Manifolds in random media: beyond the variational approximation

(1)

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Manifolds in random media: beyond the variational approximation

Yadin Goldschmidt

To cite this version:

Yadin Goldschmidt. Manifolds in random media: beyond the variational approximation. Journal de

Physique I, EDP Sciences, 1994, 4 (1), pp.87-100. �10.1051/jp1:1994122�. �jpa-00246893�

(2)

J. Phys. I IFance 4

(1994)

87-100 JANUARY1994, PAGE 87

Classification Physics Abstracts

05.40 05.20 05.50

Manifolds in random media: beyond the variational

approximation

Yadin Y. Goldschmidt

Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260, U-S-A-

(Received

26 August 1993, accepted in final form 29 September

1993)

Abstract. In this _paper _we give _a closed form _expression for the

1/d

corrections to the self- energy characterizing the correlation function of _a mamfold in tandem media. This amounts to the first correction beyond the variational approximation. At this time _we _were able to

evaluate these corrections _m the high temperature "phase" of the notorious toy-mortel describing

a classical partiale subject to the influence of both

a harmonie potential and _a ramdam potential.

Although in this phase the correct solution is rephca symmetric the calculation _is non-trivial.

The _outcome

is compared with previous analytical and numencal results. The corrections diverge

at the "transition" temperature.

1. Introduction.

The

problem

of directed manifolds in _a disordered medium _is _an

interesting problem [1-17]

which possesses many similanties to trie

spin-glass problem although

the

hope

has been that it is _more tractable. In

particular,

when trie manifold is _one dimensional it is referred _to _as trie

"directed

polymer" problem.

Trie latter

problem

_can be described _as _a walker

(or polymer)

who walks

randomly

_on _a d-dimensional lattice

(with

d

= N

+1),

where _one coordinate

(referred

to as trie

"time") always

increases. This _is trie _reason trie walk

(or polymer)

is called "directed".

The walker is attracted

(or repelled) by

_a random

potential

which is associated with each site on trie

lattice;

thus each

possible

walk

(or patin)

has _a

weight

which is the

product

of

weights acquired

at each site which is visited where trie

weight

is determined

by

the

potential

_on that site

Higher

dimensional manifolds

(in

^N+D ₌ d

dimensions,

where d is trie

embedding

dimension and D is trie intrinsic dimension of trie

manifold)

_pertain for

example

to trie

problem

^of _an

interface between two

phases

of _a system ^which contains random

impurities

_[3], and _even to such _a

problem

_as flux-fines in _a disordered

superconductor.

_[9]

The

"simplest"

^{of ail} such

problems

is trie sc-called

"toy-model"

for which D

= 0

[18, 19, 14, 16]. Usually

N is taken to be one for

simphcity,

but in this _paper _we will consider the

general

N _case. This model descnbes _a classical partiale which _can _move in d

= N dimensions,

(3)

subject

to trie influence of both _a harmonic

restoring

force and _a random

potential

which has

long ranged

correlations in space whose

magnitude

faits off like _a _power law.

Recently

Mezard and

Parisi,

in _a beautiful

piece

of work [8],

applied

the Hartree variational

approach

to trie

problem

of directed manifolds in _a random

media,

and showed that this method becomes exact when d _- _oo, where d is the number of

embedding

dimensions. Some

interesting

results bave been obtained

by

this

method,

in

particular

various relevant exponents ^have been calculated which characterize the

roughening

of the manifold _or the behavior of trie free _energy fluctuations. More

recently

trie

properties

of trie

spatial probability

distribution bave also been derived

by

trie variational method [16].

An

interesting project

is _to go

beyond

trie variational

approximation

and incorporate _sys- tematic corrections _to it. One _way to try to go

beyond

trie variational

approximation

is to calculate

1Id

_or

equivalently 1IN

corrections to the d

= oo results.

Recently

_we bave

begun

such _a

project

[15]

by calculating

trie

1Id

corrections to trie

free-energy

for trie _case of directed

polymers

with short

ranged

correlations of trie random

potential.

Trie _case of _one step

replica

symmetry

breaking (RSB) pertaining

to trie _case of short _range correlations of trie ramdam

potential

bas been

successfully

addressed. This _paper [15] will be referred _to in trie

sequel

_as I.

In trie present ^work we

proceed

to derive trie form of trie

1IN

corrections to trie

self-energy

for

a

general

manifold which is relevant to trie behavior of correlation

functions)

and calculate them

explicitly

for trie

toy-mortel

in trie

high

temperature

"phase".

Recently

Mézard and Parisi [14] ^derived trie form of _a

partial piece

^{of trie}

1IN

corrections.

It turns out

(see

Sect. 2

below),

that their

expression

does not include ail trie

1IN

corrections.

In addition

they

_were

only

able to calculate trie contribution of _Due

diagram

Dut of trie infinite set that contribute to this

1IN

term. Trie last

difficulty

_anses because of

replica

symmetry

breaking.

On trie other

baud,

_Dur calculation derives ail trie

1IN

contributions to trie self

energy, and _we could evaluate them

completely

for trie

toy-model

in the _case of _no

replica

symmetry

breaking (valid

for

high temperature).

We bave not yet ^tackled trie _case of RSB

(with exception

of trie

particular

_case of one-step RSB considered in

I).

We

hope

that trie results

presented

below will pave trie way for _a future calculation which will incorporate a

finite number and

eventually

_an infinite number of steps ^of replica symmetry

breaking,

but _so far this issue remains unresolved.

Other aspects of trie lIiAT expansion, obtained in _a somewhat different context,

by using

a

different _set of

tools,

_can be found in _recent work of Cook and

Derrida,

_[7] and Balents and Fisher

il?].

Trie latter authors raise trie

possibility

of

non-analytic

contributions

(in N)

to trie

roughening

exponent.

2.

IIN

corrections for manifolds in random media.

We first review trie method for

obtaining

the

IIN expansion (see

also [8] and

I).

Since _we

are

mainly

interested in random manifolds

(as opposed

to trie

quantum-mechanical n-body problem

which _was discussed in detail in

I)

_we _use here trie Euclidean version of trie

patin integrals.

Trie hamiltonian

descnbing

trie system of manifolds _in _a random medium is:

hiwi

=

1/

_d~

( _Iii

^~

+

i /

d~

iW(~))~

+

/

_d~

_{v(i~ W(~))} _ii)

where

w(~)

is _an N component vector field

representing

trie

position

of trie manifold and

~ is trie D-dimensional internai coordinate of trie manifold. Trie first term represents trie surface tension, The second is _a small _mass term introduced for

regularization

_purposes and

(4)

N°1

1IN

EXPANSION FOR MANIFOLDS IN RANDOM MEDIA 89 trie third term is trie

quenched

random

potential

with _a Gaussian distribution and _mean _zero.

Its

averaged

correlations

satisfy:

(V(l~, W)Vll«',W'))

=

-ôl~~l« «')

N

f 'j'

12)

The

scaling

with N is chosen such _as to obtain _a

meaningful large

N limit. For

large

distances the function f is taken to be described

by

_a _power law:

~~~~'~

2(l~ _y)

^~~ ^~ ^~~~

In order to average over trie

quenched

random

potential,

trie

replica

method is used. After

introducing

_n

copies

of trie fields and

averaging

_over trie

potential

one obtains:

(Z")

=

d[wi) d[wn] exp(-flHn)

,

(4)

with trie

replicated n-body

hanliltonian

given by:

We derive trie

1IN expansion using

the

functional-integral

formalism

[20, 21, 15]. Introducing

two

auxiliary

fields

qaô(x)

and

aaô(x)

the

partition

function _can be

expressed

in trie form:

(z")

=

/ _fl _dlw«(«)1 / _IJ diqaô(«)idi«aôlx)1

x exp

~ ~j /

_dz

aaô(x)(Nqaô(x) wa(x) ô(x))1

~

ah

~ ~~~

~ /

_~~

_l~ _~ÎÎÎ

^~

~

~'~~~~~j

~( _£ /

_dX

_N/i~aa(X)

+ ~bb(X)

ab(X))1.

i~)

~#~

The

integral

_over _«ah amounts to _a à-function _constraint

imposing:

~abiX)

"

~'aiX) '~'b(X)/N i~)

Since the coordinate _wa _appears

only quadratically

it _can be

integrated

_upon to

yield:

(~~)

"

/ _fl ~i~abiX)i~i°abiX)i ~~PiN~i~ab> °ab)i i~)

with

fl

fl2

A(qab, °ab)

_"

j £ /

_dz

aab(X)~ab(X) ~ £ /

_dz

_/jqaalX)

+ ~bb(X)

2qab(X))

+

sjaab(X))

ah ah

(9)

JOUR~AL DE PHYSIQUE T 4, N' JANUARY 1994 4

(5)

where

el"°bl~

=

/ _fld[wa(x)j ^xp1-~ _~j /

dz

wa(x) ((-i7~

₊ _{~t) ôaô}

_aaô(x)) wô(z)

a

~

ah

(10)

The

large-N

limit is determined

by

_a saddle

point

of trie functional

integral, equation (8).

We look for _an

z-independent saddle-point

solution and hence _we _express trie variables _a and q as

°ab(X)

_"

°Îb

+

~ab(~) (Il)

~ab(Z)

_" ~Îb + ~lab(X),

(12)

where _e and

i~ ^are trie

(x-dependent)

fluctuations about trie

saddle-point

values a° _and

q°.

Trie

x variable is taken _to be confined _to _a box of volume V.

Substituting equations iii)

and

(12)

in equations

(9)

and

(10)

and

expanding

to third order in _e and _q _we find:

)A(q°

+1Ji

«°

₊ _e)

=

£ _«Îôqlô ( _~ _fiqla

+ qlô

2qÎô)

ah

~

a#à

~

~ j

^~~

~

_°'~~~~ °'b~~~~~b~~~

ah

fl2

+-

dXdX'~ ~j

WaIX) Wb(X)Wc

IX') Wd(X')rab(X)fcd(X~)

~

ah cd

fl3

+

~ /

dXdX'dX"

~j ~j ~j

_Wa(X) Wb(X)Wc(X')

ah cd _ef

wd(x')we(x") wf(x")eaô(x)ecd(x')eef(x")

+

fl2

w ~ f'(qla

₊ _qlô

_2qÎô) /

_dz

_(~aai~)

+

~àô(x) 2~laô(x))

a#b

fl2

~ f"'(q(~

₊ _q)à

_2q(à)

dz

(~laa(x)

+ i~ôô(x)

2~laô(x))~

+

(13)

12V

/

a#b

The

saddle-point equations

_are determined

by

trie

vanishing

of trie linear terms in _~/ and _e.

They

_are:

a(à

= 2

f'(q(~

₊ _q)à

_2q$à),

_a

#

b

(14)

(6)

N°1

1IN

EXPANSION FOR MANIFOLDS IN RANDOM MEDIA 91

a~~

+

~

_a~b

#

(15)

b(#a)

~Îb "

/dX (~'a(Xl'Wb(X))C

"

/ _Glb(Î') ₍₁₆₎

p

with

GÎb(Î~)

_" _[lP~ +

Jl)1 °~)~~]

ah' 1~7)

In equation

(16)

trie notation

)c

stands for trie connected _average with respect to trie

quadratic

action in _w.

Taking

into account trie saddle

point

solutions and

rescaling

_e,~/ _-

ellR,~ll/R

trie _ex-

pression

for

(Z")

becomes:

(z")

=

exPlNAo1 / di~aôldifaôi

exP

lA21

lfl2

~ i

~~ ~~fl /2

~ f"'lqla

₊ _qlb

_2qÎbl /

dX(~laa(X) + ~/bb(X) 2~lablX))~

ah

~ fi

iÎ1/2 /

~~~~'~~"

~Î ~Î ~Î _~©cd,e

f

(~'~" ~")Gb (~)~Cd(~')~ef (~")

_~

l' _(~~)

ah cd _ef

were

Ao

is the fluctuation

independent

_part of

equation (9)

and _we defined

A2

=

/

_dxdx'

_{£ £}

Haô

cd(x x')eaô(x)e~à(x') ^~ ~j /

_dz

eaô(x)qaô(x)

~

ah cd

~ ~

ah

fl2

i ~ f"lqla

₊ _qlb

_2qÎb) /

dX

l~laalX)

+ ~/bb(~) 2~labl~))~>

(19)

a#b and

~

flaô,cd(~ ~')

=

$ _(wa(~) wô(~)w~(x') wd(~'))c, (20)

flllcd,e

_f

IX'X" X")

_"

$(~'a(X) ^'~'à(X)~'c ^IX') ~'d(X')~'e(X") '~'f(X"))c (21) Equations (18)

and

(19)

_are trie main equations needed to calculate the e-propagator and the

1IN

corrections to trie self energy. To obtain trie e-propagator it is

enough

to consider the

quadratic

action

A2

in

(18)

and

integrate

out _over

~/(x)

to generate the

quadratic

effective action for _e. We find:

/dX ~~~~~l~ab(X)~cdlo))

_~

~2rab,cd(Î~)> 122)

r(P)

=

Ii

₊ A

fl(P))~~ A, 123)

with

Aab,cd "

ci" _(~$u

_~

_~lv _~~$vl

~ab,uvtcd,Uv'

l~~)

uu

tab,uv " ôauôbu + ôavôb~ ôauôbv ôavôbu,

(25)

flab,cdik)

"

/i~iciP)~tdik ^P)

⁺

~idiP)~tc(~

P)))

^12~)

(7)

by

_a

product

of _two

n~

_x

n~

_matrices like A H _we _mean

(A Il)ab,cd

_"

£

Aab,1~1'II~t~,cd

(27)

UV

In I _we

proceeded

to calculate the

1/N

corrections to the free energy. Here _we _are interesteà

m the self _energy, defined _as the matrix

a(p)

in trie _expression

G(P)

= ((P~ +

/1)i «(P))~~ (28)

If _we put

a=a°+ ~a~+..., ₍₂₉₎

It is easy ^to

verify

that

j°ÎblÎ')

"

Î/2

^(~ab(o)1 ⁺

/

_dX _~~~~~

lfl~aclX)GÎdIX)Edb(o))>

130)

~~

were

G°(x)

_is the Fourier transform of

G°(p)

and the averages are taken

using

trie fuit functional

mtegral represented

in

equation (18) indudiug

trie cubic ternis in _e and ~/.

We

display

now trie final result and make _some comments on trie derivation later.

°Îb(Î')

" ~2

£ / Gldlk)tac,db(Î' k) j _{£ £ £ £ £}

f~~~(~Îu + ~Îv

~Îv)

cd ~

UV cd ef gh lm

x

Î t~d,w Ill

₊ A

fl(k))~~]~d,e ffle f,~h(k)t~h,wtim,~»[(1

+

II(0) _A)~~]im,aô

+ 2

/ _jÎ _{£ £ £}

G)~

(k)G(~ (k)G )~ lé')r~b,cd (0)re

f,~h

(k k') ₍₃₁₎

~ ~'

cd _e f gh

The first term, which is momentum

dependent,

bas been denved in _a different form _in

[14],

however trie two other terms are missing ^there. Note that trie term

proportional

to

f"'

does net exist in _a

#~-field theory

where

f

rw

x~. _In

figure

_we present some

typical graphs (Dut

^of

many

possible others)

that contribute to trie _vanous terms in

equation (31).

Trie first term in

equation (31) originates

^from trie second term _on the r-h-s- of

equation (30).

Trie other two terms result from thé first _term

in

equation (30)

combined with trie two cubic terms in ~/ and _e in

equation (18).

To obtain trie second term

(proportional

to

f"')

it is easier to first

integrate

_on _e to find trie effective ~/-propagator which is

given by:

l~(P)~l-P))

₌

jfllP) Ii

₊ A

fl(P))~~ (32)

and the term linear in _e _m equation

(30)

becomes

) /

dÙ

~ l(nl~Y))~~lab,cd

~/cdlY).

(33)

rd

This _in tutu v»11

multiply

trie cubic term in ~/ ⁱⁿ equation

(18)

which _can also ù~ written _in trie form:

fl2 r

i~ff1/2

~

~~~~~~~"

~~~~

~~"~

~ ~ ~

~~f>""~9~'~"~~~'~~

j

~~ ~~f~~~~9~~~~~~~~~~' ~~~~

u~ ef gh lm

(8)

N°1

1IN

'

Qj

~ ~

'

j, ^/

~ _1--

~

/~ ~

k k

(a) ~b) (c)

~--,

U

/

'1

~

,(

/~ _/~~~~~

(d) (f) (h)

~~~j Î

',~/ ~

/Îj"

~~I

(e) (g) ii)

Fig.

l. Sample Feynman diagrams representing _some typical contributions to order

1/N: la)

Faithful representation of _a

(#~)~

interaction

(of

order

1/N). (b)

Faithful representation of _a

(#~)~

interaction

(of

order

1/N~).

_(c) Effective four point interaction _in _a

(#~)~

theory

(of

order

1/N).

(d),(e)

Sample graphs contributing to the first term _on the r-h-s- of equation

(31). (f),(g)

Sample graphs contributing to the second term _on the r-h-s- of equation

(31)

in _a

(#~)~

theory.

(h),(1)

Sample graphs contributing to trie third term _on the r-h-s- of equation

(31).

More graphs _cari be obtain by replacing bare interactions by dressed interactions, etc.

3. The

toy-mortel:

evaluation of the

1/N

corrections.

In order to evaluate the

1IN

corrections derived in the

simplest

case of D ₌ 0. We consider _a

generalization

of the standard

toy~model

_{[18, 19,}

14],

^where ^the

partide

_is embedded in N dimensions. At the end _we _compare _our results with simulations

performed

for N

= 1

[14],

^and find that _even for such _a small value of N trie corrections to trie

(9)

leading

N

= oo result _{go in} trie

right

direction. Trie Hamiltonian is given

by:

H =

~w~

₊

V(w) (35)

which describe _a dassical

partide

_moving in N-dimensional space and feels _a

potential

which is _a _sum of both _a harmonic part and _a random paIt V. The random

potential

has a Gaussian

distribution with _mean _zero and variance given

by

the expression

lVlW)VIW'))

=

-Nfliw W'i~/N), j36)

where

f(y)

is given

by equation (3).

We _are interested in the

long

_range _case _y < 1, smce the

case ~ =

l/2

is of

particular

importance

[4-6j,

as it _cari be

mapped

into the directed

polymer problem

in 1+1 dimensions.

Trie N

=

toy-model

bas been solved

by

Mezard and Parisi [14j using the variational _approx- imation. Within this

approximation they

fourra _a

"phase

transition" from _a

high-tempeIature phase

characterized by

rephca

symmetry, to _a

low-temperature phase

characterized

by

mfinite- step

(continuous)

RSB. This

phase

transition _is _an artificial property ^of ^trie variational

(or

equivalently large-N) approximation

but trie results obtained make otherwise much

physical

sense

[14j.

We

proceed by noting

that fol trie

toy-rnodel,

trie Iesults of trie previous section still

froid,

but _now without trie internai variables _x _or _p. Thus

°Îb

"

flg ~~~

~

~) ^~~~~

^~ £l

#

_b> 137)

a(~

=

~j _a(à, ₍₃₈₎

b(#aj

~~

"

i(/l °)

_~~iab ₁₃₉₎

If _one

applies

the variational approximation, instead of trie

large-N approach,

trie

only

diiference for finite N is that [8, 14]

~ ~

à

" ~

~~~

ij)li'~~ Ill

^~~~

~

l l~°~

The factor

p(N) multiplying

_g

approaches

1 for

large

N but indudes corrections for finite N:

PIN)

" 1-

~~~~

_'~~

+

(41)

Since _we _are

performing

_a strict

1IN

expansion we will net indude trie factor

p(N).

Let _us introduce trie reduced temperature ^variable

t ₌

fl /1M _(y2g)

~~

(42)

Note that this variable differs from that used in [14] ^since we use g and net in trie definition of t. For t > 1 trie

replica symmetric

solution _to equation

(37)

is trie correct solution, _smce trie

broken replica solution does net exist in this

region.

This _means that

a$à

₌ s =

~

t~(~+~~ (a _{# b),}

_G$à

=

ôabÔ

+

il ôab)G (43)

7

(10)

N°1

1IN

with

Ô

_G

=

~,

G

=

(. ₍₄₄₎

l~ l~

We

proceed

to evaluate _ai tram

equation (31).

We need first _a

parametrization

of the n~ _x n~

matrices. In trie

replica symmetric

_case it is

enough

to consider matrices

parametrized by

_mine

distinct value

(this

is _a _more

general

^form ^than ^that considered

by

Almeida and Thouless [22].

For _a matrix

Mjaôj,jcdj,

^where ^the square brackets _mean that the order of trie indices _a, b _or b,c is

unimportant)

_we set:

~laal>laal

" ~A ~laal>lbbl "

~B

~laal>laôj

"

MG Mjaôj,jaaj

"

Mû

~lCCl>laôj " MD

, Mjaôj,jccj "

Mû

,

~labl>laôj " MP Mjaôj,jacj "

MQ

Mjabj,jcdj "

MR. 145)

In

equation (45)

_a

#

b

#

_c

#

d.

In

Appendix

A _we

give

trie expression ^{for the}

product

of _two matrices which _can be paramet- nzed in this way. The

product

is aise a matrix of this kind. Note that the unit matrix is net

actually

of this form since

laô,aô

₌ 1

# laô,ôa

_# 0 and thus ares net have the symmetry property ^of

interchange

of indices in the _same

subgroup

[ab] or [bc]. ^However ^this poses no

difficulty

_as will be _seen in the

sequel.

For the

toy-mortel

it _easy to

verify

that

ql»

+

qÎ» 2ql»

=

~j ^

⁼

) ₁₄₆₎

f"lql»

₊

_qÎ» _2qÎ»)

=

-(~ ())

% -c

(47)

The matrix A of

equation (24)

_can be

parametrized by

A~ ⁼

j-2c)jn i),

A_~

=

j-2c)

Ac ₌

j-2c)j-i)

₌

Ao,

Ap ₌

j-2c), j48)

ail other parameters

being

_zero, and the matrix II of

equation (26)

is

given by

HA ₌

Ô~,

,

DB =

G~,

IIc ₌

ÔG

=

Ho,

DD

#

G~

=

Dû,

HP =

(G~

+

Ô~), H~

=

(G~

+

ÔG),

DR

=

G~, (49)

The

product

of A with fl is

given by

X=A.H=H.A=(Ô-G)~A=

_~~A

₍₅₀₎

l~

We would like to calculate the inverse of the matrix 1 + X.

Denoting

trie result

by

1 +

Y,

Y satisfies trie

equation

Y ₊ _{X + YX}

= o

(51)

(11)

The matrix Y _can aise be

parametrized

as trie matrix M of

equation (45).

Trie value of the parameters ^obtained

by solving

equation

(51)

_are

given

in the

Appendix,

where _we defined

=

~(

=

jt~l~+~) ⁽⁵²⁾

l~

The _matrix r defined in equation

(23)

_cari _now be

easily

calculated _as A + Y A using again

the

product formula,

and the result is

simply

r =

-~2Y. ₍₅₃₎

The

replicon eigenvalue

[22] ^associated ^with e-propagatoI

(trie

minus

sign

_is because of the definition of r in

Eq. (23))

is

given by:

2(Tp 2r~

+

rR)

= 2/~~

~ ^,

(54)

and it becomes

negative

for 2À _> 1, or

equivalently

t < 1,

signahng

the

instability

of trie

replica-symmetric

solution in that _Iegion.

We are now in _a position to evaluate trie first _term

contributing

to

a(~ô

as given by equation

(31).

After _some

algebra

_we find:

~ ÎÎÎ

Il

2À) ~

~

il nÎÎÎI ^~ÎÎÀ) ^~ ^ÎÎÎ ^~/À'

^~~~~

We

proceed

to evaluate trie second term on the r-h-s- of equation

(31).

We _use trie fact that

trie

Orly

_non-zero parameters

characterizing

trie matrix t defined in equation

(25)

_are:

tc = 1, tp ₌ -1.

(56)

After _some

algebra

_we ^find that this terni contribute:

T2 _" ^~

~Ti ₍₅₇₎

Finally

we evaluate trie third _term

contributing

to

a(~ô.

^Trie calculation _is rather

straight-

forward but tedious and trie final _answer _is

~

~ _~ _~ ~~

Il

~

Î~l

~ÎÎ)Îl~~ ^~À)2 ^~

^~' ^~~~~

Thus this term does net contribute in trie _n _- 0 limit.

It is

inteIesting

to note that in ail three terms

contributing

to a~

Orly

the combination G G appears which

depend only

_on _/~ but net _on _s _as defined in

equation (44).

Trie total contribution to

a(~ô

^is ^thus

~ _~ ~ ~ ~ 2

ji

_~ _~ j ^~

~

(59)

1 ~

~~~à

_l-2À'

We bave verified that trie

diagonal

element of a~ satisfies trie relation:

a$~

+

In 1)a(~ô

= o. (Go)

Collecting

ail trie different

contributions,

trie total _expression for _«ah is given

by:

(12)

N°1

1IN

6 ,',

,' ,' ,' ,' ,' ,' ,' ,'

,' ,

5 ,'

,,' ,'

,' , ,'

à ."

oe ," ^'

8 ." ,'

4 .," ,

V .' ,

," ,

,." ,

-.-.-.;" ,'

, ,

3 ,'

' ' ' / /

2

2 3 4 5

1/

Fig.

2. Plot of w~

versus

1/fl.

The dotted-dashed fine _are the result of the N

= cc calculation.

The continuous fine _is the result of the variational calculation for N

= 1. [14j ^The ^dashed ^fine is ouf

result to order

1IN

in _a strict

1IN

expansion. The plus marks represent data points from simulation of N

= 1 _as reported in [14].

aa#à #

~t~(~+~)

Il ^~~~~

^'~~ ^~ ^~~

j~~~

+

,

(fil)

Tf 1 t~ ~

which is the main result of this section. Note that trie variational result for _a which indudes corrections for finite N because of the

replacement

of _g

by § (see Eq. (40) ),

is

given by:

Î°a#blHartree

"

t~l~+~~p(N)

#

t~~~+~~

Il ^~~~~

^'~~ ⁺

,

(62)

and thus differs from trie exact

1/N

result

by

trie

replacement

1+

lt~(i+~)

l

-Î~li+~)

^~ ^l>

⁽⁶³⁾

which coincide _m trie

high

t limit. Note aise that trie

1IN

corrections to a

diverge

for _t _-

as

expected,

because of trie

change

of

sign

of trie

replicon eigenvalue.

In terms of this result trie _mean _square

displacement

of trie

toy-model partide

is

given by:

liw~)

- -

Ii

⁺

VI

164)

In

figure

2 _we

plot

trie _mean _square

displacement

versus trie temperature variable

llfl.

In order to compare with trie results of reference

[14j,

_we choose ₇

#

1/2,N

= 1,/1

= 1 and

g =

2~/(nfp(1)).

This choice makes trie relation between trie reduced temperature t defined in eq.

land

trie temperature

llfl

become:

t =

[p(1)j~/~(llfl)

=

o.86(llfl). (65)

(13)

Three

plots

_are shown: The infinite N result, The result of trie vanational

approximation

for N = 1, [14] ^and ^trie ^result ^{of trie}

1IN

calculation _as evaluated for N

= 1. Trie data points _are results of simulations

repoIted

in reference [14]. ^These ^lie ^between ^trie ^result ^of trie variational

calculation and trie

1IN

result. Of _course better results _are

expected

for

larger

values of N.

As mentioned in trie introduction _we

frape

that this calculation _con be extended to trie _case of RSB below t

= 1, _as well _as to

higher

dimensional manifolds. It indudes trie prerequisites needed to start

thinking

^about

performing

such involved calculations. It is encouragmg the

our final

1IN

result tumed _eut

quite simple

_at the end. In the

replica

symmetric _case the

expressions

_on the

right

baud side of equations

(37, 31)

do Dot

depend

on

a°,

_since

Orly

trie combination G turn up in trie calculation. Thus _a is obtained

directly

without trie need for _a self consistent solution. This is Dot trie _case for trie RSB solution _to

leading

order in

N,

[14] ^where one bas to salve for a° self

consistently.

We expect ^that ^aise to

O(1IN)

it

might

be

possible

to

replace

trie correlation function G°

by

G in the total

expression

for _a to

O(1IN)

and salve for _a

self-consistently,

which is the

equivalent

of trie

Bray

self-consistent screening approximation. [23] ^If ^this

approach

will lead to

meaningful

results remains to be _seen _in trie future.

Acknowledgements.

This work _was

supported by

trie National Science Foundation under Grant number DMR- 9ol69o7.

Appendix.

Here _we

give

trie formula for trie

product

of _two matrices Z

= X Y each of trie type ^M

descnbed in section 3,

equation (45).

ZA=XAYA+(n-1)XBYB+2(n-1)XcYô+(n-1)(n-2)XDYb,

ZB

₌

XAYB

+

XBYA

+

2XcYô

+

2(n 2)XCY~

+2(n 2)XDYô

+

In 2)XBYB

₊

In 2)(n 3)XDYô, Zc#XAYC+XBYC+2XcYp+(n-2)XBYD

+2(n 2)XCY~

+

2(n 2)XDY~

+

in 2)(n 3)XDYR>

Z~

₌

XôYA

+

X~YB

+

2XpY~

+

In 2)XôYB +2(n 2)XQY~

+

2(n 2)XQYô

+

In 2)(n 3)XRYô,

ZD ₌

2XBYC

+ XAYD + 2XDYp +

4XCYQ

+

In 3)XBYD +2(n 3)XCYR

+

4(n 3)XDY~

+

In 3)(n 4)XDYR, Zô

=

2X©YB

+

XôYA

+

2XpYô

+

4XQYô

+

In 3)XôYB

+2In 3)XRYô

+

4In 3)X~Yô

+

In 3)In 4)XRYô,

Zp ₌

2XôYc

+ 2XpYp +

In 2)X~YD

+

4(n 2)X~Y~

+

In 2) In 3)XRYR>

(14)

N°1

1IN

Z~

=

XôYc

+

XôYD

+

XbYc

+

2XpY~

+

2X~Yp

+

In 3)XbYD +2(n 2)X~Y~

+

2(n 3)X~YR

+

2(n 3)XRY~

+

In 3)In 4)XRYR,

ZR _#

2XôYD

+

2X~Yc

+ 2XpYR +

BX~Y~

+ 2XRYp +

In 4)XbYD

+4(n 4)X~YR

+

4(n 4)XRY~

+

In 4) In 5)XRYR. (66)

The inverse of trie matrix 1+ X where X

= A H is

expressed

in trie form 1+

Y,

where Y is found

by solving

trie

equation

X ₊ Y ₊ X Y

= o. Trie matrix Y is found _to be

parametrized

as follows:

~

in i)>ji 2jn 1)>)

A

ii n>)ji 2n>)

~ À(1 2À)

~

(l nÀ)(1 2nÀ)

~ ~

Àjl 2jn 1)À)

~ °

~jl _nÀ)jl _2nÀ)

~ ~

2À~

~ ~

(l nÀ)(1 2nÀ)

2À~(1

2(n _1)À)

~~ Il

2À) ^~

Il

_2À)

Il nÀ) Il 2nÀ)

~ À~(1

2(n 2)À)

~

(l

2À)

(1- nÀ)(1- 2nÀ)

~~

(l

2À) (1

nÀ)(1 2nÀ)

^~~~~

References

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