AD-AlIl 616 MAINE UNIV ORONO DEPT OF PHYSICS AND ASTRONOMY F/S 20/13DIRECT CALCULATION -OF THE DERIVATIVES OF THE FREE ENERGY FOR IS--ETC(U)FEB 62 C HU. P KLEBAN N00014-79-C-0441

UNCLASSIFIED R-6 N6.

EEU4hEhh2

2.I 1112.1111. 2 12 .

11111_L25Rl-6IIIA

MICROCOPY RESOLUTION ILSI CHARI

03

00 DIRECT CALCULATION OF THE DERIVATIVES OF THE FREE ENERGY

FOR ISING MODELS BY A MODIFIED KADANOFF'S VARIATIONAL METHOD*

Chin-Kun Hu and P. Kleban

Department of Physics and Astronomy

and

Laboratory for Surface Science and Technology

University of Maine

Orono, Maine 04469, USA

* rTIG5

*W$ork Supported by the Office of Naval Research

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4. TITLE (and Subtle) 5 TYPE Or f4LPORT G PERIOD COVERED

Direct Calculation of the Derivatives of the Free Technical ReportEnergy for Ising Models by a Modified Kadanoff'sVariaiona Metod - PERFORMING ORG. F.EPORT NUMBERVariational Method

7. AUTHOR(eJ 6. CONTRACT OR GRANT NUMUER()

Chin-Kun Hu & P. Kleban Contract#NO0014-79-C-0441

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Department of Physics and Astronomy Project

University of Maine, Orono, ME 04469 PrR392-32

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To appear in Journal of Computational Physics

19. KEY WORDS (Cownue on rever e.a° d If nec' eery - ,md Identify by block numbe, )

Ising model, transition temperature, critical properties, renormalizationgroup, Kadanoff variational method, correction to scaling, two dimensionalmodels, three dimensional models

20 ABSTRACT (Contfnint on re-eree side It necesory end Identify by lock number)

DD IiA7 1473 EDITION OF I NOV 5IS OS;OLETES/N n102-014- 6601 I

SEICU'lITY CL ASV1FICATION OF THIS P AGE (W eI. Sfl ezed)

Abstract

-We present a systematic procedure for the direct calculation of the free

energy and its first and second derivatives for the Ising model in one to

three dimensions with a wide class of symmetry properties. This renormalization

group method is based on the modified Kadanoff's variational method (MKUM).

This method is not only general but also very accurate numerically both near

-! and far from the critical point. Further it includes correction to scaling

effects not present in the standard linearized renormalization group treatment.

This work describes the techniques and presents some illustrative results for

the square lattice and body-centered cubic lattice with nearest neighbor inter-

actions. A full treatment of our results will be given elsewhere.

ACOOSion3 ror

U 0

-7* " - -" " lea ' "

II

-2-

I. Introduction

In recent years, the position space renormalization group (PSRG) method

and its application to phase transitions of various spin models has been

widely studied. One important example is Kadanoff's lower bound renormaliza-

tion group transformation (LBRGT) [1].

In Kadanoff's LBRGT, one has variational parameters in the renormaliza-

tion group transformation (RGT) equation relating transformed and original

unit cell potentials. The optimum variational parameters must be chosen in

order to get the best lower bound free energy from the transformed Hamiltonian.

Several different methods with different degrees of complexity [2,3,4] have

been proposed to determine these variational parameters. One such method is

the Modified Kadanoff's Variational Method (MKVM) [3].

In the MIKVM, the variational parameter p is determined by minimizing the

single cell free energy. This leads to an analytic nonlinear equation for p.

In previous papers [3,5], it was found that the MKVM is very accurate for two-

valued spin models without external magnetic field in one or two dimensions. It

has also been applied to the three dimensional Ising model on a BCC lattice

where the derivatives of free energy were calculated by numerical differentia-

tion of the free energy [6].

In this paper, .,e formulate a systematic procedure to directly calculate

the free energy and its first and second derivatives with respect to temperature

and external magntic fields for a wide class of Ising models based on the MKVVi.

In this way, we avoid the numerical errors arising from numerical differentia-

tion and thus have more accurate derivatives of the free energy for further

analysis.

f -3-

In the application of RG theory to phase transitions, one usually formu-

lates the RGT equations for the coupling constants, then solves them for the

fixed point solution(s) and finally expands the RGT equations around the fixed

points to find the linearized RGT equation and its eigenvalues and hence critical

exponents. Instead of that, we apply the RGT to the Hamiltonian of the given

system and calculate the free energy and its first and second derivatives directly.

From the behavior of the variational parameters and coupling constants (which

tend to larger or smaller values), we can determine the critical point of the

given system, which has more direct physical significance than the fixed points.

By analyzing the values of the first and second derivatives of the free energy

near the critical point, we can determine critical exponents, critical amplitudes

and other critical parameters including their less divergent corrections which:are

of considerable current interest [7,83. This analysis will be presented elsewhere.

Since the quantities of direct interest here are feee energies and their

derivatives, instead of the fixed point and linearized RGT, it is not necessary

to carry out the calculation in coupling constant space. In fact, to write a

single computer program applicable to many systems with different symmetry

properties, it is more convenient to carry out the calculation in cell potential

space. This is related to coupling constant space by a linear transformation.

The present paper is organized as follows: In Section II, we derive

formulas for free energies and their first and second derivatives with respect

to temperature and external magnetic field. These formulas are written in cell

putential space instead of coupling constant space. In Section III, we con-

sider some typical systems with various symmetry properties in I to 3 space

dimensions. We show that we may use very simple criteria to classify all

possible spin configurations on a unit cell into different degeneracy groups,

each of which corresponds to an independent cell potential. The systems con-

sidered are listed in Table I.

-4-

We illustrate our formalism in section IV by calculating the free

energy and its first and second derivatives with respect to temperature and

external magnetic field for Ising models on two dimensional square and

three dimensional BCC lattices. Our results are compared with exact [9,10]

and series expansion values [11,12] and are shown in Fig. 1 and 2. In

section V, we give a brief discussion of possible extensions our method.

t -5-

11. General Formalism

Let us consider N Ising spins a.i (=±-) on a general d dimensional

lattice, which interact with each other via a Hamiltonian of the form [1):

where £ is a sum over d dimensional hypercube unit cells of the lattice.RA

aR' = ) with Z=2 t are spins at the corners of the unit cell R' and

U (aR') is the interaction potential of the spins aR ' within the unit cell R'.

After a Kadanoff's LBRGT, the transformed Hamiltonian for NW Ising spins

(=±) on lattice with double lattice spacing has the same form as Eq. (1),

i.e.:

o " z V(2)

where N' is the number of spins on the new lattice (N'=N/Z), uR' (VR) is the

interaction potential for the Ising spin 1JR=iUl ,--,-'izon a hypercube unit cell

R with doubled the original lattice spacing., and uR. is related to uR(OR-) by

the RGT equation (hereafter we will drop the subscripts on vR (oR.) and uR(PR

I4'A~)~ z (3)

with u (u,p) given by:

U el )=S 1r- (4)

where and p is a parameter. It follows from Eq. (3) and (4) thati=1

u (-) and u(o) have the same symetry properties with respect to the point

group transformation of spins within the hypercube unit cell. Thus t and

u' can be expressed in terms of the same invariant functions of Ising spins

on the hypercube unit cell:#4

=.T rc (r (5a)L= 0

, (Sb)

r -6-

where go)=go) The other gi are defined and discussed in Appendix A.

Let -(KK .---K,) and '=(K',K',---K') denote vectors of the coupling

constants. Then Eq. (3) may be considered as a transformation of the coupling

constants: _, . -

K F k P) .(6)

For Z Ising spins on a hypercube unit cell, there are 2z possible spin

configurations and thus 2z possible hypercube cell potential iY(p). However,

many u'(p) are equal due to the symmetries of the spins on a unit cell. In

fact, the number of independent u'(i) is the same as the number of independent

coupling constants in Eq. (5), ie k+l. Let -" denote the vector of these

independent cell potentials (ti,u2 ,---U 1 l). Then u' is related to

K'(oY,---K;) by the linear transformation of the (z+l) x (z+l) matrix T:0I

'J '," )=t ( kK I

or briefly denoted as:

(8)

T can be calculated from Eq. (5).

We also have the inverse transformation

(9)

where T- 1 T=I.

Now the free energy calculated from H'( I,--PN-) is always a lower bound

for free energy calculated from H(ol,a 2,--CN) and we must vary p to obtain

the optimum lower bound free energy. In the MKVM, the variational parameter

p is determined by minimizing the single cell free energy. This leads to

the nonlinear equation:

____ ____ ____ ___ ____ ____ ___(10)

-7-

which is solved at each iterative step to determine P. From Eq. (3), (4)

and (10), it is easy to show that Eq. (10) ma be rewritten as:

t+ 1

With F(S Pi,p) given by Y- ( 12

F~s,£ ; >)= z )(12)

Where S is E aj evaluated at the i th degeneracy group. Di and ui areI l j=1

the number of elements and cell potential for the i th degeneracy group respec-

tively.

To begin with, we use uR ' (aR' ) of Eq. (1) as u(a) in the right hand side

of Eq. (3) with p given by Eq. (11) and calculate the cell potential k'(11)-

from Eq. (3). This constitutes the first step of the RG transformation. We

then use u'(P) thus obtained as input in the same procedure to calculate the

transformed potential u"(11). This RG transformation is iterated further so

that a series of cell potentials v'(P), u"(p),---u,") (t) (and hence coupling

constants K', K",---K(a) and the corresponding variational parameters

pp".---p(c))are obtained. In this step by step RG transformation Ki (a)

for ij and p(a) tend to diminish to zero with increasing a when the system

is above the critical temperature and tend to grow when the system is below

the critical temperature. Thus, after a large number, say a, of RG transforma-

tions, times the free energy per spin for the original lattice may be

approximated by:

for T 7tc C() e < and

--

7-

,(.4) J- ('o ,A >(13b)

for T < Tc (or B > Bc), where do is the spin configuration at T=o.

To calculate the internal energy and spontaneous magnetization of the

system, we must take the first derivative of f(a) with respect to 0 and

h(=B) respectively. Let q denote either a or h. By the chain rule, it is

easy to see from Eq. (13) that:---- _% ,, 1% (14)

i - ° "' "'

where

C Cs I) o ) (15a)

for T > T. and. T

(15b)

for T < Tc -

it will becorme clear later that in order to deal with systems with

different symnetry properties in a unified way, it is convenient to carry

out the calculation in the cell potential space u( (p) instead of the coupling

constant space K(). From Eq. (8) and (9), we have______ --I,__ -1h (06a)

______ (16b)- T

Thus Eq. (14) may be rewritten as:

V (17)

C 1j' ~ Vt(at-i) C,

*iV

-9-

In Appendix B, we derive very simple general formulas for T C that allow us toavoid explicitly calculating the matrix T.

It should be noted that for q=h in Eq. (14), 1(m) for m=l,---, must

include all odd spin coupling constants even if we evaluate the row vector

06 (m)6 and matrix 6-(m) points where the odd spin coupling constants vanish

These points are equivalent to the points in the cell potential space with

u ( ,..a Z ) = U (-al,.. -z). Thus to evaluate the spontaneous magnetizationin cell potential space, we must consider u(a a2,a...a) and u (- -a2,.. - )

1 . zi ( , '_Zas independent cell potentials.

To calculate the specific heat and susceptibility of the system, we must

take the second derivative of f( ) with respect to P and h. From Eq. (17),

it follows that-

/ .. . . .. .. .- .. .. ... _ _ _ _ _ _ _ . .•_ _ _ lI

-I-

-5. - ll l

give by

4--.V

L ___ g 6

~ (Eli

-I10-

Thus to calculate T and , we must

evaluate 3 an 1 and

for i, j, k = 1. --- , +l. It is straightforward to calculate the first two

quantities from the initial cell potential of Eq. (1). From Eq. (3), we have

3(20)

where (Jl,--1Jz) is one of the spin configuration belonging to the j th

degeneracy group and u(a,p) is given by Eq. (4). The variational parameter

p of Eq. (20) is essentially determined by.the initial cell potentials

" .. ) This fact must be taken into account in the calculation of-

Thus we have

____ _____ - 4,) 4~~Y "'~A - ~(21)

Where

1' (22)

* I ,a-

_____ ~ ~ ~ ( (T) t Z5~t 4 .AcIF

$ (23)

and

- z )~7 (~I ;P)(24)

ev z

-11-

where F(S1,1 ;p) is given by Eq. (12). In Eq. (22), is a sum over all con-

figurations of a corresponding to the ce]j potential ui(m'l)((a). Eq. (24)

is derived from Eq. (11) and the notation is the same as that of Eq. (11).

From Eq. (21), it follows that

-e1 (- 1) ~~.~)fi -fe 4a '-' I

r~ C) X~ P

PP ar a

It is straightforward to derive equations for the partial derivatives on the

right hand side of Eq. (25). They are very involved so we do not report them

here. It is obvious that Eq. (25) is symmetric with respect to the indices

K and i. We have used this fact to check our expressions for the right hand

side of Eq. (25).

This completes our formulation of the equations for the direct calculation

of the first and second derivatives of the free energy.

IA

-12-

III. Symmetry Properties

In order to carry out the configuration sum E' in Eq. (22) and also the

equations for the partial derivatives with respect to u(m-1), u(r-1) on theK i

right side of Eq. (25), we must establish the correspondence between spin

configurations (al, --az) and the independent cell potentials ui(a I , --az).

That is, we must classify the 2z possible spin configurations into different

groups, such that all configurations of the same group have the same cell

potential ui(al,---oz), where i runs from 1 to Z+l. This correspondence

depends on the space dimension and symmetry properties of the system. However,

we can establish very simple general criteria for the purpose of such con-

figuration classification.

In Appendix A, we list invariant functions for certain systems in one

to three dimensions. We also list the relations between invariants of a

given system. From such relations, it is easy to see that all invariants for

a given system may be expressed in terms of certain basic invariants. These

are also given Table I. Thus by Eq. (5), any cell potential u(al,---a z )

can be expressed as a function of these basic invariants. So we may use

these basic invariants as criteria to classify the spin configurations.

The generation of all possible configurations and their classification into

different groups based on their basic invariants may be carried out simply

by computer. This scheme is briefly described in Appendix B.

In Table I, we also list the values of Z+l and possible applications of the

considered models.

Table 1. Basic Invariants and possible applications for the considered models.

Note that the definition of g, (see Appendix A) may differ from case to case.

Model Dimension Basic Invariants .L+l applicationa

$1 1 3

S2 2 92 5 Isotropic SQ lattice with nninteraction only.

S3 2 g1 ' 92 6 Isotropic SQ lattice with nn,nnn and 4 spin interactions.

S 2 g1, g2, g3 7 Anisotropic SQ lattice with nn,,4 nnn, and 4 spin interactions.

S5 2 g' 92 9 Triangular lattice with nn.5 and nnn interactions.

S6 3 g1 9 Isotropic BCC lattice withnn interaction only.

Sg7 3 g1' 2' g3 22 Isotropic SC lattice with allpossible even spin interactions

within the primitive unit cell.

$8 g1' g2 ' g3 ' g4 ' g5 34 Anisotropic SC (tetragonal)lattice. Crossover from d=2to d=3.

S9 3 g1, g2 ' g3 ' g4 ' g5 46 Isotropic FCC lattice will allpossible even spin interactionswithin the primitive unit cell.

a. nn = nearest neighbor, nnn next nearest neighbor.

-14-

IV. An Example: Permulation Synmmetry

In this section, we apply the formalism developed in the previous sections

to an Ising model with isotropic nearest neighbor interactions on 1 dimensional,

2 dimensional square, and 3 dimensional body-centered cubic lattices. The

Hamiltonian in these cases may be written as:

(- 6(26b)

where R is a unit cell with one spin acin the center, Z spins G1'---z at

the corners and the factor B(= ) has already been included in K and h.

Performing a decimation calculation which sums over the central spin in each

cell, we obtain an effective Hamiltonian for the remaining spins. The result-

ing cell potential is

_ h (27)

Eq. (27) has permutation symmetry with respect to al , 02, --az and thus accord-

ing to Appendix A can be classified as S1, S2, or S6 for d equals 1, 2 or 3

respectively. We can use gl=1 +-- +az as the basic invariant to classify

the 2z possible configurations of al, -- tz into different degeneracy groups,

each of which corresponds to the same cell potential. We then use UeffR of

Eq. ( 7) as io in Sec. 2 to calculate the free energy per spin and its first

and second derivatives with respect to k and h. We must divide all these

quantities by a factor 2 because Eq. (27) was obtained after a decimation

calculation. The results for d=l agree extremely well with the exact results

-15-

obtained by transfer matrix method [9]. These results also provide a check

on our computer program.

A few results for d=2 and 3 are shown in Fig. 1 and Fig. 2 respectively.

Fig. lb and Fig. Ic should be compared with Fig. 6 and 7 in Kadanoff et al.'s

paper [1]. Their results were obtained by numerical differentiation for the free

energy and thus involve more numerical error than the present work.

In future work, we will present the results of this method applied to

several models with different symmetry properties. Preliminary analysis

indicates that we obtain very accurate values both close to and far from the

transition temperature, as is clear from Fig. 1 and 2. The application of

this method near the transition temperature is of particular interest since

it allows an evaluation of correction to scaling effects due to nonlinear

terms and all the (linearized renormalization group) eigenvalues.

e

F$

-16-

V. Discussion

From the previous sections, it is clear that to calculate the free energy

and its derivatives for given model. It is sufficient to use the basic invariants

of such model and not necessary to use the relations among all invariants, i.e.

the even numbered equations of Appendix A.. For a given' ystem, to obtain the

former is much easier than the latter. However, if we desire to calculate the

flow of the coupling constants (kl, k2, -- k ) in parameter space as the step by

step RG transformation is carried out, we may use the even numbered equations

of Appendix A to calculate the (z+l) x (A+I) matrix T and hence T- of Eq. (8)

and (9) respectively.

The method used in this paper may be easily extended and applied to systems

with symmetry properties different from those listed in Table I. In particular,

we may combine the ideas of this paper and our previous paper [14] to calculate

derivatives of free energy for antiferromagnetic systems.

-17-

Figure Captions

Fig. 1. Derivatives of the free energy for the Ising Model on the SQ lattice.

(a )-k as function of K. (b) -2-f as function of K. (c) M8 as

function of K, where M (d) In Yas a function of k, where

62f

The solid curves in Fig. la, lb and lc represent Onsager [9] or Yang's

[10] exact solution. The solid curve in Fig. Id is obtained from high

temperature [12] and low temperature Fll] serious expansion for K_0.4407

and K>0.4407 respectively.

Fig. 2. Derivatives of the free energy for the Ising model on the BCC lattice.

(a) -L as function of K. (b) "J4 as function of K. (c) M as

function of k, where M = Af (d) In as a function of k, where2 Th 6

?C = -k-2 . The solid curves are obtained from high temperature [12)

or low temperature [11] series expansion for K<0.1574 respectively.

Fig. Al. Location of Ising spin al, 02 on one dimensional unit cell.

ia

Fig. A2. Location of Ising spins 01, 02, O3' 04 on the two dimensional unit cell.

a and t are the primitive translation vectors and a is the angle between

them.

Fig. A3. Location of Ising spins al, 02' ---08 on the three dimensional unit cell.

a, t, and c are the primitive translation vector and a, s, y are the

angles between them.

- .8 -

APPENDIX A

Symmetry Properties and Invariant Functions for Ising Spins on Hypercube Unit Cell.

The location of the z (=2d) Ising spins on a hypercube unit cell are shown

in Fig. Al, Fig. A2 and Fig. A3 for d=l, 2 and 3 respectively. We shall use

Ki to denote the two-spin coupling constant between o and c which appears

on the right hand side of Eq. (5).

Now we consider the following possible systems and write down the invariant

functions on the right hand side of Eq. (5) for them.

Sl d=l

g1 = 0I+a22

g 2 =o~o 2 .(Al)92 = ala2" AI

They satisfy the relation:

g2 = 12 -1 (A2)

S2 d=2 u(ol,--o 4 ) has permutation symmetry.

gl l+2+a3+a4'

g2 = ala2+02a3+03a4+04al+03+"204"

g3 ala2a3+a2a3 4 +a3a 4 a1 +a4al02"

g4 = la2a304 (A3)

They satisfy the relations

g2 = gl 2 -2,

= -"g 1 /3,

94 = [g1 g3 -292] /4. (A4)

S3 d=2 u(cl,--o4 ) has rotation, reflection and inversion

symmetries. K12=K23=K34=K41, K1 3=K2 4

but K12 K 13 , For the unit cell of Fig. A2,1 -"I 161,

= al+a23+04

g2=a a3+aa 4,92 Y~3 '2 49

g3 = ala+aZ +a a4+a~a

94 =10a 3+0203 4+o3 04 a1 +a4al a2 '

g5 = 01a2a3a4 , (A5)

They satisfy the relations

9 3 = g12/2 - g2 -2,

94 gg2 - g1

g 5 = g22 /2 -1. (A6)

S4 d=2 U(al,--o4) has reflection and inversion symnmetries.

K12=K3 4, K2 3 =K4 1, K1 2 7K2 3.

K 3=K24. For the unit cell of Fig. A2, a , oL=90 °

g I = 0+02+03+04"

92 = '0 3+02a4$g3 02o +0

9= 102 3041

g4 'c°1 4 ' '2'3,

95 = '1'2'3 + '2'3'4 + 030401 + 4a12,

(07)g6 = 01020304

They satisfy the relations:

g4 2= g1 2/2-g 2 -9 3 -2,

g5 =g1 g2-1

g 2_.(A8)

.............

-20-

S5 d=2 u(al,--a4 ) has inversion symmetry.

K12=K23 =K3 4=K41' K.3 13 K2 4 " for the unit cell of Fig. A2,

lal = ItI, a 60.0

g=al +a3,B • g2=o2+04s

g4 =o2 o4 ,

g 5=la2+a2 a3+a3a4 +a4a 1 ,

g6=01 02'3+0 1o3a4,

g7=1 l 2' 4+o2a3 04,

g8=aul 2a3a4 . (A9)

They satisfy the relations:

93=g12/2-1,

94=92 2/2_1,

95=9192'

g6 =g2g 3'

g7=g1g4,g8=g3g4 (AlO)

98.'939

-21-

S6 d=3 u(al,--a8 ) has permutation symmetry.

Let g. denote the sum of all possible products of i different

a's. g, for i=1,..8 are the basic invariants of the permutation group.

It is easy to show that g. satisfy for relation:

g1 =ol+0 +--+a8 , (All)

gi+l=[glgi - (8-i+l) gi-l ] /(i.+), (A12)

for i=l,...7. Thus each gi may be expressed in terms of gl.

S7 d=3 U(ol,--as) of Eq. (5) has symmetry properties of the

simple cubic unit cell, i.e. 0h point group.

g1 =o l+a2+a3+--+a8 ,

g2=l02+--- (12 terms),

93=a.7 +--- (4 terms),

g4=alo3+--- (12 terms),

g5= 1a 2a3a4+--- (6 terms),

g612038 +- -- (24 terms),

g7="1 2'306+--- (8 terms),

g8:al020307- (24 terms),

g9='1 2'708+--- (6 terms),

gl a3a608 + -- - (2 terms),

9 l=a a2G3a4050 6+---(12 terms),

g9=a a20304ay7+---(.12 terms),

gl3= I020305a708+---(4 terms),

g 1 4 1 a2 3 40 50 6a 7 0l term),

g i=o203+-- (24 terms),

g 1 6 1 a 2 7 + - - -20 (24 terms),

910l a306+- - (8 terms),

g18=01 c a4 cS+- '- (24 terms),

-22-

gl 9 2- 1a2c3 '7'+--" (24 terms),

g20 a1a3'4'6'8+--- (8 terms),

g21-ala 2a3a4a~a 7+--- (8 terms), (A13)

Where, in order to save space, only one typical term for each invariant is

written. It is easy to show that the g4of Eq. (A13) are related to each other

by the equations:

.4 = g12.-g 2-g 3,

g6 g3 g4 -2g 2 ,

9g8 -g2 g3 -2g 4 .

999 "2 gg3/2-2,

Sg 5=g2 2/2-6-2g4 -g8 -g 9

g7=(j 2g4-4g 2-6g 3 -2g6 )/3,

glo g1 - -2-

g1' =g2g14'912:-g3gl4 '

g1 3 g9 14,

g9 6=gl g3- g I,

g1 5= (g g2 -3g, -g1 6)/2,

g 7 =(g g4-3g, -g15 -g16 ) /3,

g18 =g14g1 5'

Wg1 9=g 14g (A14)6,

g2Cg 1 4g17,

g21:-g1491

-23-

S8 d=3 u(alo o2 --a8) of Eq (5) has symmetry properties of the tetragonal

unit cell. For the unit cell of Fig. A3, M = I # I, = = y =90.

g 1 I +0 2+ 3+a4+ 5+a 6+a7 +0 8,

Sg2=o o2a 2 3+C 3 4 4 1 +a506+c6"7+78+08'l

g3=la 5+ 2o6+ 3o 7+0408,

g4 1 07+0208+03 5+04 6,

g5= ] 3+o2 4 +05 7 +0608s

, g 6=OlO6a +Oaa2 +a a6+a3a8+a4(,.a 8aa

96 ='1 ' 6 2 5 2 7 3 6 0308+0407 1 8 4 5,g7=o1020304+05060708

g8 =a a6o5+a2a

3 a7 6 +o3 a4a8o7+al 040508,

ag9 =oIa 2a3a8+--- (8 terms)

g10='1' 2'6'8+--- (16 terms)

g1l='la 2'3a6+-- - (8 terms)

g12=la'2o 3o 7 +- - - (16 terms)

g1 3=la'2' 6a7+ (8 terms)

g14=-a 2a7a8+--- (4 terms)

g. 5 a1a +a oa8,

g1 6=01 3o6o8+ 2a4 0507,

g17= 1 2 a30 4a5" 6+- - - (8 terms),

918='l°2'3''6'7+ (4 terms),

g19='1' 2'3a5 7 8+-- (4 terms),

g20='1'2a3a4'5'7+--- (4 terms),

g21='1'2'4'5a6'7+--- (8 terms),

g2 2 =aI a20 304 05a 6a7 a8 ,

g23=a"aIa3 +--- (8 terms),

924-':' 1 2'6+--- (16 terms ),

925=a12 07 +--- (16 terms),

g26=al c5c3+--- (8 terms),

-24-

g27=1'36 - - (8 terms),

g28= I a2'3 'a5+ --- (8 terms),

g29=a'1 2'3'5a6+--- (16 terms),

g30 =aa 3 4 5a 6+- - - (16 terms),

g3 1=a'1a 2 a3 5a7+ (8 terms),

g32 a'l2a3 6a8+--- (8 terms),

g 33 oa 2a3o 4a5 a 6a 7+- - - (8 terms), (A15)

Where in order to save space, only one typical term for some invariants is

written. Invariants of Eq. (A15) satisfy the relations:

g6= g1

2/2-g2-g3-g4 -g5

g9= g4g5-2g3

g10 =g4 96 -292Sgl =g3g5-2g4

g1 2=g293-2g69g13=g 3g4-2g524 2 2 2 26

g16 (g52+ 6 -4-g13-2g5-g4 2g 2 )/6

g7=(g2 +4g16-g 62 )/4

g1 5=g5 2/2-2-g7-g1 6

g8 =g3 2/2-2-g1 5

g14=g 4 2/2-2-gi5

22=97/2-

.g9 22'i- 15 for i=17,---21,

g23=g1 (2g2-g3-g4+2g5-g6-2)/6

g2 4=29 2 3 -g 1 (g2 +96 -2g 5+2g 4 )/2,

g2 5=2g 2 3 -g1 (g2 -g 6 429 5-2g 4 )/2,

g26=-2g 23 g91 (g2-g6+2g5-2)/2,

929=g23-g1 (g2-g6 )/2,

-25-

gi-g22gi-5 for i 28, 29, ---32,

g33=g22g1 (AI6)

S9 d=3 U(l,-c 8) has the symmetry properties of the primitive unit cell for

the FCC lattice. Thelongest diagonal is in the O , a7 direction. For the

unit cell of Fig. A3, j-l = l" = Cj*, a = 0= o .

gl =01+Ca7

g2=a2+c4+G5+03+c6+0,

j9 g=a2a +CT4a5+a2a5+a a6+a8a+a a83 2 4 52 5 3 38 6 8,

4'9 g=l('3+a6+8) +'7 (a2+a4+5),

g5=a2a8 +o3a5+a4 6,

96=g17,

g7=a I(o 2 +o4 +a5)+ 7 (' 3 +o6 +cS),

g8 =0 2 03 +02 a6 +0 506+a5a8+c48+03a4,

g9='1°2a3'4+-- - (6 terms),

910 = 1 2 0 2-3-8+ - (12 terms),

gl l =o1 2047+- - - (6 terms),

912=(2a3a68+ --- (6 terms),

g13= 1 203'6+- -- (6 terms),

g14=o1'2"4'5+a3o6 7 8,

915='1 2'3'7 (6 terms),

916='2'3'4'8+--- (6 terms),

g17= 2°3'7 8+--- (12 terms),

Sg 18=1oa 2a7a8+a 1 40607+ 1050307,

g19=

3'4'5'6+'20305 8 4I2G604 08,

g20='1 30608+a2a405a7,

921='1'2"304057+ -- (6 terms),

g2 2 =o'1 2'3 4'6a8+ - -- (6 terms),

g23=al2a3a50708+ --- (3 terms)

924:-2'3'4'5'6a8,

925=al2345a6+ (6 terms)

g26 '1a2030406'7+ (6 terms)

927=al02030405060708

928=al2a4+ (6 terms),

Sg29=oa 2oa6+_(2_ers)9(6 terms),

g30=al 2o6 +--- (12 terms),

9301 67(a2+(4+a5+a3+a6+ r),

- g32=o o082a 45

36l 0 -2 5 (6 terms),

9331=a 20468+ -- -8 (12 terms),

9390a0304+ -6+ (6 terms),

g3=(3Yas+a Ca Ca5,

g3 6=l2 40 5+--- (6 terms)

g37=a1037 4007+ (6 terms)

:%- g38=ala203'4'6 - - (12 terms),

939=a2a3'4'5'6 + -- - (6 terms),

• g40='1a3'4'5'6+ - (6 terms),

941= ola 3'4c5a7+ - -- (12 terms),

942='1 2'3'6'8+--- (6 terms),

g4 3 ='1'2'4'507+01y3a608a7,

94 4 =' 1 ' 2 a304 a5 ' 6 ' 8 +02 o 3 04 0 5 6 a7 a8,

945='ia 2a3a450607 + - - - (6 terms), (A17)

Where in order to save space, only one typical term for some invariants is

written. Invariants of Eq(A15) satisfy the relations:

g6=g12/2-1

97= 1 g2-g4

g8 =g22 /2-3-g 3 -g 5

9lO= 949 5-g7

gl 2"Ag- 2g8

g1(gg-2g7 g g 0)/3

2_2

91Cg3 /23 g3-gl9 '

A g 17 (g7+g8) (90+6)-9(3+94 )-g15-916 '

9 =g7g8/2-gO-1 7/21

920"1(g394 -2g4-gl7-g9 )/39

g 9g20212-1 ,

g~~g2 g~18 for i=21,22, .. .26,

g30'*"gl 8 ,

7 931'"2g6 '

g32=g1g5,933"g2g5-g2 '

9309(2g4 -3g, -g30-g32 )/2

g28=g1 g3-g34

9W 29 g8-2 2- 33-9:28

g1=g297 5i- for i 36, 37, . ..45. (AI8)

-28-

APPENDIX B

Details of the Calculation

In this Appendix, we outline some techniques used in our computer

program to obtain the numerical results. Our computer program can be applied

to all systems listed in Appendix A with space dimension and symmetry properties

as input parameters. The important programming techniques are as follows:

1. Generation of all possible spin configurations. Let z denote the number

of spins on a hypercube unit cell of the d dimensional lattice and NCF denote

the number of all possible configuration of such z spins al,--az , then

Z=2d, NCF=2Z. Let I be an integer running from o to NCF-l. We express I

as a z digit binary number and let I correspond to spin up (+1) and 0

correspond spin down (-l), then each integer from 0 to NCF-l correspond

to one spin cofiguration where integer 0 corresponds to all spin down

configuration and NCF-l correspond to all spin up configuration which is

the ground state of the ferromagnetic system. Thus, with d as input parameter,

we can generate all possible spin configurations.

2. Classification of spin configurations. From Table Al, it is clear that

for space dimension d>2 there are some systems with different symmetry

properties. Thus, for each spin configuration generated in the manner

described above, we calculate the values of the basic invariants based

on the parameter "sym" and then classify these spin configurations into

different degeneracy groups such that the configurations in the same

group have the same values for all basic invariants (and hence cell

potential) and in different groups have at least one different value. We

label the degeneracy group in such a way that the ground state configuration,

which is all spin up configuration for ferromagnetic system, belongs to the

final group. For each group, we also store the degeneracy Di (ie the number

I

-30-

References

1. L. P. Kadanoff, A. Houghton, and M. C. Yalabik, J. Stat. Phys. (N.Y.)

14 (1976), 171-203.

2. S. L. Katz, and J. D. Gunton, Phys. Rev. B (N.Y.) 16 (1977), 2163-2167.

3. Y. M. Shih, et. al., Phys. Rev. B. (N.Y.) 19 (1979), 529 - 532

4. M. N. Barber, J. Computational Phys. (Cal.) 34 (1980), 414-434.

5. C. K. Hu, et. al. (unpublished).

6. Y. M. Shih, et. al., Phys. Rev. B (N.Y.) 21 (1980), 299-303.

7. M. C. Chang, and A. Houghton, Phys. Rev. B. (N.Y.) 21 (1980), 1881-1892.

8. M. C. Chang, and A. Houghton, Phys. Rev. Lett. (N.Y.) 44 (1980), 785-788.

9. L. Onsager, Phys. Rev. (N.Y.) 65 (1944), 117.

10. C. N. Yang, Phys. Rev. (N.Y.) 85 (1952), 808.

11. J. W. Essam and M. E. Fisher, J. Chem. Phys. (N.Y.) 38 (1963), 802.

12. C. Domb, in "Phase Transitions and Critical Pheonomena Vol. 3" (C. Domb andM. S. Green, Eds.), p 357, Academic Press, London/New York, 1974.

13. H. E. Stanley, "Introduction to Phase Transitions and Cirtical Phenomena",p 131, Oxford University Press, New York/Oxford, 1971.

14. C. K. Hu and P. Kleban (unpublished).

-29-

of configurations in the group) and the configurations belonging to

this group. We will use t to denote the vector whose elements are

degeneracy, ie t=(D I,D2 ,---D t+l

3. Calculation of the column vector.T-lt.

In this section, we will derive formulas for the vector T'It which

appears at the end of Eq. (17) and (18). From Eq. (5b), it is easy

to show that____C F Di. (B)

Thus, Eq (13) may be rewritten as:

- 1/za/NCF _ ui(a)Di+ln2) (B2a)

for T>Tc and

f(a)= _ I/Za ( +l + in 2)

for T<Tc, where u() L+l is the ground state cell potential. We now

* take the first derivative of Eq (B2) with respect to q. By the chain

rule, it is easy to show that:

6f 61 .... rn-) 1 I/Z o' (B3)4 6q "--) a-1 )

where

0o= (D 1,$D 2,--- D +l - (B4a'

for T>T and

C o:(0 ,0 , --- , , 1) (B4b'

for T<T c. Comiparing Eq (17) and (B3), it is obvious that T-lt of

Eq (17) is jsut 0 of Eq (B3) and (B4).

NNW=.ii

C~s

040

co

0-

P-4

-

.tCD

c2.2 .4 .00801 01 2 ' L

T0(;

-

m

* CD

C3 0-9. 0

11;

0~ .0o8 . .2 ' i

xx

0 x

xx

C3

'=3 x

0

C3

0.00.00.00.01'0 12x14

0

C3,

0.0x

C3

0%J

0r

0~

.2 .006008 ' u .014

*1K

0-

co xxx

Il o xt, x

xx

C) x

x

Ln x

0

0

xx.lxx

0

E- xxX

90. 12 0 1.16 0 1 20 0 1.24 . 28 0 1.32 0 1.36

K

C,

P 0y

.1C0C3

0;

9 x

o3 xC; xW- x

0x

x

0 30 x

CS

xC) x

0, 2-X X x x

cb 2 0'.16 0. 20 0 1 24 0' .28 0 1.32 0 1.36

o

0

'-4

co

CD °N x-0x

o x

(3

1.

SF_•0'- ×

0x

0 r

o --.. . _. ... . . .. .I . . . . ..x I

o x= , I

cu

0303

03

Co x

x0x

C3

xX

C3 0

xx

CoC0

0.6 02 02 5 . 2 '.3

C4 0"3

a1 a5

C6

11

0

1