Spin flip probabilities in the 3-D diluted Ising model

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Spin flip probabilities in the 3-D diluted Ising model

Uwe Gropengiesser

To cite this version:

Uwe Gropengiesser. Spin flip probabilities in the 3-D diluted Ising model. Journal de Physique I,

EDP Sciences, 1994, 4 (8), pp.1133-1138. �10.1051/jp1:1994243�. �jpa-00246973�

Classification Physics Abstracts 05.50

Spin flip probabilities in the 3-D diluted Ising mortel

Uwe

Gropengiesser

Institute ofTheoretical Physics, University _zu KôIn, Zülpicher Sir. 77, D-50937 KôIn, Germany

(Received

6 April 1994, accepted19 April

1994)

Abstract, According to Ray and Jan the cluster development in _a spin glass _can be de- scribed by calculating _spin flip probabilities and by association of rarely flipping spins. We show that for the _pure ferromagnetic Ising mortel the predicted Ray-Jan percolation transition temper-

ature does not agree with the phase transition of the Ising system. Both diluted and undiluted

Ising mortel show critical percolation spin flipping rates bigger that the Ray-Jan values.

1. Introduction.

Clusters in the

Ising

model

(H

=

£~ JS~Sj

_are

usually

described

by

the definition of

Coniglio

and Klein [1, 2]

forming

bonds

betileen _{parallel spins}

_with

probability

I

exp(-2J/kT).

An alternative

description involving

sites instead of bonds is to observe the

spin flip probabilities.

Using

invasion

percolation

_{[2] we} calculate the

flipping

rate

Rc

_nessessary for

building

_a

spanning

duster oui of

parallel spins

with

flipping

rates R < R~,

According

to _a theoretical

argument ^of

Ray

and Jan [3] the transition temperature ^of an Edwards-Anderson spin

glass (H

=

£~

~

J~jSjSj equals

the temperature ^{where the}

Rc(T)

and the

Pfrozen(T)

_{curves cross.}

Pfrozen is

ihe flipping

rate of the "weakest linked" spins, ^{1-e- the} spins with two _more

(or

D ₊

I)

nearest

neighbours ^parallel

^thon

antiparallel (D 1):

~~°~~~

1 ₊

expÎ4J/kT)

^~~~

Simulation data in 2 to 6 dimensions show

good

agreement ^of transition temperatures ^and crossing points for _spm

glasses

[3, 4].

We examine the behaviour of the _pure

ferromagnetic Ising

model _neon the transition _{as a} function of system size and dilution

using

various innervai

lengths

for

calculating

the

flipping

rates.

i134 JOURNAL DE PHYSIQUE I N°8

2. Calculation of

spin flipping

rates.

After

equilibrating

the

Ising

system

using

Glauber

kinetics,

_{we get} the

spin flip probability separately

for each site

by adding

_up the spin

flips during

_an innervai of

length

I,

getting

rates

through

division

by

I, The literature

gives

several definitions whether _a

spin flip

takes

place

or non

[3-5].

.

(A)

A spm

flip

is counted each lime _a

spin changes

its orientation.

.

(B)

A spin

flip

is

counted,

if this

spin changes

orientation in the actual lime step but has been

unchanged

in the

previous

lime step.

.

(C)

A spin

flip

is

counted,

if _a

spin changes

its orientation at _an odd

(even)

lime step.

.

(D)

A

spin flip

is counted if _a

spin changes

its orientation at an even lime step

compared

with the

previous

_even lime step.

To get ^the correct

flip probabiliy 1In

for _{a sequence} like

(iii)

of _n time steps _{up, n} time steps ^clown etc. it is _nessessary to divide

through

I for version

(D)

and to divide

through 1/2

for version

(C).

Table I compares the

flip

numbers from diflerent versions for various spin

sequences:

sequence

(A) (B) (C) (D)

lime _-

Ii) ii iii i i i ii1

2 0 _or 2

(ii)

ii i

iii ii ii1

2 2 0 _or 2 2

(iii) ÎÎÎÎÎÎÎÎÎÎÎ

3 3 1 _or 2 3

(iv) i ii iii if i i1

6 2 2 _or 4 2

Obviously,

versions

(B)

and

(D) give

the _same results for ail _sequences in statistical _average.

This fact has been verified

by

simulation data of Staufler and Jan.

Also, remembering

that the

flipping

_sums of version

(A)

_are divided

by

I while the _sums of

(C)

_are

only

divided

by 1/2,

these two versions

give equal

results in statistical _{average, too. Even} for small innervai

lengths

we found _no diflerences

using (C)

instead of

(A). According

to this

only

versions

(A)

and

(B)

will be considered further.

Within the observed temperature area

(T

<

Tc)

_{sequence (1) appears} most

frequently,

which

means that

by

_{using version}

(B)

_one gets

approximately

half the

flipping

rate of version

(A),

1-e- the

Ray-Jan freezing

formula for _version

(B)

should be modified.

jB) o-à

~ ~

p(A) (2)

ir°z~~

I +

exp(4J/kT)

^~~°~~~

3. Undiluted

Ising

mortel.

The undiluted _pure

ferromagnetic Ising

model is the

special

_case of _a spin

glass

with ail correlation coefficients

J~

₌ I. The system is uniform and ail sites have

equal rights. According

to that _one expects ^{in the}

limiting

_case of innervai

length

I _{- oc} the

flipping

rate to

equal

the average of the

flip probability

of ail spin

configurations

with respect to the temperature

dependent probability

of their appearance. In

particular

ail _spins at ail sites have _an identical

flip probability.

According

to

this,

the

Ray-Jan

_argument is not

expected

to be correct for the

Ising mortel,

because _no spin or no site _is

distinguishable, especially

the "weakest linked" _spins do not

(Ai every flip counted (Ray-Jan-formu la 0/(exp(4J/kTi+1ii +-

(Bi. only separate fhps counted (Ray-Jan-formula 0 5/(exp(4J/kTi+i ii --

i

o 99

1

«

~

0.98 _a

~ * w

(

₀₉₇ ',

"',

+ ',,

( _{)~-,____}

~ ~~~

'~"'Î---,_

~_~

095

~~~' ~

""~""-fl---_

""~

0 94

0.93

100 1000 10000

Innervai length

Fig. 1. Crossmgs of Ray-Jan _curve ^{with simulation} data

(undiluted

Ising

mortel).

05

RAY-JAN i/(exp(4J/kT)+1)

0 45

p=06 p_o~

0 4 p=08 p=09

0 35

P=0 5

o

~

0.3

'~'~

;,~~'

é

,f

"

,~'

i

,j~'

'

fl

0 25

'/

~

,Î'

0 2

>Î'

/~'

o 15

o.l ,~'+ _,,1'

~,É

o 05 ~~'

0

0 0 2 03 0A 05 0 6 0 7 0 8 09

Temperaiure T/Tc(p=11

Fig. 2. Percolation flip probabilities of diluted Ising model, version

(A),

L

= 40.

i136 JOURNAL DE PHYSIQUE I N°8

prefer special

sites and therefore the _average

flipping

rates do _not include information of the

percolation

structure of such

spins.

Instead _our simulations of finite innervai

lengths give simply

Gaussian

probability

distribu- tions. Data from 1

= 50 to1

= 50000

shows,

that the _variance of the

flipping

rate distribution vanishes with

1" ~

~~~

For

large

innervai

length (1

>

1000)

variation of the system ^size does non influence the

flipping

rates, ^{1-e- for L} > 40 _no finite size eflects _are observable.

Figure

I shows the temperatures ^{where the}

percolation flipping

rates _cross the

Ray-Jan

curves. The

crossing points

_are below Tc for ail I and decrease

monotonically

with

inceasing

I.

For the exarnined innervai

lengths

_up to 1

= 10000 _no convergence of the

crossing

temperature

can be _seen.

These results

confirm,

that the

Ray-Jan description

is non exact for the undiluted

Ising

mortel. On the other

hard,

the _crossing temperature ^{and the} transition temperature ^ailler

only slightly.

To examine, if this coincidence is

only by

chance _or Dot, we compare

flipping

rates and

Ray-Jan

transition of the diluted

Ising

mortel ai various dilution fractions.

4. Diluted

Ising

mortel.

The introduction of dilution into the

Ising

model

changes

the situation

fundamentally.

To stay in the

spin glass

formulation _we set _a fraction of I -p correlation coefficients

Jjj

of the

spin glass

Hamiltonian _{to zero.}

Figure

^{2 shows the}

percolation flipping

rates of version

(A)

at various dilution fractions _p. For each dilution innervais from 1= 100 to 1 ₌ 1000 _are examined. The

arrows

point

to the

predicted crossing

points

by Ray

and Jan at the transition temperatures

of the diluted

Ising

^{model. We refer} to critical temperatures calculated

by

Heuer [fil.

Similar to the undiluted _case the

flip probabilities

seem to cross the

Ray-

Jan _curve non

exactly

at

T~(p)

^but

slightly

^below the transition temperature. Simulation data gets ^{less reliable with}

increasing

dilution fraction and thus _very low concentrations p were avoided.

While the

flipping

rates of the undiluted

Ising

mortel

(p

₌

I.o)

_grow with

increasing

I

(below Tc),

the _curves of the diluted systems show _no

significant

variation of

percolation flip probability

_as a function of the interval

length,

at least not _near

Tc(p).

Especially

for intervals

longer

than1

= 1000 we examined _up to 1 ₌ 20000

only

statistical fluctuations of the

percolation flip probabilities

_can be found. For smaller I _some systems show

similarly

to the undiluted _case

flip probabilities growing

with

increasing

innervai

length

for T <

Tc.

Above Tc the situation tums around and

longer

innervais _cause smaller

flipping

rates. But fluctuations do _not allow _to

verify

^this effect

generally.

On the other

hard,

_a different property of the

percolation flip probability

_curves is strik-

ing:

the

percolation flipping

_curves have _an inflection point at or near

Tc(p)

for ail dilution fractions _p.

To _examine these eflects _more

precisely

_we ^observe _a system ^of size L ₌ 80 and with dilution

p = 0.9.

Figures

3 and 4 show the

flipping

rates of _versions

(A)

and

(B)

in comparison to the

corresponding Ray-Jan

_{curves near}

Tc.

The simulation data confirms that the

crossing

is

slightly

below Tc. Similar to the undiluted _case the agreement ^{of version}

(A)

with the

Ray-Jan prediction

is much berner than that of _version

(B)

but still non

perfect.

Instead the inflection point of the

flipping

_curve seems to agree with the transition temper-

ature for both versions. We found similar data for ail _p, but _near Tc statistical deviations _are

=

jjj j

0 32

,( IÎÎÎ Î

+ ^~' l=2000 _+-

j

^{1=5000 M--}

+ ,

~

_'&

~ ~~~~~~~

D

~'

^o

0 3 ;

~

jÎ

( ^./~

+

~ ^{~ ~~} rÎ'~

~~

f

+

1."'+ ^{o °}

~."~+

0 26 _{o o}

Î~

''

0.24 b

0 87 0 88 0.89 09 0.91 0 92

Temperaiure T/Tc(p=i

Fig. 3. Percolation flip probabilities (A), _p

= 0.9, L

= 80.

02

1= 100 _o

~

l= 200 ₊

+

1=500 _D

0 19 1=1000 _X

o ^-' 1=2000 +-

j ^{-1' 1' ~'~ R~ÎÎÎÎ}

0 18

~

jÎ l''

~

j

^~ o

fl

^{° ~} _~

~

oe ~

Î f

^{~ o}

~

; o

# 0 16

f'

,'Î'

o +

0 15 o

Î'

0.14 o

0 13

0 87 0 88 089 0 9 0.91 0 92

Temperature T/Tc(p=1)

Fig. 4. Percolation flip probabilities

(B),

_p

= 0.9, L

= 80.

JOURNAL DE PHYSIQUE T 4 N'8 AUGUST 1994

i138 JOURNAL DE PHYSIQUE I N°8

significant

and

especially

for

high

dilutions _no

precise

data _can be

expected,

_even non

using

very

long

innervais.

5. Conclusions.

We hâve shown that the

Ray-Jan

formula for spin

glasses, applied

here to

ferromagnets,

_{is a}

good approximation

but not exact. The

crossing

temperatures of the

Ray-Jan

_curve with the

percolation flipping

rates _are

systematically

lower than the

Ising

transition temperatures. ^At least for dilution concentrations _p > o.8 statistical fluctuations _are small

enough

to state this

rehably.

It would be desirable that similar _accuracy could be achieved in

testing

the

Ray-Jan

formula for

spin glasses.

Further, the inflection points ^{of the} percolation

flipping

_{curves agree very}

good

with the

Ising

transition. This agreement ^is

plausible

because

raising flipping

rates _are identified with _a

weakening

_{or a} destruction of bonds. At the inflection

point

_as well _{as at} the

phase

transition the number of bonds

breaking

reaches _a maximum.

Acknowledgments.

The author thanks D. Staufler and A.

Aharony

for

suggesting

this work and

Graduiertenkolleg

Scientific

Computing

and German-Israeli Foundation for support.

ILeferences

[1] Coniglio A. and Klem W., J. Phys. A 13

(1980)

2775.

[2] Staufser D. and Aharony A., Introduction to Percolation Theory

(Taylor

and Francis, London,

1992).

[3] Ray T.S. and Jan N., J. Phys. I France 4

(1993)

2125.

[4] ^Stevens M., Cleary M. and StauRer D., ta be published

[Si ^StauRer D., J. Phys. A 26

(1993)

L525.

[6] Heuer H.-O., Phys. Rev. B 42

(1990)

6476; Europhys. Lent. 12

(1990)

551.