On the glassy nature of random directed polymers in two dimensions

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Submitted on 1 Jan 1990

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On the glassy nature of random directed polymers in

two dimensions

Marc Mézard

To cite this version:

On the

_glassy

nature

of random directed

_polymers

in

two

dimensions

Marc Mézard

Laboratoire de

_{Physique Théorique}

de l’Ecole Normale

_{Supérieure (*),}

24 rue _Lhomond, 75231

Paris _{Cedex 05,} France

(Received

on

_April

_{2, 1990,}

_accepted

on

_May

16,

1990)

Résumé. 2014 Nous étudions

numériquement

des

polymères dirigés

dans un

_potentiel

aléatoire en

1 + 1 dimensions. Nous introduisons deux

copies

du

_{polymère, couplées}

par une interaction

thermodynamique

locale. Nous montrons _que le

système

est instable sous l’effet d’une

_répulsion

arbitrairement faible entre les deux

copies.

Ceci

_suggère

une similitude avec une

_phase

verre de

spin,

avec

_plusieurs

« vallées », où les différences

d’énergie

libre entre vallées croissent comme

t03C9,

où t est la

longueur

du

polymère

et 03C9 est

_{probablement égal}

à

_1/3.

L’effet d’un

_champ

électrique

transverse est étudié en détails et on démontre l’existence _pour la

susceptibilité

correspondante d’importantes

fluctuations d’un échantillon à l’autre. Les résultats des simulations

sont

_comparés

à des calculs

_analytiques

utilisant la

_{représentation}

de ce

_problème

en termes de

mécanique quantique

et l’Ansatz de Bethe.

Abstract. 2014 We

study numerically

directed

polymers

in a random

_potential

in 1 + 1 dimensions.

We introduce two

_copies

of the

_{polymer, coupled through}

a

_{thermodynamic}

local interaction. We

show that the _system is unstable versus an

_arbitrary

weak

repulsion

of the two

_copies.

This

suggests a

_similarity

with a

_{spin glass phase,}

with several «

_valleys

_», where the

_typical

differences of the free

energies

of the

_valleys

grow like

t03C9,

where t is the

length

of the

polymer

and

03C9 is

_{probably equal}

to

_1/3.

The effect of a transverse electric field is studied in details

_showing

the existence of strong fluctuations from

_sample

to

_sample

in the

_{corresponding susceptibility.}

The results of the simulations are

_compared

to

_{analytic computations using}

the quantum mechanical formulation of the

_problem

and the Bethe Ansatz.

Classification

Physics

Abstracts

75.ION - 05.50

1. Introduction.

The

_study

of directed

_polymers

in a random

_potential

is an active research

_topic

which is

important

in several

_respects

_[1-6].

Besides the

_polymers

themselves it is related to interface

fluctuations and

_{pinning [2], spin glasses [3], crystal growth [4],}

the random stirred

_Burgers

equation

in fluid

_{dynamics [5],}

and

_quantum

mechanics in a

time-dependent

random

_potential

[6].

In this _paper we shall be

_particularly

interested in its relation with

_{spin glasses,}

since this

system

might

well constitute a kind of «

_{baby spin glass » problem.}

This relation with

_spin

glasses

was

_{particularly emphasized}

_by

Derrida and

_Spohn

_[3].

_Solving

the mean field

theory

of the

_{polymer problem,}

i.e. the

_problem

of

_polymers

on the

_Cayley

_tree,

_they

found a rather

simple spin glass

structure. This

_system

has a low

_temperature

_phase

with broken

_ergodicity

and many pure states, where the

_overlaps

between different states

_(the

fraction of common monomer

_positions)

vanish. This behaviour is

_exactly

identical to the random energy model

[7]

which has been shown to be a kind of

_{simplest spin glass}

model

_{[8], displaying}

most of the

properties

of the

_{Sherrington-Kirkpatrick}

model

_[9].

For finite dimensional random directed

polymers

in d + 1 dimensions the situation is somewhat less clear. A

_phase

transition has been shown to exist for d :> 2

[10]

and is found

_numerically

for d = 2

[11]

but

the nature of the low

temperature

phase

has been

_{explored only through}

_{1/ d}

_{expansions [12].}

Here we want to

_study

this

_problem

_starting

from the other

_end,

i.e. in 1 + 1 dimensions.

The

_problem

is

_easily

formulated : in a two dimensional _{« space} time » with coordinates

(x, T )

we consider all oriented walks

_{x(T ),}

T E

_{[0, t]}

_]

with

_x(O)

=

0,

and

x(t)

=

y. The

partition

function of this

_system

at

_temperature

T =

_{1/ f3}

is :

where we have added to the usual Wiener measure for random walks an extra _{space time}

dependent

random

_potential

_{V (x, T).}

We want to

_study

the case where the

_potential

is of the

white noise

_type

with local correlations

_(throughout

this paper a bar denotes the _average over

various

_samples

_{and ( )}

denotes the thermal

_{averages) :}

(A

simpler

model with nonlocal correlations can be

_analysed

in

great

details

[13].)

The total

_partition

function for walks

_originating

at

_{(0, 0)}

is :

The

_{thermodynamics}

of

_the

_problem

is

_{given by}

the

_quenched

free _{energy :}

and another

_{interesting quantity}

is the

_{probability density}

of the

_position

of the end

_{point :}

This

_problem

can be handled

_through

numerical

simulations,

where transfer matrix

techniques

enable an exact

_{computation of Z(x, t)}

for

_systems

of sizes _{up to} several thousands

time

_steps

_[14].

It can also be studied

analytically through

a

_mapping

to the

_quantum

mechanical

_problem

of which

₍₁₎

is a

_{path integral}

representation

and

through

the use of the

replica

method

_[6].

_Many

efforts have

_already

been devoted to this

_{study ;}

it is

_expected

that

this

_system

does not have a

_phase

transition and is

_always

in its low

_temperature

phase.

is known to scale

_{as (x2) t2 JI}

where several

_arguments,

both numerical

_[2,

14,

1]

]

and

analytical [5,

6,

15],

indicate that the

_{exponent v}

is

_equal

to

_2/3.

Our aim is to

_study

the nature of this

_phase,

and

_especially

to

_try

to

_analyse

if it shares some

aspects

of a

_{spin glass phase.}

The

_approach

will be

_mostly

numerical. In the next section we

analyse

a

_system

of two

_copies

of the

_{polymer coupled through}

a

_repulsive

local interaction.

We also

study

the effect of weak transverse electric

_field,

_coupled

to the

_{endpoint position}

of the

_polymer.

Section 3 draws some

_comparison

with some

_analytic

Ansatz for

Z(x,

t ) recently

proposed by

Parisi

[15].

Section 4 contains some

_speculations

on the

physical

nature of this

phase

as it

_might

be

guessed

from the

_preceeding

numerical simulations.

2. Numerical simulations.

In the mean field

_theory

of

_{spin glasses,}

a standard method to characterize the

_{spin glass phase}

is to

_compute

the distribution of

_{overlaps P (q )}

[16,

9].

To

_compute

_{P (q )}

for one

_given

sample

we introduce two identical

copies x( T )

and

_{y( T)}

of the

system

(with

the same

realization of the

_potential)

which are

_uncoupled,

and

compute

the distribution of

Numerically

it is easier to

_compute

the first few moments of

_{P (q) ;}

we find that

Tfi)

goes to a finite and nonzero limit _qo

(which depends

on the

temperature)

when

t --+ _oo, but the second connected

moment (q 2) - (q) 2 seems

to _go to zero in the

large

time

limit,

suggesting

a trivial

_{P (q)}

function :

_{P (q) & (q - q 0).}

However it is well known that a

_system

can have several pure states and a trivial

q

function

_{[17, 18].}

This

_{happens typically}

when the free energy differences between the

states grow like

t’’,

with 0 y

1,

in the limit of

large

volumes.

(To

get

a nontrivial

P (q)

one needs y =

0.) Recently

Parisi and Virasoro

[ 18]

have

pointed

out that the existence

of several states and of

_replica

_symmetry

_breaking

effects in

_{spin glasses}

should be

_analysed

through

the

_reactivity

to a

_{thermodynamic coupling}

of several

_copies

of the

_system.

This kind

of

_approach

has

_already

been used in the numerical

_study

of three dimensional

_{spin glasses}

[22]. Specifically

we introduce two identical

_copies

of the

_system,

which are

_{coupled through}

a

delta function

_potential

of

_strength

e. The Hamiltonian we consider is therefore :

(for

E > 0 the

coupling

between the two

_copies

is attractive while for E 0 it is

_repulsive).

The

averaged

free energy is

and the

_overlap

between the two

_copies

is :

We have

_{computed numerically}

_{the average}

_{overlap q (E)}

for various values of

two dimensional

_grid

_(x,

_{t )}

E Z x N and the

partition

function for

pairs of polymers coupled

as in

_{(8) arriving}

at

_points

_{x, y} at

_{time t,}

satisfies the recursion relation :

In order to

_{compute q}

we introduce the

_partition

function

Z(x,

_{y, q, t )}

which sums over

pairs

of

_{paths arriving respectively}

at the

_{points x}

and _y, with a fixed

_overlap

_q. The first

00

moment

_{in q}

of

_Z,

Y(X,

y, t)

=

E

qZ{x,

y,

q, t )

satisfies :

Eventually

one

_{gets :}

Equations (11), (12)

have been iterated with

_gaussian

local

_potential

of mean 0 and variance 1. Most of the simulations were carried out at

₁₃

=

7,

and some of them at

,8

= 10. The sizes

ranged

up to t = 255. The results _{are as} follows

(see Fig. 1) :

for

E 0 there are _very little finite size effects and

_{nothing spectacular happens,}

in the sense that

lim

_{qt( E )}

is a smooth function

_{q(e). (For}

₁₃

_{= 7,}

_{q(O) - 0.87.)}

For £ 0

_(repulsive

t-+oo

interaction)

the

_{overlap q t ( £)}

decreases. But there is a

_strong

finite size effect : the decrease is

much

_sharper

for

_large

_systems.

In fact the data for

_qt(£)

can be

_plotted

with the

_scaling

_form

qt(£)

=

q (--t 1 -

Figure

1 b shows this

scaling

form,

with w =

1/3.

This value has been chosen because of some

_arguments

which will be

_exposed

in the last section. From the numerical data

_{of qt(£)}

one cannot determine w with

_{high precision,}

but it is found to be in the range 0.2-0.4. There are still some finite size effects at t = 255 which make it difficult to

give

a much more

_precise

determination of w. From this

_scaling

we find that :

Therefore the free energy

F 2 ( B)

is

_nonanalytic

at E = 0. This shows that this

phase

has _a nontrivial structure with _{the coexistence of several pure} states. It has

_recently

been

_proposed

that such an effect could be

_present

_[15],

this will be discussed in the next section. We shall

expand

on a

_possible

_{interpretation}

of these results in the last section.

Besides the

_coupling

between two

_copies,

another kind of

_perturbation

one can add to the pure

system

is a field

_conjugate

to the

_{endpoint position x(t)}

of the

_polymer.

This is useful in

Fig.

1

_{- a) Average overlap}

between two

_copies

of the

polymer

in the same realization of the

potential, coupled through

a local

_potential

of

_strength

e, versus E

_{(-- negative corresponds}

to a

repulsion). Polymers

of different

lengths

are

_{represented by}

different

_symbols.

The

_lengths

studied are t = 31

(cross),

63

_(diamond),

127

(octogon),

255

_(square).

The number

of samples

studied to

_perform

the average is

respectively,

for each of the sizes : 1 000, 200, 200, 200. The inverse _temperature is

f3 = 7.

b)

Same data

plotted

versus the variable

et2/3 .

energy of the

system.

It is like a transverse uniform electric field

_acting

on the

_charged

head

(endpoint)

of the

_polymer.

In the mean field

_{theory (on}

the

_tree)

it is _easy to see that there

exists a critical

(de

Almeida Thouless

_like)

line

_Tc(h)

such that for T «--

_Tc(h)

the

_system

is in its

_{glassy phase.}

The

_equation

for

_Tc(h)

is identical to that found in the REM -

_{simplest spin}

glass [7, 8].

In one dimension one does not

_expect

such an

effect,

but we will see that

interesting

behaviour takes

_place

in the limit of small values of the field. As we have seen the

typical lateral

extension scales like

_{(x) ’"}

_{t JI,}

with v -

_{2/3 ;}

furthermore the

_previous

results

on

coupled

_{systems suggest}

that the free energy of relevant fluctuations scales like

t6j,

with say i -

_1/3.

As a _{consequence it is natural} to scale this field like h =

htP

in such a _way

that

htP tJl ’"

tW,

i.e.

_{p = cd - v ’" -}

_1/3.

_Plotting

_(X)

_/t2/3as

a function of

_ht1/3,

we find a linear

behaviour which is reasonable since it

_corresponds

to a linear response

W -

aht for small h. This linear behaviour in h and t is proven in

appendix

1

_[25].

The

_surprise

comes from the

study

of each individual

_sample.

In

_figure

2 we

_plot

_(x)

/ t2/3

and

( X2> _ X> 2)/t

versus

ht 1/3

for one

_sample,

with

_/3

= 7 and t = 1 023. We _see that the behaviour is much richer.

Inside some intervals of

h

the extension

_(x)

_/t2/3

is

_nearly

constant, and it

_{jumps suddenly}

to some other

_plateau

at some critical values of

h,

which fluctuate from

sample

to

_sample.

The

susceptibility

«x2) - (x)2)/t

is

_basically

zero inside the

plateaux,

and _possesses

_{high peaks}

at the critical values of

h ;

when

_averaging

over

_samples

these

_{high peaks}

dominate the average

leading

to

«X2) - (x) 2) ’"

t. The width of the

_peaks

scale as t - a and the

_height

like

t1 + a in

order to ensure this linear behaviour in t for the average

susceptibility.

As we shall see

Fig.

2

_{- a)}

Rescaled average

position

of the

_endpoint

of the

_polymer,

_X>

_/t2/3,

, versus a rescaled

transverse electric

field, h

= ht

1/3 .

The data is from one

_{single sample}

of

_{length t}

= 1 023, at inverse

temperature 8 = 7.

b)

Rescaled

susceptibility

( X2> _ (x) 2) /t,

versus the

rescaled

_field, for the same

sample.

This

_analysis

_{of the response} to a field is

_interesting

since it

_gives

information to what

happens

in zero field.

_Concerning

the

_{susceptibility X =}

_{(X2> (X> 2@}

we

_expect

that for

almost every

sample (i.e.

with a

_{probability going}

to one when t

00),

Ylt

will be zero, but

occasionally (with

a

_{probability t-}

cd),

there will exist a

_sample

where h = 0 is a critical value of

h,

and therefore

_{X ’" t 1 + cd,}

, so that the average

susceptibility y

is linear in t when

t --+ oo. In order to check

this,

and to

_compute

the

_{typical susceptibility (which}

must be

_quite

different _{from the average}

_one),

we have

_{computed numerically}

the values of

_{(x 2> _ (X> 2}

for several thousands of

_samples

with sizes

_ranging

from t = 31 to 1 023.

Letting

aside the tail of

the distribution

_{corresponding}

to rare

_{samples (on}

which the statistics in

_{naturally poor),}

we

have found that the

_{typical susceptibility}

scales like

_{t2 IL,}

where 1£ is in the range 0-0.2. In

figure

3 we

_plot

the

_probability

that X

Ito.22

be less than c, for various sizes t. For c not too

large

this

_probability

is

_independent

of t. Of course when c becomes

large

the finite size effects

become

_important.

From

_figure

3 we

_conjecture

that the limit when t --+ o0 of the

_probability

00

that

_{( (x 2> _ (X> 2)/t2 y}

is a function

_{P (y),}

with

P ( y ) dy = 1.

The fact that

0

X is

dominated

_by

rare events

_corresponds

to a tail of P at

_large

_y, which will behave like

P (y) - y - P,

p =

(5 -

3 (4 -

3

J.L).

A more

precise

determination

of »

is difficult

Y - lm

because of

_large

fluctuations in

_{Log (,y). Actually}

when

_plotting

the distribution of

Log

(X ),

we find a

_peak

of the distribution which is

_independent

of t, which would indicate

that » -

0. But the tail of the distribution at

_large

values of

_Log

_{(X )}

increases

_slightly

with t,

resulting

in the effective

_growth

of

_{exp (Log (X)) like t 0.22}

. Much more statistics and

larger

sizes will be needed to

_give

an accurate value of », but in any case it is much smaller than

_0.5,

Fig.

3 -

Probability

that the rescaled

susceptibility

(X2) -

(x)

2)/tO.22

be less than a

_given

constant c, versus c. The data was taken at inverse _temperature f3 = 7, in _zero field. The sizes are t = 127, 255, 511, 1 023. The number

of samples

studied to compute the

probability

is

respectively,

for

each size : 8 000, 4 000, 2 _000, 1 000.

3.

_Comparison

with the

_{analysis through}

the

_replica

method.

As is well

known,

in order to

_compute

_{the free energy}

_{F ( t )}

and the

_{averaged probability}

density

of the

_{endpoint P (x, t),}

defined in

_(4),

_(5),

one can use the

_replica

method and the

quantum

mechanical

_{representation [6, 19]. Starting}

from the

_partition

function

Z(x, t)

defined in

(1)

let us introduce :

then we have :

and :

On the other

hand,

performing explicitly

the

_integration

on the random

_potential

in

_(15),

we

find for

_{Z(xl, ..., x}

_{n’ t)}

=

Z(xl, - - -, X

n,

t )

exp (-

{3 2 nt

(0)/2)

a

path integral representation

which shows that

Z

satisfies the

_following

_Schrôdinger

_equation

_{(1) :}

(1)

Here we use the continuous formulation which is more _compact. A

_{precise meaning}

can be

_given

to all

_{quantities by going}

to a discretized version of the model, similar to the one used in

(11)-(12).

Then the infinite constant 8

₍₀₎

_{which appears in the definition of}

Z

_gets

_regularized.

This constant leads to a

with the limit condition

_{Î(xl, ...,}

je

_{n’ t}

=

0)

=

n 8

(x,,).

Therefore

Z

is the Green’s function

a = 1

of a

_system

of n

_{particles interacting through}

an attractive 6 function

_potential.

This

_suggests

to write

where f/Ja(Xb ..., x n)

are

_eigenstates

of the n

_body

Hamiltonian

_(18),

and

_Ea

are the

corresponding energies.

For t

_{large (19)}

should be dominated

_by

the

_ground

state

_(2),

which is

at least for n

_integer

and nonzero the Bethe Ansatz wave function :

This fact has been used

_by

Kardar to show that the fluctuations of free

_energies

from

_sample

to

_sample

scale as :

which in turn

_{implies through}

an indirect

_argument

that v =

2/3.

In order to obtain

_{P (x, t )}

_directly,

we need to

_compute

the

_asymptotic

Green’s function at

large

times and therefore to sum over low

_lying

states in

(19).

A natural

_suggestion

would be

to sum over the continuous excitation

_spectrum

corresponding

to the centre of mass motion :

The

_{corresponding}

distribution

_{of endpoint}

_positions,

_{P (x,}

_t),

is

_computed

in the

_appendix

2. For _any non zero n and _any

_positive

_time,

we can

_compute

_(see

the

_{appendix 2)}

the

_typical

drift :

unfortunately

the result is :

and cannot be continued to n = 0 in a

straightforward

_way

_(an

_attempt

of such a continuation can be found in

_{[23]). Actually}

the result

_(24),

which is derived

_exactly

from the assumed Green’s function

_{(22), imposes}

a relation between the time t and the number n of

_{replicas :}

for

n

small,

one needs

tn3:>

1. The existence of a crossover

_regime

where t --+ _{00, n -+}

0,

and

tn 3 _ l@

suggests

that there should be some relevant excited states of the Hamiltonian

_(18),

with an excitation energy of order

n 3,

which will contribute to

₍₁₉₎

in the limit of

a state _{built of two groups}

_{of n /2}

_particles,

where the wave function inside each group is of the

Bethe

_type

_(20),

while the center of masses of each group are

widely separated.

For n small

the excitation energy of such a state is indeed of order

n 3.

This state is also the relevant one

when one uses two

_copies

of the

_polymer

with

_repulsive

interactions. The

_{n /2}

_replicas

of the first copy build one _{group and the}

_{n /2}

_replicas

of the second copy build the second group. As

the center of masses _{of the groups need} to be

_{widely separated}

in order to have an

_eigenstate

of the

_n-body

Hamiltonian,

this

_explains

the fact

(14)

that the

_overlap

between the

_copies

vanishes in the case of

_repulsive

interaction.

Clearly,

to

_get

the correct

_asymptotic

form of the Green’s

function,

we need to include in

(19)

such excited states. So far we have not succeeded in

getting

an

_{explicit expression}

for

P(x,

t )

from this resummation. On the other

hand,

Parisi has

_{recently proposed}

a

_simple

Ansatz for the Green’s function

_[15],

which takes the

_following

form :

This

_{predicts :}

with :

Furthermore,

the

_{joint probability}

for

_arriving

at

_positions

_{xl, ..., X k is found in this Ansatz}

equal

to :

Therefore to each

_sample

there should

_correspond

a realization of the random

_potential

- j (d4/ dy) dy

0

(x)

drawn with the measure

e - 26 1

2 f (dO/

dy)dy

, and the

corresponding probability density

of

x for this

given sample

should be

_{given by}

formula

_(27).

Actually

one can

_change

variables in

_{(27) : defining x by x}

=

xt2/3 and cÎJ

(.î) = 0 ( x)

t1/3,

one

gets :

1 fA 2

and

₄

has the same

distribution,

e - 2

(d4>/dy)dy

, as the

original

/J.

The distribution

(29)

implies

that,

at

_large

_times,

Je

_gets

_trapped

into the minimum of the function

_Sê/2

+

4

(x),

has been introduced as a

_toy

model for the

study

of random field

_problems,

and the relations like

_{M -}

t 4/3

and

_{(X2) - (x) 2 ’" t}

can be deduced

exactly

from the

_{rigorous analysis}

of this

toy model

_performed

in

_[20,

_21].

We have tried to test Parisi’s Ansatz further

_{by comparing}

(29)

with numerical results. First of all we

argued

in the

_previous

section that the

_typical

value of

_{(x2) - (x) 2}

is

t2 p" IL "" 0.0-0.2

rather than t. Simulations of

₍₂₉₎

favour a

_scaling

of

(.xl) - (x) 2 ""

t4/3 - 29 ’

t4/3 -

2 F with

g

in the range 0-0.16.

(Again

there are

large

fluctuations in the distribution of

Log

(X ),

similar to the case of the

_{polymer problem.)}

This is

_compatible

with the

_equality

fi =

IL, but the uncertainties on the values of these two

_exponents

are too

_large

for this to be

fully

conclusive. We have also

_{computed Z(x, t )}

for several hundreds of

_samples

of sizes

between 63 and 2 047.

_Figure

4a contains a

_plot

of

_{[Log (Z(x, t ) ) -}

_{Log (Z(0,}

t»] It 1/3

versus

x/t2/3 =

x.

_According

to both Ansâtze

₍₂₂₎

and

₍₂₉₎

this should behave like

ct.xl. (The

c’

comes from the discretization

_procedure.)

This is

_precisely

what is found : for

large

times the data

_points

sit on a

_parabola

_for

_{1 Sc 1}

«-- 5.

(Of

course for

_larger

values of

(X the finite size effects

are

much more

_{important.) Figure 4b}

contains a

_plot

of

([Log

Z(x, t) -

Log

Z(0,

t )]2_ [Log

Z(x,

t) -

Log

Z(0,

t )’2

_{/ (,8 tl/3i}

versus x. From

Parisi’s Ansatz

(29)

this should behave like

_{c’l Sc 1}

This behaviour is obtained for small values

of

1 Sc 1

( 1 Sc 1 -- 2 ),

but it saturates for

larger

values

_of

_Ixl.

This saturation becomes more

apparent

when

_increasing

the size of the

_system

_(the

data in

_Fig.

4b

_correspond

to sizes 511 and 1 023 for which the finite size effects in the

_range

_{1 Sc}

5 become

_quite

small with

_respect

to the

_{fluctuations).}

Therefore the Ansatz

₍₂₉₎

seems to be an

approximation

which _may be valid

only

at

relatively

small values

_of

_{1 xl.}

.

Fig. 4 - a) - (LogZ(x, t ) - LogZ(0,

t))/({3t1/3)

versus x=

X/t2/3.

Data taken at inverse

tempera-ture {3 = 7, in zero field. From _top to _bottom, the curves

_correspond

to

_polymers

of

_lengths

t = 63, 127, 255, 511, 1 023. The number

of samples

studied

range between 200 to 1 000. The full curve

is the best

_quadratic

fit to the

_experimental

data of t = 1 023,

given by :

0.0039 + 0.014 x +

4. Some

_conjectures

on the

_{physical interprétation}

of the numerical results.

One of the main results of the

_previous

sections is the

_instability

of the

_polymer

system

with

respect

to an

arbitrarily

small

_repulsion

between

_copies.

This indicates the existence of several

pure states in this

_system.

Here we shall not

_attempt

to

_give

a

precise

mathematical definition

of these pure states, but we shall use this notion in a

phenomenological

_way

(4).

A state

should

_correspond

to a bunch of

_locally

favoured

_{paths, separated}

from the other states

_by

free _energy barriers.

_(The

existence of families of local

_{optimal paths}

has also been

conjectured by Zhang [24],

on the basis of an

_analysis

of the fist excited

path.)

We thus

conjecture

that there exist such states ; the state « has a free _energy

_Fa

which scales as :

where F _{is the free energy}

_density

which is the same for all states and the state

_dependent

corrections scale as

tw. (Such

a

_picture

is known to exist in mean

_field,

with ci =

0.)

The

Gibbs measure should then be

_decomposed

into the contributions of the states ; for any observable 0 :

Let us first show how this can

_explain

the results obtained when

_coupling

two

_copies.

In the limit of a small

_coupling,

_{the average}

overlap

between the

_copies

can be written as :

where :

The crossover

_regime

is found when et -

tW,

i.e. £,..., êtW - 1.

Such a crossover was

_precisely

found in the simulations of section

_2,

_{and these gave} an

_exponent

J in the _range 0.2-0.4. The

assumption

that i =

1/3

which we have used in this paper is the

_simplest

one since the

_sample

to

_sample

fluctuations of the free _energy are known

to

scale with the

exponent

w =

1/3

and we are

_{just making}

the reasonable

_assumption

that w = i

(another

argument

indicating

that J = 2 v - 1 will be

given

in the _next

paragraph).

However a more

_precise

direct determination of ci would be welcome. It is worth

_exploring

further how the above

equation (32)

can account for the data of

_figure

1. This

_equation

holds for one

_{given sample.}

For ê --+ 0 the sum over states is dominated

_by

the

ground

state, say a =

0,

with

self-overlap

of the state, qoo. When ê is

negative (repulsive interaction)

and

large enough,

the

overlap

will

_jump

to

_qo,,

where y is the state such

that 1 ê 1 (qoo - qo,) - f ,

is maximal

(assuming

that the self

_overlap

of the states are all

_equal,

otherwise the

_pair

of states which dominates

₍₃₂₎

_might

not contain the lowest

_state).

Therefore when ê decreases we

_expect

for

q(ê)

a discontinuous _curve, with

_{jumps taking place}

whenever the

_gain

in free _energy from

the two

_copies

of the

_polymer

_taking

_paths

which have smaller

_overlap

_compensates

for the loss from at least one

_{polymer taking}

an excited

_path.

Such a discontinuous curve is found when

_simulating

one

sample (although

the

plateaux

are not

_really

_constant, which

_might

signal

more subtle

effects).

But the curve in

_figure

1 was obtained

_{by averaging}

over _many

samples,

and as the values of the

_{f y}

at least fluctuate from one

_sample

to

_another,

the average

gives

back the continuous curve of

_figure

1 b. The nonzero

_slope

of the curve at 6 - 0- tells that there is a finite

probability of having

an

_arbitrarily

small _gap between the first

excited pure state and the fundamental one. It is difficult to extract more information from

this curve without some further

_assumption

on the

_overlaps

of the states. We shall not

_expand

here on the various

possibilities.

Let us

_just

mention a

_{simple possibility,}

which is the one

which takes

place

in the mean field

_theory,

and in

_expansions

around the mean field for

large

dimensions

[12] :

if the

_overlaps

can

_just

take two

values,

with qaa - q and

qay

=

0,

a :0 y, then the curve

_{q ( é )}

of

_figure

1 b is

_simply

related to the distribution

Il(I1I)

of the gap free energy

(rescaled

difference between the free energy of the first excited state and the free

energy of the lowest

lying

pure state

_Af

=

(FI -

F 0) / t¿j) :

As for the results on the

_{susceptibility (Figs.}

_2,

_{3), they}

can also be accounted for if one

supposes that the

susceptibility

inside each state scales as

12 IL, J.L

_{1/2 :}

For most

_samples,

the Gibbs average is dominated

by

the lowest pure state _{a o} and therefore

the full Gibbs

_{susceptibility}

is

_equal

to

X a . However

occasionally

it may

happen

that a

sample

will have two

_{nearly degenerate}

_pure

_ground

_states, with

_{Af -}

t-

w.

This will be the case for a

fraction of the

_samples

of order

t- £0 ;

for such a

_sample

the Gibbs

_{susceptibility}

must be of order

t2 JI "" t4/3 (assuming

that the

_typical

transverse fluctuation for each _{pure state} scales as

t’).

These rare

_samples

will therefore dominate the _average

_{susceptibility, leading}

to a Gibbs

susceptibility _

t2 JI -

w,

which

_gives

the correct

_scaling

demonstrated in

_appendix

1

(,f - t) if 2 v i = 1.

5.

_Summary

and

_{perspectives.}

To summarize in a few

_words,

we have found

_numerically

two main results : -

an infinitesimal

(but

extensive in the

_{thermodynamic sense)}

local

_repulsion

between two

identical

_copies

of the

_polymer

_system

_yields

an

_{instability ;}

-

coupling

a transverse electric field to the

_endpoint

of the

_polymer,

the

_{corresponding}

susceptibility

scales

as t2 ,

for

_{generic samples,}

_{with li -}

0.0-0.2. The

_sample

_averaged

susceptibility

is dominated

_by

rare events and it has been shown to scale as t.

One

_possible

_{interpretation}

of these results would be the existence of several pure states,

such that : - the

typical

differences between the free

_energies

of the states differ

_by

terms _{which grow}

- within _a

given

state a the transverse extension scales as

_(X>

t2l"

and the

susceptibility

scales as

(x2) a -

( (x) a)2 --

t2

p,.

If this

_{interpretation}

is correct, the situation is as follows : for almost all

_samples,

there is

one state _{which dominates the Gibbs average. For}

_{exceptional samples (a}

fraction

- t- 1/3 of all the

samples),

there _are two

quasi degenerate

states, the free

_energies

of which

differ

_{only by}

terms of order 1. These

_{samples give}

the dominant contribution to _{the average}

susceptibility.

Nevertheless,

for all

_samples

there are several states and the

_{resulting phase}

is very similar to a

_{spin glass phase.}

The

_major

difference is the fact that cd ±

0,

while in the mean field

_theory

of

_{spin glasses}

and of random directed

_polymers

J = 0. But these states

exist,

they

have the same free energy

density

and each of them can be obtained and stabilized

through

a

_quench

from

_high

_{temperatures.}

From this

_point

of

view,

it seems that the

_major

effect of

_having

J > 0 is the technical fact that the Gibbs _{average is}

generically

dominated

by

one

single

state.

The

_remaining

open

problems

are numerous and

_fascinating.

From the

_{analytical point}

of view we have seen that

_replica

_symmetry

_{breaking (i.e.}

the

_breaking

of Bose

_symmetry)

is

necessary to describe the above effects. However a full

_study

of the Green’s function of a

system

of zero

_{particles interacting through}

delta function

_potential

in one dimension is

necessary to solve the

problem.

Another

challenge

is of course to

_try

to _prove or

_disprove

the

above

_speculations

on the

_physical

nature of this

_phase.

The notion of several

_coexisting

pure

states seems to be

_quite

useful at least from a

phenomenological point

of view. A

_precise

mathematical definition of the pure states in this context would be

_{quite important.}

The extension of the

_type

of

_analysis

carried out here to

_higher

dimensions would also be welcome. One

_interesting

effect which one could

_try

to check

_numerically

in 2 + 1 and

3 + 1 dimensions is the existence of an

_instability

line of the de Almeida Thouless

_type

when

coupling

the

_endpoint

of the

_polymer

to a transverse electric field.

Acknowledgments.

1 am indebted to B. Derrida for numerous

_{helpful suggestions}

and comments, and to G. Parisi for an

illuminating

discussion. 1 would also like to thank J. P.

Bouchaud,

D. S. Fisher and D. J. Gross for useful interactions. The numerical simulations were carried out on the Convex Cl at ENS and on SUN workstations. 1 thank J. P. Nadal and G. Weisbuch for

_providing

me

plenty

computer

time on their

sparcstation.

’

Appendix

1.

We prove that the average

susceptibility

is (X2) - (x)2 --

t. Let us start from the

_partition

function in the presence of a field

_h,

for one

_sample

with

_{potential V :}

From which one deduces :

where

_{Î (x,}

T )

=

Y (x -

6 hr,

T

_{). As 9}

has the same distribution

as V,

we

_get

after

_sample

averaging :

The second derivative of this

_expression

with

_respect

_{to 6 h}

is :

This proves the result for the continuous model. It would be

interesting

to use

_(Al.3)

to

compute

the

_{typical susceptibility,}

but we have not succeeded in

_doing

so.

Appendix

2.

We

_compute

the distribution

_{P (x,}

_{t )}

obtained from the

_ground

state wave function

_(20),

including

the

degree

of freedom of the center of mass motion. The

_asymptotic

behaviour of

the Green’s function at

_large

times is then

_{approximated by :}

(with À ==

/3

2/2),

and we

_{compute :}

The average distribution of the end

point position,

f(x,

t),

should be obtained

_eventually

from the n - 0 limit of the

_previous

formula. However because of the

_problems

involved with this limit we first

_compute

_Pn.

The normalization constant _{cn will be obtained from the}

condition :

Choosing

an

_ordering

of the

_{particles’ positions}

we

_{get :}

where :

where :

In order to

_compute

_Sn

we first rewrite

_(with

k’ = k - 1

_{integer) :}

. , , 1

From

_(A2.7)

and

_{(A2.8), introducing integral representations}

of the F functions which remain

at the numerator, we

get :

In the

_integral

we

change

variables

from t,

u to x, z, where

_{t = xl ( 1 + e’)}

and

,

u = t e Z. This

_{gives eventually :}

From

_(A2.6)

and

_(A2.10)

we can now

_compute

the final form of

_Pn.

The normalisation

constant is found

_equal

_{to : cn} =

r (n) À n - 1/2

_1Tn, and :

Clearly

the

_limiting

distribution for n --+ 0 is ill-defïned.

_Actually

we can

_compute

the

moments

_{through :}

This

_gives

for instance :

From

_{(A2.13), using}

the fact that

_{ep’(n ) - n - 0}

_{’r2/6 - Iln2,}

we find that the distribution

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