Directed walks with complex random weights: phase diagram and replica symmetry breaking

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Directed walks with complex random weights: phase diagram and replica symmetry breaking

Yadin Goldschmidt, Thomas Blum

To cite this version:

Yadin Goldschmidt, Thomas Blum. Directed walks with complex random weights: phase diagram and replica symmetry breaking. Journal de Physique I, EDP Sciences, 1992, 2 (8), pp.1607-1619.

�10.1051/jp1:1992230�. �jpa-00246643�

Classification Physics Abstracts

05.40 72.20 71.55J

Directed walks with complex random weights: phase diagram

and replica _symmetry breaking

Yadin Y. Goldschmidt and Thomas Blum

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

(Received

26 February1992, accepted in final form 21 April

1992)

Abstract. We consider the problem of directed walks

(or polymers)

in _a random potential with both real and imaginary _parts. The correlations of the potential _are of short range. The problem is solved using _a variational approximation which is expected to become _exact in the limit of _a large embedding dimension. A rich phase diagram is found which contains five phases.

In addition to the simple high temperature phase, there _are four ordered phases with diiTer- ent degrees of

replica-symmetry-breaking.

Three of the phases _are dominated by interference

(quantum)

eiTects, and in two phases _a pairing interaction between replicas plays _an important role.

1 Introduction.

The

problem

of directed

polymers

in _a random medium has attracted considerable interest

recently [1-8].

In addition _to

modeling

various

physical phenomena [1-3],

it _{serves as a} prototype of _a system which

displays

the consequences of local frustration and is somewhat

simpler

than

(although

similar in many respects

to)

the

spin-glass problem [4-8]. By

"directed"

polymers,

one

loosely

_means '~no

overhangs"

and _more

precisely

that the walker

(polymer) proceeds always

in _a

positive

direction

along

_one chosen coordinate

(called "time")

in _an

(N

+

I)-dimensional

space.

Recently,

Mdzard and Parisi

(MP)

_[6] have found _a solution _to directed

polymers

in _a random real

potential

in the limit of

large embedding

dimension

(N

-

oo);

^{it exhibits} _many

properties

similar to those of the mean-field solution of the

spin-glass problem

[9]. In

particular,

this solution is characterized

by

the existence of _many

"valleys"

_or

thermodynamic

"states" _a fact that manifests itself in _a mathematical property of the solution called

"replica

symmetry

breaking" (RSB).

A different

problem,

related to the

above,

is that of directed

Feynman paths

which thread

through

_a disordered

medium, acquiring

random

phases [10-19].

This

problem

arises in the

study

of interference effects in Mott

Variable-Range Hopping

in

doped

insulators and semicon- ductors

[10-12].

^The transmission

amplitude

^for the

hopping

of electrons from

one location _to another _can be described _as the

superposition

^of

paths

which

pick

_up random

phases

_as

they

pass

through impurity

sites. For

strongly

localized states, the dominant contribution to the

1608 JOURNAL DE PHYSIQUE I N°8

transmission

amplitude

is from the shortest and hence directed

paths.

There _are several

conjec-

tures and numerical simulations

concerning

this

problem

in I+ I dimensions

[10-15,19].

Here _we consider _a

generalization

which includes both the

directed-polymer

_case and the

random-phase

case, viz. directed walks in _a

complex

random

potential, incurring

both random

amplitudes

and random

phases

at each site. Cook and Derrida

(CD)

_[16] have

investigated

_a

special

_case of this

general problem by considering

walks _{on a}

Cayley

tree with random

phases

restricted to 0 _{or ~r}

(random signs)

with

probabilities

I _p and _p,

respectively.

In that

work,

CD have

obtained

expressions

^{for the "free}

energy" by relating

^it to another

model,

^the Generalized Ran- dom

Energy

Model

(GREM),

which has been shown to coincide with the model _on the tree for the _{case p}

= 0

(random amplitudes only)

_[4]. CD have

conjectured

that GREM describes

polymers

with

complex weights

_on the tree _even for _p

# 0,

when the latter could _not be solved

directly.

The

resulting phase diagram

consists of three

phases

_a

high-temperature phase,

_a

low-temperature

frozen

phase,

and _{a new}

phase (as compared

to the

random-amplitude model)

in which interference effects become

important.

In this

work,

_our aim is to

apply

the

replica

method _to this

generalized

model in order to obtain _more detailed information about the structure of the various "states" in the different

phases

that

might

_occur. We have used _a variational

(Hartree-type) approximation

which is

expected

to become _exact in the limit of _a

large embedding

dimension [6]. ^To our

surprise,

the

phase

structure _we obtain is richer than that of CD

[16],

^{in the} sense that _we find two new

phases

which _are characterized

by

_a

"pairini'

of

replicas (See Fig. I).

Medina _et al. [12] ^have

argued

^that

pairing

of

replicas

is

expected

in the

random-phase

model and that "doublets"

of

replicas

will

play

_a role similar _to that of

singlets

in the

random-amplitude

_case.

Thus,

in addition _to the

regular

scheme of

RSB,

_we allow for _an extra

pairing

interaction when

searching

for the best variational solution. The

"simple" high-temperature phase (I)

is

separated

from two of the other

phases (II

and

III) by

first order transition

lines,

similar to what has been found

by

CD. These two

phases

_are

separated

from each other and from the

"pairing" phases (IV

^and

V) by

second order transition lines. Details will be

given

below. We have obtained the

wave functions and pattern of RSB in each

phase

which sheds

light

_on their

physical properties.

It _seems

plausible

to _us that the _reason that CD did _not find the two _new

phases (IV

and

V)

is because the GREM

(or REM)

allows

only

for _a tree-like structure of RSB and does

not accommodate the _extra

pairing

interaction. Since it has _never been shown that GREM

actually

describes the random walk _{on a} tree for the _case of random

signs (or phases),

^it may describe

faithfully only

part ^of the

phases.

For

completeness,

_we should mention that it is

possible

in

principle

that the

phase diagram

of _a walk _{on a}

regular

^{lattice in the limit} of

large embedding

dimension _may be different than that of _a walk _on the

Cayley

tree.

Furthermore,

there is

always

the

possibility

that _a solution to the saddle

point equations

with _a different _or

higher

level of RSB

(I.e.

_a different

parametrization

_{or more} variational

parameters)

than the

one we have obtained exists.

Lastly,

there is

possibly

_a ^difference between the

phase diagram

of the

random-sign

^{model and} a model with _a continuous distribution of random

imaginary phases,

^{but this} is not very

likely according

to CD

(see

_p. 984 of Ref.

[16]).

Our method differs from that of MP in the _sense that _{we use} the Hamiltonian

approach

rather than the

Lagrangian (Green's function) approach

that

they

_use, but the results _are identical for the _case of _a real

potential

which

they

have considered.

2.

Description

of the model.

In the continuum

limit,

the model _can be described

by

_a

path (x(t), t)

in _an

(N+ I)-dimensional

space-time,

^with _x an N-dimensional _vector and t the "time" coordinate which

always

increases

o

°'~

iii iv

i,o

i

is

~

2.O

2.5

O.5 1-o 1.5 2.O 2.5

I

Fig, I. The phase diagram containing five phases.

for _a directed walk. Each

path

has _a

weight

which

depends

_on the local value of the random

potential (both

its

amplitude

and

phase).

The _{sum over} all

paths,

each

weighted appropriately,

is

given by

_a

Feynman path integral (which

_one

might

call _a

partition function, though

it is not

necessarily real)

of the form:

Z(y, T)

₌

~~~~~

_~X

Xpl- f

_dt

^~°

(~~)

~

+

~X~

₊

iy8(X, t)

+

flV(X, )j

). ^(2,I)

(o o) 2 fit 2

The first term in the

weight,

which resembles

a kinetic energy, has _a coefficient _no which _we call the tension. It is _a soft version of the constraint found in lattice versions which allows _a

walker to move from _a

given

site

only

to its immediate

neighbors and,

in the _case of

polymers,

is related _to their

flexibility

_or

rigidity. Depending

_on the

model,

_one

might

choose _no to be

1610 JOURNAL DE PHYSIQUE I N°8

temperature

dependent,

but here _we do not make _any such

special

choice. The _mass term

(+-

p)

_{serves as a}

regulator

and will

eventually

be taken to _zero. The

term17e(x,t)

_represents

the random

phase,

while

flV(x, t)

denotes the random

amplitude

at each site. We consider the

case in which the variables fl and V have normal

(Gaussian)

distributions with _{zero means} and the

following

correlations:

v(x,i)v(x>,i>)

=

-J(t-t>)Ni~[(xjx')~j ^(2.2)

e(x, t)e(x', t')

=

-6(t t')Nf~ ~~

j~'~~ j, ^(2.3)

where

g(z)

indicates

averaging g(z)

with respect to the

quenched

random variables. The N-

dependence

has been chosen

(following MP)

such that the

large

N limit of the

theory

is well defined. For

large

^distances _{we assume} ^that both

f~

and

f~

vanish like

f(z~)

«

-((z~)~+; (2.4)

whereas,

for short distances the

f's

_are

regular

at the

origin (see below).

This form follows from dimensional considerations in which _one seeks _a "smoother"

representation

of the correlation than _an N-dimensional delta function. [6]

We _use the

replica

method [9] to carry ^out the

quenched

_{averages over} the random

potent1als.

Upon replicating

and

averaging,

one obtains:

(Z*Z)

^~

^((ya),

^T) =

~~~~~~~~ ~xi ~x2n

xpl- f

_dt

it _{~j (~~}

^~

+ ~

~j xi

(joj,o) ²

~

2

~

(

2

£ _{eoepN f~} (j(xa ^p)~)

~,p

+

£

_N

_iv (j(xa _{xp)~)j ),} (2.5)

~ p

WheTe

+I, if _{a =}

I,

, n;

(2.6)

~« "

-l, if _{a = n}

+1,.. ,2n.

Note that _one

replicates (Z*Z)

in the

complex

_case; this choice is apparent ^{in the} context of electron

hopping conductivity

where the quantum mechanical

probability

is the _square of the absolute value of the transmission

amplitude. Consequently,

there _are 2n

replicas:

the first set of _n from the

Z*'s,

and the second set from the Z's. The

averaging

results in interactions

among the

replicas.

The interactions

stemming

from the

random-amplitude averaging

_are

all

attractive; while,

those

arising

from the

random-phase averaging

_are

"charge-dependent".

Their

sign depends

_on whether the two

replicas

in

question belong

to the _{same or} different

sets.

Finally,

_{one can} extract the

following

associated

N-dimensional, (2n)-body

Hamiltonian:

fi2

~2

1

~

2

~ [

0zf~

2

( ~°~~~~~ Nno

~~° ~~~j

+i~jxa~

₊

_Nf~ (£(xa ^p)~)

,

(2.7)

2

~ ~ p Ko

where xa's have been rescaled

by n)~/~

_{and where}

p has been rescaled

by

_no.

Note that G

=

(Z*Z)"

of

equation (2.5)

^{satisfies the}

Schr6dinger-like equation

_[5]:

)

= -HG.

(2.8)

In the limit of

large

_T,

G _"

'vt(1011'v0(iYal)~~~°~, (2.9)

where

Eo

is the

ground-state

_energy of H and ivo is the

corresponding

_wave function. Thus

~ljn~ -(In f _{dyi dynG(yi,}

, Yn Yi,

, Yn T) "

(2.1°)

is the 'free

energy" density.

3. The variational

approximation.

In their

study

of manifolds in _a random

(real) potential,

MP have used

a variational

approach

to surpass standard

perturbation theory. They

have

employed

_a

general quadratic

action with tunable parameters to _{serve as a} variational

approximation

to the exact action and have

argued

that the

result,

which is

equivalent

in this _case to _a

Hartree-type approximation,

becomes exact for

large

N [6]. ^Their

proof

involves

representing

the action in terms of

auxiliary

fields and

using

a steepest ^descent

approximation.

Such _an

approach

is

equivalent

to the Hartree method

and becomes exact when

N,

which

multiplies

the

action,

becomes

infinitely large.

We have been able to

generalize

their

proof

to the

complex potential

_case.

For the one-dimensional manifolds of interest

here,

_{one can recover} the results of MP

(the

case 7 =

0) by choosing

a Gaussian _wave function _{as a} variational candidate for the

ground

state of the Hamiltonian

(Eq. (2.7)) [20].

^{We thus start with} a Gaussian variational _wave function of the form:

'V = fit _exP

I-j _L£'~~O,p

z;o

z;p), ^(3.i)

a,fl "

where1i1 is _a 2n _x 2n matrix. It is the exact

ground

state of _a

quadratic

Hamiltonian of the form:

~ "

~)[[(

+

'[~°~

+

)[~"P~"

^{~P' ~~'~~}

where the matrices 1i1 and

fir

are related

by:

1i1 ₌

(pi

₊

_{fir)~/~, (3.3)}

where

I

_{is the}

identity

matrix. Note that _even for the best Gaussian _wave function the

ground-

state energy of this Hamiltonian differs from that of the exact Hamiltonian. In order to have

some

physical

intuition into the

meaning

of the variational

parameters

as

representing

effective

"interactions" _among

replicas

and also in order to make _a connection with

MP,

we will _start with the

parametrization

^of

fir (and

ih will then be

given by (3.3)), although

in _our

approach

it is

technically

unnecessary ^to introduce h

(3.2)

at all.

For

practical

_reasons, it is

impossible

to choose the most

general

matrix

Ai (since

_one

requires

_an

expression

for

general

n to enable continuation to _{n =}

0);

^rather it is _necessary

1612 JOURNAL DE PHYSIQUE I N°8.

to select _an

appropriate parametrization

of

k.

_{For the}

short-range

_case, it is reasonable to

believe, following

MP for _a real

potential,

that _a

finite-step

RSB scheme is

appropriate.

Such

a choice is further

supported by

the finite-n GREM considered

by

CD. Our

parametrization

scheme first breaks the 2n x 2n matrix

li

_{into two}

n x n matrices _as follows:

fir

=

(~ (~

,

(3.4)

2

where

ii

_{and 1j2}

are shown in

figure

2. The matrix

ii

_{represents the} interactions _among the _set of _n

equally charged "particles"

and 1j2 the interactions _among

oppositely charged

particles (recau Eq. (2.6)).

This choice of

li

_{breaks the 2n}

replicas

into ^~ _groups of size

2k,

k

each

containing equal

^{numbers of}

oppositely charged particles.

A

given particle

has _non-zero interaction

only

^with the

(2k-1) particles

within its _group. The

strength

of interaction between

equally charged replicas

is _ae, and that between

oppositely charged replicas

is _ao.

Furthermore,

there _can be _an additional interaction between _a

given replica

and _one

particular replica

of _an

opposite charge;

the

strength

of this extra interaction is _ad

Finally,

the center-of-mass of each group is free.

Z -ae

ae~

-ae Z -ae

0 0

-ae ~ae Z

~ ~°e ~ae

-ae Z -ae

0 0

-ae ~ae ~

~ -ae ~ae

-ae ~ -ae

0 0

-ae ~ae ~

Fig.

2. The matrix

ii

_is

an n x n

(hierarchical)

matrix with one-step

breaking.

It consists of

n/k

matrices

along

the diagonal. The

diagonal

elements of these k _x k matrices _{are ~}

=

(k I)(«o

+

«e)

+ _ad, this choice _ensures translational invariance. The matrix

12

is obtained by

letting

_{ae - ao,}

which parametrizes interactions between oppositely-charged particles, ^and _{~ - -«o ad} where _ad parametrizes the

(extra)

pairing interaction.

Along

^with the aforementioned basis for

adopting

_a

finite-step

BSB

scheme,

this choice has been further motivated

by

^the

following

scenario. The

charged

interaction induces

oppositely charged replicas

to form

pairs (hence ad)(

while the overall attractive interaction elicits _a

clustering

of

pairs (hence

_ma and

ae).

This

clustering

amounts to the fact that

replica

_symmetry

is broken in the _{n -} 0 limit

(which

_can ^be

thought

of _as

resulting

from the existence of _many

"states"

energetically

^close to each other and unrelated

by symmetry) _[6,9].

Some additional variational parameters have been considered but

proved inconsequential.

For

example,

_we have

performed

calculations which allow for _non-zero interactions between

replicas belonging

to distinct _groups with different

strengths

for

equally-

and

oppositely-charged pairs,

I-e-, we have put in two parameters in

place

of all the _zeros in

iii

_and

_{1j2, (with only}

an overall free

center-of-mass). However,

these additional parameters ^become zero under vari- ation. The _same

phenomenon

_occurs in the real

potential

_case considered

by

MP [6]. ^In a

separate

calculation,

_we have allowed for the

possibility

of _groups

containing unequal

numbers of

oppositely charged particles,

but the variational

procedure

leads to

"charge

neutral" _groups

whenever ₇

#

0.

The matrix

li

can be written _as follows:

fir

=

[k(«o

+ me) +

2«d]

12 lP

ij

_©

ik -(ae-«o) i2lPij©fk

-«o

J219

ii ©

fj,

^gnifies the direct

Calculating he

of ^he Eq.

(2.7) with

the ^ave iV (Eq.

.I))

equires

owing

the

elements of

fli

and 1i1~~ ^he

interaction

erms,

be

^omputed

in

the^ollowing manner:

l~j(xa -xp)~j) _

~ f~i) _{j~~ ~} ~~>j

noN

~

_ jj/Ni

"

_~

~ >

1614 JOURNAL DE PHYSIQUE I N°8

The matrix fli and its inverse have the _same form _as

k;

_{in terms of its}

eigenvalues,

fli _can be written:

fli=

Ii I24lI~©Ik

+

)(12-li) i2lPij4lfk

+

j(13 ^{Ii) f2}

^4l

ij

_©

ik

+

(14 13

+

Ii 12) )2

_4l

i~

_©

fki _(3.9)

and the inverse fli~~ is obtained

simply by replacing

the l's with

l~~'s.

_{Note that the}

regulator

p is

required

to

perform

this inversion.

Calculating (H)w

^leads to the

following expression:

)$ (~~~ ^~4k~

^{~~ ~}

/k~~

^~

~4k~

~~ '~~~~~~ ~

~~~~~~

~'~~~~ ~

k

~~

^{~ j~o~ ~}

^{~~~ ~}

_k

~~

^~

~~

~~

~~~~~ ~

~o

^~

~o

^{~ ~~}

^~~~~~

~

~o

^~

~o

~ ~~ ~~~~~~

~~2k~~

~o

^~

~o

^~

~o

~ ~~ ~~~~~"

~~2k~~

~o

^~

~o

^~

~o I'

^~~'~~~

We will focus _on

short-range correlations; therefore,

_any interaction

resultin~

ⁱⁿ a term of the form

/[£a;1/~

_{+ p~}

ii

has been _set

equal

to _zero in the _{p -} 0 limit

(f[oo]

₌

0).

This step is where the difference between short and

long

_range sets in. In

fact, henceforth,

_we will

concentrate on a

particular short-ranged

correlation:

/v[z]

=

([xl

=

(f°

^~

^fl~'

^~~°~~~

)~~

(3,ll)

0, otherwise,

where

lo

is

negative. Higher

_powers of _z, if present, will not

change

_any of the

physics

and _may have _{a very} minor effect _on the location of the

phase

boundaries. Since the l's

are

independent

functions of the

a's,

_{one can} find the

stationarity

conditions

by extremizing

with respect to the l's and k

(instead

of a's and

k).

The variable k

originally

lies between I and _n: I < k < n, which in the _n

- 0 limit becomes 0 < k < I.

When

looking

for the best variational parameters, _one must restrict the variations to remain within the

physical

_range. In this

work,

there _are two

important

constraints _on the allowed values of the l's and

k,

besides the usual

requirement

that

1;

> 0 which is _necessary for the

normalizability

of the _wave function. The first constraint is:

(k

_k

ll~_i

1 ~

^1~_~

k ^~ ~ ~'

(3.12)

Table I. Listed here _are

expressions for l's, a's,

F and k that result

from e~tremizing

F.

Phase l's a's F and k

li"o «o=o

I

l~=0 «e=0 F=7~-fl~

13 _" o _ad

" o

Ii

=

2kl fl

_{ma =} 2fl~ +

272

F ₌

4klfl 6kfl2

II 12 ₌

2ki(fl2

₊

7~)I

_me

=

2fl2 27~

l~

=

2kl fl

_ad

= 0

-4ki

_{fl2 +}

(2k

+

2)fl

₌

(fl2

+

72)1

Ii

₌ I _{ma =}

k/2

F ₌

(1+ y2 3fl2)/2

III

12

₌ 2 _{ae =}

-k/4

13=1

ad=0

k=(3fl2-7~+1)/4fl~

Ii

₌

(I k)/2k~

_ao

=

1/4k4

F ₌

(1 8k3fl2)/4k2

IV 12 ₌

1/2k2

_ae

=

(k 1)/4k4

13

₌

1/2k

_{ad =}

(1- 2k)/8k4 4k2(72

+

fl2 fl~/k)I

= I

Ii

= ma =

2fl2

+

272

F

= +

(2 2(2

V 12 ₌

27/(2 _k)

^I _me

=

2fl2 272

>3 ₌

2k~fl

_ad

=

~~~ ~~~~~

2kfl~

^~ + ~~

ll~~ ^4fl~

= 0

Thh condition follows from the

requirement

that the arguments ^{of the}

l's

_{must be}

_positive,

since these functions _are

only

defined for

positive

arguments.

Taking

into account that _md > 0

(I.e.,

the

pairing

is

always attractive),

_one establishes that

Ii

> 13 in all

phases,

and thus

equation (3.12)

is sufficient to _ensure that all

arguments

^of

l's

are

positive.

The second constraint _we have found _necessary to

implement

is

Ii l~

+

13

2 0.

(3.13)

This constraint _ensures that the kinetic _energy, and hence the total _energy, remains bounded from above _as k

-

0,

and

subsequently

the extremum is

always

a

global

^maximum as a function of k. We have found that this

requirement

is _necessary in order _to find

a consistent solution

throughout

the entire

fl

₇

plane.

When these conditions _{are no}

longer satisfied,

_we enforce the constraints _as

equalities (either

or both _as

necessary)

and search for the extremum of the

1616 JOURNAL DE PHYSIQUE I N°8

free _energy

subject

to the constraints. We also check _e

posteriori

that the arguments ^of

/

are

always

less than

~~~,

_as

required by equation (3.ll).

It

happens

that the extremal

point

is

ii

actually

_a saddle

point,

since the best _wave function results from _a delicate balance between the attractive and

repulsive

interactions. In all

phases,

the _energy is maximal with respect to the variation in k. It is also _a maximum with respect to Ii and

13 (when they

_are not related to

l~ through

_a

constraint)

and _a minimum with respect to

l~.

Table II. The

e~pressions for

the

phase

bovndelies.

Phase Boundaries

I II

(6fl~

+ 4

67~)1

+

54fl(fl~

+

72)I

₊ 8

187~ 90fl~

₌ 0

1- III

y~+p~

= I

II III _y

=

lip

II V

4fl~

1

~fi 4fl

+

fl£fi

= 0

(P?+72) (P?+7?)

III Iv y~

p~

= i

Iv v

(p2

₊

72)

^'

=

8vfp4

4. The

phase diagram.

Before

proceeding

to the

results,

let _us rescale variables _as follows:

>;

_-

lfol-~fini~>;,

₇

-

lfol-~f/Ki17,

_and

_p

-

j/ol~~f)Kilfl.

Furthermore,

let

~

~i~~ ~~~

~~'~~

Extremizing

F

(Eq. (3,10)) subject

to the constraints

(Eqs. (3.12)

and

(3.13)) yields

five

phases

_as shown in

figure

I. The

resulting

values for

l's,

a's and F's _are

provided

in table

I, along

with the

accompanying stationarity equation

for k.

The

phase

boundaries _are found in table II. Phases I,

II,

and III _meet at the

point (7

"

~,fl

=

~);

while

phases

II, III,

IV,

and V meet at the

point (y

₌

~, fl

=

~).

The

2 2 2 2

transitions between

phase

I and

phases

II and III _are of first

order,

with the

exception

of the

point

at 7 = 0. All other

phase

boundaries _are continuous transitions. One _can understand

some of the

properties

of these

phases

in _terms of three order parameters: ma

(the "glass"

order

parameter),

ao+2ae +2ad

(the

"thermal" order

parameter)

and _ad

(the "pairing"

order

parameter).

The

"glass"

^order parameter ^is zero in

phase

I

(the high-temperature,

weak- interaction

phase)

and _non-zero in all other

phases.

The "thermal" order parameter is _zero in

phases I,

III and

IV,

which _are all

high-temperature phases.

Notice that in these three

phases,

the

kinetic-energy

constraint

(Eq. (3.13))

holds _{as an}

equality.

The

"pairing"

order parameter is _zero in

phases I,

II and

III,

in which the

charged

interaction is weak. This time the other

constraint

(Eq. (3.12))

holds _{as an}

equality

in

phases

IV and

V,

where the

'§~airini'

order parameter is _non-zero. Phases

III,

IV and V do _not exist for ₇

= 0

(no

random

phases),

and thus _{we can say} that interference effects _are strong ^{in these}

phases,

and _we

might

call them

"quantum" phases.

For _y

=

0,

_we have checked that _our results lead to the _same

stationarity

conditions _as derived

by

MP

(note

that

they

have made the choice _no

=

fl)

_[6].

Actually,

because _we have two sets of

replicas,

_we find

degenerate

^solutions at 7 =

0;

in

fact,

_we find _{an even} greater

degeneracy

when _our

parametrization

does not _assume

"charge neutrality." However,

_{as soon}

as 7

# 0, only

the

charge-neutral

solution survives. On the

fl

₌ 0

axis,

there is

similarly

a

high degeneracy

of the solutions with respect ^{to k. It follows from the absence of} attraction between bound

pairs

when

fl

₌ 0. We believe that this behavior is _a feature of the variational

approximation

used here and is _true in the N

- oo limit and that this

degeneracy

would

probably

be lifted in finite dimensions. This

degeneracy

is

again

lifted _{as one turns on}

fl,

I-e-, the real part ^{of the random}

potent1al.

In that _case bound

pairs

_are attracted to each other and tend to form _groups of 2k

replicas.

It is

tempting

to

identify

_our

phases I,

II and III with the

corresponding phases

found

by

CD

[16].

^{To make the}

comparison,

we first subtracted the free _energy of

phase

I from that of

phases

II and III both in _our

expressions

and in those of CD.

Expanding

to

leading

order in p, ^we have found that the _energy of

phase

III becomes identical _to that found

by

CD if _we

identify

_p

=

7~/4 _(where

_p is the

probability

for _a

negative sign)

and also take

In(I()

= I for

their tree constant

(I.e.,

_a tree with

branching

ratio of

e).

In

phase II,

_our free _energy is very close to that of CD for

fl

not too

large (we

have checked that

by comparing

contour

plots

of the

corresponding

free

energies).

We attribute the difference to the fact that their model is _on

a lattice and _ours in the

continuum,

and _{one may} need to tune the tension _no to make the two models

truly

coincide. Note further that had _we

plotted

the

phase diagram

in the variables T

versus

7~,

the I-II

phase boundary

would

approach

^the _{y =} 0 axis

tangentially

as in CD. See

figure

3 for _a detailed

comparison

of the free _energy between the results of CD and those of this _paper.

We should

emphasize

that it is

impossible

for _our

phase

III to extend into the

higher

₇

region,

since _we have found that first the constraint

(3.12)

is violated and further into the

large

7

region (when 7~

_> 1+

3fl~),

^{k becomes}

negative

which is also

unacceptable.

It turns out that _on the transition line between

phases

III and

IV,

k

=

),

^{and this is where the} constraint

(3.12)

^has to be

implemented.

5. Discussion.

In this paper, we have calculated the

phase diagram

for directed walks in _a random

potential having

both real and

imaginary

parts. We have used _a variational

approach

that should become exact in the limit of

large embedding

dimension. The

advantage

of the variational method is that _{one can}

identify

the different parameters ^{in the trial} wave function _as effective interactions

among

replicas (or "particles"), providing

_an intuitive

physical picture

for the

properties

of the different

phases,

like

grouping, pairing, interference,

etc. Since _we have not proven that there

1618 JOURNAL DE PHYSIQUE I N°8

o.o o-o

I-O I-O

fi

f _k

~ f k n o

d

2.O ^~ 2.O

a a

o-O I.O 2.o o-O I-o _{2 o}

y 7

a) b)

Fig.

3. A comparison of contourplots of the function

f(fl,7)

_"

F(fl,7) FI(fl,7)

between the results of CD

(a)

and the present _paper 16). Fi ^{stands for the} free _energy in phase I. In

figure

3a _we

use the ad hoc correspondence

7~/2

=

-In(1 2p)

in order to plot CD's results in the

(fl, 7)

Plane.

The lines stand for contours of fixed value. These _are

(a)

0.3

(b)

0.2

(c)

0.I

(d)

0.0

(e)

-o.05

(f)

-o.15

(g)

-0.25

(h)

-0A (I) -0.6

(j)

-0.8

(k)

-1.2 (1) ^-1.6

(m)

-2.0

in)

-2.8

(o)

-3.6. In phase I, f is

identically _zero.

cannot be other solutions to the saddle

point equations,

we cannot rule out that

ultimately

a more elaborate variational solution _may be found. We believe

though

that _any such solution will retain the essential

physical

features of the various

phases

which have been found in this

work.

We have found _a rich

phase diagram

with five

phases

_as

depicted

in

figure I,

with all

phases

except I

displaying

_some

degree

of RSB. We have verified that the value of the exponent v

[12,

19] ^{is in all}

phases,

^similar to what has been found

by

MP in the real

potential

_{case [6].} Our

2

phase diagram

includes two new

phases

as

compared

to the

Cayley-tree

calculation _of CD

[16].

We would like to comment about the Monte Carlo simulations done

by

CD.

They

have checked for the location of the first order transitions between

phases

I and II and between

phases

I

and III

(Fig.

6 in Ref.

[16]),

and also verified that there is _a transition between

phases

II and III

(Fig.

2c in [16] ^shows one

point

_on the

boundary).

These results _are also in agreement with _ours and _are not

enough

to rule out _or establish the

possibility

of other transition lines _as

displayed

in

figure

I. It is _a

challenge

for future research to

verify

the

phase diagram

of

figure

I

by

additional simulations.

A number of _open issues

concerning

the behavior in lower

embedding

dimensions

remain;

for instance: In what dimension do _some of these

phases

obtained

disappear?

And how does the exponent v

depend

on dimension? Some of these

questions

_may

possibly

be addressed

by

a

I/N expansion

[7].

Acknowledgements.

We thank NSF grant DMR-9016907 for

support.

^{YYG thanks} M. Mdzard for useful dhcussions and his

hospitality

at ENS Paris. TB thanks Y.

Shapir

and D-S- Koltun.

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