A percolation explanation for the ± J spin-glass critical temperature

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A percolation explanation for the ± J spin-glass critical temperature

Tane Ray, Naeem Jan

To cite this version:

Tane Ray, Naeem Jan. A percolation explanation for the ± J spin-glass critical temperature. Journal

de Physique I, EDP Sciences, 1993, 3 (11), pp.2125-2130. �10.1051/jp1:1993235�. �jpa-00246856�

Classification Physics Abstracts

05.50 05.70F

Short Communication

A percolation explanation for the +

J

spin-glass critical temperature

Tane S.

Ray

and Naeem Jan

Department of Physics, Saint Francis Xavier University, Antigonish, Nova Scotia, B2G 1C0, Canada

(Received

6 August 1993, accepted 16 August

1993)

Abstract We present _a simple procedure using invasion percolation for estimating the crit- ical temperatures of the +J nearest neighbor spin glass in two, three and four dimensions. The critical temperatures are found to be 0, 1.35 and 2.0 in the square, cubic and four-dimensional hypercubic lattices, which agrees with standard numerical techniques. We also estimate the crit- ical temperature for the Sherrington-Kirkpatrick ^+J model with _an analogous method which is in rough agreement ^{with the} exact calculations.

1 Introduction.

The

spin glass problem

is _one that has received continuous attention _over the last several decades. The mean-field behavior is well understood

mainly

due to the solution of the

Sherrington-Kirkpatrick

model

ill.

Much less is known

concerning nearest-neigbor spin glass

models where there is _some controversy as to the nature of the transition: is it

roughly

anal- ogous ^to that of the mean-field model _or is it _{more or} less described

by

the

droplet

model [2-

ii

?

Insight

into the

problem

_may be

gained by examining

the detailed structure of the

spin glass

as it freezes into the ordered _state. If the system ^{freezes with the}

ordering

spins

forming

_a

structure similar _{to a}

spanning

cluster in _a

percolation problem,

_one expects a critical

depen-

dence of the

freezing

_{process on} the behavior of the

spins linking large

pieces ^{of the}

spanning

cluster. These

spins

_are

analogous

to the red bonds in

percolation

clusters [8].

Based _on this

idea,

_we

develop

_an

approach

to the determination of the critical temperature of the

nearest-neighbor

+ J

spin-glass.

The Harniltonian for this model is

His]

=

~j _J,js~sj

«j

and the

J,j's

_are selected

randomly,

with

probability 1/2: J,,j

is

equal

to J and with

probability 1/2, J,j

is

equal

to -J. This model is

commonly

referred to in the literature _as the + J model.

The critical temperatures of this model _on

two-(SQ), three-(SC)

and four-

(HQ)

dimensional systems have been determined

by

various

techniques

and the values

reported

are

roughly

0, 1.3 and 2.o

respectively

_[9]. We present our method below and

apply

it to the various models and show that _our results _are in agreement ^{with those}

reported

in the literature.

2126 JOURNAL DE PHYSIQUE I N°11

2

Ordering.

The

spin-glass

transition has been described _{as a} "frozen" transition in that the

spin

at a

particular

site is correlated with itself at _a later time. One _may label

spins roughly

in terms of their

degree

of

"ordering"

of their

neighbors.

The term

"ordering"

is _{a measure} of the spin state of the

neighbors

^{times the} bond. Thus

£~ J,jsj

₌ 4J indicates that the

spin

at the central site has _a

high probability

of

being

_up while -4J _means that the

spin

has _a

high probability

of

being

down if _we consider the

SQ.

Of _{course, most}

spins

will have the

ordering

_sum less that

these extreme values.

Consider the random distribution of

ferromagnetic

and

antiferromagnetic

^bonds _on the _var- ious lattices. In two dimensions there will be _an infinite fractal cluster of

ferromagnetic

bonds and _a similar fractal cluster of

antiferromagnetic

bonds

(because

_pc

=

1/2

for bond

percolation

in two

dimensions).

In three dimensions and also four dimensions there will be

spanning

clus- ters which will be compact _{as we} are

high

above the

respective

bond

percolation

limit for these lattices. Thus within these spanning clusters there will be sites that have

only ferromagnetic

bonds down to sites with two _more

ferromagnetic

than

antiferromagnetic

^bonds. We shall label all these sites _as

belonging

to the

ferromagnetic

cluster. There will be sites that have _an

equal

number of ferro- and

antiferromagnetic

^bonds. These _we may consider _as "free

spirits"

and _as such not part of the cluster of interest.

We _{now pose} the question ^{when is such} a cluster frozen? It _seems

plausible

to label this cluster _as frozen when its weakest link is frozen. It may be _a fair indicator to consider the

probability

of

breaking

two bonds _as the definition of "frozen"

l§roz~~ ₌

exp(-2J/kT)/(exp(-2J/kT)

₊

exp(2J/kT))

in _our standard heat bath simulations.

Therefore,

when the system is frozen to the extent that

on average the

ordering

for all sites is

+2J,

the above expression ^{will be}

equal

to the measured rate at which

spins flip

out of their

preferred

orientation.

3. Critical temperature of the

spin-glass.

We consider the system ^with a set of

randomly

distributed bonds and for _a

specified

tempera-

ture _we

equilibrate by performing

_a

large

number of Monte Carlo steps with Glauber

dynamics.

A _counter is set at each lattice site and _we compare the state of the

spins

at even Monte Carlo time steps. ^If the state at _a

particular

site is different from the

previous

even

timestep

_we

increment the counter

by

_one. We considered _every other step as we wish not to double count those

spins

which

flip

and then return at the next step to their

previous

state. The data _are collected at these sites for at least 40,coo Monte Carlo time steps at the

high temperatures

^and about 200,000 Monte Carlo

timesteps

at the low temperatures. The number of Monte Carlo steps was sufficient to _ensure the system _was

equilibrated

_as evidenced

by

the measurement of

static quantities

(e.g. specific heat).

At _every site _we compute ^the

probability

of

flipping.

This is defined _as the number of

flips

divided

by

the number of

configurations

checked. Every site is

assigned

_a number identical to the

probability

of

flipping and, using

an invasion

percolation

method _on these numbers [9,

lo],

the

probability

threshold to obtain _a

spanning

cluster of sites is determined.

Thus,

all sites which

flip

with _a

probability

less than the threshold _are part

of

the

spanning

cInster. This is shown

by

the diamonds in

figures

I to 3. We have also

plotted

_on the _same

graph

the

above-mentioned theoretical

probability

Pfraz~n.

For the

SQ

two-dimensional system we note that the

percolation

threshold is

always

above p~~~~~~

(see Fig. I) although

_we cannot exclude that these _{curves may} intersect below T ₌ o.3

(jn

units where

k/J

=

I).

^Thus we conclude that the "frozen" _or

spin-glass

transition in two dimensions is about T

= o. In three dimensions the results _are rather clear cut

(Fig. 2).

At

high

Pc _vs T 2-d spin-glass I

, o o

, o

~

0. I *

o

~ + _:.'"

o _:"

0. 01

~

;.'

+ ;"

+ ."

0. 001 ./

+ ;.'

o. oooi

~

le-05

le-06

0 0.5 1 1.5 2 2.5 3 3.5 4

Temperature

Fig. 1. Probability threshold for the spanning cluster _versus temperature for the two-dimensional Ising model. Data for

a system with length 32 _are given by diamonds; system with length 64 _are given by plusses. ^{The dashed line is the} freezing criterion.

temperatures the

spanning

threshold is above the Pfraz~n while at low temperatures ^the reverse

is true. The cross-over occurs at about 1.35. At this temperature the

flipping probability

within _a

spanning

cluster is

equal

to the weakest link in _our

ferromagnetic

_or

antiferromagnetic spanning

clusters.

Figure

3 shows the

equivalent

data for the four-dimensional spin

glass.

Here the

corresponding

_curves intersect at about 2. Both the three and the four-dimensional critical

temperature _are ⁱⁿ

good

agreement with other numerical data [11].

We have also tried _to

apply

the method to the +J version of the mean-field

Sherrington- Kirkpatrick

model

ill.

The correct

limiting

behavior for the model _occurs when the bond

strength equals N~~/~

_{where N is the total} number of

spins

in the system. In these

units,

the

critical temperature Tc ₌ I. The fact that all

pairs

of spins ⁱⁿ the system interact

implies

that the

previous

^{criterion for the}

freezing

transition in

nearest-neighbor

models must be modified.

Instead of

taking

the

ordering

_energy to be +2J

(representing

_{an excess} of two bonds and

neighboring

spins

dictating

the

preferred

orientation of _a

spin),

_we find that the

ordering

scales _as

N~/~J/2

with J

=

N~~/~

_This

quantity

^is a first

approximation

to the

typical

number of _excess

ferromagnetic

_or

antiferromagnetic

bonds in the

paramagnetic phase which,

when the

respective spins

are

correctly oriented,

will

give

_a

preferred

orientation to the

spin

in

question.

In

addition,

_we must

modify

the

percolative analysis

because site

percolation

is not defined in this model. On the other

hand,

bond

percolation

will

give

a nontrivial

connectivity

in

this model

[12],

so we

assign breaking probabilities

to the bonds rather than the sites. To be precise, we take the

product J,js,sj

for every bond and check _{every even} Monte Carlo step

to _see whether the status has

changed

in _{a manner}

analogous

to the

nearest-neighbor

studies.

The number of

changes

to the bond status is then divided

by

the number of times the bond

was thecked. Then the

probability

threshold value is determined which connects half of the

2128 JOURNAL DE PHYSIQUE I N°11

Pc vs T 3-d spin-glass

I

, o

* .

________

,,,..-.-.--..

o __,,,..-...

."""""""'

+ ,,:...."""

0. I ,,.."

f__:..

~.""

_____~6

£ 0 01

.""~

o.ooi S

0.0001

0.5 1 1.5 2 2.5 3 3.5 4 4.5

Temperature

Fig. 2. Probability threshold for the spanning cluster _versus temperature for the three-dimensional Ising model. Data for _{a run} time of100,000 Monte Carlo steps are given by plusses; _run time of

200,000 MCS _are given by diamonds. The length of the system _was 32 spins. The dashed line gives

the freezing criterion.

Pc vs T for 4-d spin-glass I

, o

~ 4 ^~

o-1

~

+

°

0 01 "'

"

/

~

l' +

~ ^'

o,ooi

+

o.oooi

le-05

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Temperature

Fig. 3. Probability threshold for the spanning cluster _versus temperature ^{for the} four-dimensional

Ising model. Data for _a system with length lo spins is indicated by plusses; length 16 is indicated by diamonds. The dashed line gives the freezing criterion.

spins

in _a

single

^{cluster. This} value of

1/2

is somewhat

arbitrary;

the critical value of _p in the infinite volume limit will be the _one which first results in _a connected network of

spins occupying

_a ^{finite fraction of} the system.

Pc _vs T for s-K spin glass

1

+ + ~_____,~___.,_~,.,.,.~..,,,.~,.,,.,~.,,., __,l...W""'*""'

,,-+"' °

__:." o

o-1 ;.'4

.~o

j +

of

I _{0.01 ./}

o.ooi

+

0.0001

0 0.5 1 1.5 2 2.5 3

Temperature

Fig. 4. Probability threshold for the spanning cluster

(bond

percolation) _versus temperature ^for the Sherrington-Kirkpatrick model

(diamonds).

Average flipping probability for sites is indicated by plusses. For this system the number of spins N

= 100

(a

larger system ^{with N} ₌ 300 yielded the

same results for

a few data

points).

The system _{was run} for 500,000 MCS. The dashed line gives the

freezing criterion.

Figure

4 shows the mean-field results for two systems

having

a different total number of

spins.

In addition to the

percolative plot

for each system, a

plot

determined from the site-

flipping

data is shown. Here, the

probability

for

flipping

sites _was measured _as in the nearest-

neighbor

_runs. A threshold

probability

greater than that of half of the sites _was calculated.

Because it has

nothing

to do with the

connectivity

of the system, this

quantity

is of interest in that it proves ^to be not too different from the bond

plots.

The _reason for the agreement _can be traced to the fact that both the site

flipping

and bond

breaking probabilities

_are distributed

with _narrow

roughly

Gaussian

peaks.

4. Discussion.

The data from the

nearest-neighbor

studies _seem to indicate that _our

freezing

criterion is

capturing

_some of the

physics

of the

phase

transition in that it _crosses the

experimental

_curves

at the

spin glass

critical

point

_as found in

previous

work. The mean-field

plots

_are not as

convincing,

and it may be that there is too much

ambiguity

in _our

procedure

to

precisely

determine the critical point. Another

possibility

is that the mean-field

spin glass

is much different in its

freezing properties

than in the finite dimensional models.

Above Tc, the data in all of the

plots

are greater than the

freezing

criterion. This indicates that the

spins

_are

changing

orientations

quite

often. These

spins

influence their

neighbors

to

2130 JOURNAL DE PHYSIQUE I N°11

flip

_more

frequently

than that

given by

the

freezing

criterion. There _are few

spins

which have

an

ordering

_energy

greater

^{than that inserted into the}

freezing

criterion. One would have _to raise the

probability

threshold

higher

in this _{case to} obtain _a

spanning

cluster of

spins,

which is what is observed. Below T~, in three and four dimensions the

spins

should

predominantly

^be

part ^of

spanning

^{clusters made} _up ^of

mostly

^either

ferromagnetic

_or

antiferromagnetic

bonds.

On _average, the

ordering

should be

higher

than the

ordering

_energy ^{value chosen} in the

freezing

criterion which will make the

probability

^threshold

generally

^lower.

In the

nearest-neighbor models,

_{we may}

classify

the

spin

environments

according

to several

categories.

^There are those which _are surrounded

by

_{as many}

indicating spin

_{up as}

spin

down.

Let _us refer to these _as the "free

spirits".

Then there _are those who _are

strongly

constrained

by

their enviroments the

"strongly-bonded";

and

finally

there _are those who _are

minimally constrained, "quasi-bonded"

I-e- those who have _an energy of 2J and these contribute _to

Pfraz~n.

In _our scenario the

spin-glass

state is attained when the

spanning

clusters of either

ferromagnetic

_or

antiferromagnetic

bonds _are frozen. This would _occur when the weakest links in them become frozen and this _{we assume occurs} when these

spins only flip

with _a

probability

equivalent

to their

neighbors

in the ordered _states.

There _are several papers in which the spin

glass

is described

along

the lines of this paper.

Recently Liang

_[13]

proposed

_a ^definition of clusters in which the

empirically

deternJined

flip- ping probability

is taken into account and this _was used to reduce the critical

slowing

down in the two-dimensional

spin glass.

We

applied

these

probabilities

here to site

percolation;

Stauffer [14] ^{used bond}

percolation

but then it _{was more} difficult to make the connection with the crit- ical temperature. The earlier work of Kinzel [15] ^is

perhaps

closest in

spirit

to our

approach.

We _were able to add _a Pfraz~n threshold that _was able _to detect the transition in two to four dimensions.

There _are several

queries

and questions ^that occur. Will the _same

properties

hold in five and six dimensions? Is it

possible

to extend this

investigation

to learn _more about the nature of the

spin-glass phase

in finite dimensions?

Acknowledgements.

We~wish to thank Gene

Stanley,

Sri

Shastry,

Antonio

Coniglio,

D. L. Hunter and

especially

Dietrich Stauffer for discussions and encouragement ^and

making

_us aware of _some of the earlier

work,

results and

pitfalls.

This research is

supported

in part

by

NSERC of Canada.

References

Ill

M6zard M., Parisi G. and Virasoro M-A-, Spin Glass Theory ^and Beyond (World Scientific Pub,

1987).

[2] ^{Binder K.} and Young A-P-, Rev. Mod. Phys. 58

(1986)

801 and references therein.

[3] Chowdhury D., Spin Glasses and Other Frustrated Systems

(Princeton

University Press,

1986).

[4] Fischer K-H- and Hertz J-A-, Spin Glasses

(Cambridge

University Press,

1991).

[5] Fisher D-S- and Huse D-A-, Phys. Rev. Lett. 56

(1986)

1601.

[6] Bray A-J- and Moore M-A-, Phys. Rev. 31

(1985)

631.

[7] Mcmillan WI., Phys. Rev. 31

(1985)

344.

[8] Coniglio A., Phys. Rev. Lett. 46

(1981)

250.

[9] Wilkinson D-J- and Willemsen J., J. Phys. A16

(1983)

3367.

[10] ^{Wilkinson D, and} Barsony M., J. Phys. A, 17

(1984)

L129.

ill]

Klein L., Adler J., Aharony A., Harris A-B- and Meir Y., Phys. Rev. B 43

(1992)

11249.

[12] Ray T-S- and Klein W., ^J. Stat. Phys. 53

(1988)

773.

[13] Liang S., Phys. Rev. Lett. 69

(1992)

2145.

[14] ^Staufler D., J. Phys. A26

(1993)

L525.

[15] ^Kinzel W., Z. Phys. 46

(1982)

59.