Ising spin glasses in the effective-interaction-approximation

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Ising spin glasses in the

effective-interaction-approximation

L. Turban, P. Guilmin

To cite this version:

L. Turban, P. Guilmin. Ising spin glasses in the effective-interaction-approximation. Journal de

Physique Lettres, Edp sciences, 1980, 41 (7), pp.145-149. �10.1051/jphyslet:01980004107014500�.

�jpa-00231746�

L-145

Ising spin glasses

in the

_{effective-interaction-approximation}

L. Turban and P. Guilmin

Laboratoire de _Physique du Solide _(*), Université de _Nancy I _{Co140, 54037 Nancy Cedex,} France

(Reçu le 6 novembre 1979, accepte le 7 _{février 1980)}

Résumé. 2014

Les _{lignes critiques} des verres de _spin de Harris et _{d’Ising (± 1)} sont obtenues dans _{l’approximation} de l’interaction effective. Pour le réseau carré nous retrouvons les résultats exacts de Harris

Tc(~)/Tc(0) = 1 -

0,3116

~2

et _{Domany Tc(p)/Tc(1)} = 1 -

3,209(1 - p). La concentration critique pour le ferromagnétisme pc1 est en bon accord avec les résultats connus et tend vers sa valeur de _champ moyen pc1 = 1/2 quand z, la coordinance du réseau, tend vers l’infini.

Les réseaux non alternés ont une _ligne

_{critique rétrograde,}

_{typique pour le désordre recuit,} dans

_{l’approximation}

à une liaison. Un

_{développement}

en amas _{pour les réseaux triangulaire} et nid d’abeilles semble _{indiquer que} ce

comportement n’est pas physique dans le cas du désordre trempé.

Abstract. 2014

Using an

_{effective-interaction-approximation}

we _study the critical lines of the Harris and (±

1)-Ising

spin

glasses.

For the square lattice we recover the exact results of Harris

_Tc(~)/Tc(0)

= 1 -

0.311 6 ~2

and _Domany

Tc(p)/Tc(1) = 1 -

3.209(1 - p). The critical concentration for _{ferromagnetism} pc1 is in good agreement with known results and tends to its mean-field value pc1 = 1/2 when z, the coordination number, tends to

_infinity.

Non-alternate lattices present a _retrograde critical _{line, typical} of annealed disorder, in the

_single-bond

approxi-mation. A cluster extension of the method on the _triangular and _honeycomb lattices indicates that this behaviour is _unphysical for _quenched disorder.

LE

JOURNAL

DE

_{PHYSIQUE -}

_LETTRES

, J. _Physique - LETTRES 41 (1980) L-145- L-149 1 er _AVRIL 1980, Classification Physics Abstracts 75.10H

One of us

_(L.T.)

_recently

introduced a new

effective-medium

_approach

to the bond-disorder

_problem

in

Ising

and Potts _systems

_{[1, 2].}

This

«Effective-Interaction-Approximation »

(EIA)

is an extension

of the

_{Effective-Medium-Approximation}

for conduc-tion in mixtures

_developed

_by

Landauer

_[3]

and

Kirkpatrick

[4].

In the bond-diluted

_{Ising problem}

the EIA

_reproduces

the results of Harris

_[5]

for the initial

_slope

of the critical line

_7~)

and

_gives

the

correct behaviour

_{(7c(/? 2013}

_{~c) ~ [In (p -}

_{p~)l-1 [6])}

near _pc, the critical concentration for bond

perco-lation ;

pc itself is exact on the selfdual _square

lattice,

within 1

_%

of the exact results in two dimensions

_(2D)

and within 5

_%

of the series results in 3D. The critical line for the bond-diluted Potts model with s = 1

(percolation

limit)

on the square lattice is exact.

In this letter we _present some results

_concerning

the ferro-

_{(antiferromagnetic)}

critical line in the Harris model and the

_(±

_{I)-Ising spin}

_glass

model where the random interaction

_strength

_{K¡j = #Jij}

is

independently

distributed on the bonds

_(ij)

_according

to the two-delta

_probability

distribution :

The method as it stands is unable to

_predict

_any

_spin

glass

transition. This should not have much influence

on the 2D-results since there is now

_mounting

evi-dence that for

_{Ising spins}

the lower critical

dimen-sionality

is two

_[7].

We

_begin

with a

_presentation

of the

_method;

then we

give

some mean-field results which are valid when the coordination number z tends to

_{infinity ;}

we determine the critical line

_{Tc (P)}

for the Harris

_[5]

and

_{(± )-Ising spin glasses}

and use a cluster extension

of the method to show the limitations of the

single-bond

_{approximation.}

1.

_{Effective-Interaction-Approximation.}

- Let the

nearest-neighbour Ising

hamiltonian be written :

where the

_spin

_{variable Ui}

is

_equal

to ± 1 and the

L-146 JOURNAL DE _{PHYSIQUE -} LETTRES

random interaction

_strength

_Kij

is distributed

accord-ing

to the

_probability

law

_P(Kij).

In the

EIA,

the random system is

_mapped

onto a pure

Ising

system with p- and nontrivial

T-dependent

interaction

_strength

Km.

The effective-medium

_(EM)

hamiltonian reads :

and eq.

(2)

may be rewritten as :

The thermal _average of _any

_quantity

_Q

is therefore

_given

_{by :}

where we used the

identity,

valid for

Ising

spins :

and where ...

~m

denotes the EM-thermal average.

In the

_{single-bond approximation :}

and : o

In this

_expression

1;,,, =

Uk Ul

>m

is the

EM-first-neighbour

spin

correlation function. After a

configu-rational _average we _{get :}

provided

that Q

does not involve the random inter-action

_K¡j

and that the EM-interaction

_strength

Km

is chosen to

_{satisfy :}

As in the

_single-site

_CPA,

it _may be shown that the first

_nonvanishing

corrections to _eq.

₍₉₎

are of the fourth order in tanh

_(x)

for an

_{interaction-independent}

Q.

When

_Q

contains the interaction

_strength

_Kij,

corrections of the second order are to be taken into

account.

The effective interaction in the

_single-bond

appro-ximation

_given

_by

_eq.

₍₁₀₎

is in fact identical to the

exact

_expression

for annealed disorder for which the bonds are allowed to move

_[8].

As we mentioned

_above,

the method is unable to

predict

a

spin glass

transition.

_Using

_eq.

_(8),

we can

show that the Edwards-Anderson order _parameter

q is

proportional

toy

cri

)~,

it follows

that q

is nonzero

_only

when the _system is in the

ferro-or

_{antiferromagnetic}

state.

2. Mean-field results. - In the mean-field

approach

the critical interaction

_{strength Kc}

for a

_{ferromagnetic}

transition is

_{given by :}

and is exact when z tends to

_{infinity since. fluctuations}

are irrelevant in this limit.

We first consider the effect of bond-dilution on

_Tc.

Let q

= 1 -

p be the concentration of

missing

bonds;

when z -+ oo _eq.

₍₁₁₎

is

_changed

into :

or :

In the Harris

_{Ising spin glass}

_[5],

_P(Kij)

is centred onK:

with 11

1. Then :

and

_{finally :}

so that :

In this last case, the

ferromagnetic phase

_disappears

(r,(~)

=

0)

when

p = Pc =

1/2.

3. Harris

_{spin glass}

in the EIA. - In the Harris

spin glass

the random interaction

_Kij

is distributed

according

to _eq.

_{(14). Using}

_eq.

₍₁₀₎

with

Km

= ~

Bm =

Bc, H

=

0,

the critical values for the _pure

Ising

system, we _get the critical line :

In the limit of small fluctuations

₍₁₁

_{1 ),.}

_expending

Kc(l1)

near

_Kc

leads to :

with :

This coefficient _ac for 2D and 3D lattices is

_given

in table

I,

where it can be seen that the results of Harris

[5]

are recovered for the _square and cubic lattices. We observe that when z

_increases,

_ac tends to the mean-field

_{value ac}

= 0

(eq. (15)).

4.

_{(± l)-Ising spin glass}

in the

_single-bond

EIA.

-For the

_{( ±}

_{I)-Ising spin glass,}

the critical line is obtained as above

_using

_eqs.

₍₁₎

and

₍₁₀₎

and the

equation

determining Kac(P)

reads :

tanh K,,c (p) =

wi th :

The critical concentration

is deduced from eq.

(22)

with tanh

(Kac(P))

= 1.

The critical concentration for

_{ferromagnetism}

pcl is

_given

in table I for the _{different lattices. The square} lattice result is in

_good

_agreement with recent

perco-lation estimates

_(Domany

_[9]

_{p~ 1=}

0.826;

Vannime-nus and Toulouse

_[10]

_Pcl =

0.91)

and Monte Carlo results

_(Kirkpatrick

_[11]

_{] p~ 1}

=

0.80-0.85 ;

Ono

[12]

p~ 1 =

0.85).

The behaviour

_{of p~ 1 with-increasing}

z is also

_correctly

_predicted.

The

_properties

of the critical line near _p = 1 _can be

determined

_using

_eq.

_(22).

The

_expansion

of

_Kac(P)

near

_Kc

leads to :

with :

where

_be

is

_given

in table I for 2D and 3D lattices. For the square

lattice,

we recover

an’,exact

result of

Domany [9],

obtained

through

a variational

calcula-tion. When z

_increases,

b c

tends to its mean-field

value,

bc

= 2

(eq. (17)),

as

_expected.

Some of the features of the

_single-bond

approxi-mation are

_typical

for annealed systems where the

Table 1. - Noi-i7ialized initial

slope of

the critical line near _p = 1

for

2D and 3D lattices in the Harris

_spin

glass

(ae)

and

_(:¡:

_I)-Ising

_spin

_{glass (be).}

Critical concentrations

_{for ferromagnetism}

in the

_single-bond

_{approximation}

L-148 JOURNAL DE PHYSIQUE - LETTRES

bonds are allowed to move in order to minimize the

free _{energy. The} line of maximum

_{magnetization}

in the

ferromagnetic phase given

by :

’

and the occurrence of

multiple

transitions

(retrograde

critical

_line)

on non-alternate lattices

_{(Fig. lb)}

are not

recovered in the three-bonds

_{approximation given}

below. The

_retrograde

critical line in annealed _systems has been discussed

_by

Kasai and

_Syosi

[13].

Fig. 1. - Critical line for

ferromagnetism in the _{Ising spin glass}

(single-bond approximation : dashed line; three-bonds

approxi-mation : heavy line) for the honeycomb lattice _(a) and the triangular

lattice (b). ,

5.

_{( ±}

_{l)-Ising spin}

_glass

in the three-bonds

approxi-mation. - The EIA is

easily

extended

_beyond

the

single-bond approximation

by working

on a cluster.

We illustrate this

_{possibility using}

a three-bonds cluster on the

_triangular

and

_honeycomb

lattices.

The treatment is

_parallel

to that used

_by

one of us

in the conduction

_{problem [14].}

Eq. (5)

remains valid but the

_perturbation

on the

effective

_{medium xij}

is now nonzero on the

three-bonds of the cluster which is chosen in _agreement with

the _symmetry of the lattice

_(Fig.

_2a,

_b).

A

_simple

expression

like _eq.

₍₈₎

for

_{any Q}

is no

_longer

obtained

and one has to

_specify

_Q.

This is done here

_{using :}

Fig. 2. - Three-bonds clusters used in the three-bonds

approxi-mation on the _{triangular (a)} and honeycomb (b) lattices.

where _Ui and _Uj are _any two- of the

_three-spins

common

to the cluster and the effective

medium. Q~

0’ j

plays

the

same role in this

_problem

as the

_potential

difference

( Vi -

Vj)

in the conduction

_problem

_[14].

The effective interaction is obtained

_by

_requiring

that :

This

_equation

involves

_only

the

_{first-neighbour spin}

correlation function ~i, on the

triangular

lattice,

whereas

_{second-neighbours}

and

_four-spins

correlation functions B2 and B4 are needed on the

_honeycomb

lattice. These are

easily

obtained

through duality

and

a

star-triangle

transformation

(details

will be

given

elsewhere)

and reads :

The critical lines

_(Fig.

_la,

_b)

are obtained as above and we _get the critical concentrations :

Pc3

(honeycomb)

= 0.908 ;

~

p~3

(triangular)

= 0.806

(31)

These values are within 2-3

_%

of the

single-bond

results. On the

_honeycomb

lattice p~3 > Pel whereas

the reverse is true on the

triangular

lattice. On intuitive

ground,

one would _{expect Pe,} critical concentration for

_quenched

disorder,

to be greater than

_p*,

critical concentration for annealed

disorder,

or Pc 3 > _pcl since

pc 1 =

p*.

In fact this

inequality

is not

always

satisfied

in the

_problem

of bond dilution where exact results are known

[15, 16]

(Honeycomb :

pe = 0.652

7,

p*

= 0.647

8;

triangular :

_Pe = 0.347

3,

p*

= 0.352

2).

Notice that for both

_problems

we

_{have Pe}

_p*

on the

Fig. 3. -

Spontaneous magnetization of the Ising spin glass in the _{single-bond approximation (dashed} line) and three-bonds approximation

(heavy line) for _{positive-bonds} concentrations p = 0.9 _{(a) and p} = 0.83 _(b).

In the three-bonds

_{approximation}

the critical line is no

longer

retrograde

on the

_triangular

lattice. The

spontaneous

magnetization

was also calculated

(Fig.

3a,

b)

and the line of maximum

_{magnetization}

at finite _temperature, which is observed in annealed

systems

[13],

no

_longer

occur in the three-bonds

approximation

for both lattices.

The difference between

_quenched

and annealed disorder is

_only

taken into account when

_working

on a cluster with more than one bond so that we _expect these

_qualitative

_changes

between the two

approxi-mations to be

_significant.

To end let us mention that in the bond dilution

problem

exact results for the bond

percolation

thresholds are obtained on the

_triangular

and

honey-comb lattices

_giving

us some confidence in the

relia-bility

of the three-bonds

_approach

in the

low-tempe-rature range.

References

[1] TURBAN, L., Phys. Lett. A (1980) (to be published). Similar

approaches were _proposed earlier _by HARRIS, A. B.,

J. Phys. C 9 (1976) 2515 and LAGE, E. J. S., J. Phys. C

10 (1977) 701.

[2] TURBAN, L., J. Phys. C 13 (1980) (to be published). [3] LANDAUER, R., J. Appl. Phys. 23 ₍₁₉₅₂₎ 779.

[4] KIRKPATRICK, S., Rev. Mod. Phys. 45 (1973) 574.

[5] HARRIS, A. _B., J. Phys. C 7 ₍₁₉₇₄₎ 1671.

[6] BERGSTRESSER, T. K., J. Phys. C 10 (1977) 3831.

[7] ANDERSON, P. W. and POND, C. M., Phys. Rev. Lett. 40 (1978)

903.

[8] THORPE, M. F. and BEEMAN, D., Phys. Rev. B 14 (1976) 188,

eq. (13).

[9] DOMANY, E., J. _Phys. C 12 (1979) L119.

[10] VANNIMENUS, J. and TOULOUSE, G., J. Phys. C 10 (1977)

L537.

[11] KIRKPATRICK, S., Phys. Rev. B 16 (1977) 4630.

[12] ONO, I., J. Phys. Soc. _Japan 41 ₍₁₉₇₆₎ 345.

[13] KASAI, Y. and SYOSI, I., Prog. Theor. Phys. 50 (1973) 1182. [14] TURBAN, L., J. _Phys. C 11 ₍₁₉₇₈₎ 449.

[15] SYKES, M. F. and ESSAM, J. W., J. Math. Phys. 5 (1964) 1117.

[16] SYOSI, I., Phase transitions and critical _phenomena, Vol. I,

C. Domb and M. S. Green eds. (London : Academic