The cumulant expansion in renormalization group transformations of ising models : cells with nine spins and anisotropic cell interactions

HAL Id: jpa-00208807

https://hal.archives-ouvertes.fr/jpa-00208807

Submitted on 1 Jan 1978

HAL

is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire

HAL, est

destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

The cumulant expansion in renormalization group transformations of ising models : cells with nine spins

and anisotropic cell interactions

E. Moline, M. G. Velarde

To cite this version:

E. Moline, M. G. Velarde. The cumulant expansion in renormalization group transformations of ising

models : cells with nine spins and anisotropic cell interactions. Journal de Physique, 1978, 39 (7),

pp.732-736. �10.1051/jphys:01978003907073200�. �jpa-00208807�

THE CUMULANT EXPANSION IN RENORMALIZATION GROUP

TRANSFORMATIONS OF ISING MODELS : CELLS WITH

NINE SPINS AND ANISOTROPIC CELL INTERACTIONS

E.

MOLINE

and M. G. VELARDE

(*)

Departamento

^{de Fisica-}

C-3,

Universidad Autónoma de

Madrid,

Cantoblanco

(Madrid), Spain (Reçu

^{le 8 août} 1977, ^{révisé le 20}

février

^1978,

accepté

^{le 14 mars}

1978)

Resume. ²⁰¹⁴ Nous discutons certains aspects de la méthode de développement ^en cumulant des

équations

du groupe de renormalisation. Pour des cellules avec

neuf spins,

^{sur un réseau}

triangulaire

plan, ^les

calculsjusqu’à

^{l’ordre deux (en} absence de champ

magnétique)

^font apparaitre ^{des constantes}

d’interactions intercellulaires anisotropes. L’anisotropie ^{est éliminée en} introduisant un paramètre

dont l’influence sur les grandeurs critiques

(point

^fixe, valeur propre relevante ^et exposant ^de Widom)

est présentée.

Abstract. ²⁰¹⁴ The cumulant

expansion

of renormalization

equations

is reexamined. The calculations for cells with nine

spins

^{on a} triangular Ising ^{model to} second-order (in the absence of

magnetic

field) generate ^different

direction-dependent

intercell interactions. The anisotropy is eliminated in a

parameter-dependent way, and the influence of the choice of this parameter ^is

investigated.

Classification

z

Physics ^Abstracts

05.50 ^- 05.70 ^- 75.1OH ^,

1. Introduction. ^-

Recently

several authors

[1-3]

have examined the

utility,

and eventual

limitations,

of the cumulant

expansion procedure proposed by Niemeijer

^{and van Leeuwen}

[4]

^{to evaluate renor-}

malization recursion relations in discrete

spin

systems.

The cumulant

expansion

^{is an} attractive calculational scheme as it is

quite simple

^to

apply and, though

^not

expected

^{to be a} convergent ^one

[3],

^it

yields fairly

accurate results with

just

the first few orders of

approximation.

Relative to earlier results of

Niemeijer

^{and van}

Leeuwen

[4],

^{Hsu et al.}

[1] improved

the cumulant

expansion by enlarging

^{the cell size to} include nine

spins

^{with the} result that in a second-order calculation for the _square

Ising

lattice the thermal and

magnetic eigenvalues

^{were accurate} within 0.2

%

^{and 1.1}

%

of their

respective

^{exact values. For}

seven-spin

^cells

on a

triangular

^{lattice a} similar conclusion was

reached

by Sudbo

and Hemmer

[2].

^{This is to be}

expected

since the intercell part ^{of the} interaction is treated as a

perturbation,

and constitutes a smaller part ^{of the} Hamiltonian when the cell size is

large.

In an exact renormalization

calculation,

however, results should be

independent

of the number of

spins

per cell. In this note we report ^{on some additional}

though

technical rather than basic difficulties ^encoun- tered with the cumulant

expansion

^{in the case of a}

(*) Author ^{to whom all} correspondence should be addressed.

triangular spin

system. ^We carry the

cumulant expansion

^to second order with cells

containing nine,

¹

spins.

2. Renormalized

coupling

constants, ^{and the aniso-}

tropy

of intercell interactions. ^- In the

Niemeijer-

van Leeuwen renormalization

scheme [ 1 -4]

^{one divides}

the

spins

^into

cells,

^each

containing m spins,

si ⁼ ± 1, and associates with each cell a cell

spin s’,

^{which we}

define here

by

^the

majority

^rule

summed over the

(m

⁼

9) spins

^{in the} cell. The cell

spins

should be located on a lattice

isomorphic

^with

the

original

^{one. Then one}

computes

the effective intercell interactions

by decomposing

^the

spin

^Hamil-

tonian

X(s)

^{into an intracell} part

Jeo(s),

^{and an inter-}

cell part

’tY(s) treating

^{the latter as a}

perturbation.

This leads to the cumulant

expansion

for the cell

spin

Hamiltonian

Je’(s’).

Thus the renormalized Hamiltonian is

generated by

the formal

expansion [1-4]

+

higher-order terms } .

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01978003907073200

733

The

averages >0

^{are with} respect ^to the Boltzmann

factor,

exp

Jeo,

^{and are} averages ^over all

spin

^confi-

gurations ( a ) compatible

^{with a}

given cell-spin configuration { s’ 1.

^{A factor}

( - fl)

is included in the

Hamiltonian and in the

coupling

^constants

[4].

^We

assume, for

simplicity,

^zero

magnetic field,

^{and a}

constant term has been omitted in

(2)

^{as done for}

convenience in references

[1-4].

For a

given sthe

^site

spins { s; }

^{can still assume 2$ = 256}

configurations. Using

^the symmetry

properties

of the

nine-spin

^cell

(see Fig.1 )

^and

according

^{to the} location of the

spin

variables, ^we have defined the

following quantities

^and averages :

FIG. 1. ^- a) ^{The cell with nine} spins. b) ^First- and second-order interactions in zero field.

where M accounts for the total number

of cells,

^{and as before}

Jeo

^{is the intracell} Hamiltonian,

To get ^the necessary

consistency,

the evaluation of

(2)

^to second-order demands consideration of ^nearest-

neighbour

interactions as well as next-nearest

neighbours.

This

yields

the non-linear

equations

for the renormalized

couplings Ki (Fig. 2)

between the

cell-spins.

It

happens

^that

anisotropic cell-interactions

^are

generated.

This situation has

already

been encountered

by Sudbo

and Hemmer

[2]

^{and Hemmer} and Velarde

[3],

but these authors

merely

considered an

equalweights

average of all contributions to a

given

^K’. All of these

anisotropic

^{effects are more}

important

for cells with a

higher

^number

of spins,

^{a case} for which the cumulant

expansion

^is

expected

^{to be better} suited. The

anisotropy

can be recast in a

parameter-dependent problem,

^{and the} influence of the choice of this

parameter, p (defined

below),

is studied here. For an

investigation

of related matters see the work

by Stanley

^and collaborators

[5,

^6,

7].

The

following ^relations

in which H, V, L and R account for

horizontal,

vertical, left-handed, ^and

right-handed

interactions

(Fig. 2) provide

^a

self-explanatory

definition of the

anisotropy

parameter, p

(0

p

1).

Using

^the parameter p, the renormalization

equations

^{to second order are :}

of which we seek trie fixed

point

coordinates

735

Linearizing

these renormalization

equations around

the fixed

point

coordinates we calculate the

redevant eigenvalue ÂT

of the transformation matrix _,

Knowing

^this

eigenvalue,

the Widom

homogeneity

exponent ^{follows from} the relation

The intersection of the critical surface in three- dimensional space with the

K21

^axis

yields

^{the inverse}

critical temperature

K21

for the model with ^nearest-

neighbour

interactions

only.

The results can be

compared

^{with the exact value}

K21

i

= 4

^{In 3 = 0.2747.:}

3. Results. ^-

Figures 3,

^{4 and 5}

provide

^a

graphie representation

^of

^the

^{results found}

^~for

^{the values of}

p

(0

p

1).

^{We have}

given

^{the results at the first-,}

and second-order

approximations

^to illustrate the

qualitative

différences between these

approximations.

The best estimates of the

eigenvalue

^and

thé

Widom exponent ^are found in the range 0.25 p 0.35, which

corresponds

^{to an almost}

equal weight,

equa- tion

(22),

of one third to each part of the interaction.

On the other hand it is to be noted

(Fig. 5)

^{that the}

critical temperature ^with the second-order

approxi-

mation varies _very little with _p.

The results found led us to conclude that the

anisotropy generated

^{in the} cell-interactions is not too

important,

and thus in view of the lack of convergence

FiG. 3. 2013 Thermal eigenvalue : a) First-order, b) Second-order

approximation in the cumulant expansion (2) ^{as a} function of

anisotropy parameter p (0 - p - 1).

of the cumulant

expansion [3]

^{a mere}

equal weight

average suffices for numerical _purposes.

FIG. 4. ^- Widom’s homogeneity exponent : a) First-order, b) Second-order approximation ^{as a function} of p.

FIG. 5. ^- Inverse critical temperature for the model with nearest-

neighbour interactions : a) First-order, b) Second-order approxi-

mation as a function of p.

,Lastly,

and for the record we

give

^{in tables} I, II and III the values known for cells with three

[3],

seven

[2],

^nine

(our

^{values at} p ⁼

0.28),

^thirteen

[8],

and nineteen

spins [8].

Acknowledgments.

^- The authors

acknowledge

fruitful discussions

with

P. C. Hemmer. We are also

grateful

^{to one of the} anonymous referees for construc- tive

suggestions

^that

helped

^{make a} substantial

improvement

^{in the} paper.

TABLE 1

Thermal

eigenvalues, AT

> 1

Referencés [1] Hsu, S., NIEMEIJER, Th. and GUNTON, J. D., Phys. ^{Rev. B 11}

(1975) 2699.

[2] SUDBØ, ^{Aa. and} HEMMER, P. C., Phys. ^{Rev. B 13} (1976) ^980.

[3] HEMMER, ^{P. C. and} VELARDE, ^M. G., ^J. Phys. ^{A 9} (1976) ^1713.

[4] NIEMEIJER, Th. and VAN LEEUWEN, ^{J. M.} J., Physica ⁷¹ (1974) ^17.

See also their more recent review article in Phase Transi- tions and Critical Phenomena, ed. C. Domb and M. S.

Green (Academic Press, New York) 1976, ^vol. 6, pp. 425- 505.

[5] HARBUS, F. and STANLEY, ^H. E., Phys. ^{Rev. B 8} (1973) ^2268.

[6] LIU, L. L. and STANLEY, ^H. E., Phys. ^{Rev. B 8} (1973) ^2279.

[7] CHANG, T. S. and STANLEY, H. E., Phys. ^{Rev. B 8} (1973) ^4435.

[8] JAN, ^{N. and} GLAZER, ^A. M., Phys. ^{Lett. 59A} (1976) ^3.