Barriers and metastable states as saddle points in the replica approach

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Barriers and metastable states as saddle points in the replica approach

J. Kurchan, G. Parisi, M. Virasoro

To cite this version:

J. Kurchan, G. Parisi, M. Virasoro. Barriers and metastable states as saddle points in the replica approach. Journal de Physique I, EDP Sciences, 1993, 3 (8), pp.1819-1838. �10.1051/jp1:1993217�.

�jpa-00246833�

J.

Phys.

I Hance 3

(1993)

1819-1838 _AUGUST

1993,

PAGE 1819

Oassification

Physics

Abstracts

64.60 75.10

Barriers and metastable states

_as

saddle points in the replica approach

J. Kurchan

(~),

^G.Parisi

(~)

^{and M-A- Virasoro}

(~)

(~)

Dipartimento

di

Fisica,

Universit£ di

Roma,

La

Sapienza,1-

00185

Roma, Italy

and INFN

Sezione di Roma

I, Roma, Italy

(~) Dipartimento

di

fisica,

Universith di Roma

II,

Tor

Vergata,

via E.

Camevale,

Roma 00173,

Italy

and INFN Sezione di Roma

II, Roma, Italy

(Received

30 December 1992,

accepted

in final form 5

ApriJ 1993)

Abstract. In the context of the

p-spin spherical

model _we show how saddle

points

of the

n - 0

replica

free energy surface _can be

interpreted

_as metastable states _or barriers.

1. Introduction.

In _a

previous

work

[I]

some of _us

proposed

the _use of the

replica

method _on constrained

systems

as a tool _to

study

the

properties

of _a

system

^off

equilibrium.

In this paper we

show,

in the _context of the

spherical p-spin model,

^that

by considering

a set of identical

systems ("real replicas"

with

adequate

constraints between

them,

_{one can}

probe

the

phase-space

structure of the

original

model and obtain information _on its metastable states and _even barriers.

Interestingly enough,

one can uncover within this framework

a

meaning

for several saddle

points (other

than the _one

yielding

the Gibbs-measure

results)

that _appear in the

replica

treatment for the model.

Furthermore,

_we will show that

quite _apart

from the

replica approach, considering coupled

"real

replicas"

is also useful from the numerical

point

of view.

The

p-spin spherical

model _was introduced in [2] where the

replica

solution _was derived.

The authors showed that for _zero

magnetic

field and below _a certain

temperature (denoted

_as

T(

in the next

sections)

there

are two stable solutions. One of them is

replica symmetric (RS)

while the other _one has _a

single

level of

replica _symmetry breaking (IRSB).

The coexistence of stable solutions is not _new in the

replica approach

and in

general

_one has to invoke ad-hoc

arguments

to decide which is the relevant

one. In this

example

the

peculiarity

of the IRSB solution is that the _z

parameter

becomes

larger

than _one in the

temperature

_range

Tc

< T <

T(. Then, by analogy

with the Random

Energy

Model for which the solution _can be

checked~

for _{z >} I the RS solution has to be chosen.

However,

the IRSB solution continues to exist and has _a

larger

free energy. Its

meaning

is not clear: We shall show that the consideration

JOURNAL DE PHYSIQUE T 3, N'S, AUGUST >993 65

of two constrained

systems

_can be

helpful

in

understanding

this solution. We shall _see in what follows that the IRSB solution for _z > I is

just

_one

example

of solutions which _even if

they

are to be dhcarded _as far _as the

computations

within the Gibbs-measure

they

still contain

physical

information _on the

system.

The

organization

of the _paper is _as follows: In section 2.I _we review the results obtained in reference [2]

using

^the standard

replica approach

with

a

single

real

replica.

In section 2.2 _we solve the TAP

equations

for this

model,

which turn out to take _{a very}

simple

form.

They

_are

used later for the

interpretation

of the solutions. In section 3 _we solve the

thermodynamics

of several real

replicas

with constraints

amongst

^them. In section 4 _{we use} all of the

previous analysis

to

try

to derive _a

complete picture

of the different saddle

points.

Finally,

in section 5 we

present

the results of numerical simulations for this model. We show how the numerical

analysis

of several constrained real

replicas

_can be very useful in the

study

of

phase

transitions.

2. The model.

The

p-spin spherical

model consists of N

spins

that take continuous values constrained

by:

£S,?=N ₍₁₎

The _energy is _a

p-spin

interaction

E =

~ J;,...; S;, S; (2)

p p

1<;,<;~...<;~<N

where the

J;,...;~

are

independent

random variables with _{zero mean} and variance

~~,j~-,.

^{We shall}

only

^{work with} _zero

magnetic

field.

For _p

= 2 the energy

(2)

is _a function

having

_a

pair

of minima

corresponding

to the

highest eigenvalue

of

J;; (and

^indeed the _{same can} be said of the TAP free

energy),

_so that _one does

not

expect

_any

interesting spin-glass

behaviour for that _case. We concentrate _on p > 2.

In this _{paper we} shall

consider,

_{as a} tool to

analyse

this

model,

the associated

system

consisting

of R "real"

replicas [Ii

with the _same interaction

R

ER

_"

£ _J;;..;~ £[S/,

S/~

^{+ +}

S~ S(] (3)

1<I,<ia...<i~<N 1

where each set

S),..

,

St

satisfies _a constraint

(I).

In addition _we fix the distance between the

configurations

of the real

replicas:

£S[S)

= N

p[~

^for r

#

t

(4)

;

Before

going

into the discussion of several real

replicas,

in the next two subsections

we review

briefly

the standard

replica-method

solution reference [2] ^{and the TAP}

equations

for _one real

replica.

2,I REPLICA SOLUTION.

Following

_{[2] we} calculate Z" with

(2).

After

averaging

_over the

interactions, uncoupling

^the

sites,

and

partially using

the saddle-

point equations,

_we arrive at

N°8 BARRIERS AND METASTABLE STATES ₁₈₂₁

the

expression

for the free energy

(here

and in what follows _we write the extensive

quantities

per

site):

~F "

L(Qofl)~ jTtl'~ol ⁽⁵⁾

off where

Q

is the usual

n x n

overlap

matrix.

The

spherical

constraint for the

spins

^induces _an additional term in the free _energy:

Fconstr

_"

£ _{Za (I Qoo) (6)}

a

where

Za

_are

Lagrange multipliers imposing Qaa

= I.

Proposing

for

Q

a Parisi ansatz with blocks of size _To, xi,...,zk,zk+i " I

corresponding

to

overlaps

qo,qi;.,qk,qk+i " I _one finds that the

only saddle-point

solution with _q;,z;

increasing

with I is _a

single-step

ansatz

(determined by

_{qi, go =}

0, z) yielding

_a free _energy and energy per site:

~'~ ~~

~~ ~'~~~~ ~

~fl~~~1 1

~))qi ^~j~~~~

^~~~

Er,b

₌

II (I z)q(] (7)

The saddle

point

values for _qi and _{z are} most

easily

obtained [2]

by

first

calculating

the

temperature-independent quantity

_y:

~

=

-2v j

^~

_)))~ (8)

and then

using:

~)~~(~ gl)

_"

)) IT ₍₉₎

flZ

"

qi~())~() ^{l) (1°)}

We find three

phases:

a)

at low

temperatures

the

replica symmetry breaking

solution has _a

higher

free _energy than the

replica symmetric

_one. The value of _z is < I for

temperatures

below _a certain

Tc, given by:

Tc

₌

y(

^~

)i(I _{y)'~~ (ll)}

2y

b)

Above

Tc

the

replica symmetric

solution

yields

the

statics,

but there is _a range of tem-

peratures Tc

< T _<

T(

whithin which there is still _a

replica symmetry-breaking solution,

but

now with

z > I.

T(

is

given by

Tl

₌

~(()l(1- ^~)l~~ (12)

at which

temperature

_qi takes its lowest and _z its

largest

value:

2

qmin " 1-

zmax =

~

~( l) (13)

c)

^Above

T(

there is

only

the

replica-symmetric

solution with _q

= 0. The free energy ^{is then:}

Fr,

=

(14)

We shall refer to these three

phases

_as the low

temperature, intermediate,

and

high temperature phases respectively.

For future

reference,

let _us write the results for _zero

temperature.

As T _- 0 _we have:

flz

_-

a~=(~~)i(~-l)

P l/

fl(I-qi)

-

aq=(~~)i

P

E -

Ea

₌

-((~)i(1-

₊

P VP

(F-E)

_- 0

(15)

the last relation because the

entropy

_goes as

-Infl (as

it should in _a continuous

system).

From this _{we can}

compute (see _[4])

the _average

logarithm

of the

"spectral density" p(E)

of local minima of the _energy

(near

its

O(N) tail):

0

^E <

Eo

lnp

₌

(16)

~

a~(E Ea) (E Ea)

>Ci 0

2.2 TAP EQUATIONS. The "naive" _mean field free _{energy per}

spin

for this model reads:

fnaive

_"

~( _£

J"> >...>'P'~'>

'~"P #'~(i ^q)

q N =

£ ml ₍₁₇₎

;

to this _one must add the

Onsager

reaction term which is the _{same as} for _a +

spin

model [5]

and has been calculated for

p-spin

interactions in

[6]:

freact.

"

[(P ~)~~ P~~~~ +11 (18)

This free energy has _a saddle

point

at m; = 0 Vi. One _{can uncover} the structure of the solutions with _q

#

0

by making

_a

change

of variables _mi,

, mN -

ii,

,

@N, _q, where:

~' _"

@

'

$

_~i ~'~

(19)

'~'

, '~

to

get:

iTAp

₌

-qP/2(11 J;,,...,;~i;, _I;~ (in(i q)

_~ j(p i)qP pqP-1

_~

ij

_~

4

~~~~~'~"°~~~ fl~~~~

~~ ~~~ ~~~~

~~~

^~ ~ ^~ ~~~~

N°8 BARRIERS AND METASTABLE STATES ₁₈₂₃

Hence,

the behaviour of the TAP

free-energy

with

respect

to the

"angular"

variables

@; is identical to the _zero

temperature

energy

landscape,

maxima _are

mapped

into

maxima,

saddle

points

into saddle

points,

etc.

The saddle

point equations

must _now be

supplemented

with the

equation

for _q:

~~)~'

^{= 0}

⁽²¹⁾

This _can be solved in _two

steps, defining:

z +

pqi-i(i _{q) (22)}

one obtains _an

equation

for _z in _terms of the energy of the local minimum _at T ₌ 0:

Z "

(P 1) (~I~T=0

⁺

(l~~=0 ^~~c)~~~j

E0c

"

-(~~~ ^{~j~~~ (~3)}

P One root for _z is smaller and _one

larger

than

~ i/2

~~

(P )P~

~~~~

We _{can now} insert _z obtained from

(23)

into

(22),

the

largest

root for _q of this

equation yields

a minimum.

If _{we assume} that _as in the TAP

approach

for the

Sherrington-Kirkpatrick

model the _con-

dition of

validity

of _a TAP solution is the _{same as} the condition for

stabilty

^with

respect

to fluctuations inside _a cluster in the

replica approach,

_we have the condition

(see [2]):

)fl~P(P ^I)q~~~

⁺

^{(1 q)~~}

^{> °}

⁽²⁵⁾

which _can be rewritten

using (22-24)

_as

z < zc

(26)

which

implies

that

only

the value of _z

given by (23)

^{with the minus}

sign

has to be considered.

We have hence the

following picture: given

_a local minimum in _energy at T ₌ 0 it maps

continuously

into at most _one local minimum for the TAP free energy at

temperature

T

# 0,

and all the TAP solutions _ale obtained this _way. For _a

given ET=o,

the value of _q of the associated minimum decreases with

temperature (cfr. (22))

until it reaches the value _qm;n at which

temperature

the solution

disappears.

The

temperature

at which _a TAP solution

disappears depends

_on the _energy it has at T ₌ 0.

Furthermore,

since for saddle

points:

(((f)

=

q'

> 0

(27)

the TAP free _energy local minima _are ordered in the _{same way as} the local minima of energy at T ₌ 0 to which

they correspond. Hence,

the lowest TAP local minimum with _q

#

0

(with

free energy e

fa(fl))

"is

mapped

from" the lowest minimum at _zero

temperature ET=o

_"

Ea.

It is also the _one survives to

higher temperature.

Note that in contradistinction with the

Sherrington-Kirkpatrick model,

there is neither

split-

ting

of roots as

temperature changes

in _a second-order

phase

transition

fashion,

_nor

change

in the

ordering

of free

energies

of TAP solutions

(I.e.

no

"chaoticity"

with

respect

to the

temperature).

To

proceed

further _we need to know the lowest _energy at T

= 0. We make _use of the

replica

results of the

preceeding

^subsection but

only

those

for T=0,

and show that both

approaches

are consistent for all

temperatures.

Putting

the value for the lowest energy

Ea

at T

= 0

(IS)

into

(23)

_we obtain

z(ET=o)

=

())~'~ ⁽²⁸⁾

so that the TAP

equation

for _q

(22)

reduces to the IRSB

equation (9).

One _can also check that the TAP

expression

for the free _energy

(20)

coincides with the

replica expression (7):

ITAP(ET=o

"

Eo)

₊

/o(p)

"

Frsb(p) (~~)

Hence,

_we have shown that the

replica symmetry-breaking

values for _qi, and F

correspond

to the lowest TAP solution

(with

_q

# 0)

for all

temperatures, including

the "intermediate"

phase

where _{z >} I.

Furthermore,

since this is the TAP minimum

(with

_q

# 0)

which

disappears

at

higher temperature,

we have that for T _>

T(

the TAP

equations only

have the

paramagnetic

q ⁼ 0 solution.

One _can

readily

calculate the tail of the

density

of TAP-states for all

temperatures

T <

T(

using

the value for T

= 0

(16)

and

multiplying by

the

"compression"

factor

(27)

of the

density

lo ^/

^<

^lo

j inp

⁼

(30)

azqi~'~(f _{fa(fl)) (f fa(fl))}

_{>Ci °}

The coefficient

multiplying f fo(fl))

in

(30)

satisfies

(cf (15)):

azqi~~~

_"

qi~~~( ))I() ₁₎ (31)

which is

nothing

but

zfl

_as obtained from the

replica

calculation

(cf (10)),

_{as was} to be

expected

[4].

We have hence established the connection with the

replica approach

for all temperatures.

We _{can now}

interpret

the three

phases:

a)

at low

temperatures

the

system

is frozen in the lowest TAP minima

b)

for T _>

Tc

the

logarithm

of the

density

of TAP solutions _grows fast

enough

with the free energy, so that the Boltzmann

weight

exp[N(

In

p(fTAP) fIfTAP]

_~~~~

has _a saddle

point

above

lo,

and the Gibbs state is _a state made of many TAP solutions. This is

why

the

replica approach

_ceases to individuate each TAP solution. The actual free energy of the Gibbs state has in this

phase

_a contribution both from the TAP free energy and from In p

(usually

called the

"complexity").

c)

above

T(

there _{are no more} TAP solutions other than the

paramagnetic.

The

phase

^{transition from}

a)

to

b)

is in strict

analogy

with the random- _energy model [7]

(with

the

p-dependent

TAP solutions

playing

the role of the random

energies)

and has also been discussed for Potts

glasses [8, 9],

^{where it is referred to} as

"entropy

crisis"

N°8 BARRIERS AND METASTABLE STATES ₁₈₂₅

3.

Analysis

with several real

replicas.

We _now turn to the

analysis

of the

system by considering

R real

replicas (3)

with the distances between the

configurations

in the real

replicas

constrained

by (4).

The effective

averaged

free energy has the _same form

(5)

as for _a

single

^real

replica;

with

Q

_a Rn _x Rn matrix:

Qll p12 plR

p12~ _Q22 p2R

Q

_"

plR~ p2R~ QRR

The constraint term _now reads:

Fconstr

_"

£ _{Za (i Qoo)} £ _ErsPil

₊

£ _{ErsPl~ (33)}

a r#s,o r#s,o

Alternatively,

_{one can} ommit the last term and consider

£~~,

~

er,P[[

as an additional inter-

action. ^'

In this work _we

only

consider real

replicas

constrained to be at the _same distance

pl'

_{= pd} for all _r

#

_s

(34)

We shall make for

Q

the ansatz:

Q~~ =

Q

forall _r

P~' ₌

(P~'l'~

₌ P for all _r

#

_s

(35)

with

Q,

P Parisi matrices with the _same block sizes 0 _{= To} < xi < < zk < zk+i " I. The free energy reads

(

"

-( £1(Qap)~+(R~l)(Pap)~l

off

~j~~~~~~~~~

^~~~~~

~R/~~~~~

^{~~~ ~~~~}

Denoting

_{qo,qi, ..,qk, qk+i "}

I,

^and _po, pi,..,pk,Pk+i " Pd the

overlap

^{values for}

Q

and P

respectively,

_we have:

(

"

1(ql

+

(R l)Pl)Zi

₊

(ql

+

(R l)Pl)(22 Zi)

+ +

+

(ql

+

(R l)I)(1 ~k)

+

(I

+

(R I)Pl)1

~R/

^~

Ii ~~~~~~

^~

2 i

~~~~~~~ ^{~ ~}

~+i k

~~~~~~~~~~

2~fl

^~

Ii ^~~~~~~

^~

_/2 Ii ^~~~~~~~

^{~ ~}

^{~+i k ~~~~~~~~~~}

(37)

with:

q~ " q0~P0

ql'

_{= qo} +

(R i)po

q~ " q;-P;

q~' = q; +

(R I)p;

k+I

Ll

=

£z>(qj-qj-i)

;=;

k+I

ii

_"

£ _{~i(~i q)-1) (38)}

j=I for R

= I _{we recover} the usual

single-system expression.

In the next subsections _we shall discuss

specially

the _case of two real

replicas

R ₌

2,

and

only

mention the behaviour for R > 2 for _a few _cases of interest.

Within this

ansatz,

at all

temperatures,

_we have

only

found solutions with _one

breaking;

defined

by: (qi,

_Pi,

z) and,since

_we work at zero

magnetic field,

_po

" qo " 0. The free _energy

reads:

FR=2

₌

(11

⁺

« ₍₁ z)(qi

₊

qi)I

)iIniz(qi ^Pi)

+

(I

_{qi Pd + Pi}

_)i

₊

Iniz(qi

_{+ Pi +}

(I

_{qi + Pd}

Pi)li

-((I _))iIn(I

qi Pd +

Pi)

+

In(I

_{qi + Pd}

Pi)1 ₍₃₉₎

In the next subsections _we discuss the results and

implications

at different

temperatures.

3.I ZERO TEMPERATURE LIMIT, TAP _BARRIERS. Let _us start

by considering

the _zero

temperature

^{limit for} two real

replicas.

Using equation (39)

_we find that in this limit _qi

-

I,

z - 0

(as expected)

and _{pi -} pd.

More

precisely

_we have

,

with

b~, bp,

_bq ^finite:

flz

_{- b~}

~(l-qi)

_{- bq}

fl(P-Pd)

_{- bp}

(F E)

_- o ~~~

Substituting

these variables in

(39)

_we

get

their values and the energy as a function of _pd

(see Fig. I).

For pd - I the _{energy per} real

replica

is the _{same as} that of _a free

system (this

iS true also for R >

2),

_as is _to be

expected

since in that _limit the _two

configurations

coincide and _can be chosen in the _same energy minimum. As _pd decreases the energy per real

replica

increases until it reaches _a maximum

Emax

at _a certain

p7~~

^when

fl(pi _p7~~)

= bp = 0. For

smaller values of _pd the _energy decreases until it reaches

again

the value for _a free

system

at

pd " 0. This tells _us that the minima of lowest energy are

typically

at

right angles

to each

other,

_a result that is not

surprising

for

entropic

_reasons. We will discuss the behaviour with pd in _more detail in the _next section.

Let _{us now} concentrate _on the

implications

this has for the barriers. Assume _one starts

on a local _energy minimum with _energy

Eo (the

lowest _energy for which there is _an

O(e/~)

N°8 BARWERS AND METASTABLE STATES ₁₈₂₇

2.51

~

-2.52

2.53

2 54

0 02 04 06 OS I

Pd

Fig.

I. Free _{energy as a} function of _pd for _p

= 4 and T ₌ 0.

number of local

minima)

and walks towards another minimum also of _energy ED We have

seen above that these other minima _are at

right angles

to the

original

_one. In

particular,

this

means that _{at a} certain moment of the

journey

one has to be _{at a} distance

p7"

^{from the}

point

of

departure.

At this

point

the energy

is,

_say,

Ei

But:

El

+

Eo 2 2Emax (41)

because

2Emax

is the lowest combined _energy of two

points

of

phase-space

at

p$~~

^{from each}

other. Since _we have found

Emax

>

ED,

we have

El

_ED

~ 2(Emax E0)

> 0

(42)

Hence _we have shown that

Emax Eo

is _a lower bound for the energy barrier between _two

typical

minima of energy

Eo.

In

particular,

this

implies

that the barrier energy grows like

N,

unlike the

Sherrington-Kirkpatrick

model for which it is believed _{to grow as a} certain _power N°

, a < I.

For finite

temperatures,

we have shown that the TAP free energy saddle

points

are obtained

by

_a

mapping

of the _zero energy saddle

points-

their free energy

being

_a smooth function of the

temperature.

Hence _{we can}

immediately

extend the

previous

discussion to finite

temperatures

and TAP free _energy barriers.

Indeed,

the TAP local minima do not

disappear

because the

barriers

separating

them

vanish,

but because

they

_cease to be minima in the q direction.

3.2 0 < T <

Tc.

^{Here and in} what follows _we refer

(unless

otherwise

stated)

to the

system

with

only

two constrained

replicas.

We denote the

parameters

^for a free

system (single

real

replica)

with the

superindex

"free".

.43

,44

.45

0 0,2 04 06 OS I

pd

Fig.2.

Free _energy

as a function of _pd for _{p =} 4 and T

= 0.3

(< Tc).

We have

found, depending

_on the value of _{pd, two}

types

of solutions:

I)

from _pd

" 0 _up to a a critical value

p(~

there is _a solution with _pi

#

_qi The value of _pi

increases with _pd from _zero at pd " 0 to pi " qi ^at the

point

_pd

=

p(~

The value _{of qi} varies little in all the _{range of pd.}

II)

_{for pd > p~~~} the solution has _pi

" qi, both

increasing slowly

with _pd. The

merging point

p(~

of both solutions satisfies:

( _{P(P I)q~~~}

"

(I _p~~~)~~ (43)

There _are three values of _pd for which _{pd " pi1}

Pi " pd " 0

(solution I)

Pi "

p7~~

_<

pi~ (solution I)

Pi " Pd " qi " q~~~~

(solution II) (44)

The free energy as a function of _pd has _a similar behaviour _to that of the

preceeding

subsection

(Fig.2)

at pd " 0 the free _{energy per}

replica

is

equal

to that of

a free

system

^and _{pi =}

0,

_{qi =}

q(~~~ As _pd increases the free _energy

increases,

until _a maximum is reached _at the

point

pd "

P7~~

" Pi- For

larger

_pd the free energy falls until the

point

_pd

= pi = qi =

q(~~

^is

reached for which

again

the free _{energy per}

replica

coincides with that of _a free

system (this

is also true for R >

2).

For

larger

values of _pd the free _energy increases

again.

The

physical interpretation

of the behaviour for _{pd <}

p(~

is much the _{same as} for the _zero

temperature

limit. For _pd of the order and

larger

than q(~~~, we are

choosing configurations

N°8 BARRIERS AND METASTABLE STATES ₁₈₂₉

inside the _same state. Since the

overwhelming majority

of the

pairs

of

configurations

inside _a state are at _a distance _q(~~~ from each

other,

if _we fix _pd

" q(~~~ ^we have for R

replicas

_{an energy}

equal

R times that of _a

single replica,

and

an intra-state

entropy

also

equal

to R times the intra-state

entropy

^of a

single

state

(the

volume of the set of R

configurations

inside _a state

mutually

at distances _pd

" q(~~~ is the volume of

configurations

to the

R~~ power).

The

logarithm

of the number of _pure states

(complexity)

remains the _same; since _{once we} have chosen in which state the first

configuration is,

_we determine that the other R I _are in the _same state.

However,

since the

system

^{is frozen in} this _range of

temperatures (see

the discussion at the end of section

2),

the

complexity

of the

equilibrium

distribution is _zero both for _one and for several real

replicas.

For pd

larger

than _{q(~~~ we} start

paying

_a

price

in intra-state

entropy

^because the

logarithm

of the volume of

configurations

inside _a state with mutual distances _pd decreases _as -N

In[(I qf~~)~ (P qi~~)~]

3.3

Tc

< T _<

T(.

We _now turn to

considering

the intermediate

phase

in which there _are

many pure states

(and

TAP

solutions),

but for which the usual

replica

trick

yields

_a

replica

symmetric

solution

corresponding

to _a Gibbs state made of _an infinite

quantity

of _pure states each with

vanishing weight.

We shall _see that

considering

several constrained real

replicas yields

useful information in this situation.

We have in this _case to

distiguish

between two

possibilities.

Let z~~~~ be the value of _z of the IRSB solution of _a free

system.

In this range of

temperatures

I < z~~~~ < zmax

(cf. (13)).

If

we consider R real

replicas,

the solution behaves in _a different _way if the

temperature

is such that

z~~~~/R

> I than if

z~~~/R

_< I. The _reason for this will be

explained

below.

For the _case

z~~~~/R

< I _we have found in this

temperature

_range, three

types

of solutions

depending

on the value of _pdi

I)

from _pd

= 0 _up to _{a a} critical value

pf~

there is the

replica-symmetric

solution with

Pi " qi = 0.

II)

Above

pf~

the

replica symmetry

is

broken,

there is _a solution with _{non-zero pi}

#

_qi At the

starting point pf~

the value of _z

=

I,

and it decreases for

higher

_pd, hence the free _energy

changes continuously

with pd.

III)

At _a

higher

value of _pd

"

P~~ satisfying (43)

the solution merges with _a solution pi = qi,

as in the low

temperature phase.

The free energy per

replica

at pd = 0 coincides with the

replica symmetric

free energy of _a free

system. Again,

as pd increases the free energy increases

(see Fig.3),

until _a maximum is

reached at the

point

_pd

=

p$~~ (but

still _qi

= pi "

0).

For

larger

_pd the free energy

falls,

it

does _so in the whole _range in which the solution

(II) holds,

and further up to when the

point

pd = pi = qi " q(~~~

(within

the range of solution

(III))

is reached. At this

point

the free

energy per real

replica

takes _a value

Fr,b

"

fa(fl).

For

higher

values of _pd the free energy grows

again.

^Note that

Fr,b

₌

fa(fl)

is in this range

la~yer

than the

replica symmetric

free

energy.

We have

found, using exclusively

the

replica trick,

the lowest free energy of _{a pure} state- in _a range for which the

ordinary replica

trick _ceases to individuate _pure states because

they

have

vanishing weight.

To understand this _we note

that,

_as described in the last

subsection,

when _pd

" qi "

q(~"

the energy and the intra state

entropy

for R

replicas

_are

just

those of _a

single

one

multiplied by R;

while the

density

of states at _a

given

free energy remains the _same. The Boltzmann

weight (32)

then reads:

exp[N(

In

p( fTAP) flR fTAP] (45)

0,76

0 78

o so

0 52

054

0 02 04 06 OS

Pd

Fig.3.

Free _{energy as a} function _{of pd} for _p

= 4 and T

= o.6

(< T(

and _>

Tc).

If the

system

^is

frozen,

as in the last

section,

the saddle

point

is

simply

the lowest value

fTAP

_"

fo(fl),

and it does not

depend

_on R. In the

phase Tc

< T <

T(

we have _seen that the

quantity

of states grows with their free _energy fast

enough,

_so that the saddle

point

of

(32)

is achieved for states of

higher

free

energies.

For R

replicas

the factor R in

(45)

^{favours the}

free-energy

in its

competition

with the

complexity,

and hence the saddle

point

is achieved _at lower free

energies.

For R

large enough

_so that

z~~~~/R

< I the

system

freezes

(see

discussion

after

(30)).

3.4 METASTABLE AND UNSTABLE SOLUTIONS. In the

preceeding

sections _we have

empha-

sized the

importance

of the solutions at values of _pd at which the free energy of the constrained

system

^is

stationary

with

respect

to the constraint value:

$(pd)

_" 0

(46)

Pd

At this

point

it is useful _to look _at the calculations _we have been

doing

from _a

slightly

different

point

of view: We consider the constraint term in the free energy

(33)

without the last

term,

^I.e. we fix the interaction _term

strength (instead

of

considering

it _{as a}

Lagrange multiplier):

F(e)

=

F(pd)

_e

£ _{P$[ (47)}

r#s,a and calculate the value of _pd obtained from _a

given

_e

by:

e =

)(Pd)

=

)(Pd) (48)

N°8 BARRIERS AND METASTABLE STATES ₁₈₃₁

Clearly,

at

stationary

values of _pd such that

(46)

holds the additional interaction needed to

impose

the constraint is _zero. One _can

imagine

a process in which the interaction _e is

changed adiabatically

until the

system

^reaches one of these

points (note, however,

that

"adiabatically"

involves

reaching equilibrium

at each

stage,

I.e. _very

long times).

We shall return to this.

Consider _now instead the _case in which _e is fixed and small. The total free _energy

(47)

in

terms of _pd

(in

the

present

context a variational

parameter) corresponds

to the _{ones we} have

calculated

(Figs 1-3),

but "tilted"

by

the term -epd. For T _<

Tc

this has the

effect,

however small _e

,

of

favouring

the minimum

corresponding

to

larger

_pd

(Fig.2).

As _{soon as we} reach

Tc

the free energy of this minimum increases with

temperature (Fig.3),

this

finally

overcomes the effect of the term -epd at a certain

temperature Tc(e),

above which the minimum with smaller pd dominates. For _e

small, Tc(e)

_ci

Tc. Hence,

_a small

perturbation

_e has the effect of

making

the hidden first order nature of the transition show _up: this will be shown in section 5 to be very useful in actual simulations.

Let _{us now} discuss the relevance of the

points

for which the free _energy is

stationary

with

respect

to pd. Since in these

stationary points

the interaction needed to

impose

the value _pd is

zero, one would

expect

that the solution

describing

them

corresponds

to _a solution for the free

system.

Given that the free _energy of _some of those

stationary points

is

larger

than that of _a free

system,

the

question

arises _{as to}

why they

_are not considered _as the

equilibrium

solutions for the free

system following

the usual criterion of

maximising

the free _energy. We show below

that these solutions with

higher

free energy indeed

correspond

to saddle

points

of the

replica

free energy for _a free

system,

but with _{z >} I for the stable

stationary point

and with two

breakings

and _{z2 < xi} for the unstable

stationary point.

Consider first the stable

stationary point

with _pd

" 0. It is _{easy to see} that the free energy

for all

temperatures

of this

point

coincides with that of _a free

system.

The fact that this

happens

for _{pd "} 0 is due to the absence of

magnetic

field.

Next,

let _us consider the stable

stationary point

^{with with} _{qi " Pi " Pd " q(~~~} ^{Within this}

subspace,

it is

easily

verified that the free _energy of the

system

of R

replicas correspond

to the free energy of _a free

system

^with

parameters #i, lo,

it _:

#1 ~ ql

fo

_- 0

it _- zR

(49)

Note that the constrained

system

^with z < I

corresponds

to a free

system

with it _< I for T _<

Tc

but with it _> I in the range

Tc

_< T _<

T(.

Hence in the intermediate range of

temperatures

the free energy of this

stationary point corresponds

to the

replica symmetry breaking solution,

which is

higher

than the

equilibrium

bee _energy.

Finally,

consider the maximum of free energy with

respect

to pd. In the low

temperature phase

_we have _seen that this

corresponds

to a value of _pd such that _{pi " Pd.} It _can be

easily

verified that _a constrained

system

under this restriction

corresponds

to _a free

system

^with two

breakings

with

parameters

1i2,

iii, #2, #1, lo

_" 0 related to that of the constrained

system by:

#2 ~ ~l

Ii

_{- Pi " Pd}

§o - 0

f2

_{~ Z}

iii

_- Rx

(50)

Note that 1i2 <

iii

At intermediate

temperatures

we have that the maximum with

respect

to pd of the _con- strained

system corresponds

to qi " pi " 0. This is related _to the free _energy functional of _a free

system

^with one

breaking

and:

Ii

_{- Pd}

#Q - 0

~ ~ R

(51)

Hence,

_{we now see}

why

these

points

have been

rejected

_as

equilibrium

solutions for

(even

if

they

_are saddle

points of)

the free

system,

but _{we now} have _an

interpretation

for them.

4. Saddle

points

of the free _energy for _a

single

real

replica.

We have _seen above that all solutions of the

replica

saddle

point equations

for _a

single

system

are

potentially interesting,

even when

they

do not

correspond

to the

equilibrium

distribution.

It is hence of interest _to be able to enumerate all such saddle

points, including

those with

z >

I, decreasing

_z; etc.

In order to do _{so we}

generalize

the

procedure

of reference [2] ^to more than _one

step

^of

replica symmetry breaking-without assuming

^either _z; < I _{or z; <} z;+i. ^We assume k-level

breaking.

We start

by

the

replica

free _energy for _a

single system. Putting

R

= I in

(37):

~

(~~~l

+

Q~(~2 Zl)

₊ +

~~(~ ~k)

~

~)

~()

~

)l~(~l)

⁺

((

))ln(~2)

^{+ +}

()

))lD(~k+1))

(52)

with:

k+I

~l" =

£ _{zi(qj qj-i)}

_I

=

I,..

,

k j=I

~k+I

_" I qk

($3)

Differentiating

with

respect

to the _q;

we

get

the

equations:

lzs

zs+1

3qs

_{zs-i zs}

qs-i~

~

l~(~l~~ ~lll)

"

)[~~)() ^(5~)

S

and

~tq(~~

qo

[Li]~~

" 0

(55)

Here and in what follows _{~J %}

fli.

A second set of

equations

is obtained

by differentiating

with

respect

to the _z;. After _some

algebra,

and

making

_use of

(54)

we

get:

((~i~ ^~i-')

⁺ lL(~~

~~-i)~iI'

=

~~'ili~~~ (ill()) ₍₅₆₎

N°8 BARRIERS AND METASTABLE STATES ₁₈₃₃

Multiplying (54) by (q, q,-i)

and

dividing by (56)

we

get

an

equation independent

of the

temperature:

~

(f~ w()- Ii- _il'~

₌

-~l/~ (i~i l~)~'

^{S "}

ⁱⁱ

^k

^~57)

where _we have

defined,

for I

=

I,..

,

k:

qs-i

w, =

qs

YS "

~#~ ⁽⁵⁸⁾

s

Another

equation independent oftemperature

is obtained

by dividing (54)

for two succesive values of the indices:

'°$il

_i

W, i

Vi](~ _{~'°s+I) (i Wi~~)} g,g,+1'

^~

^{~'' '~}

^~~~~

These last two sets of

equations

have _to be

supplemented

with the

equation

for _qk:

lL~i~~

=

[

₊

_~llrl°

_{+ +}

~ qk-i qk-2

~ qk qk-i

LkLk-i Lk+iLk

=

(1

~~)2Y (6°)

where _we have defined:

ow, the to

solve the

a)

_assume

a

value

of

_wi

b) using

₍₅₇₎ this yields a

value of

_vi

c)

nowing wr,yr _{one can} calculate _using

value for _{wr+i .}

'°'+~"°i+2.. wk

Lk

~~

~

^{Yi'Yi+I.. yk}

^(~~~

from which _{one can}

compute

^Y.

e) Using (60)

^{and the value of Y} one obtains qk and

Lk+i,

and hence all the _q; and the _z;.

f)

_one must now check that the value of _go obtained

by:

go " qk wi .w2.. wk

(63)

satisfies

(55).

This is

trivially

true if the root qo = 0 of

(55)

is used.

Note that the

steps (a) (d)

are

independent

of the

temperature.

There is also the

possibility

of _a continuous

change

in

thq

_q;,z;. In this _{case one} does not need to minimi2e with

respect

to intermediate _z;.

Taking

the _square root of

equation (54) assuming

_{q; q;-i} and _{q;-i q;-2}

small,

and

subtracting

two succesive

equations,

_one

easily

arrives at

~(~)

_"

(i~(P ~))~~/~(( ^{~)~~~/~ (6~l}

z decreases with _q.

In order to _see how this continuous branch is

matched,

_we have solved the

equations

with I

breakings

for

large

values of k. Note first that the map

(wr,yr)

_-

(wr+i, l/r+i)

has _a fixec

point

at _{w =} y = I.

Equation (57)

reads for _{w, Ci} y, ci I:

~j~(i-w,) =(i-v~) ₍₆₅

Now,

the successions _{wo, wi, w2,} and go, Vi, l/2, are both

increasing

and

approach

_{w = y =}

I. Near this fixed

point,

since the

change

_{in q and z} become

infinitesimal,

_{we are on} th<

continuous solution.

We have from

equation (58)

that:

~_

q;

_~ ~ ~

i=-- s+I. s+2.. k

~k

§" _"

~fl~

" Ys+I.Ys+2...Yk

(66)

s

One _can check that the

approach

_{wr, yr -} I is slow

enough,

_so the

tbr, §r

defined in

(66)

_go

to _zero for finite _r and k _{- cc.}

Hence,

for _an infinite number of

breakings, only

the values of

tbr, §r

for _{r near} k _{are non-zero.} But these values _are

products

of _{w;, y; near} to _one

(I.e.

_{q, z}

varying continuously).

For

large

values of _{r we ale} entitled to _use

(65)

to obtain:

d

n~

p 2 ~~~~

from which _we

get:

16 #

jiF-3 (68)

where _we have set the _{constant to one} because

§k

"

tbk

" I.

One _can also write the

expression

for Y

(61)

_{as an}

integral,

to

get:

Y =

f~ ^dtb§~

=

(69)

o P

The maximum value qk for k _{- cc} is obtained from

(60)

^{with the} value

(69)

of Y:

q©/1x~(1 qmax)

=

(~~~[

~~

)~/~T (7°)

This

completes

the solution of the limit k _{- cc.} One _{can now}

easily

check that

equation (68) together

with _qmax from

(70)

is indeed the continuous solution

(64) palamefrized

_as follows:

starting

from _{z - cc, q -} 0 the value of _z decreases while that of _q increases _{up to q}

= qmax

where the solution ends.

It is

interesting

to note that that the

equation

for _qmax is

exactly

the

expression

obtained in the TAP

equations

for _q of the

highest

_energy

saddle-points~

I-e-

inserting ET=o

₌

Ec

in

(22- 24).

One _can also check

(after

_some

algebra)

that the free energy of this continuous solution

also coincides with the free _energy of such saddle

points.