUBIC ISING MODEL WITH FOUR-SPIN INTERACTIONS
where the first sum is over all NN pairs of spins, the second sum is over all squares on the surfaces of the
primitive unit cells, and the third sum is over all sites of the lattice. A typical primitive unit cell 05 the
simple cubic lattice is shown in Fig. 1. In terms of cell potentials, the H normalized by P(being k~,
where k is the Boltzmann constant) may be -written as:
/JH = - E uR(uR1 uR2 . . . . , uR8 ) ,
Fig. 1. Location of lsing Spins (ur , u2. . . us) on a
typical cube of the simple cubic lattice.
where the sum is over all primitive unit cells R of the SC lattice. ORI, uR2, . . and URN are eight
spins located at corners of the primitive unit cell R. and uR is given by:
UR(“Rl> OR2, t.. , OR.8 > = u( 01 ,a,. . , 08 )
=- +-g, +-g1 >
4 g2 ;4 8
Ks = /3Js, k4 =/JJ4, B = fib,g2. gs, a n d g, are two-spin, four-spin: and on
given by Eq. (A13) of Ref. (1 l),
g, = u1u2 + u2u3 + u3u4 + u4u* + u*ucJ + 0206 + 0307
t u4u, + usua + ue,u, + U./Us + oaos ,
g1 = o1 t u2+ u3+ u4 + us + Qj + 01 + % (6)
C.K. HU AND H.Y. LEE 13
In the last two expressions of Eq. (3) and Eqs. (4) to (6), the subscripts “R” on the spin variables have
been suppressed for the sake of simplicity.
Now we use the cell potential u(ui, UZ, . , us) of Eq. (3) as the initial cell potential u” of Ref.
(1 1) to calculate the free energy and its first and second derivatives with respect to B and f3 for J, = I
and J4 = 0, 1, 2, 3 and 4. It is well known that M, x, U, and Ch are related to the derivatives of the
normalized free energy f by the equations:
x = - -
ch= -kp2 - f (10)
In the step by step renormalization group (RG) transformations, we find that f, M, x, and
U approach smoothly to their final values but Ch does not, perhaps, $ue to numerical truncations. Thus
we use the numerical differentiation of U with respect to /3 to obtaina of Eq. (10). TO test the reliabi-
lity of the numercial differentiation, we use Ap/&.= 10e7 and 4.0~10-~ near acritical point and find that
the resulting 7 values differ only 1.0%. The calculated results for f, M, x, U and Ch are shown in
Fig. 2 to Fig. 6, respectively. The curves in FIG. 3, 4 and 6 for J 4 = 3 to 5 are undistinguishable from
each other. The calculated f, U and Ch for Jq =O are identical with those of a previous paper”. Note
that the calculated f values are lower bound of the exact free energies. Thus they could be used to test
the reliability of the free energies obtained by other approximate methods. We have also used
the behavior of the variational parameter p (becoming smaller or larger values) in the step by step RG
transformations, to locate the critical point for given J z and J4. The results are shown in Table 1. It
is clear from Fig. 4a and Fig. 6a that In x and Ch go to infinite at critical points. The calculated data
for U and M also indicate that the phase transitions are second-order for J2 = 1 and 0 S J4 <4. In
another paper, we will use the data near the critical points to analyze the critical behavior including the
correction to scaling.
This work was supported by the National Science Council of the Republic of China under contract
14 RENORMALIZATIONGROUP STUDY OF A SIMPLE CUBIC ISING MODEL WITH FOUR-SPIN INTERACTIONS
_._ , . _.:-
C.K.HU AND H.Y. LEE 1s
16 RENORMALIZATION-GROUP STUDY OF A SIMPLE CUBIC ISING MODEL WITH FOUR-SPIN INTERACTIONS
I I 1 I I I I I I 1 I 1 I I I
0o.t I 00’21 00’0 I 00.8 00.9 00-t 00.2 00
C.K. HU AND H.Y. LEE 17
T- 1 I 1 I I t I I I
‘0 .85 0.91 0.97 I .03 I .09 1.15
Fig. 6 (b) data of FIG. 6a near critical points as function of T/T,.
Table 1 Critical points & of the simple cubic Ising model,
with the NN interactions of strength Jz=l and
four-spin interactions of strength Jz
i_. --_ .
18 RENORMALIZATIONGROUP STUDY OI- A SIMPLE CUBIC ISING MODEL WITH FOUR-SPIN INTERACTIONS
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