Constrained Annealing for Spin Glasses

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Constrained Annealing for Spin Glasses

Giovanni Paladin, Michele Pasquini, Maurizio Serva

To cite this version:

Giovanni Paladin, Michele Pasquini, Maurizio Serva. Constrained Annealing for Spin Glasses. Journal

de Physique I, EDP Sciences, 1995, 5 (3), pp.337-354. �10.1051/jp1:1995130�. �jpa-00247059�

Classification Physics Abstracts

05.50 02.50 75.10N

Constrained AnneaEng for Spin Glasses

Giovanni

Paladin(~),

Michele

Pasquini(1)

and Maurizio

Serva(2)

(~) Dipartimento di Fisica, Università dell'Aquila, 1-67100 Coppito, L'Aquila, Italy (~) Dipartimento di Matematica, Università dell'Aquùa, 1-67100 Coppito, L'Aquila, Italy

(Received

3 October 1994, accepted 29 November 1994)

Abstract. The quenched free energy of spin glasses _con be estimated by _means of annealed averages where the frustration _or other self-averaging variables of disorder _are constrained _to their average value. We discuss the _case of d-dimensional Ising mortels with random nearest

neighbour coupling, and for +J spin glasses _we introduce _{a new} method to obtain constrained annealed _averages without recurring ^to Lagrange multipliers- It _requires to perform quenched

averages either

on small volumes in

an analytic _{way, or on} finite _size strips via products of random transfer matrices. We thus give a sequence of converging lower bounds for the quenched

free energy of 2d spm glasses-

l. Introduction

Disordered systems are made of random variables which relax _to

equilibrium

_{on very} diiferent time scales. For instance in the standard models of

spin glass,

the spins _arrange ^themselves

to minimize the free _energy while the random

couplings

_are assumed to be frozen in their initial

values,

since their evolution time is much

longer.

As _a consequence, one should consider diiferent kinds of _averages:

quenched

_averages for the frozen

variables,

and annealed _averages for the fast variables.

However,

it,is much easier to compute annealed _averages, which

#ve

lower bounds of the

quenched

free _energy, the

physically

relevant

quantity.

These bounds

are often _{very poor}

estimates,

since in annealed _averages the main contribution _comes from _a set of disorder realizations which bave _zero

probability

in trie

thermodynarniclimit.

In those

realizations,

trie

quenched

random variables _are

arranged

in order to minimize trie free _energy, at diiference with _a

typical sample

of trie system where

they

_are ^frozen in _a

given

realization.

Trie idea of constrained

annealing

is to

perform

annealed _averages where trie random _vari- ables _are

partly

frozen

by

trie requirement of

satisfying appropriate

constraints, ^related to some

disorder

self-avera#ng

variables. Trie standard _{way to}

impose

_a constraint is

by

_a

Lagrange multiplier. By

that

method,

trie constrained annealed _average allows _one to introduce _a sort of Gibbs

thermodynamic potential, depending

_on trie temperature ^and on trie

Lagrange

^mul-

tipliers

which

play

the role of trie chernical

potential

in

ordinary

statistical mechanics [1]. ^In the

thermodynamic

limit, the values of the

multipliers

that maximize the Gibbs

potential

_are

those ones that select the realizations with _a correct value of the disorder _intensive

variables,

@ Les Editions de Physique 1995

and,

at trie _same

time,

that minimize the diiference between trie _mean free _energy

density

and its annealed

approximations.

Constrained

annealing

for disordered systems bas been

proposed

and

applied

in

particular

contexts

by

_many authors

(see [2-6]).

A

general

formulation of trie method and its

physical

meaning

_can be found in [1].

Trie _purpose of this paper is to show that it is

possible

to

perform

constrained annealed averages without

Lagrange multipliers,

_m d-dimensional

spin glasses

with dichotomic random

couplings

in terms of _a

quenched

_{average over a} finite size

sub-system.

This is often

possible

either _{m an}

analytic

_{or m a} numeric way.

Using

_our

method,

_{we are} able to obtain _very

good

lower bounds of trie

quenched

free _energy of _two dimensional spin

glasses.

Trie summary of trie _{paper is} trie

following:

In section 2, we recall _some basic facts about spm

glasses.

We also

classify

the set of trie constraints necessary to recover trie

quenched

free _energy

by

annealed averages. In section 3,

we

study

trie

Ising

model with

couplings

which _{con assume} trie values J~j = +1 with

equal probability (the

sc-called +J spin

glass).

In section 4, we introduce our method to obtain

constrained annealed _averages without

recurring

to

Lagrange multipliers

in trie +J spin

glass.

In section 5, _we

apply

trie method to

loops

of infinite

length using

trie

Lyapunov

exponent of

product

of random transfer matrices. We _are thus able to get accurate estimates of trie free

energy in two dimensions.

2. Constraints for d-Dimensional

Spin

Glasses

Spin glasses

_are trie

challenge

of constrained

annealing.

In that _case, trie relevant variables to be constrained should be related to frustration [7], as first remarked

by

Toulouse and Vannimenus in 1976 [5]. As far as we

know,

there _are

only

^Montecarlo

simulations,

_{see e.g.}

[8,9],

which used

that idea while _no

analytic

calculations bave been

performed

_on that

ground.

However, it is

possible

to get

analytic

results _m dimension d > 2 with constraints connected to frustration.

Let _us consider d-dimensional

Ising

models with nearest

neighbour couplings

_J~j which _are

independent identically

distributed

(1.i.d.)

random

variables,

in absence of externat

magnetic

field. _Trie

partition

function is

zN

=

~ fl expjpJ~jaiajj (2.1)

lai i J

where trie

product

_runs

over1, j

which _are nearest

neighbours. Using

standard relations of trie

hyperbohc functions, (2.1)

becomes

ZN "

~ fl cosh(flJ~j) il

+ a~aj

tanin(flJ~j) (2.2)

lai '>J

This form is useful _since shows that trie non-trivial part ^of trie partition function is

given by

fl,

~

il

+ a~aj

tanin(flJ~j

)]. ^A

typical

term m that

product

is a~ajaj ai

tanin(flJ,j tanin(flJji)

where trie bonds

(1, j ), ( j,1)

_can form either _a closed _{or an} open line _on trie lattice. If trie bonds form _an open

line,

at least two distinct spin ^variables survive _as factor of trie

product

of the various

tanh(flJ,j ),

and the related term

gives

_{a zero} contribution to the partition function,

after

jerformmg

the _{sum over} the _a

configurations.

As _a consequence, one bas _to consider

only

trie terms

corresponding

to bonds which form _a closed fine

(where

trie

product

of trie a's is

equal

to

unity

_smce

they

_appear

twice)

_so that

(2.2)

_con be written _{as a sum over} all trie

possible loops

LÎ~~, where _T > 4 indicates trie

length

of trie

loop

_{(T bas} to be

even)

and _s labels

diiferent

loops

with the _same

length:

ZN

"

eÉ.,>

^~" ~°~~~~"~

2'~

l

+

~j fl tanh(flJ,j) (2.3)

lL[~>j j~,jjeL[~>

It follows that in _a system ^with a finite number N of

spins,

the free _energy of _a disorder realization is

given by

y~ ⁼ in z~ ₌ B +

fl~ _(t(4)

~(Rj _~~

~)

flN fl

^{' '}

where R is the maximum

possible length

of _a

loop

in the system

(R

_{< const}

N),

and

B = In 2 +

~j

In

cosh(flJ~j (2.4a)

~

>,J

and

~~~~~~~' '~~~~~

Î

_~~

~

~

~Î~~

(2.4b)

jL[~>j This function

depends

_on the _new set of variables

t(~) ₌

fl tanh(flJ~j) (2.4c)

j~ ~j~~(r>

given by products

which _{run on} the

loops

LÎ~~, 1-e-

plaquettes

of

larger

and

larger perimeter

at

increasing

_T. For instance, the

loop

^L(~) is the

elementary plaquette

of four bonds. Let _us

stress that _a

loop

^L(~) _can be built up as a set of _two

(or more)

distinct and separate

loops

L(~'l

_and L(~2) _with

Ti + _{T2 "} T. In the

following

_we often label _a

coupling by only

_one label

mstead of _{two ones,} 1.e. Jki is denoted

by Jk,

since there is _no risk of

ambiguity

_on a closed line.

As the number of the nearest

neighbour couplmgs

in _a

hypercubic

d-dimensional lattice is dN, in the

thermodynamic

limit the

quenched

free _energy reads

f

e lim @ _" d In

cosh(flJ~j

+ F ₊ In 2

(2.5)

N-CC

fl

where

F = lim

j _lui

₁

+

~j

_tÎ~~

_(2.6)

~~~°

j~(r>j

The reduction of the

partition

^function

(2.1)

into

(2.3)

is the

procedure

used in the Kac and Ward solution of the 2d

Ising

model without disorder.

Equation (2.5)

is fundamental since it shows that the

quenched

free energy

depends only

on the distributions of the variables

t(~),

apart the _mean value In

cosh(flJ,j).

The relevant variables to be fixed to their _~r~ean value _{m a} constrained annealed _average of the free _energy

are

related,

with

increasing difliculty 1)

In cosh

flJ~j

2) t(~l

=

tanh(flJi) tanh(flJ2) tanh(flJ3) tanh(flJ4)

3)

^t~~l with _T > 6

(considering only

connected

loops).

Fixint

all the constraints

corresponding

to those

quantities

in _an annealed _{average is}

equiv-

alent to solve the system ^with

quenched

disorder. This _can be understood

noting

that in

(2.6)

we have _{a sum over} ail the

possible

tÎ~~. Because of the law of the

large

numbers the rescaled

sum of the t-variables

corresponding

to the _same

topological

kind of

loops self-averages

_to its

quenched

_mean value.

Moreover,

in the

appendix

we shall _prove that this result holds _even in _a finite volume system ^for a +J

spin glass:

the

quenched

free _{energy can} be obtained

by

a minimization of the

logarithm

^{of the} annealed average of _a

Grand-partition

function with respect to the

Lagrange multipliers

related _to the variables tÎ~~ over ail the

possible loops

of the finite system. For _a

generic

distribution of the

couplings,

an extension of this statement is

possible.

In _one dimension there _{are no}

loops,

_so that the constraint related to

point 1)

is suflicient to obtain the exact solution. In

general,

_we expect that many

physical

features of _a

spin glass

can be

reproduced already by considering

the first two

points.

The _case of random

couplings

_{J~j =} +1 with

equal probability

_is

particularly simple

in this respect, since the variable In cosh

flJ,j

is _a constant and thus _can be

ignored. Moreover,

_one

immediately

_sees that

(2.4c)

becomes

t(~) =

tanh~(fl) fl

_J~j

_(2.7)

j, jj~ ~(r>

implying

that the

quenched

free _energy

only depends

_on the values of the

product

of

J,j

_on

closed fines of bonds. The _non trivial part of the free _{energy yN m}

(2.4)

is thus

FN

_"

ln(1

+ x4

tanh~ là)

+ x6

tanh~(fl)

_{+ .)}

(2.8)

N

where the coefficients

(x4,

_x6;.

)

_are

non-mdependent

random

quantities.

^As _a consequence, the

quenched

free _{energy is}

f

=

_fl (d

^{In cosh}

^fl

^{+ In 2 + lim}_{N-no éÎ}

^ln(1

^{+ x4}

^{tanh~ (pi}

^{+ x6}

^{tanh~ (fl)}

⁺

^(2.9)

In the annealed model, the last non-trivial term of

(2.9)

vanishes since n

= o for ail i's

(each

x, ^is a sum of

products

of

independent

random

couphngs).

In the limit of

high

temperature

là

_-

o),

the diiference between annealed free _energy and

quenched

free _energy thus is of order

tanh~ fl

~J

fl4.

More

important,

^the

simplest

relevant constraint in the +J

spin glass

is the frustration

Jp

_on the

elementary

_square

plaquette

^{of four}

couphngs Ji, J2, J3, J4,

that is

Jp

₌ -JiJ2J3J4

(2.lo)

since t(~)

=

-Jp tanh~ fl.

3. +J

Spin

Glasses

In this section _we consider _a d-dimensional spin

glass

with 1.i.d. nearest

neighbour couplings

which _assume two

possible

value

J,j

₌ +1 with

equal probability.

From

(2.9),

_{one sees} that the annealed free _energy

density

is

In _one

dimension,

there _{are no}

loops

and

fa

is

equal

to the

quenched

free _energy

density f,

since In

coshflJ,j

is _a constant, at diiference with the Gaussian _case where that variable _is random and has to be constrained in order to _recover

f [loi.

In section

2,

_we have recalled standard results which show that it is _necessary to constrain variables associated to the

loop

variable

t(~),

in order to

improve (3.1).

^{As first} noted

by

Toulouse and Vannimenus [Si, one

has to consider the intensive variable

a~r=

j _{j (}

13.2) j~j,>j

which has _{zero mean}

value,

Wi _" o. Let _us remark that it is _a

self-averaging quantity,

that is

limN-co

_ON

" o for almost ail disorder realizations. The related constrained annealed _average

con be

regarded

_as the

Grand-partition

function

iiN(à>

_ÎL) ~

ZN

e~'~""N

In the

thermodynamic

limit the

Lagrange multiplier

_p that maximizes

~~~'~~

ÀÎCO flÎÎ

^{~~ ~~}

^~'~~

fixes the frustration _on square

plaquettes

to its _mean value. Moreover it is

simple

to show the

foiiowing inequality:

f

~

Ill

_{~ ~°~~}

girl, PI

In _{our case, we} have to compute

QN(fl, _Al

"

~exp(~j _flJ,ja,aj ~j _{pJp) (3.3)}

(a) _i,j j

~l')j

Taking

into account that the

coupiings

_are dichotomic with

equal probabiiity,

_{one con}

perform

the gauge transformation J~j -

Jjja~aj

which leaves

unchanged

the free energy of _a disorder realization of the system as well _as

Jp,

1-e-

Jp

_-

Jp.

As _a consequence,

(3.3)

becomes

iiN(à,

_ÎL)

"

~~ eXp(~ flÀj ~ _{pJp) (3.4)}

1,j

j~l')j

This expression

(which

is

meaningful

for _any dimension d >

2)

cannot be

computed exactly including

all the

possible

_square

plaquettes

LÎ~I of the d-dimensional lattice. It _can be shown that the

problem

is

equivalent

to solve _a d-dimensional _gauge model without

disorder,

whose

solution _is not known for d > 2.

However, _an exact solution _can be achieved if _we restrict the _sum

(3.4) only

_{on a} part

(of

order

N)

of the

plaquettes.

In other terms the constraint is

imposed

_{over a} fraction of the

plaquettes,

that is _{we are} able to

partially

fix the

plaquette

frustration to its _mean value.

As _a consequence, the annealed system exhibits _a "residual" frustration related to the _non- constrained

plaquettes.

Let _us consider the 2d model. It is convenient to limit ourselves to consider _one half of the

plaquettes,

which must be chosen in such _a way that

they

do not share _any

coupling,

as it

happens

for the black _squares of _a chessboard. With this restriction

(3.4)

contains _{a sum over}

-1

_j .-.-.-.-.-.-.-. d=2

j=±1

~=--,

f'~ /~

i _/

~ /

Î

~ /

/ /

/ f ^sq

Î / f~~

~

/ f ^cr

ac

0 0.5 1-5 2

Fig. 1. +J spin glass _m 2d: the annealed free energy fa

(dashed

fine) and the constrained free _ener- gies fil

(fuit

fine) and fil

(dot-dashed

fine) _versus temperature T

=

1là.

The constraints

are imposed, respectively, _on

N/2

independent plaquettes and _on

N/8

independent _crosses of rive plaquettes.

products

of

N/2 independent

variables

(corresponding

to trie

N/2

black

plaquettes)

and trie

Grand-partition

function _can be rewritten _as:

QN

(à)

_"

^2'~ (exp(fl(Ji

^{+ J2 +}

^J3

⁺

^J4)

⁺

^{pJi J2J3J4} ))

^~~~

^(3.5)

where Ji

,

J2

,

J3

,

J4 _are trie

couplings

of _one of trie

plaquettes.

After

maximizing

with respect

to p trie Gibbs

potential

_{[1], one} obtains

f])

₌ In 2 + In cosh

2fl

₊

In(

3 +

cosh4fl ) (3.6)

fl

2 4 4

The functions

fa

and

fil

of trie 2d spin

glass

_are shown in

Figure

1. At diiference with trie annealed _case trie entropy of trie constrained annealed system _never becomes

negative.

We bave thus obtained _a real

qualitative

improvement.

It is also

interesting

to consider trie heat capacity ^{C which} can be

explicitly computed.

For

fl

_{- co one} has

C

~J

fl~e~~P (3.7)

The argument ^of the

exponential

has been

conjectured

in

Ill]

for the

quenched

model and it is diiferent from the exact one-dimension result where the argument is

-2fl.

Note that the

annealed system also

gives

C

~J

exp(-2fl).

Finally,

since _we have fixed

only

_one half of the

single plaquette frustration,

it is reasonable to ask what is the residual frustration #a~ Since the black

plaquettes

have _zero

frustration,

this

quantity

will be _one half of the frustration of _a white

plaquette.

The four

couphngs forming

a white

plaquette

_are

independent

_smce

they

_are not

coupled by

the

Lagrange multiplier.

Therefore, the white

plaquette

frustration _can be written as « /

»~,

and the

global

frustration #a~ reads:

#a~ ₌ ^~ « Jj

»~ (3.8)

2

/ / /

/

~

'( ^o.

(

_/

j

É j j]1

/

/

~

/ 0

ig. . -

+J pin

lass

for the mortel. The

constraint is imposed on N/2

plaquettes.

where « . » is an verage on _{the Gibbs} measure associated

(3-3).

One ~ _~~~ ~~ ~~~

~~

~ ~~ ~

2~ôÎ~ ^~'~~

where p* is

alue

which

^aximizes

total umber of ^ouplings is 2N. Since ri(4) " _9(4,

Îl~(4)1

and

since

the

_derivative

Using the xpression (3.6) for fil we

can

compute the _residual

~~~

~ Î

~~

~ ~~~ ~ Î Î~~~Î~~ ~ 4 ~ÎÎÎ~Î2fI)1~

At

^ture #a~ =

3~ o.158.

^This

esult

indicates ^trie

non _vanishing residual

frustration.

An alogous

xpression can be found from trie

annealed

free

energy

fa. In this

case ail

plaquettes can bave _negative rustration and

#a =

-

« Ji _»~ =

-th~(fl)

(3.12) here

trie is in

trie

Gibbs

measure

_generated

#a(T = o) = -1 since at _zero temperature trie^{spins are} aligned and trie

is

Both #a and _{#a~ are} reported ⁱⁿ Figure 2.

Trie

fact that in

trie

constrained annealed

^model trie residual rustration does

not

anish indicates that trie

free energy estimate could be improved by further

constraints

as we shall

discuss section.

4. Constraints without

Lagrange Multipliers

As far _{as we}

know,

_up to _now constrained

annealing

bas been

performed only using Lagrange multipliers.

This is _a strong

limitation,

since in most _cases it is too diflicult _to derive

analitically

an annealed _average

using

_more ^than one constraint. Trie

spin glass

^{with dichotomic}

couplings

bas _some

algebraic

features which

permit

_us to introduce _an alternative

method,

in order to obtain constrained annealed _averages

avoiding

in this way trie

problem

of trie minimization of trie

Lagrange multipliers.

Let _us consider _a d-dimensional lattice of N points: trie total number of trie

couplings

is

dN,

but trie system can be

fully

described

by

_a lower number of

independent

random variables. In fact trie non-trivial part ^{of trie}

quenched

free _energy

(2.6)

is _a function of trie variables (tÎ~~

),

see

(2.4c)

,

defined _on trie closed

loops

of trie lattice.

Indeed,

from _a

topological point

of view any closed

loop

LÎ~~ _can be

thought

_as trie union of _some

elementary

_square

plaquettes,

_{say a}

number

k,

_so that trie related variable

tl~l

_can be written _as:

k

t(~) =

tanh~(fl) fl

_J)~)

_(4.1)

1=1

where _we bave introduced trie frustrations

Jp

₌

fl)~~

J, ₌ +1 with

equal probability

of trie k

plaquettes (trie

index1 _{runs over} trie four sites of each

square).

Let _us stress that _a

loop

_can

be either connected _or

disconnected,

1-e- it is not necessary that trie

plaquettes

_are

neighbours.

Notice that trie

plaquette

frustration _was

previously

defined with _a minus

sign (sec (2.lo)),

but

in this context we

neglect

it to

simplify

trie notation.

Moreover,

we dot not _assume

periodic

boundary

conditions in this section.

In other terms, we con consider trie

plaquette

frustrations _as trie random variables of trie system, ^{instead of trie}

coùphngs,

with _no loss of

generality.

We shall see that trie number of

frustrations _necessary for _a fuit

description

of trie system is

always

lower than

dN,

trie total number of

couplings.

For instance, ⁱⁿ

2d,

there _are 2N

couplings,

while trie number of

plaquette

frustrations is of order N when N _{- co.}

However,

trie 2d _case exhibits _a

special feature,

since ail trie

possible plaquette

frustrations turn out to be a set of

independent

random variables

([et

_us recall that _we do not _assume

periodic boundary condition).

In trie

general

_case d > 3, this _statement is _no

longer

valid.

Indeed,

^[et _us consider _a cube in _a d-dimensionai iattice: _a moment of reflection shows that trie

product

of trie reiated six

plaquette

frustrations bas _to be 1, _so that

only

rive frustrations _are

mdependent

random variables. As _a consequence, for d > 3 trie

change

of random variables from trie

couplings

to trie

plaquette

frustrations invoives

oniy

_a subset of

them,

[et _{us say}

N*,

chosen in such _a way that

they

_are

independent

random variables and that

they

_can build up every

possible

closed

loop

_on trie lattice.

Let _us illustrate this

change

of random variables for _a 3d lattice of N

points,

where _{we can} introduce three

integer

coordinates

(x,y, z).

A square

plaquette

has three diiferent orienta- tions,

belonging

to a x = const

plane,

_{or y}

= const, or z = const. In the

thermodynamic

limit

one has

Nl/~ planes

for each orientation, and each

plane

is a set of

N~/~ plaquettes,

for _a total number of 3N

plaquettes

in the whole lattice.

Consider _a cube of _six

plaquettes,

with _one of them _on the _z

= o

plane.

As

previously discussed,

_one of the J's _cannot

belong

to the subset of

plaquette

frustrations that describe trie whole system, ^[et us say trie

Îrustration

_{of trie}

_plaquette

on trie z = 1

plane.

Let _us

now consider trie other cube

sharing

this

"neglected" plaquette. By introducing

_as random variables of the system the frustrations which _are related _to the four

plaquettes

that share _a

coupling

with the

plaquette

_on the _z

= 1

plane,

_we have to

neglect

the sixth frustration of the

plaquette

_on the _z

= 2

plane,

since the

product

of ail the

plaquette

frustrations of these two

nearest

neighbour

cubes _is fixed to

(the

_presence of the

plaquette

_{on z}

= 1 in trie

product

is

irrelevant,

_since it appears

twice).

These arguments con be

easily repeated,

_so that trie subset of

plaquette

frustrations that

fully

describes trie whole system can be found _as follows: _one bas to consider _two distinct orientations and to take into account ail trie

plaquettes

with these

orientations,

for _a total number of 2N.

Moreover,

trie third orientation

gives only

_a

plane

of

N~/~ plaquettes (trie

_z

= o

plane

in _our

example),

_so that in the

thermodynamic

limit trie number of

independent

random frustrations N* that

fully

describe trie systems _{grows as} 2N.

Trie

generalization

of this result is immediate: in trie

thermodynamic

limit trie d-dimensional lattice of N points

(with

d >

3)

is built _up

by

N distinct

plaquettes,

since is trie number of diiferent orientations for _a

plane

_m trie

lattice,

but

only

_a number N*

= 2N

of

plaquette

frustrations is necessary ^to describe trie system. ^{In fact} trie

change

of random variables involves

only

the frustrations of the

plaquettes

with two well-defined orientations

(2N plaquettes), together

with _a

plane

of

plaquettes

for each _one of trie other

orientations,

but

m trie

thermodynamic

limit trie main contribution _to N* _comes from trie former _term.

Using

this subset of

plaquettes,

whose related frustrations _are all

independent

random

variables,

it

is

possible

to build _{up every}

possible

closed

loop

_on trie d-dimensional lattice.

In conclusion the described

change

of variables

permits

_us to reduce trie number of

indepen-

dent random variables _necessary to describe trie system, ^{from dN}

couplings

to N*

frustrations,

with N*

= N for d

= 2 _or N*

= 2N for d > 3. In trie

following,

if _not

diiferently specified,

trie term "frustration"

always

indicates _one of trie N* random variables of trie system.

Recalling (4.1),

equation

(2-6)

becomes:

N*

F ₌ lim

In[1

+

~j ~j $"~ _$'~~ tanh~~'" '"k)(fl)] (4.2)

~~~° ~

k=i _i,.. i~

where _one considers all trie

products

of k E

Il, N*] elementary plaquettes (trie

indices ii

~

12

~ ~

ik _run from 1 to

N*)

and

T(ii,

,

ik)

is trie

length

of trie

loop

built _up

by

trie k

plaquettes,

while trie overline represents now trie _{average over} trie frustrations.

In trie

following paragraph,

_we consider trie

plaquettes

of trie lattice

only

from _a

topolo#cal point

of view, 1.e., trie fact that _a frustration of _a

plaquette

is _a random variable of trie system

(or not)

is irrelevant. At this

point,

_we

perform

_a

decomposition

of trie _set of

plaquettes

into subsets such that each

coupling J,j belongs

to _one and

only

_one of these subsets. It follows that two distinct subsets _can bave in _common

only

isolated lattice

points.

For instance, three diiferent

decompositions

of trie bonds of _a 2d lattice

(trie

black

regions)

_are illustrated in

Figure

3. It is worth

stressing that,

^after the

decomposition,

_we get a collection of

sub-systems

which

do not cover trie whole

original lattice,

_e.g. in

Figure

3a

only

_one half of the 2d lattice is

covered. This kind of

decomposition

divides trie

plaquettes

into two classes: trie "black" _ones, which _are

organized

in groups

correspoiiding

_to trie

sub-systems (e.g.

trie _crosses in

Fig. 3b),

and trie "white" _ones, which do _not

belong

to any

sub-system-

Coming

back _to trie

frustrations,

note that in

general

there _are many "white" _or "black"

plaquettes

whose frustration _is not _a random variable of the system.

At this

point

we can derive _{an upper} estimate Fa~ of the function

(4.2), by treating

ail the random variables of frustration associated to the "white"

plaquettes

_as annealed

variables,

1.e.

by averaging

_over these variables

only

^trie argument ^{of trie}

logarithm

_m

(4.2),

instead of the

ai

hi

Cl

Fig. 3- Ising mortel with nearest neighbour interactions _{on a} 2d lattice: decompositions of the set of the couplings _in independent subsets

(the

black

areas): a)

_squares,

b)

_crosses and c) elongated _crosses.

logarithm

itself:

~~ lôj

Fa~ ₌ lim

ln[1

+

~j ~j 1"~ _l'~~~'~~ tanh~~'" "~)(fl)] (4.3)

~~~° ~

k=i _{i, i~}

where R~'°) _and R~~) represent trie averages over trie frustrations related to,

respectively,

trie "white" and the "black" squares. In

(4.3) only

the terms

l~~~ l'~~

_{with all} "black"

frustrations do not vanish after

performmg

the "white" _average:

p~~

(b)

Fa~ = lim

jIn[1

+

~j ~j _{l~~~ l~~~} tanh~~~'

~~)

(fl)] (4.4)

~~~°

k=i _{ii. .>k}

where _now trie indices ii, .,ik run

only

_over trie N~ "black" frustrations. It should be _re- marked that after

performmg

trie "white" _average, trie

surviving loops

^do not connect diiferent

sub-systems.

In other terms, any

loop appearing

in trie average

(4.4)

_can be

decomposed

into _a set of

"sub-loops",

each of them limited to _a

single sub-system.

This

implies

that tanh~~~i> >~k~(fl) ^con be

factorized,

_1-e-,

T(ii,

.,

ik)

is trie _sum of trie

lengths

of trie _vanous

"sub-loops".

As _a consequence, trie whole argument of trie

logarithm

_m

(4.4)

is factorized

among ail trie

sub-systems,

and it

immediately

follows:

Fa~ =

~

In

il

+

~j ~j _l~~~ l~~~ tanh~~~"""~~)(fl)] (4.5)

'lJ

Îi

i~ _~~

where trie _average is

performed

_over trie np "black"

plaquette

frustrations that

belong

to a

single sub-system,

and where nj ^is trie number of

couplings

in trie

sub-system,

_so that

é

represents trie total number of

sub-systems,

rescaled with N in trie

thermodynamic limil.

Recalling (2.5),

from

(4.5)

trie lower estimate

fa~

of trie

quenched

free _energy

yields:

fl fa~

₌ d In

cosh(fl)

₊ ^~

ln[1

+

~j ~ l~'~ _l~~~ tanh~~'i'

""k)

(fl)]

+ lu 2

(4.6)

"J

~i

1, .>~

It is

preferable

to choose _a

topolo#cal decomposition

of trie system ^{such that trie} _np "black"

frustrations

fully

describe trie

single sub-system.

In this context Fa~ is

practically equivalent

to the

quenched

free _energy of the

sub-system,

_a part _a ^factor

proportional

to In cosh

fl.

_{If np}

is not too

large,

the

computation

of Fa~ _can be

performed analitically

_or

numerically.

Trie _correct

multiplicative

factors to transform trie

quenched

free _energy of _a

sub-system

to

trie

global

constrained free energy

(4.6)

_can also be obtained

by

_a

simple

argument. Indeed trie

partition

^function _Z~z~ ^of a

sub-system

made of _ns

spins

and nj

couplings

is trie _sum of 2'~S

terms which _{are given}

by

trie

product

_{of nj}

exponentials

while trie

global

constrained Grand-

partition

^function

QN

of _a system ^with N

spins

and dN

couplings

is trie _sum of 2'~ _terms which _{are given}

by

trie

product

of dN

exponentials.

In order to compare two

quantities

^of

order _{one, one} bas to write trie

followmg equality:

2~~ilN _)~/~~~~

= 2~"~

z»~ )~/"'

so that

1 _{~ ns}

(4

7)

/

₌

--(1-

d

")

ln2 + d

-fsub

~~

fl

_{"J "J}

This formula is

completely equivalent

to

(4.6),

but bas trie

davantage

that trie

quenched

free energy

fsu~

of trie

sub-system explicitly

_appears. Trie

ground

state energy can be estimated

by

trie saddle

point

method when

fl

_{- co m} trie

quenched

_average of Zn~

Ea~(T

=

o)

₌ ^~ max

H((J)) (4.8)

nj j«j

where trie maximum is taken _over ail trie _a

configurations

for each disorder realization. One

can also obtains trie residual entropy as

Sa~(T

=

o)

=

(1 ~d)

_{In 2 + In}

_{deg((J)) (4.9)}

"J

where

deg((J))

is trie number of disorder

configurations

which bave energy

H((J)) equal

to

Ea~(T

₌

o).

Let _us

briefly

_resume trie

discussion,

_m order to

clarify

trie

meaning

of trie result.

Using

a

topological decomposition

of trie system into

independent sub-systems,

we bave derived _a lower estimate of trie

quenched

free energy that

depends

from trie

quenched

free energy of trie

single sub-system.

As discussed in trie

appendix,

trie

quenched

^free energy of _a

sub-system

can be obtained

by

_a minimization of _an annealed _average of trie

Grand-partition

^function

with

Lagrange multipliers

_over ^ail its

possible loops. Using

_our

procedure,

_{we are} able to get trie annealed _average

fa~

^{of trie}

global

system ^{where trie} constraints _are

imposed

_over ail trie

possible

rescaled _sums of frustration _on ail trie

loops

that appear in trie various

sub-systems.

To illustrate _our

method,

_we

apply

it in trie _case of Section 3, 1-e- trie

decomposition

^of a

2d

spin glass

in "white" and "black" _square

plaquettes,

like in _a

chessboard,

_see

Figure

^3a.

In this _case ail trie N

plaquette

frustrations represent a set of

independent

random

variables,

i e. N* ₌ N. One half of them is related to "white"

plaquettes,

so that

they

_are treated _as annealed

variables,

while trie other half related to "black"

plaquettes

_as

quenched

variables.

We also bave nj = 4 and np = 1, so that

(4.6)

becomes:

fl f])

₌ 2 In

cosh(fl)

+

ln[1 tanh~(fl)]

+ In 2

(4.lo)

It is _easy to check that after trivial

algebraic manipulations

_one gets

again (3.6).

Trie

ground

state energy and trie _zero temperature entropy can be

directly computed

from

(4.lo),

and

respectively

_are:

E])(T

=

o)

₌ -1.5

S])(T

=

o)

₌ o

(4.Il)

The method of constrained annealed _averages without

multipliers

_can be

applied

to _non-

elementary sub-systems

in _an easy numerical _way, since _one bas not to introduce _a set of

corresponding Lagrange multipliers.

In order to improve

(4-la)

and

(4.Il),

_we bave considered

a

partition

of trie 2d lattice into

independent

_{crosses of np =} 5 square

plaquettes,

as shown

m

Figure

^3b. We thus obtain _a constrained annealed _average where ail trie relevant variables

on trie _non connected

loops

inside trie _{crosses are} frozen. Since nj = 16, ^from

(4.6)

trie free energy is

easily evaluated,

and is shown in

Figure

1. Trie

ground

state energy and trie residual entropy are

respectively

E([(T

=

0)

₌ -1.484375

S([(T

₌

o)

₌ 0.00882...

(4.12)

The next step is to considered trie

elongated

_cross of trie type ^shown in

Figure

3c

(np

= 8 and nj =

24).

In this _{case we} get

E[[~(T

=

o)

₌ -1.477865

S~~(T

=

o)

₌ 0.0130...

(4.13)

At

increasing

trie size of trie subsets of trie

decomposition,

trie convergence ^to trie

quenched ground

state energy Eo is rather slow _as trie numerical result of

Ill] #ves

Eo _" -1.404 + 0.002 and So

" o.075 +

o.oo4). However,

trie main

qualitative

feature

(positive

residual

entropy)

is

reproduced by

_our approximations.

We bave also

applied

_our

technique

to trie +J spin

glass

in three dimensions. From _a

topological point

^of view, a 3d lattice with N

points

_can be

thought

_as trie union of distinct cubes distributed in such _a way that two of them bave

only

_one lattice

point

in _common

(trie

3d

equivalent

of trie 2d

chessboard).

In trie

thermodynamic

limit their number is N. Let _us stress that in this _case trie six

plaquettes

of each cube represent ^trie "black"

re#on)for

a total of

~N,

_{while there}

are

~N

"white"

plaquettes

between cubes.

2 2

It is easy to realize that there _{exists a}

diiferent,

but

equivalent, change

of random variables with respect to trie _one

previously described,

which involves np = 5 "black"

plaquette

frustra- tions for each cube

(trie neglected plaquette

must be

always

trie

same), together

with other

plaquette

frustrations _{in trie} "white" areas, for _a total of 2N

independent

random variables.

After

performing

trie "white" annealed average, trie

problem

is reduced to trie calculation of

/ / / / / /

~ -2.

t /

~ /

/ / /

/

-f~

/ ~ ^cube

~

0 0.5 1-5 2 2.5 3

T

Fig. 4- +J _spin glass _in 3d: the annealed free _energy fa

(dashed fine)

and constrained free energy

f(i~~ (fuit

fine _versus T

=

1là.

the

quenched

free _energy of _a

single

cube. In this _{case we} obtain _a constrained annealed _av- erage of the free energy of the

global

system

f(f~~,

where the constraints _are

imposed

_over the

products

of the

J,j's

_on all the closed

loops

of bonds in the

~N independent

cubes. The

4

resulting f(f~~

has _a very

long analytic expression

and is

plotted

in

Figure

4, where it is also drawn for comparison ^{the annealed free} _energy.

Unfortunately,

the residual entropy is still

negative,

^thus

indicating

that in three dimensions _more constraints _are necessary ^to get a fair

approximation

of the

quenched

system at low temperature- However, trie free _energy should be

a

non-decreasing

function of trie temperature, _so that trie

ground

state energy can be estimated

as trie supremum of _an annealed

approximation although

trie free _energy bas

negative

deriva- tive at low temperature [5]. ^Our constraints _on all trie

loops

inside trie alternated cubes allows to estimate

Eo(d

=

3)

> -1.917 which

improves

trie lower bounds

given by

trie _supremum of trie annealed free _energy, Eo > -1.956.

5. Constrained Annealed

Averages

_on Infinite

Loops

The method introduced in trie previous section could allow _us to obtain very ^accurate estimates

(lower bounds)

of trie

quenched

free _energy. Indeed trie main limitation _we bave met is that trie number of

plaquettes

_np of trie

sub-system

should be not too

large

_m order to

perform

an

analytic

calculation of its

quenched

free _energy. For instance _m 2d _we bave

stopped

trie estimates at trie

elongated

_cross with np = 8

plaquettes.

However, we can consider infinite

loops by estimating

trie free _energy of

sub-systems

of infinite _size

only

in _one

direction,

via trie

Lyapunov

exponent ^of trie

product

of random transfer matrices [12].

To

simplify,

[et _us consider

again

trie two-dimensional

Ising

model with

J;j

₌ +1 with

equal

probability, although

_our discussion _can be extended to three and

higher

dimensions. Trie idea is to find _an

independent decomposition

of trie 2d lattice _m

strips

^of size L. We then compute trie

quenched

free energy of trie strip _as trie

Lyapunov

exponent ^of trie infinite

product

of