Antiferromagnetic Potts model on Sierpinski carpets

(1)

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Antiferromagnetic Potts model on Sierpinski carpets

A. Bakchich, A. Benyoussef, N. Boccara

To cite this version:

A. Bakchich, A. Benyoussef, N. Boccara. Antiferromagnetic Potts model on Sierpinski carpets. Journal

de Physique I, EDP Sciences, 1992, 2 (1), pp.41-54. �10.1051/jp1:1992117�. �jpa-00246461�

(2)

Classification

Physics

Abstracts

05.50 05.70F 75.10H

Antiferromagnetic Potts model

_on

Sierpinski _carpets

A. Bakchich (1.

2),

A.

Benyoussef (2)

and N. Boccara (3, 4)

(1)

Ddpartement

de

Physique,

Facultd des Sciences, El-Jadida, Morocco

(2) Laboratoire de

Magndtisme,

Facult£ des Sciences, B.P, 1014, Rabat, Morocco

(3) Institut de Recherche Fondan~entale, DPh-G/PSRM,

CEN-Saclay,

91191 Gif-sur-Yvette Cedex, France

(4)

Dept,

of

Physics,

Box 4348, UIC,

Chicago,

IL 60680, U-S-A-

(Received 14 January 1991, revised 29

August1991, accepted

20

September

1991)

Abstract. An

approximate real-space

renormalisation group method, based _on the

Migdal-

Kadanoff recursion relations, is used _to

study

tile critical

properties

of the pure and diluted q-state

anfiferromagnetic

Potts model _on

Sierpinski

_carpets. Fixed

points

and

phase diagrams

_are

calculated for botll large and small

lacunarity

_carpets family. ^The pure model has phase transitions at finite temperatures ^and critical behaviour is observed for q less than a cutoff value qo. We detern~ine tile value of qo, tile

percolation

concentration _{p~ for} tile diluted model, and the critical exponents, ^and we

investigate

tlleir

dependence

on various

geometrical

characteristics of the fractal.

1. Introduction.

In the past ^few years, considerable progress has been occurred in

understanding

various

physical phenomena

described

by

different statistical models on fractal lattices

(I,e,

lattices

lacking

translational invariance but

revealing

self-similar

structure).

These models include for

example Ising

and Potts models _as well _as different

types

^of ^random processes.

However,

to our

knowledge

here is no non-trivial model which has been solved

exactly

for _a

complete

class of fractals embedded in two-dimensional Euclidean space. One of the best known

examples

of such _a

family

is the

Sierpinski carpet (SC) type

^of fractals

[Il.

These lattices _are

characterised,

in addition to their

non-integer

fractal

dimensionality D, by

_an infinite order of ramification which exhibits

phase

transitions at finite temperatures, as

opposed

to fractal lattices with finite order of ramification which

display

_a transition

only

at zero

temperature [2, 3].

The critical

properties

of

spin

_systems have been studied in the non-trivial _case

provided by

SC

using

various theoretical

methods, including approximate real-space

renormalisation

group

techniques [4-11],

numerical simulations

[12-15],

and _more

recently

with

high-

temperature

expansions [16].

It has been

claimed,

from

universality

classes, that certain

properties

of

spin

models _on fractals may

depend,

_apart from the fractal

dimension,

_on other

geometrical

_parameters such _as the

connectivity Q

and the

lacunarity L,

which have been

suggested

_as additional characterisations of the

fractal,

and he

question

is how these

(3)

characteristics affect the critical behaviour of

spin

systems on these lattices. A

complete

answer to this

question

has _not been

given.

Real-space

renorrnalisation group

(RG) techniques

_are

widely

used in

determining

the critical

properties

^of lattice models _on

SC,

and in the

particular

within the

Migdal-Kadanoff (MK) bond-moving approximation [17, 18].

The main

advantage

of this

scheme,

which

preserve the

symmetry

^of ^the

lattice,

is its

simplicity

and has been found _to

give

_a

reasonably quantitative description

of the critical behaviour.

In the _most

technique

has been concentrated on the influence of fractal

geometries

on the critical behaviour of

ferromagnetic

_systems. The situation is much less clear for

systems

^with

non-vanishing

residual entropy, ^such as the q-state

antiferromagnetic (AF)

Potts

model,

which

might

exhibit _a rather

interesting phase

transition. This model has

attracted

increasing

attention after _a

rescaling prediction by

Berker and Kadanoff

[19],

that a

distinctive

low-temperature

critical

phase,

characterised

by

_a

algebraic decay

of

correlations,

can exist when _q is less than _a cutoff value _qo. This cutoff value

depends

on the

topological

characteristics of the lattice and is _not known for the model _on fractals.

The AF Potts model _on the carpet ^has ^been ^considered ⁱⁿ the past

[20],

but the author used

the MKRG method based _on the break

collapse technique [21]

for the pure

model,

to be

distinguished

from _our method for the pure and also the diluted Potts model which has _not been

investigated previously.

However

following

_a

previous study

of the

ferromagnetic

_case

on these lattices

[22],

_we _are

going

to

perform

_a renorrnalisation group

analysis

_to

investigate

the influence of fractal

geometries (fractal dimension, connectivity

and

lacunarity)

_on the critical behaviour of the AF q-state ^Potts ^model on SC. We _use the MKRG scheme since _an

exact or

analytic

method is _not available _on these fractals. In the

particular

_case of the pure

system

we recover the results

reported

in reference

[20].

For the pure model, our

analysis

enables _us to

study

all

possible phase diagrams

for both

large

and

small-lacunarity

_carpets

family,

and for

arbitrary

values of _q, in _order _to put ⁱⁿ evidence the existence of _an ordered critical

phase

at

low-temperature,

_as

suggested by

Berker and Kadanoff for

hypercubic

lattices. This

picture

is obtained for _q less than _a cutoff value qo above which there is

only

the

paramagnetic phase.

We determine the critical value

qo for various carpets, ^and we

investigate

how qo and how critical exponents ^for

q ~ qo,

depend

on various

geometrical

characteristics of the fractal. The results

certainly depend

on the fractal dimension and the

connectivity,

but the influence of the

lacunarity

is

much less clear. We find that qo tends _to increase very

slightly

and the exponent

v tends _to decrease, with

increasing lacunarity.

On the other hand, we describe the dilution

by introducing

two concentrations p ^and

p~, where p~ denotes the concentration of

occupied

bonds which border _an eliminated

subsquare,

and p the concentration of all the other

occupied bonds,

and _we

analyse

the critical

properties

of the model within

Migdal's approximation.

The

percolation

behaviour is described

by

a bond

percolation

concentration, which is identified

by

the critical value

p~ = p = p~. We determine p~ for various carpets ^and we

investigate

its

dependence

_on

geometrical

factors. However

through investigating

the carpets ^with ^different

(b, I),

_where b and

I

are the structure

parameters

^of

carpets (I xi subsquares

are eliminated from

b _x b

subsquares),

_we find that, under

given b,

_p~ decreases with

I increasing

and with D and

Q decreasing

for carpets ^with ^central cutouts

(large-lacunarity

carpets

family).

Besides

depending

on D and

Q,

_we find that p~

depends

also on

lacunarity.

Indeed for fixed b and

I,

and therefore for the _same values of D and

Q,

_p~ ^increases ^when ^L ^decreases.

In _an other

respect,

we calculate the

phase diagrams

within the two-dimensional parameter space

~p, p~)

ⁱⁿ ^order to calculate the critical

exponents

^associated ^with ^the non-trivial fixed

points

of the RG

transformation,

and _we

investigate

their variation with fractal

geometries.

(4)

The outline of the present paper is _as follows. In section 2 _we describe the construction of lkactal lattices and introduce the

geometrical

parameters

characterising

these systems.

Section 3 is devoted _to _an

analysis

of the pure ^and ^diluted ^AF ^Potts model _on SC, We draw

our final conclusions in section 4.

2. Construction of the fractal lattices.

In _our _case the lattice is _a standard SC characterised

by

two

integer

numbers b and

f

(I St

w b

2),

which _can be used to describe the structures.

They

_are constructed

by

_a

subdivision of _a unit square into

b~ subsquares,

out of which

f~

_squares

are cut. This

construction is

repeated iteratively

for the smaller squares, in _a self-similar way, until _one obtains the fractal lattice in the limit of _an infinite number of iterations.

Thus,

each stage ^of

iteration creates a lattice whose

length

scale is reduced

by

_a factor of

b,

and whose number of

elementary

_squares is increased

by

a factor of

b~- f~.

_This

procedure

defines _a lattice with _a fractal

dimensionality

and _a

connectivity,

^which are scaled variables defined

respectively

_as

D

= Ln

(b~- i~)/Ln _(b) _(I)

Q

=

Ln

(b I)/Ln _(b). ₍₂₎

On the other

hand,

two

carpets

with the _same values of b and

f (and

therefore the _same D and

Q)

can have different distributions of the eliminated squares, which result in different

degrees

of non-translational invariance.

Thus,

in order _to characterise the

geometry

of these systems, one needs _an additional parameter, ^the

lacunarity L,

which indicates the

degree

of

translational variance of the fractal. In the

following,

_we shall be interested in _cases in which the

f~

_holes _made _at _each _step _of construction of the fractal lattices _are chosen in two ways,

leading

to

geometries

with different lacunarities _: either

they

_are condensed at the centre of the squares

(corresponding

to a

large-lacunarity family),

_or

they

_are distributed

throughout

each square

(small-lacunarity family). Examples

of these _are

presented

in

figures

la and

16, respectively

for b

=

7 and

f

= 3 after _two

steps

^of ^rue ^iterative

procedure.

Mandelbrot and co-workers

[6]

_gave an

approximate expression

^for ^the

lacunarity

of SC.

L ^~~~

=

(l/n(I _>) z in, (I fi(I

_>i~

_(3>

where

n(f>

=

(i/n(f» z ^n,(f> (4>

m .

mmm m

. m

(a) (b>

Fig. I.

ndensed

at

squares

family).

(5)

with

n(f)

is the total number of

ix f

_cells contained in _a square of b _x b cells, and

n,(f)

_the number of uneliminated cells in the I-th

f

_x

f covering.

The above formula is

only approximate

and it

actually gives

L

=

0 for _some carpets ^which

are

obviously

not

translationally

invariant. The _reason for the

inaccuracy

of the definition is that it _uses

only

_one _stage in the construction of the

carpet.

^A ^better ^definition ^would ^take

many stages ^into consideration. H6wever this

expression

_was

improved [23-26]

_to make its

zero value _to be _a necessary and sufficient condition for _a

translationally

invariant

fractal,

and to reflect the relative

homogeneity

of the carpets. ^In

particular, by

virtue of

bond-moving

RG

results,

SC of

small-lacunarity

L

- 0

(nearly

translational

invariant)

have been

suggested

to possess identical critical

properties

_as abstract

hypercubic lattices,

continued

analytically

to

non-integer

dimensionalities

[5, 27]. Thus, lacunarity

is defined _as the _mean square deviation divided

by

the square of the _mean. One _can define

[8]

L ^~~~

=

(l/n(f )) £ i(n, (f )/R(f ))

_{I j~}

(5)

In the

following

_we shall refer to these _two definitions of

lacunarity.

The second definition _seems _more

appropriate

^for ^the

following

reason : if _one considers SC with scattered cutout

(as depected

in

Fig. lb)

with b

=

2f +1,

one expects that, as

b- _co, the carpet ^looks more uniform and L- 0. But numerical results

give

L

~~~-

co

whereas L

~~~-

₀

as b becomes

large

^which suggests that L ^~~~ is _not

properly

norrnalised.

3.

Model,

recursion relations and results.

3.I MODEL. It is

possible

to consider _a Potts model _on the

SC,

considered _as _a

lattice, by putting

a q-state ^Potts ^variable « on each lattice site of the

microscopic

_one,

including

those

which border the eliminated _areas. Thus, the Hamiltonian of the Potts model is now written

as

PH

"

£ Kij(q3"< _"j

^l

£ (Kw)mn (q3"m

_"n l

(6)

~iJ> ~mn>

where

(mn)

means the

nearest-neighbour

sites that are in the border of the eliminated _areas,

and

(ij)

means the

remaining nearest-neighbour

ones.

For the Potts model in the presence of bond

inhomogeneity,

_K,~ and

(K~)~~

_are assumed to be

independent

random variables

according

to the

following probability

distributions.

P

(K,~

=

p3

_(K~~ K ₊ I _p 3

(K,~ (7a)

P

( (K~

_>~n

~ Pw ^~

( (K~

_>~n

K~

+

(

I p~ ³

( (K~ )~n ) (7b>

where p and p~ are the concentrations of _an

occupied

bond with

coupling

_K~~ and

(K~)~~, respectively.

We

employ

the _same renorrnalisation scheme _as Gefen _et al.

[6]

used for the

Ising model, Migdal-Kadanoff's bond-moving

renorrnalisation, to

produce

the recursion relations for the

pure and bond-diluted AF Potts model.

Following

Yeomans and Stinchcombe

[28],

it is

convenient to introduce _new variables defined _as follows

t,~ ⁼

ii exp(qK,~> ⁱⁱⁱ

+

(q

i

exp(qK,j>j-

ⁱ ₌

f(K,~> (8a>

(tw

)mn "

Ii

_eXp

(q (Kw

_)mn

) Ii

+

(q

i eXp

(q (Kw

_)mn

l~ f (Kw

_)mn

(8b)

Thus,

one has for the AF model,

(- I/q 1)

_~ _t,~,

(t~)~~

_~

0,

for 0

~

K,~, (K~)~~

< co.

(6)

3.2 RECURSION _RELATIONS _AND _RESULTS FOR THE PURE SYSTEM. -When the system ^is

homogeneous,

the

generalisation

of the recursion relations for the _case in which _a

single large

square, of size

f

x

f,

_is _eliminated _in _the _centre _of each

larger

_square

(Fig. la) yields

t'

=

f I(b f

i

f i(tb>

₊ 2

f- i(tb t[>

+

(f

i

f ~(t~

~>i

(9a>

t[

₌

f lf~ ~(t$>

+

((b

^f

2>/2> f-

~(t~> +

f- ~(t~- t[>

₊

((f 1>/2> f- i(tb

_I>1

_(9b>

These relations have been obtained

by

first

performing

the decimation and then

moving

the

bonds. We could have first moved the bonds and then

performed

the decimation. This last

procedure

would have

given

recursion relations of the

following

form

t'

=

fb-ijbf- i(t>j _f ij(b f

_i

f- i(t>

₊ 2

f- i(t~>i _(ioa>

t[

=

f~~~i((b 1>/2> f- i(t>

₊

_f- i(t~>i. f~i((b f 2>/2> f- ~(t)

₊ 2

f- ~(t~>i. (lob>

We shall refer to these _as XY and YX recursion

relations, respectively.

As mentioned above _we _can construct various carpets ^where we

keep

D and

Q

constant and vary L. We

present

^here ^relations

analogous

to

(9)

for the _case of

small-lacunarity

_carpets,

where the eliminated

subsquares

are

uniformly

distributed

throughout

each square

(Fig. lb).

As _an

example

we consider the _case b

= 2 f ₊ I. The recursion relations _are, t'

=

flf~

_~(tb> ₊

(b i> f- ~(tb-~ t[>i _(ha>

tl

=

flf~~(t$)

₊

(b ^f i) f-~(t~-~t[)i. _(lib)

If _we had focused _on the YX

transformation,

then _we should have

found,

instead of

(

II

a)

and

(I16),

the

following

recursion relations

t'

=

f~ibf- i(t>j. ff- i(t>

₊

(b i> f- i(t~>j _(12a>

t[

=

f~ ~iff~ _~(t>

₊

_f- ~(t~>i. f~i(f

+

i> f- i(t~>j. _(12b>

Notice that within

Migdal's approximation

the recursion relations _are within the _two-

dimensional parameter space

(t, t~).

In the

particular

case

f

=

0 _we _recover the known

d

= 2 recursion relations

[29].

A first result obtained from _our calculation is that for the AF Potts model _on

SC,

the RG recursive relations exhibit _a cutoff value q~

lying

^between two and

three,

above which there is

only

the

paramagnetic phase,

whereas for q ~qo the system exhibits _a

low-temperature phase,

characterised

by

_an

algebraic decay

^of correlation functions. This result is consistent with the fact that, _{for q} _m

3,

the system ^is characterised

by

a

high degeneracy

and _a very

high degree

of

complexity

of the

ground

state which exhibits _a finite residual entropy.

In the

particular

_case

q=2 (Ising model),

the

qualitative phase diagrams

in the

(t, t~)

_space for the SC

family

with central cutouts _are shown in

figures

2a and

2b,

associated with XY and YX recursion

relations, respectively.

We find three distinct flow

diagrams. Figure (A)

shows the

special

case b

=

3 and

f

=

I which differs from all the _cases with b

=

f ₊ ₂

~ 3.

Figures (B)

and

(C)

characterise carpets ^with structure parameters ^b _~

f

₊ ₂ _and b

=

f

₊ 2 with

f

_~ _l,

respectively.

There _are

always

two non-trivial fixed

points

^E ^and ^F ^whose

(t, t~)

coordinates _are

E _:

(tE, i>

F _:

(tF, tj>.

In all _cases the full _curves EF represent ^the ^critical ^line

separating

the disordered

phase (flows

to

A)

from the ordered

phase (flows

to

C).

The critical

point characterising

this transition is

(7)

'

-o.5

/

_-o.5

/

_-o~

F A A A

F -o.5

c c

~

/w

(Al (E3) (Cl

a)

t

_-o.5

/

_-o.5

/

-o.5

F

c ~ ~

/~

j

_/w

(A) (E3) (C)

(b>

Fig.

2. Phase diagrams in the (t, t~) space for the

antiferromagnetic

Ising model _on

Sierpinski

_carpets with central cutouts: (a) within XY recursion relations; (b) within YX recursion relations. (A) b

= 3~ f

= I (B) b

~ f ₊ 2 (b

= 7, f ₌ 3) (C) b

=

f ₊ 2 f _# I (b 7, f 5).

represented by

the fixed

point

F, except at t~ = I line which is characterised

by

the fixed

point

E. We _note that the

phase diagrams

and fixed

points

of the AF

Ising

model _are

symmetrically

located in relation _to the _ones for the

ferromagnetic

_case in the

positive

_part of the

(t, t~)

_space. For the

types

of

carpets

studied here _our results agree with those obtained

previously by

the _same

bond-moving

scheme

[25, 7]. Comparison

with the results

reported

in table I of reference

[6]

indicates _some

discrepancies namely

for the _case b

=

7

f

=

3 and b

=

Ii

f

= 3, where the authors obtain

respectively

t~ ₌ 0.013 instead of t~ ₌

0.00453,

and t~ = 0.00917 instead of t~ ₌ ^0.000407. ^The

origin

of these

discrepancies

is

probably

of _a

numerical _nature.

For q ~ 2 the RG flow

diagrams

for XY and YX recursion relations _are shown

respectively

in

figures

3a and 3b. These

phase diagrams

_present _no

phase

transition _at t~ ₌ I

line,

_even

at T

=

0,

unlike the q =

2 _case. However the RG

trajectory originating

at T

=

0 flows _to _a stable fixed

point G,

in the parameter space, at non-zero, non infinite temperature. ^This new

non-trivial fixed

point

characterises an ordered

phase

at

low-temperature.

As q increases from

2,

the two fixed

points

F and G merge ât â critical value go above which the system îs

always

in the

paramagnetic phase.

This _can be understood from _a

ground

state RG argument

(8)

/

_-o _s

t

_o.5

/

_-o.5

A A

F

G

/w ^w

(A> (B) (C)

(a)

/

-o.5

/

_os

/

_-o.5

A A A

F

[ [ [

(A) _(£Y) (C)

(b)

Fig. ^3. ^Phase

diagrams

in the (t,

t~)

_space for the

antiferromagnetic

Potts model _on

Sierpinski

carpets with central cutouts for

2<q~qo:

(a) within XY recursions relations _; (b) within YX recursion

relations. (A) b

= 3, f

=

I _; (B) b

= 7, f

= 3 (C) b

=

7, f

=

5.

[19].

However it is well known that _at _a fixed

point

the correlation

length

f is _zero _or infinite

[30].

The former holds in the _case, for

example,

of the trivial fixed

point

at T

= co, but _a fixed

point

at non-zero, non-infinite temperature ^is ^associated ^with ^critical ^behaviour.

Therefore,

in this

interpretation, f

= co at G

and, consequently, throughout

the temperature

region

which renormalises _to G. This _means that the correlations

decay algebraically, throughout

^this

distinct

low-temperature phase [19, 31].

For

small-lacunarity

_carpets the above

picture

does _not

change.

There _are two

possible phase diagrams

shown in

figures

4a and 4b associated with XY and YX recursion relations. In

both _cases the fixed

point

E

=

(t~, _I)

is the _same for both

types

^of

carpets,

^but ^the ^fixed

point

F is _now _at

(- I, t$)

similar to

phase diagrams

such _as in

figure

2b

(C)

for q =

2. For 2 _< q < go the renormalisation flow stops at _some finite

temperature

^fixed

point

G in the parameter space, which

corresponds

to an ordered critical

phase

_as discussed above.

In addition _to that

analysis

we iterated the recursion relations

numerically

and identified

the locations of the fixed

points

E and

F,

and their associated critical exponents. ^The

numerical results _are summarised in tables I and II. A notable remark is for the

special

_case

b

=

f

₊ ₂

=

3,

where the _two schemes of decimation

give

^different results for the critical exponent

Y(.

This _case is also

special

in the

topology

of the flow

diagram.

(9)

/

-o.s

/

_-os

A f3 A

D

1

_/~

(4) (B)

t

E3 -o.s

/

_-o.s

A A

s

E

/w

/

w

(A) (B)

(b)

Fig.

4.

Typical

flow

diagrams

in the (t,

t~)

_space for the

antiferromagnetic

Potts model _on

Sierpinski

carpets ^with scattered cutouts _: (a) within XY recursion relations (b) within YX recursion relations.

(A) _q

=

2 (B) ² _~ _q _~ q~.

The critical

properties

of the model

certainly depend

_on the fractal dimension and the

connectivity.

To

investigate

the effect of

lacunarity

_on the critical

exponents

^of ^the carpets,

one has to fix b and

f (and

therefore to

keep

D and

Q constant)

and to look _at carpets ^with a

different distribution of cutouts, as shown in

figures

la and 16. We _note that the exponents at E _are

equal

for both types ^of carpets ^with ^the same b and

I,

but at F the exponent v is smaller for the central _cutout type.

Thus,

_v decreases with

increasing

L.

On the other

hand,

_we determined the cutoff value go for _a

given (b, f) characterising

various

carpets.

^Notice ^that

only integer

values of g are

physical.

However the _case

b

=

II

f

= 9 and

large-lacunarity

is of

particular

interest since the _two values

predicted by

^XY

and YX recursion relations _are

respectively

_go ₌ 2.006 and go =

2.999. These _two values _are

almost

g=2 (Ising model)

and

g=3 (three

state Potts

model).

To

investigate

the

dependence

_{of go} on the

geometrical factors,

_{go is}

plotted against

the fractal dimension D in

figure 5,

for various values of b and

f.

_We find that go is ^not a

monotically decreasing

function

of D. One _reason is due to the inaccuracies associated with the MK scheme. The influence of

lacunarity

on the cutoff value is much less clear. We note that go shows ^a very

slight

rise as the

lacunarity

increases. It is _not clear if this rise is

intrinsically

related _to

lacunarity

_or is associated with the

approximations

involved in the method. This

suggests

^that go is rather insensitive _to

lacunarity.

(10)

Table I. -Results

for

the

antiferromagnetic

Potts model _on

Sierpinski

carpets ^within ^XY recursion relations.

b I D Q L^~~l L^~~l qo q tE

Y(

^tF t$

Y(

t~

t$

(a) Large-lacunarity

3 1.893 0.631 2.137 2 -0.4354 0.5976 -0.9999 -0.l192E-6 0.9999

2.1 -0.9999 -0.l135E-6 0.9999 -0.9065 -0.8778

5 1.975 0.861 2.175 2 -0.6685 0.6985 -0.7551 -0.3564 0.6320

2.1 -0.7753 -0.4176 0.4376 -0.9085 -0.8922

2.15 -0.7939 -0.4841 0.2242 -0.8644 -0.7913

5 3 1.722 0.431 2.416 2 -0.2542 0.4084 -0.4991 -0.2539 0.3843

2.1 -0.5121 -0.2681 0.3514 -0.9090 -0.9055

2.3 -0.5522 -0.3148 0.2376 -0.7619 -0.6762

71 J.989 0.92J 2.1662 -0.7413 0.6995 -0.7761-0.4086 0.6703

2.1 -0.7943 -0.4735 0.4801 -0.9087 -0.8946

2.15 -0.8132 -0.5488 0.2446 -0.8641 -0.7754

7 3 1.896 0.712 3.9424 0.l188 2.205 2 -0.6267 0.6015 -0.7254 -0.3865 0.5842

2.1 -0.7437 -0.4314 0.4606 -0.9090 -0.9022

2.15 -0.7579 -0.4696 0.3499 -0.8683 -0.8330

7 5 1.633 0.356 2.913 2 -0.1682 0.3466 -0.2552 -0.1297 0.3348

2.1 -0.2585 -0.1326 0.3262 -0.9090 -0.9087

2.5 -0.2755 -0.1476 0.2824 -0.6642 -0.6174

11 3 1.967 0.867 2.145 2 -0.7838 0.6629 -0.8119 -0.4745 0.6500

2.05 -0.8190 -0.5064 0.5687 -0.9523 -0.7521

2.1 -0.8295 -0.5559 0.4342 -0.9089 -0.8957

11 5 1.903 0.747 24.040 0.l192 2.189 2 -0.7098 0.5943 -0.7552 -0.4380 0.5857

2.1 -0.7719 -0.4893 0.4625 -0.9090 -0.9044

2.15 -0.7859 -0.5358 0.3406 -0.8684 -0.8286

11 9 1.538 0.289 2.999 2 -0.1003 0.2862 -0.1256 -0.6307E-1 0.2847

2.1 -0.1264 -0.6370E-1 0.2818 -0.9090 -0.9090

2.5 -0.1298 -0.6646E-1 0.2692 -0.6666 -0.6636

(b) Small-lacunarity

5 2 1.892 0.682 2.122 2 -0.5074 0.5889 -1.0 -0.5063 0.3845

2.05 -0.9340 -0.6214 0.3377 -0.9523 -0.9503

2.1 -0.8958 -0.7228 0.2192 -0.9082 -0.8881

7 3 1.896 0.712 0.9984 0.0201 2.ill 2 -0.5675 0.5922 -1.0 -0.6031 0.4504

2.05 -0.9399 -0.6972 0.3740 -0.9523 -0.9514

2.1 -0.9027 -0.7930 0.1879 -0.9085 -0.8886

II 5 1.903 0.747 2.949 0.0080 2.094 2 -0.6519 0.5960 -1.0 -0.6998 0.4987

2.05 -0.9452 -0.7796 0.3899 -0.9523 -0.9519

2.1

(11)

Table II. -Results

for

the

antiferromagnetic

Potts model _on

Sierpinski

_carpets within XY recursion relations.

b I _D Q L~~l L^~~l go q tE

Y(

tF t$

Y(

i~

tj

(al Large-Iacunarity

3 1.893 0.631 2.136 2 -0.lI97 0.5619 -0.6992 -0.1568E-1 0.2240

2.1 -0.6383 -0.21I3 0.2602 -0.7446 -0.6764

5 1.975 0.861 2.179 2 -0.1420 0.6707 -0.2048 -0.5378E-2 0.6742

2.1 -0.2407 -0.II23E-1 0.4980 -0.6189 -0.5654

2.15 -0.2778 -0.2195E-1 0.3133 -0.4822 -0.3118

5 3 1.722 0.431 2.042 2 -0.4108E-1 0.4139 -1.0 -0.4858 0.3836

2.03 -0.8367 -0.5718 0.2405 -0.8606 -0.8303

2.J

7 1.789 0.921 2.169 2 -0.1253 0.6828 -0.1500 -0.1782E-2 0.6936

2.1 -0.1797 -0.4756E-2 0.5165 -0.5119 -0.4586

2.15 -0.2145 -0.1231E-1 0.3084 -0.3596 -0.1717

7 3 1.896 0.712 3.94240.l188 0.099 2 -0.8658E-1 0.5942 -0.2682 -0.4530E-2 0.5151

2.05 -0.3147 -0.lll0E-1 0.3604 -0.7103 -0 6918

2.01

7 5 1.633 0.356 2.017 2 -0.2068E-1 0.3491 -1.0 -0.6629 0.3424

2.01 -0.9239 -0.7095 0.2451 -0.9323 -0.9202

2.I

II 3 1.967 0.867 2.122 2 -0.8521E-1 0.6475 -0.1240 -0.4072E-3 0.6513

2.05 -0.1393 -0.9647E-3 0.5505 -0.5846 -0.5797

2.1 -0.1697 -0.3932E-2 0.3530 -0.3463 -0.2386

11 5 1.903 0.747 24.040 0.l192 2.086 2 -0.6884E-1 0.5993 -0.1885 -0.1096E-2 0.5487

2.05 -0.2277 -04030E-2 0.3911 -0.5843 -0.5599

2.I

11 9 1.538 0.289 2.006 2 -0.8310E-2 0.2867 -1.0 -0.7992 0.2857

2.005 -0.9441 -0 8589 0.l147 -0.9459 -0.9224

2.1

5 2 1.892 0.682 2.l17 2 -0.1003 0.5852 -1.0 -0.l197 0.3835

2.05 -0.6647 -0.1764 0.3372 -0.7834 -0.7753

2.1 -0.5693 -0.2740 0.2032 -0.6178 -0.5508

7 3 1.896 0.712 0.99840.02012.106 2 -0.8658E-1 0.5942 -1.0 -0.1448 0.4599

2.05 -0.6258 -0.1819 0.3920 -0.7106 -0.7057

2.1 -0.4916 -0.2852 0.1590 -0 5104 -0.4322

11 5 1903 0.747 2.949 0.0080 2.090 2 -0.6884E-1 0.5993 -1.0 -0.1322 0.4984

2.05 -0.5349 -0.1657 0.4106 -0.5846 -0.5819

2.1

(12)

+

3 .ni,9>

.<7,5>

2.5

. <5,3>

,t7,31 (1>5)

qj5,lj (3,1)e~~~~

^~~~~

~~~~~ o(11,3j

' °(11,5j (5,3)

(7,5> ^°

2 ^°

1.5 2 D

Fig. 5. qo

plotted against

lkactal

geometries

((a) fractal dimension (b) connectivity) for

Sierpinski

carpets ^with ^central ^cutouts ^within ^XY (. ) and YX (o) recursion relations.

3.3 RECURSION _RELATIONS _AND _RESULTS FOR THE DILUTED SYSTEM. If the

t;j

^and

(t~)~~

_are

independently

distributed

according

to

(7),

then the

probability

distributions

P'(t()

and

P'( (t[)~~ )

of the renorrnalised

couplings

_are of _a _more

complicated

form than the

initial _ones.

Thus,

in order _to find the recursion relations which _we

need,

_we make _an

additional

approximation

at each iteration

by forcing

^the transformed distributions back to a

two-peak

form.

Namely,

Pippmx(tjj>

=

p'

3

(tjj t'>

+

(I p'>

3

(tjj) (13a) Pipprox((ti)mn>

"

Pi

~

((ti)mn t'>

₊

(I p')

3

((ti)mn) (13b)

where

t', p'

and

t[, p[

are determined

by setting

^the first two moments of

P(~~~~~(t,j)

^and

P(~~~~~

( ( t[

_)~~

)

_to be

equal

to those of the P '

(t)j ⁾

^and ^P ^'

⁽ ^(t[

)~~

).

(13)

For the

problem

of

bond-dilution,

it is

quite

obvious that when the bond concentration is below the

percolation concentration,

which characterises the critical

behaviour,

the system

consists of _an aggregate ^of ^finite connected clusters and therefore _no

phase

transition is

expected.

The

percolation

concentration _can be determined

by

the fixed

point

of the RG transformation _at T

= 0. For fractal lattices considered here the

percolation

by

^the ^RG ^recursion ^relations

determining p'

^and

p[.

We

specialize

the

following

to the XY transformation.

Then,

for

large-lacunarity

carpets

family,

the recursion relations _are

P'

=

i

(i pb)b-~-i _(i pb-~p[)2 _(1- pb-~)~-i _(14a)

p[

=

I

(I p$)(I p~)~~~~~~~'~ (l p~ p$)(I _p~)~~

~~'~

(14b)

If _we had focused _on

small-lacunarity

carpets, ^then we would have

found,

instead of

(14a)

and

(14b),

recursion relations of the

following

form

PI

~

l

(I -P~l(i p~p[)~

~~

(]5a)

P[

"

I

(I -P$)(I p~p$)~~~~~ (lsb)

These relations have been obtained for the _case b

=

2 f ₊ I, f odd.

The

corresponding

flow

diagrams,

ⁱⁿ the parameter space

~p, p~),

are

represented

in

figures

6a and 6b for

carpets

^with ^central cutouts and _cutouts

spread, respectively.

The

~p, p~)

coordinates of the fixed

points

are

E:

~p§ i) F:~pLp$>.

We _note that

percolation

concentration p~ is identified

by

the critical value _at which the critical line EF _crosses the

diagonal

_~p~

= p =

p~).

It is reasonable _to expect ^that ^for ^fractal

lattices,

_p~ is greater ^than ^the ^bond

percolation

critical

point

of the square lattice

~p~ =

0.5),

because there _are _now holes in the lattice.

i~

_E

i~

E

~ E

~

s c s C c

F

A F A A ~

o~P o~p o~P

(A) (s) (cl

(a)

~

E c

s

F

~ D

o~p

l~) (b)

Fig.

6.- Phase

diagrams

in the ~p,p~) _space: (a) large-lacunarity; (b) small-lacunarity. (A) b

= 3, f

=

I (B) b

= 7, f

= 3 _; (C) b

= 7, f 5.

(14)

Table III, Fixed

points

and critical exponents

characterising

bond

percolation

behaviour in the

~p, p~)

_space.

b f D Q L ^(ij ~ 12) pE yE ~F

~j

_yF p~

(a) Large-lacunarity

3 1.893 O.631 O.4893 0.4987 1.0 0 0.9999 0.7508

7 1.989 0.921 0.7839 0.5888 0.8148 0.5113 0.5837 0.8008

7 3 1.896 0.712 3. 9424 0.1188 0.6741 0. 5 lo1 0.7699 0.4795 0.51 17 0.7480

7 5 1.633 0.356 0.1808 0.3127 0.2816 0.1524 0.2908 0.2815

II 3 1.967 0.867 0.8177 0.5663 0.8432 0.5706 0.5638 0.8338

II 5 1.903 0.747 24.040 0.l192 0.7494 0.5098 0.7934 0.5269 0.5068 0.7851

II 9 1.538 0.289 0.1050 0.2680 0.1329 6.8838E-2 0.2626 0.1329

7 3 1.896 0.712 0.9984 0.0201 0.6136 0.5001 1-o 0.6618 0.3783 0.8420

II 5 1.903 0.747 2.949 0.0080 0.6915 0.5071 1-o 0.7483 0.4199 0.8775

The numerical results _are summarised in table III. For each

pair

of b and

f, characterising

various

cawets,

^the ^table

lists,

in addition to the values of

geometrical parameters,

p~

and the exposents near

E,

the coordinates of F and the

exposents

near F, and the

percolation

concentration p~.

It _can be _seen from table

III,

that for the _same

b,

_p~ decreases with

f increasing

and with D and

Q decreasing

for

carpets

^with central cutouts. For the _same D and

Q

but different

lacunarity

L, _{p~ in} the

family

^of carpets ^with ^central cutouts is smaller than that for the

family

of

carpets

^with

evenly

scattered _cutouts.

Table III also shows the critical exponents

Y~

^associated ^with ^the non trivial fixed

points

for both

types

^of

cawets. By inspection

_we note their variation with fractal dimension and

connectivity.

Indeed for b fixed and

varying f, _Y~

_decreases _as _D _and

_Q

_decrease. On the other hand the results

suggest

^that

lacunarity

has litle effect _on the exponents near the fixed

point E,

while

Y)

decreases with

decreasing lacunarity.

4. Conclusions.

We constructed _a

bond-moving

RG based _on the MK recursive

relations,

and used it _to

study

the critical

properties

of rue pure and diluted q-state ^AF ^Potts ^model on a fractal lattice with

an infinite order of

ramification, namely

the

Sierpinski

carpets. ^Fixed

points,

critical

exponents

^and

phase diagrams

_are

presented.

^For the pure model _we found that there is _a cutoff value go, such that for 2 _< g < go, the system ^exhibits a

low-temperature

critical

phase

characterised

by

_an infinite correlation

length.

^The ^value of go and of critical exponents were

obtained for different values of b and

f characterising

^the carpet, ^and ^for ^different

distributions of _cutouts. The results

certainly depend

_on the fractal dimension and the

connectivity

of the fractal. In

addition,

_go tends to increase very

slightly

and the exponent v

tends _to decrease with

increasing lacunarity.

For the diluted

model,

the

percolation

by

a bond

percolation

concentration p~ which decreases with

f increasing

and with

(15)

D and

Q decreasing,

while its associated critical exponent v~ ^decreases ^with

decreasing lacunarity.

References

[I] MANDELBROT B. B., The Fractal

Geometry

of Nature (San Francisco _: Freeman, 1982).

[2] GEFEN Y., AHARONY A.'and MANDELBROT B. B., J.

Phys.

A16 (1983) 1267.

[3] GEFEN Y., AHARONY A., SHAPiR Y. and MANDELBROT B. B., J.

Phys.

A17 (1983) 435.

[4] GEFEN Y., MANDELBROT B. B. and AHARONY A., Phys. Rev. Lett. 45 (1980) 855.

[5] GEFEN Y., MEIR Y., MANDELBROT B. B. and AHARONY A., Phys. Rev. Lett. 50 (1983) 145.

[6] GEFEN Y., AHARONY A. and MANDELBROT B. B., J.

Phys.

A17 (1984) 1277.

[7] RIERA R. and SHAVES C. M., Z. Phys. B 62 (1986) 387.

[8] LAI P. Y. and GOLDSCHMIDT Y. Y., J. Phys. A 20 (1987) 2159.

[9] LIN B., J. Phys. A 20 (1987) L163.

[10] HAO L. and YANG Z. R., J.

Phys.

A 20 (1987) 1627.

[ll] BOCCARA N. and HAVLIN S., J. Phys. A17 (1984) L547.

[12] BHANOT G., NEUBERGER H. and SHAPIRO J. A.,

Phys.

Rev. Lett. 53 (1984) 2277.

[13] BHANOT G., DUKE D, and SALVADOR R.,

Phys.

Lett. 165B (1985) 355.

[14] ANGLES D'AURJAC J. C. and RAMMAL R., J.

Phys.

A19

(1986)

L655.

[15] BONNIER B., LEROYER Y. and MEYERS C., J. Phys. France 48 (1987) 553.

[16] BONNIER B., LEROYER Y. and MEYERS C., Phys. Rev. 840 (1989) 8961.

[17] MIGDAL A. A., Zh.

Eksp.

Teor. Fiz. 69 (1975) 810 Sov.

Phys.-JETP

42 (1975) 743.

[18] KADANOFF L. P., Ann. Phys. (NY) 100 (1976) 359.

[19] BERKER A. N. and KADANOFF L. P,, J.

Phys.

^A13 (1980) L259.

[20] RIERA R., J. Phys. A19 (1986) 3395.

[21] TSALLIS C. and LEVY S. V. F., Phys. Rev. Lett. 47 (1981) 950.

[22] BAKCHICH A., BENYOUSSEF A. and BOCCARA N., J.

Phys..

Cond. Matter 3 (1991) 1727.

[23] LIN B. and YANG Z. R., J.

Phys.

A19 (1986) L49.

[24] TAGUCHI Y., J. Phys. A 20 (1987) 6611.

[25] Wu Y. K. and Hu B.,

Phys.

Rev. A 35 (1987) 1404.

[26] Wu Y. K., J. Phys. A 21(1988) 4251.

[27] LE GUILLOU J. C. and ZINN-JUSTIN J., J.

Phys.

France 48 (1987) 19.

[28] YEOMANS J. M. and STINCHCOMBE R. B., J.

Phys.

C13 (1980) L239.

[29] BAKCHICH A., BENYOUSSEF A., BJAz T. and LAANAJT L.,

Phys.

Rev. B 37 (1988) 9443.

[30] WILSON K. G.,

Phys.

Rev. B 4 (1971) 3174.

[31] RUJAN P., WILLIAMS G. O., FRISH H. L. and FORGACS G., Phys. ^Rev. ⁸²³

(1981)

1362.