LOW TEMPERATURE CRITICAL BEHAVIOUR OF THE ISING MODEL OF FERROMAGNETISM

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LOW TEMPERATURE CRITICAL BEHAVIOUR OF THE ISING MODEL OF FERROMAGNETISM

A. Guttmann, C. Domb, P. Fox

To cite this version:

A. Guttmann, C. Domb, P. Fox. LOW TEMPERATURE CRITICAL BEHAVIOUR OF THE ISING

MODEL OF FERROMAGNETISM. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-354-C1-

355. �10.1051/jphyscol:19711121�. �jpa-00213938�

JOURNAL DE PHYSIQUE

Colloque C 1, supplbment au no 2-3, Tome 32, Fkurier-Mars 1971, page C 1 - 354

LOW TEMPERATURE CRITICAL BEHAVIOUR OF THE ISING MODEL OF FERROMAGNETISM

A. J. GUTTMANN, C. DOMB and P. F. FOX Physics Department, King's College, London, England

RBsum6.

- On a obtenu des d6veloppernents en skrie du modele d'Ising dans le cas oh le spin S

=

1 pour divers rkseaux a deux et

a

trois dimensions. On trouve par extrapolation que les indices critiques B,

^y'

et

6

semblent &re les m&mes que dans le cas du spin S

⁼

t .

Pour le modele d'Ising avec spin

S = &

on a trouv6 que la catkgorie de complexions qui sont des arbres (graphes finis connexes sans cycles) fait une contribution sensible dans le d6veloppement aux basses tempkratures. En particulier on trouve que la distribution des singularitks dans la region

I u

I < I

uc

I

(uc =

exp[-

_- - 4 J / k T c ] , Tc

est le point critique) est bien reprksentk en tenant compte seulement des arb?es.

Abstract. -

Low temperature series expansions for the spin

S = 1

Ising model on various two and three dimensio- nal lattices have been generated. By extrapolation of these series, it is found that the critical exponents P,

y'

and

6

appear to be the same as for the spin S

=

t Ising model.

For the spin

S =

t Ising model, we have found that the class of configurations represented by Cayley trees make

a

significant contribution to the low temperature series coefficients. In particular, we find that the distribution of singu- larities within the disk I

u I

<

I uc

I

^{(uc =}

exp[- 4 J / k T c ] , T c is the critical temperature) is well represented by including only the Cayley trees in the partition function.

We consider here two topics that arise in the study of the low temperature behaviour of the Ising model of ferromagnetism. Initially we dis,cuss the variation of thermodynamic behaviour near to and below the Curie temperature T , with spin S, as determined by extrapolation of power series expansions. Our second topic is an approximation which accounts for certain complex aspects of low temperature series expansions of the Ising model.

The Ising model Hamiltonian may be written

where

si

may take the ( 2 s + 1) values s,

=

-

S, 1

- S, ..., S - 1,

S

( H is the external magnetic field, m the magnetic moment of a single spin and < ^i,

^j

>

denotes summation over nearest neighbour pairs).

For a regular space lattice, the thermodynamic pro- perties may be expanded around T

=

0 as a power series in the variable

forJinite spin

S.

(When both the coordination number of the lattice and 2S are odd, the expansion variable is z(s)

=

u ( s j X . ) We have obtained such expansions for the

S =

1 Ising model on a number of two and three dimensional lattices

-

the series for S

=

3

having been known for some time [I]. These series expansions have been studied by the method of Pad6 approximants [2] and the method of N point fits [3].

Transformations have been applied to the series so that they converge up to the Curie temperature, and these new series have also been studied by the same methods. The estimates of the critical exponents obtained as a result of this study are summarised in Table I, where the critical exponents take their usual meaning [4].

As for the S

=

5 model, it has not been possible to obtain reliable estimates for the specific heat expo- nent

ccr

by direct series analysis. The spontaneous magnetization exponent p may be estimated quite

Estimates of critical exponents for the Zsing model of spin

S =

3 ^and

^{S =}

1 obtained

for

a number

of two and three dimensional lattices

Dimen- Spin Critical Exponents

sionality

B

Y' 6

-

- -

-

-

2 S = + 0 . 1 2 5 1.75 1 0.04 15.00 f 0.08 S = 1 0.125 f 0.004 1.72 1 0 . 0 7 14.7 1 0 . 7 3 S = ) 0.313 i 0.002 1.28 f 0.04 5.0 f 0.1 S = 1 0.313

+

^0.003 1.24 1 0.05 5.02 i 0.13

accurately however, particularly for three dimensional lattices, and as can be seen appear to be spin inde- pendent. The estimates of the exponent y' are less accurate, but again are consistent with the S

=

5

values. In fact the results support high-low temperature exponent symmetry rather better than do the spin S

=+

series. Estimates of the exponent

6

have been taken'from Fox and Gaunt

^[5],

and are again consis- tent with the spin

S =

$ values.

Using the Pad6 method [2] we have obtained repre- sentations of the thermodynamic functions in the entire temperature range 0 < t < 1, where t

=

TIT,.

In figure 1 we show the variation of the spontaneous magnetization with temperature for the two spin values S

=

Q and

S =

1. For comparison we aIso show the molecular field predictions. It can be seen that for fixed relative temperature

t =

TIT, < 1, the spontaneous magnetization M,(S, t ) is a decreasing function of

S,

while for fixed spin

S,

M,,(S,

t )

in decreasing function of t. This behaviour is predicted by molecular field theory.

We therefore conclude that for finite spin S the critical exponents appear to be independent of S.

This agrees with, and complements, the earlier studies of high temperature series expansions for various values of spin S by Domb and Sykes [6] and more recently by Jasnow and Wortis

[7].

We now consider our second topic. From the above study of low temperature series of spin S on a lattice

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711121

LOW TEMPERATURE CRITICAL BEHAVIOUR OF THE ISING MODEL OF FERROMAGNETISM C 1

-

³⁵⁵

of coordination number q we have found that there tree approximation have been published elsewhere [lo].

are (qS - 1) singularities in the disk 1 ^u(S) I < u,(S), We have calculated the distribution of singularities where uc(S)

=

exp(- JILTc S2). (For the case q and within the disk ] u I <

^u,

for the spin + Ising model 2 s both odd, the result is 2(qS - 1) singularities on a number of lattices from this Cayley tree approxi- in the disc I z(S) I < z,(S)). It is the presence of mation. In figure 2 we show the results for the face- these singularities closer to the origin than the physical

singularity that causes the series coefficients to behave in a very irregular manner, varying apparently ran- domly in sign and magnitude in many cases. The

FIG. 2. - Distribution of singularities in the complex u plane for the spin S =

+

Ising model on a face-centred cubic lattice.

Exact >> position of singularities, obtained from series ana- lysis. A Position of singularities obtained from Cayley tree

approximation.

FIG. 1. - Variation of the spontaneous magnetization of the

~~i~~ model on a triangular and on a faceqcentred cubic lattice

centred cubic lattice, with the results obtained from

with spin S and relative temperature t = TIT,. Molecular field

analysis of series expallsions [3] shown for comparison.

theory results are also shown. Key :

-1-(-1-

Triangular

It can be seen that the agreement is surprisingly good.

lattice. - - - Face-centred cubic lattice.

---

field theory.

Another aspect of the approximation is that it gives rise t o a svinodal curve as found bv Gaunt and Baker [l 11, but the validity of this predikion depends position of these singularities has been estimated the investigation of two and three dimensional for the low temperature Ising model series for various configurations to be undertaken subsequently.

values of spin S

r3,

81. For the S

=

t model we can

see the origin of these singularities-by considering Acknowledgements. - Two of us (A. J. G.) and an approximation that takes into account only those (P. F. F.) would like to acknowledge the financial graphs known as Cayley trees. These configurations support of the Science Research Council. Numerous differ from the droplets of Fisher's model

[9]

by taking helpful conversations with Drs. M. F. Sykes and into account volume exclusion. Details of the Cayley D. S. Gaunt are gratefully acknowledged.

References

[I]

SYKES (M. F.), ESSAM (J. W.) and GAUNT (D.

S.), [7]

JASNOW (D.) and WORTIS (M.),

Phys.

Rev.,

1968;

J. Math. Phys., 1965, 6, 283. 176,739.

[2]

BAKER (G.

A.) Jr, Adv. theov. Phys., 1965,

1,

1.

[3]

GUTTMANN

(A. J.), J. Phys. C : Solid State Physics, [8]

Fox (P. F.) and GUTTMANN

(A. J.), Phys. Letts,

1969, 2, 1900. 1970, 31 A, 234.

[4]

See for example FISHER (M. E.), Rep.

Prog. Phys.,

+ (to be published).

1967, 30, 615. [9]

FISHER (M. E.),

Physics, 1967, 3, 255.

[51 'Ox

F.) and

GAUNT

(D.

J . Phys. : "lid [lo]

DOMB (C.) and GUTTMANN

(A. J.),

J.

Phys.

C

: Solid State Physics, 1970, 3, L 88.

+ ^(to

^be

published).

State Physics, 1970, 3, 1652.

[6]

DOMB (C.) and SYKES (M. F.),

Phys. Rev., 1962, [ l l ]

GAUNT (D. S.) and BAKER

(G. A.) Jr, Phys. Rev. B,

128, 168. 1970, 1, 1184.