Replica trick and fluctuations in disordered systems

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Replica trick and fluctuations in disordered systems

A. Crisanti, G. Paladin, H.-J. Sommers A. Vulpiani

To cite this version:

A. Crisanti, G. Paladin, H.-J. Sommers A. Vulpiani. Replica trick and fluctuations in disordered systems. Journal de Physique I, EDP Sciences, 1992, 2 (7), pp.1325-1332. �10.1051/jp1:1992213�.

�jpa-00246624�

Classification Physics Abstracts

75.10N 05.40 02.50

Replica trick and fluctuations in disordered systems

A. Crisanti

(~),

G. Paladin

(~),

H.-J. Sommers

(~)

and A.

Vulpiani (~)

(~) Dipartimento di Fisica, Universith di Roma "La Sapienza", 1-00185 Roma, Italy (~) Dipartimento di Fisica, Universit£ dell'Aquila,1-67100 Coppito, L'Aquila, Italy

(~)

Fachbereich Physik, Universitit-Gesamthochschule Essen, D-4300 Essen I, Germany

(Received

6 February 1992, accepted 24 March

1992)

Abstract. The finite volume fluctuations of the free _energy in disordered systems can be

characterized by _means of the replica method. This is achieved by introducing _a canonical

ensemble in the _space f1of disorder realizations. In this _space the role of energy is played by the

free energy of the original system, and that of the temperature by

n~~,

where _n is the number

of replicas. We apply this method to analyze the fluctuations in random directed polymers and in the Sherrington-Rirkpatrick model for spin glasses. Anomalous fluctuations _are found

in both _cases. For the low temperature zero field phase of the Sherrington-Kirkpatrick model

this approach predicts _a

N~~/~

_{law for the relative} fluctuations of the free _{energy per} spin ^of a

N-spin system.

The

analysis

of finite volume fluctuations in disordered systems is _a

subject

which has attracted _an

increasing

attention in _recent times. One of the central issues is the

comprehension

of

mesoscopic

effects which _can become

quite

relevant in _many

experimental

situations. The

replica

method is _one of the most

important

tools for the

study

of this

problem ill.

As far _{as we}

know,

the first result _on this

subject

has been obtained

by

Toulouse and Derrida [2] who reconstruct the

probability

distribution of the free _energy in the

high

temperature

phase

of the

Sherrington-Kirkpatrick (SK)

^model _{[3] from} the moments of the

partition function,

I-e-, from the free _{energy per}

spin f(n)

of _n

non-interacting replicas.

The aim of this paper is to show how the

replica

formalism _can

give

informations _on the finite size fluctuations of the free energy among different realizations of

couplings

in

spin glasses

and other disordered systems.

Consider _a

generic spin

^model with Hamiltonian:

1,N I,N

HN

(J]

₌

£ _J;j

_{«;«~ h}

£

«;

(I)

I<j I

where _«;

= + I is the value of the

spin

_on the site I, ^the

couplings J;j

are

independent

random variables with

probability

distribution

P[J],

and h is _an external

magnetic

field.

1326 JOURNAL DE PHYSIQUE I N°7

For _any fixed

coupling

realization J, the

partition

function of the

N-spin

system at temper- ature

I/fl

is

given by

ZN(J]

₌

£ exp(-fIHN(J]). (2)

la)

The

quenched

free energy per

spin

is

fN

_"

-lnZN /Nfl,

where

t

_{indicates the average}

over the

couplings

realizations. We _assume that the

thermodynamic

limit of the free _energy, limN-oc In ZN

(J] / Nfl

exists and is

equal

to the

quenched

free energy

f

₌ limN-oc

fN

for almost all

coupling

real12ations J

(self-average property).

It is worth

noting

that this property

corresponds

in the context of

dynamical

systems, _or of random matrix

products,

to the Oseledec theorem for the characteristic

Lyapunov

exponents [4]. ^{Indeed if the}

partition

function _can be

expressed

with the

help

of transfer

matrices,

then

f

is the maximal

Lyapunov

exponent of the

product

of these matrices [5].

The

analytic computation

of the

quenched

free energy,

I.e.,

of the _average of the

logarithm

of the

partition function,

is _a

quite

difficult

problem,

_even in

simple

_{cases as} nearest

neighbour

interaction in _one dimensional models.

However,

since the

integer

moments of the

partition

function _are easier

computed,

the standard method _uses the _so called

"replica

trick~'

by

_con-

sidering

the annealed free energy

F(n)

₌

nf(n)

of _n

non-interacting 'replicas'

of the system

Ill,

~(~)

" ~ffl~

~

~ll

((~N@])"j ⁽³⁾

°C

The

quenched

free _energy of the

original

system is then recovered _as the continuation of

f(n)

to n ₌

0,

~ ~$ _{/f~oc ~~~~~~ ~$~~~~'}

_~~~

In real

experiments

_or numerical

simulations,

the number of

spins

N is finite and the real- ization J fixed. Therefore what _one computes is

The

elf-average property ensures _that foralmost allrealizations

J

yN

~

f

when

N

large

but

_finite

systems

yN is _still a

andom

quantity whose

obability

distribution^depends

in

a omplicate way _on that ^of

J.

It

of

the

free energy (5) in large

systems.

In general one may address two ifferent The first_concerns the

fluctuations

_of

yN about its average value

w

in

a

eviations

of

yN from

the

asymptotic value f.

i~ ~-pNVn i~@

£

^~k ~k

(~)

N

~

~j

k=I

where

Ck

is the kth cumulant of In ZN>

I.e., Cl

_" In

ZN,

C2 ₌

(lnZN)~ (ln ZN)~,

etc. The factors

fl

and N _are introduced for convenience. From

(3)

and

(7)

we see that

L (( n~

=

-fINFN(n) (7)

~l

where

FN(n)

for

integer

and

positive

_n is the free energy of _n

replicas

_on the system of size N. Thus if

FN(n)

is known the cumulants _are identified with the coefficients of the

Taylor expansion

of

FN(n)

in _powers of _n. In

particular

the fluctuations of _yN about its _average value

are

given by

^the coefficient of n~

as

C2

_"

fl~N~

_(gj~

§N~). (8)

Unfortunately

what _one

usually

evaluates is _not

FN(n)

but its

thermodynamic

limit

F(n)

for N ~ oo.

Consequently (7)

has to be

replaced by

-fl>(n)

₌

Ii ) _{~ i) n~. (9)}

~l

From this

equation

^{it follows that if the} cumulant Ck is

o(N)

then the term

n~

will be

missing

in the

Taylor expansion

of

F(n).

In

particular if,

^for

example, C2

~'

N~~?

with1

> 0 the

expansion

does not contain the

n~

_{term. Vice}

versa if this term is present, then

)- _w~ ~'N~~ (10)

I-e- normal fluctuations. As _a consequence the absence of the n~ _{term in the}

expansion

of

F(n) signals

anomalous fluctuations. In this _case to have the correct

scaling

of C2 with N _one has to _use

FN(n)

which includes finite N corrections. In

general,

however, the calculation of

FN(n)

is

difficult,

and hence the

study

^of

C2

in _presence of anomalous fluctuations is

quite unpractical.

Informations _on the deviations of _yN from its

asymptotic

value

f,

_on the contrary, _can be obtained

directly

from

F(n) by using

_a method

largely

used in the last years ^to

study

the finite time fluctuations of the

Lyapunov

exponent in

dynamical

systems and of the localization

length

in the Anderson model [6].

The idea is to

perform

statistical mechanics in the _space Q of all the

possible

realizations of disorder J. For _any fixed temperature

fl~~,

_{a state} in this _space

corresponds

_{to a} well defined system with _a

given

free energy yN. The states in Q _can be classified

according

to the value _y of _yN, and the _space divided in classes

w(y)

labelled

by

_y. The

probability

distribution

P(y)

is hence

proportional

to the number of states in

w(y),

I-e-,

P(y)

_c~

(#

of states in

w(y))

c~

e~/~~(V) (11)

where

by

definition -N

S(y)

is the entropy of the class

w(y).

We have

explicitly displayed

the extensive

dependence

_on the system ^{size N} so that in the

termodynamic

limit

S(y)

-J

O(I).

We _are indeed

using

_a microcanonical ensemble in the

Q-space

where the role of the energy is

played by

the free _{energy y} of the

original

system. In

particular,

the

self-average

property ^of the free _energy is

equivalent

_to the existence of the

thermodynamic

limit in the

Q-space-

As _a

consequence,

-S(y)

must be maximal for _y

=

f.

^Since the entropy ^{is defined} _a part ^from an

additive constant, we can take

S(y

₌

f)

₌ 0 and

S(y)

> 0 if _y

# f.

A direct calculation of

S(y)

is _{not easy,} but from

thermodynamics

_we know that it _can be obtained from the free _energy, which in the

Q-space

reads

f~ _~-N

(S(V)+flvn)

~-Nflnf(n)

_~i~~

~

where

(fin)~~

is the temperature in the

Q-space-

We have introduced the factor

fl

for _conve- nience. This does not

change

the results since _we work _at fixed temperature.

1328 JOURNAL DE PHYSIQUE I N°7

We _are interested in the

leading

term of N

S(y)

in

N,

thus _{we can assume} N _very

large

in

(12)

and evaluate the

integral by

saddle

point.

This

yields

the usual relation between free energy and entropy

fin f(n)

₌ min

(S(y)

+

tiny) (13)

v

and the well known relation

~~~

~. ~~~

~~

_~~~~

among the entropy

-S(y),

the _{energy y} and the temperature

(fin)~~

The relation

(14)

shows that each _n selects _a class

w(y*)

with free energy y*. The

higher

_n, the less

probable

is the

corresponding

class of realizations. We remark that in this _way

we consider

only

the extensive part of the entropy. In other words _we

neglect

_any term

o(N)

in the exponent of

II).

^In usual

statistical mechanics terms

o(N),

in

general, correspond

to surface contributions.

They

do not affect the

thermodynamic limit,

but _one should control their effects _on the finite volume

fluctuations. In

generic

systems, these terms _are

expected

to be

unimportant.

We do not consider them _on the _same

ground

of what is done in the standard _treatment of finite volume fluctuations in statistical mechanics.

By

this

assumption,

_{we can use}

(11)

and

(13)

to derive

some

predictions

_on the finite volume behaviour of disordered systems.

The entropy can be obtained

by inverting

the

Legendre

transformation

(13-14),

s(v)

=

fl>(n) tiny (15)

where

n(y)

has to be eliminated with respect to _y with the

help

of

~

(~~~l=

_y

(16)

'~ n(v)

The

important point

is that to evaluate

S(y)

_one needs

just

the extensive part of

F(n) which,

for

integer positive

_n, is what _one

usually

computes with the

replica

method.

Standard

probability theory

theorems

iii imply

that

f(n)

is _a

monotonic,

non

increasing

function of _n. For small _{n we can}

expand F(n)

in _powers of _n, I-e- _an

high

temperature

expansion

in the

Q-space-

Let _us suppose that

?(n)

=

fn )n~

⁺

^{O(n~). (ii)}

with _{/t >} 0.

Inserting

^this form into

(15)

and

(16)

_one gets

~~ ~~2

S(v)

= fl

~~ (18)

so that the variance of the fluctuations is

given by,

(v f)~

₌

). ₍₁₉₎

In _{some cases} it _may

happen

that the n~ _term is

missing.

In this _case

(19)

is not valid

anymore. Assume for

example

that for _n > 0

F(n)

=

fn

^~

^n~

+

O(n~) (20)

3

with _{a >} 0 for

convexity.

Note that the condition _n > 0 is _necessary since

(20)

with _{a >} 0 is

not _convex for _{n <} 0. An easy calculation shows that this leads to,

The restriction on the

sign of

f ^- y follows from the

n

> 0

what

_one

has

_in

ordinary

magnetic systems here an ^xternal field _{selects a}

phase of

magnetization,

e-g-

ositive or

negative. here _n can be

also

seen as _an

external

magnetic field,

in

_which

case

y ecomes a sort _of agnetization.

As

for an

ordinary

magnetic

ield, sign

of

n

_leads to

a

on the ^ign of

the

magnetization,

which in

our

case

is

f - y. _In

(21) we

took f -

_y >

0

because _S(y) was

evaluated

hence in

the

n > _{0 hase.}

S(y)

distribution of y in the

y

< f phase.

Note

that

if _the

moments

@

grows

more

than an

exponential

with n

the

probability

istribution is not _{uniquely termined}

by

F(n)

[8].

evertheless (21) gives

the ^mall

from

f.

Therefore, ^from

f y)

c~

N~~/~, f y)2

_c~

N~~/~

However,

to calculate the

cumulants,

_e-g-

C2,

_we need the full

probability

distribution.

Unfortunately

_{we are} not able to evaluate

directly S(y)

for _y

f

> 0 since this would

require

the calculation of

%

_for

negative

_n.

By looking

at

(21)

_{we see} that the second derivative of

S(y)

with respect to y

diverges

_as

y ~

f~,

I-e- _n

~ 0+ This

implies

_a second order

phase

^transition in the

Q-space

at _n

= 0. If

we exclude _{some very}

patological

_cases, in

general

the critical exponents are

equal

from both sides of the critical

point. By

this

assumption

it

readly

follows that

or,

equivalently,

F(n)

₌

fn

₊

~n~

₊

O(n~),

_n < 0

(23)

3 with b _> 0. In

general

_a

#

b.

A free energy of the form

(20)

_appears ^{in the}

study

of random directed

polymers

_[9]. In fact

by employing

_a

replica

method _one has for the

integer positive

moments of the

partition

function [9]

%

= exp

(- f

L

(n n~)),

^L »

(24)

where L is the total

length

of the walk _on the

polymer.

From the above

result, therefore,

_one readily concludes that the

probability

distribution of the free energy F for small deviations from the

asymptotic

value L

f

is

P(F)

_{c~ exp}

(-a&

_(L

f F(~/~ L~~/~) (25)

where _we take _a+ if

Lf

F _> 0 and _a- if

Lf

F _< 0. A similar form with a+ = a- was

proposed by Zhang

_[10], and later

by

Bouchaud and Orland

ill].

However, in contrast with their

formula, (25) leads,

ⁱⁿ

general,

to _an

asymmetric probability

distribution. This is in

agreement

^with recent numerical results

[12].

1330 JOURNAL DE PHYSIQUE I N°7

From

(25)

one can conclude that if a+

#

_a-

(L f F)

_c~

^L~/~ (26)

in accord with references [10] ^and

ill],

and with the numerical results of Mdzard

[13]. By using (25)

_a

straightforward

calculation leads to the the Kardar

scaling

_[9]

Cj/~

-~

L~/~ (27)

Sinfilarly

_{one can} find that

C(/~

-~

L~/~

_{The available numerical data [12]}

are in

quite good

accord with these

scalings

with the exponents 0.333 + 0.003 for the second

cumulant,

and 0.37 + 0.02 for the third.

The above arguments can be

repeated

for any powers of _{n, as}

long

_as there is _{a power} n~

with _{m >} I in the expansion ^of the extensive part of

F(n)

nf. If the first

non-vanishing

power is n~, then for _{m >} 2

(f v)

CK

N~~~~~~/~ (28)

f y)2

_~

N-2(m-i)/m (~g)

Note that if _a+

= a- then

(f y)

₌ 0. In this _case

(28)

is valid

only

if _we restrict to the

f

_y > 0

(< 0) phase.

As last

example,

_{we now}

apply

this method _to

investigate

the fluctuations in the

Sherrington- Kirkpatrick (SK)

model [3]. ^Before

addressing

the

problem

of the low temperature

phase,

^let

us

spend

_a few words about the

replica

symmetric

phase.

In the

high

temperature

phase,

I-e-, above the de Almeida and Thouless line there is

no

replica

_symmetry

breaking [14].

^An

explicit

calculation shows that for h

#

0, the extensive part of

F(n)

contains the term n2 [3]. ^{On the} other hand if h

= 0 the latter is

missing

and

(19)

is not valid. In this _case

F(n)

= n

f,

which

can be _seen

formally

as the _m

~ oo limit of the above _case. Therefore from

(28)

^and

(29)

it follows

(f

_Y)

+~

N-i, _{f y)2}

-~

N-2 (30)

Indeed _an

explicit

calculation leads to [2]

fl f

₌

~

ln2

ln(I fl~)

~~

(31)

fl/t

₌

-j

^{(fl~ +}

In(I _fl~))

where _{/J is} the coefficient of the

n~/2

_term, which _can be related _to

N(f y)2.

Note that /J

diverges

at the

spin glass

transition

fl

₌ I. This is

a

general

result. The

I/N

terms contain the fluctuations about the saddle

point,

and hence it is not difficult _to understand that the

coefficient of

n~/N

should

diverge

at the

spin glass

transition where the

replica symmetric

solution becomes unstable [14].

The _more

interesting phase is, nevertheless,

the low temperature

phase

where the

replica

symmetry is

completely

broken and _many different minima of the free _{energy appear}

ill.

We limit ourselves to the _case h ₌ 0. At the

spin glass

transition the

overlap

matrix _qap between

different

replicas

is _zero and _{one may}

perform

_an

expansion

in _powers of qap. This leads _to the _so called "reduced" _or "truncated" model.

Using

this model Kondor [15] ^{found that the}

replica symmetric

solution is stable

only

for _n > n~ =

)T

+

,

where _T

=

(Tg T)/Tg

is

the reduced temperature ^and Tg ^{is the}

spin glass

transition temperature. ^For n < ns a stable solution is found

using

the Parisi ansatz. This leads to [15]

f(n)

=

fp(T)

^~

n~

₊ for _n <

n~(T) (32)

5120

with

~3 ~4 ~s

~~~~~

6 ^~ 12 ~

10'

(33)

From

(32)

^{it follows that} the first

non-vanishing

_power

(excluding

the linear

term)

in the

expansion

of

F(n)

is n~ _{and hence from}

(28)

and

(29)

_we have

(f

_{Y) '~}

N~~~~, (f v)~

'~

N~~~~ (34)

Assuming

_an

asymmetric probability distribution,

I-e- a+

#

_a-, this result

implies

that the correction to the extensive part of the free _energy is of order

N~/~,

_{in agreement} with recent

suggestions [16].

^A

straigthforward

calculation leads to the

scaling

law for the second cumulant C2

+~

N~/~ (35)

Numerical data

iii,

_18] for the _zero temperature _energy fluctuations for systems of sizes 6 <

N < 200 leads _to C2

~' N? with _{i ci}

1/2.

We note,

however,

that if

only

the data for 50 < N < 200 _are taken into account these _can be fitted with _{i m}

1/3 (see Fig.

9 Ref.

[17]).

We conclude

by noting

that _our arguments ^allows one to

interpret

the

"searching

for the

maximum",

which in the standard

replica

trick follows from the fact that the space dimen-

sionality

of the order parameter qap becomes

negative

for

n ~ 0, in _{a very} natural _way. In

fact,

the

physics

is

given by

the infinite temperature

limit,

in the

Q-space,

and hence

by

the

largest

free energy. This also

explains why

in

spin glasses

the metastable states have lower free

energies [19],

^which at first

glance

_may ^look

strange.

Acknowledgements.

We thank E.

Marinari,

M.

Mdzard,

G. Parisi and M. A. Virasoro for useful discussions. AC thanks the

Sonderforschungsbereich

237 for financial support ^{and the} Universitit-

Gesamthochschule for kind

hospitality

where part ^{of this work} was done.

References

ill

M6zard M., Parisi G. and Virasoro M-A-, Spin Glass Theory and Beyound

(World

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1988);

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University Press,

1991).

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