Exact solution of the 2D « brickwork » Ising model with second neighbour interactions in the horizontal direction

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Exact solution of the 2D “ brickwork ” Ising model with second neighbour interactions in the horizontal direction

R. Bidaux, L. de Seze

To cite this version:

R. Bidaux, L. de Seze. Exact solution of the 2D “ brickwork ” Ising model with second neigh- bour interactions in the horizontal direction. Journal de Physique, 1981, 42 (3), pp.371-379.

�10.1051/jphys:01981004203037100�. �jpa-00209020�

371

LE JOURNAL ^DE PHYSIQUE

Exact solution of the 2D « brickwork » Ising ^model

with second neighbour interactions in the horizontal direction

R. Bidaux and L. de Seze

DPh-G/PSRM, ^CEN Saclay, ^{B.P. N°} 2, ⁹¹¹⁹⁰ Gif-sur-Yvette, ^France

(Reçu ^{le 21 août} 1980, accepté le 13 novembre 1980)

Résumé.

Boehm et Bak [11] ^ont analysé, ^dans l’approximation de champ moléculaire, ^{un modèle} qui ^conduit

à un diagramme ^de phases ^où figurent des transitions multiples ^entre phases commensurables. Il se trouve qu’un

modèle d’Ising bidimensionnel, présentant ^{les mêmes} propriétés ^en champ moléculaire, peut être résolu exactement.

Ce modèle est construit sur un réseau en mur de briques, ^{et sa} fonction de partition ^est calculée dans le cadre des

hypothèses suivantes :

i) ^Dans la direction verticale, ^les premiers ^{voisins sont} couplés par des interactions ferromagnétiques ^{2 J’} (J’ > 0).

ii) ^{Dans la direction} horizontale, ^les premiers ^{voisins sont} couplés par des interactions ferromagnétiques ^J (J > 0)

et les seconds voisins sont couplés par des interactions arbitraires J2.

Les résultats sont discutés en fonction du paramètre J2/J. ^{On trouve en} particulier que :

1) ^Pour J2/J > - 1/2, ^le système ^s’ordonne ferromagnétiquement ^{à une} température critique Tc ⁼ 1/kB 03B2c

définie par

tanh 2 03B2c ^{J’. sinh} 2 03B2c J. exp 4 03B2c J2 ^{= 1.}

2) ^Pour J2/J ~ 2014 1/2, ^aucune phase ^ordonnée n’apparaît, ^{même à} température nulle. Les états fondamentaux et leur dégénérescence ^sont examinés. Les résultats obtenus sont comparés ^{avec ceux} donnés par traitement de

champ moléculaire qui prédit l’existence d’un escalier du diable, ^{et avec ceux} fournis par ^une étude Monte-Carlo récente pour le modèle ANNNI bidimensionnel [13].

Abstract.

Boehm and Bak [11] ^have analysed, within the mean field approximation, ^a model which leads to a

phase diagram including multiple phase transitions between commensurate phases. ^It happens ^{that an} exactly

soluble two-dimensional Ising ^model exhibiting ^{similar mean field} features, is available. This model is built on a

brickwork lattice, ^{and its exact} partition ^{function is obtained} within the following assumptions : i) ^{In the vertical} direction, ^nearest neighbours ^are coupled by ferromagnetic interactions 2 J’ (J’ > 0).

ii) ^{In the} horizontal direction, ^nearest neighbours ^are coupled by ferromagnetic interactions J (J > 0) ^{while next-}

nearest neighbours ^are coupled by arbitrary interactions J2.

Results are discussed with respect ^{to the value of the ratio} J2/J. It is found in particular ^{that :}

1) ^For J2/J > 2014 1/2, ^the system ^orders ferromagnetically ^{at a critical} temperature Tc ⁼ 1/kB 03B2c ^defined by

tanh 2 03B2c ^{J’. sinh} 2 03B2c J. exp 4 03B2c J2 ^{= 1.}

2) ^For J2/J ~ - 1/2, ^no ordering occurs even at zero temperature. ^{Ground states and their} degeneracy ^are

examined. These results are compared with those obtained for the same system ^{in the mean field} approximation,

which predicts the existence of a devil’s staircase behaviour for the periodicity ^versus temperature curve, and with those given ^{for the} two-dimensional ANNNI model by ^{a recent} Monte-Carlo study [13].

J. Physique ⁴² (1981) ^{371-379 MARS} 1981,

Classification

Physics ^Abstracts

75.10H

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01981004203037100

372

1. Introduction.

Increasing interest has focused

recently ^on Ising ^{models with} competing interactions,

in connection with other attempts ^{to obtain a better} theoretical understanding ^of ill-condensed magnetic

systems. ^When competing interactions are suitably introduced, ^the corresponding magnetic system ^beco-

mes frustrated, i.e. it is impossible ^{to find a} configura-

tion which enables each pair ^of interacting spins ^to

minimize separately its energy. As ^a result one can

hopefully (and ^one does) ^observe phase diagrams ^with

a richer variety ^{of order} parameters.

Among frustrated systems, ^those characterized

by ^a periodic distribution of the interactions deserve

special ^{interest due to the fact that,} according ^{to the} dimensionality and the range of interactions, ^exact solutions _may be obtained for thé partition ^function

at zero applied field and for the pair correlations [1-3J.

Competition between interactions can, ^{as a} rough description, ^be introduced i) ^with [4-7] ^or ii) ^without [9]

calling upon interactions involving further than nearest neighbours (the degree ^of neighbourhood,

when defined by comparing ^{mutual distances between} points ^of space ^at dimensions higher ^{than one, has}

an ambiguous definition since these distances _may have their hierarchy ^modified by homotopic ^defor-

mations of the lattice). ^Exact solutions at d ⁼ 1 (i))

and d ⁼ 2 (i) ^and ü)) have been obtained [4, 6], ⁱⁿ

which the seed of pathological features is already

present.

A special version of case i), ^{the axial} next-nearest

neighbour Ising (ANNNI) model, ^{first invented} by

Elliott [7], ^{has known a recent revival} owing ^{to its} display ^{of an} extremely ^rich phase diagram including

modulated magnetic structures and incommensurate,

commensurate transitions. This model has been

examined in the case d ⁼ 3 [10, 11 ] using ^{mean field} approximation ^and Landau-Ginzburg ^free _energy expansions, then studied for d > 2 by ^{means of}

Monte-Carlo methods [12] ^and low-temperature expansions [8].

A recent study [13] of the ANNNI model for d ⁼ 2

(uniaxial model) making ^use primarily ^{of Monte-}

Carlo data has been performed. ^This model consists of Ising spins, ^{s = ± 1, on a} square lattice with ferro-

magnetic interactions J > 0 between nearest neigh-

Fig. 1.

Sketch of the phase diagram ^{for the} uniaxial model accord-

ing ^{to W.} Selke and M. Fisher.

bours and arbitrary interactions J2 ^{between next-}

nearest neighbours along ^one direction. The cor-

responding phase diagram ^obtained by the authors is sketched in figure ^1.

Although the uniaxial mode( quite unfortunately,

cannot be solved exactly, ^at least with the techniques

available to date, ^{it turns out} that another mode(

apparently very similar, ^{can be solved} exactly. ^{In the}

next section, ^we describe this model and determine its

partition ^{function. In the third} section, ^the ground

states of the system will be examined. In the fourth and last section, the similarities and discrepancies

between our model and the uniaxial model will be discussed.

2. The uniaxial brickwork lattice model.

2.1 DESCRIPTION.

The uniaxial brickwork lattice model is pictured ⁱⁿ figure ^{2. The} spins (s ^{= ±} 1) ^located

on the lattice interact according ^{to the} following ^{rules :} i) ^In the vertical direction, ^{connected nearest} neighbours ^are coupled by ferromagnetic ^inter-

actions 2 J’ (J’ > 0).

ii) In the horizontal direction, ^nearest neighbours

are coupled by ferromagnetic interactions J (J > 0)

while next-nearest neighbours ^are coupled by ^arbi- trary interactions J2.

As can be seen from figure 2, bonds between inter-

acting spins ^can be drawn without crossing points ;

this is a necessary condition for being ^{able to evaluate}

the partition ^function, which fails to be met by ^the

uniaxial model.

Fig. ^2.

^The uniaxial brickwork lattice.

The partition function of the uniaxial brickwork lattice model (referred ^{to as UBL model hereafter}

for the sake of simplicity) ^{will be now evaluated} using

the Pfaffian technique. ^{To do so, we start from a} rectangular lattice with periodic interactions described in figure ^{3a. The} corresponding partition ^function

can be determined in terms of the four interactions X, J, J2, J’ and ^the temperature, ^{but its} calculation is too lengthy and cumbersome to be reproduced here,

since the analytical expansion ^{of a 16 x} 16 determi-

nant is involved; ^{the main} steps ^{and results are}

indicated in the Appendix. ^{If now} interaction X is

assumed positive ^{and tends to} infinity, ^each pair ^of

spins ^connected by ^{a X bond} will be constraint to

Fig. ^3.

Step by step transformation of the primitive rectangular

lattice into the uniaxial brickwork lattice.

remain parallel and will be equivalent ^{to a} single,

two-state fictitious spin. ^This condensation of the X bonds amounts to be left with the lattice shown in

figure ^3b, for which the number of spins is half the number of spins ^{in the} original lattice of figure ^3a.

Figure ^{3c is a} simple transcription ^of figure ^{3b which}

shows how effective bonds 2 J’ are created. After

flattening ^each strip ^of zigzag ^{J bonds one} finally

obtains in figure 3d the UBL model just ^described except ^{for an} interchange ^{of the} crystallographic

axes.

2.2 PARTITION FUNCTION.

The partition ^function per spin of the UBL model is obtained from _expres- sions (A. 6) ^and (A. 7) ^{of the} Appendix. ^One finally ^{finds :}

where

and A, B, C, ^{D are defined as follows :}

2. 3 CRITICAL CONDITIONS.

One now looks for the zeroes of A(0, (p). ^{As a first} insight ^{one notes that}

so that a critical temperature corresponding ^{to a} paramagnetic-ferromagnetic transition (0 = cp = 0) may be temporarily registered provided ^the equation A + B + C + D = 0 has a solution.

Complete inspection ^of A(0, (p) ^{shows that no other}

solution can be obtained ; ⁱⁿ particular, ^the interaction

-

and temperature

dependent angular ^solutions of (DAID9 ⁼ o ; ôA/60 ⁼ 0) ^never yield A ^{= 0. Since A}

is a continuous function with respect ^to fl, 0, cp, ^no

singularity ^{in the} partition ^function except ^that arising from (0 = 0; cp = 0; A + B + C + D = 0)

can take place. ^From (2.3) ^the transition condition

can be defined by :

As can easily ^{be seen,} given ^three interactions J, JZ

and J’ satisfying ^the assumptions ^listed in section 2. 1,

no solution of (2.4) ^can be found if J2 ^- 2 J, ^i.e.

when the ground ^{state of an isolated} horizontal chain of spins ^is a 22 ) antiphase. ^If JZ > - 2 J ^(which

conditions ^a ferromagnetic ground ^{state for} the isolat- ed horizontal chain), ^a unique solution fi,,, ^{is found.}

For J2 = - 2 ^J, Pc ^{- + oo can be} considered as a limiting solution and expresses the fact that

(J 21J ^{= -} 1/2, ^{T =} 0) ^will play ^{the role of a multi-}

critical point ^{in the} phase diagram.

Figure ⁴ displays ^{a set} of transition curves in the

(J21J; kB T/J ) coordinates, corresponding ^{to succes-}

sive given values of the ratio J’/J. ^One may ^note that several standard transition conditions pertaining

to familiar models can be obtained from (2.4) by

374

Fig. ^4.

Transition curves of the UBL model.

considering peculiar ^{choices of the} interactions : for example, ^if JZ ^{= 0 the} critical condition tanh 2 j8e J’. sinh 2 Pc J ^{= 1} for the usual brickwork

(honeycomb) lattice with three interactions J, ^{J and}

2 J’ is recovered ; ^on the other hand, by letting ^J’

go to + oo, the UBL model maps ^{onto a} triangular

lattice with three interactions J, ^{J and 2} J2 ^{for which}

the critical condition is given by

3. Ground states of the UBL model.

Due to the nature of the vertical coupling ^{in the brickwork} lattice, ^the minimization of the total energy ^can be achieved in two successive steps which consist in

minimizing separately the self energy of each hori- zontal chain, ^then ensuring ^{that the} corresponding respective ground ^{states of two} adjacent ^horizontal

chains are compatible, ^i.e. any ^two spins coupled by ^a

vertical bond 2 J’ should be parallel.

3. 1 J2 ^{= -} 1/2 J.

^{In this case,} ground ^states

of an isolated horizontal chain are characterized

by ^the interdiction, ^for any block of three consecutive

spins, ^to adopt ^a (1 1 1) ^{or a} (i 1 J,) configuration.

When condition JZ ^{= -} 2 J ^{is taken into} account, ^the partition ^function (2.1) ^{of the UBL model becomes :}

with

As mentioned previously, ^no transition is found. In the low temperature ^limit (f3 --+ ⁺ co), ^{one can} easily ^show

that

and a residual ^zero point entropy per spin

is found, ^which corresponds ^to half that of the _pure antiferromagnetic triangular ^lattice.

The situation at T ⁼ 0 can be better understood

by retuming ^{to the} description ^{of the lattice as} given

in figure 3c. This is done in figure ^{5a where} only

horizontal zigzag strips have been displayed (the

choice of the horizontal direction is the same as in

figure ² which is taken as a reference for the brickwork

construction). For obvious reasons, ^a ground ^state

of the UBL model will be obtained if and only ^if

the following conditions are simultaneously satisfied :

i) ^{no vertical 2 J’ bond should be} frustrated ; ii) ^every elementary triangular plaquette should have ^one and

only ^one frustrated bond (although each of these

plaquettes ^{has two J bonds and one} ( - J/2) bond,

the frustration _{energy is} identical for both types ^of bonds since a frustrated J bond is involved in two

adjacent plaquettes ^{whereas a} frustrated (- J/2)

bond is involved in only ^on plaquette). ^{Therefore the} ground ^{states of the UBL model, when} J2 ^{= -} J/2,

can be deduced by ^a one-to-one correspondence,

from those of the lattice obtained by condensing ^each

of the 2 J’ bonds into a point while each pair of parallel

spins previously ^{connected via a} 2 J’ bond becomes

Fig. ^5.

J 2/J ^{= -} 1/2. a) ^A horizontal chain of the UBL model with hanging ^vertical bonds, ^after returning ^{to the} zigzag repre- sentation ; ^an elementary frustrated triangle ; ^{contours of} adjacent elementary frustrated triangles. b) ^Dimer representation ^{of the} ferromagnetic ground ^state. c) ^Dimer representation ^of a 22 ) antiphase ground ^state. d) ^A ground ^{state with wave} vector n/3

along ^the zigzag path (2 n/3 along ^the horizontal direction) : heavy

circles correspond ^{to zero local field and} respective spins ^{can choose} arbitrary directions ; ^{the same} one, completed, in dimer represen- tation.

a single fictitious spin. ^{One is then left with the} problem

of the ground ^{states of a} triangular ^lattice containing

half the number of spins ^{of the} original lattice, ^and

characterized by interactions (J, J, - J ) ; ^this problem

in tum reduces to that of the isotropic antiferroma-

gnetic triangular ^{net with} interactions (- J, - J,

-

J) by ^{means of a} Mattis transform. Result (3.2)

is a trivial consequence of this equivalence.

A few typical ground ^{states limited to a} zigzag strip

are shown in figures, 5b, ^5c, 5d, ^{where the associated}

dimer equivalence ^{has been used. In} figure 5d, special

attention has been paid ^{to a} spin distribution ^corres-

ponding ^{to a wave vector} n/3 propagating along ^the zigzag ^{chain of} spins ^connected by ^{J bonds, which}

amounts to a wave vector qx = 2 n/3 ^{in the horizontal} direction; ^{the reason} for this interest is because,

in expression (3.2), ^the argument ^{of the} logarithm

vanishes for 0 ⁼ 0, cp ⁼ n/3 and, ^{as well} known,

the corresponding ^{wave vector will show} up if long

range correlations at T ⁼ 0 are to be examined, ^as

turns out in the ^case of the isotropic antiferroma-

gnetic triangular ^lattice.

3.2 J2 ^- 1/2 ^J.

^{In this} case, ^an ideal ground

state of an isolated horizontal chain is a 22 ) antiphase, with eventual corrections (walls) ^{due to}

the choice of boundary conditions and the number of

spins ^on the chain. According ^{to this choice the num-}

ber of ground ^states may be either equal ^{to 4} (free ^ends

or suitable cyclic conditions) ^or of the order of the

length n ^{of the chain. As a} consequence, and owing ^to

the markovian procedure ^for constructing ^the ground

states of the complete ^lattice by piling up successively compatible ground ^{states of} adjacent ^horizontal chains, ^an upper bound to the number of ground

states of the complete lattice is n"’ where m is the number of horizontal chains. The residual entropy

per spin ^{therefore vanishes, as can be seen} directly

From (2.1) when the low temperature ^{limit is examined.}

,

In order to avoid having ^{to bother with} edge ^effects

in the following discussion, ^{we assume} that, ^for any

ground ^{state of the} system, ^{it is almost sure} (in ^the probabilistic meaning) that any group of four ^conse- cutive spins ^{on a row will} adopt ^one of the four follow-

ing configurations :

Suppose the first spin ^on the left of the _group is located at the top of a 2 J ‘ bond, ^{in a} given ^{row which}

will be named row 1; ^the downward transfer matrix from row 1 to row 2 below is

and the transfer matrix from row 2 to row 3 below is

so that the transfer matrix from ^row 1 to row 3 is

with the following properties :

376

As an immediate consequence, the number of distinct ground ^states nGS for a lattice with m rows

(m even) ^and cyclic conditions between the first and the mth row is given by

The properties of the transfer matrices Vl, V2

and W show clearly that, starting ^{with a} given ^confi- guration ^{in row} 1, ^all configurations will show _up with equal proportions ^{in row 3 and the} following

rows (Fig. 6). Therefore the quantity SR SR, ) averaged ^over ground states, when locations R and R’

do not belong ^to adjacent ^{rows, is} strictly ^zero.

Fig. ^6.

J2/J ^- 1/2 : ^four possible ground ^states, with initial

configuration (+ + - -) imposed ^on the first row.

It is a striking feature of the UBL model that for

J2 ^- 1/2 J ^{and T =} 0, correlations in a row

(where ^{a is} the lattice spacing) ^are given by

so that the correlation length is infinité in the hori- zontal direction, while it is limited to one lattice

spacing in the vertical direction. Thus the UBL model is not ordered at T ⁼ 0 if J2 ^- 1/2 ^{J, whence we}

better understand the discrepancies with the uni- axial model to be discussed now.

4. Comparison with the uniaxial model [ 13].

^From

the point ^{of view of} phase transitions, the thermo-

dynamic properties of the UBL model turn out (quite

not surprisingly) ^{to be less} rich than those of the

complete uniaxial model. This is due to the lack of

an ordered structure at any temperature, ^{even at}

T ⁼ 0, ^{for the UBL} model, ^when J2 - 1/2 ^J.

As far as predictions derived from mean field

techniques ^are concerned, they become the more

unreliable as the dimension is lower. Figure ^{8 recalls}

the mean field phase diagram of the UBL model;

the fine structure of the modulated phase ^{has not}

been specified ^{since the exact results} have shown that this is ^a mock phase. ^{It is} worthy ^{of note that the}

formalism of solitons and phasons, ^{as described} by

Bak and von Boehm, adapted ^{to the case of dimen-}

sion d ⁼ 2, applies ^{as well to the case} of the uniaxial model as to the case of the UBL model ; ^{now at last}

the conclusions to which such a formalism could lead in discussing ^the phase diagram presented by

Selke and Fisher (Fig. 1) ^{should not deserve an}

unreserved adherence, taking ^into consideration its

erroneous prédictions ^{for thé} phase diagram ^{of the}

UBL model.

As a set-off to these rather negative conclusions,

the UBL model could be used as a good testing ^stand

in the Monte-Carlo methods, ^{in order to ascertain}

that no modulation in the horizontal direction can

be stabilized, ^at any temperature, by ^a particular

choice of size or of boundary conditions for the system. ^{Such a} preliminary checking, ^if proving positive, ^could only give ^more weight ^to the results obtained by Monte-Carlo simulations for the complete

uniaxial model (1).

APPENDIX

Partition function of the UBL model.

The stan- dard Pfaffian procedure [16] ^is applied ^{to the lattice}

shown in figure 3a, ^which will be referred to as pri-

mitive (and ^indexed by P) ^{in this} Appendix ^when

necessary. Although ^{a more direct method can} certainly be used if the partition function of the sole UBL model is needed, ^{we found more} convenient to

deal with the primitive ^{lattice which has the rectan-} gular symmetry, ^{because it leads to several} interesting particular ^{cases after} specific choices of the interac-

tions, ^{one of them} being ^{the UBL model.}

The primitive ^lattice (n ^{rows and m} columns,

mn ⁼ N), ^is helically ^{wound on a torus. The unit}

cell of interactions (Fig. 7) ^recurs N/4 ^times along

(1) ^This point ^{of view has received an} unexpected support ^from

a preprint ^{of a work to be} published by ^J. Villain, where the author

daims that the Lifshitz point ^{of the} two-dimensional ANNNI

model should take place ^{at zero} temperature, ^{so that the} ferro-

magnetic-sinusoidal transition deduced from computer simulation

should not exist.

Fig. ^7.

Unit cell of interactions in the primitive ^lattice.

the helix ; ^{m and n are chosen so that} (m - 2)/4 ^and n/2 ^{should be} integers. ^The partition ^function Z’ ^for

the N spins ^{of the} primitive ^{lattice is} given by

Fig. ^8.

^{Sketch of the mean field} phase diagram ^{for the UBL model.}

Fig. ^9.

Structure of the matrix M. (Transposition operation ^is represented by ^a bar.)

378

where D(M) ^{is the} anti-symmetric determinant asso-

ciated with the 4 N x 4 N matrix M shown in figure ⁹ (no ^confusion should arise from the fact that, ^for

simplicity, ^some of the 4 x 4 elementary submatrices defined have been named after the bond which is involved in their unique ^non-zero element). ^{Q is the} antipetmutation matrix with dimension N/4, ^i.e.

QN/4 = - I (7 ^{is the} identity matrix with the same

dimension as Q). Then, ^as well-known from the usual factorization procedure,

where m runs through ^{all the} (N/4)th ^{roots of - 1,}

and Li (Mw) ^is the lb x 16 determinant associated with the matrix Mw ^defined by

Going ^{to the new variables col/2 =} exp(iO),

it is found after some algebra ^that

Expressions for a, b, c, d, e ^are found in table I, ^where notations x ⁼ tanh 03B2X, y ⁼ tanh 03B2J, ^{z = tanh} 03B2J’,

t ⁼ tanh pJ 2 have been used.

Retuming ^to (A. 1), we can now write the partition

function _per spin ^{of the} primitive ^{lattice :}

As X - _{+ oo,} the partition function per spin ^{of the}

UBL model can now be reached from (A. 6) ^after

the following successive steps :

1) ^{Let tanh} 03B2X ^{= 1 in} (A. 5) (in table I, ^{do x =} 1).

Notice that doing ^so, expression e ^{in table 1 vanishes.}

Table 1.

Detailed expressions for ^a, b, ^c, d, ^{e, in} (A. 5).

Table Il.

Detailed expressions for ^{a, b, c,} d, ^{e in} (A. 5) ^{when x = 1.}

Table Il.

Detailed expressions for ^{a, b, c,} d, ^{e in} (A. 5) ^{when x = 1.}

2) Subtract from the term t ln cosh pX ⁱⁿ (A. 6)

the bond energy per spin 2 PX ^{due to the X bonds.}

3) Multiply the final result by ^a factor 2 in order to take into account the fact that if the primitive

model contains N spins, ^{the model} obtained after condensation of the X bonds contains N/2 spins.

The previous operations may be summarized by

where z is the partition ^function per spin ^{of the UBL}

model.

The previous procedure, ^followed by insertion of the remaining hyperbolic ^{cosines into the} integral

term in (A. 6) for which the expressions ^of a, b, c, d, e

when x = 1 listed in table II have been used, yields

In z as given by (2.1) ^and (2.2).

References [1] WANNIER, G. H., Phys. ^Rev. 78 (1950) ^341.

[2] VILLAIN, J., ^J. Phys. C : Solid State Phys. ¹⁰ (1977) ^1717.

[3] ANDRÉ, G., BIDAUX, R., CARTON, J.-P., CONTE, R., ^DE SEZE, L., J. Physique ⁴⁰ (1979) ^479.

[4] STEPHENSON, J., ^Can. J. Phys. ⁴⁸ (1970) ^1724.

[5] STEPHENSON, J., Phys. ^{Rev. B 1} (1970) ^4405.

[6] STEPHENSON, J., BETTS, ^D. D., Phys. ^{Rev. B 2} (1970) ^2702.

[7] ELLIOTT, J., Phys. ^{Rev. 124} (1961) ^346.

[8] FISHER, M. E., SELKE, W., Phys. ^{Rev. Lett. 44} (1980) ^1502.

[9] ANDRÉ, G., BIDAUX, R., CARTON, J.-P., CONTE, R., ^DE SEZE, L.,

J. Appl. Phys. ⁵⁰ (1979) ^7345.

[10] BAK, P., ^VON BOEHM, J., ^to be published.

[11] ^VON BOEHM, J., BAK, P., Phys. Rev. Lett. 42 (1979) ^122.

[12] SELKE, W., FISHER, M. E., Phys. ^{Rev. B 20} (1979) ^257.

[13] SELKE, W., FISHER, ^M. E., ^{to be} published.

[14] HOUTAPPEL, ^{R. M.} F., Physica ¹⁶ (1950) ^425.

[15] STEPHENSON, J., ^{Can. J.} Phys. ⁴⁷ (1969) ^2621.

[16] GREEN, H. S. and HURST, C. A., Order-disorder phenomena,

Monographs in Statistical Physics ^and Thermodynamics

(Interscience Publishers).