Replica approach to directed Feynman paths with random phases

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Replica approach to directed Feynman paths with random phases

Thomas Blum, Yonathan Shapir, Daniel Koltun

To cite this version:

Thomas Blum, Yonathan Shapir, Daniel Koltun. Replica approach to directed Feynman paths with random phases. Journal de Physique I, EDP Sciences, 1991, 1 (5), pp.613-620. �10.1051/jp1:1991156�.

�jpa-00246355�

Classification

Physics

^Abstrticts

05.40 7120

Shod Communication

Replica approach to directed Feynman paths

v4th random phases

Thomas

Blum,

Yonathan

Shapb

and Daniel S. Koltun

Department

of

Physics

and

Astronomy University

of

Rochester, Rochester,

NY

lM27-l0ll,

U.s.A.

(Received

19

Febmmy 1991, ticcepted11

March

1991)

R6sum6. Los chemins de

Feynman dirigds _qui

accumulent des

phases

^aldatoires font

partie

des 6tudes r6centes de la

propagation

d'61ectrons _ou d'ondes

dlectromagnetiques

_en milieux d6sordon- nds. Nous examinons leurs

propridtds

d'£chelle _en utilisant la th£orie des

r£pliques.

Deux mod61es

avec corrdlations de _courte et de tr6s

longue port6e

en I+ I dimensions sent 6tud16s. Nous 6valuons le

comportement

^b

grandes

et

petites

valeurs

(finies)

de _n des modHes

quantiques

k 2n corps avec in-

teractions attractives _et

r6pulsives qui

_en resultent. Nous

soulignons

(es diflicult6s

qui surgissent

dans la Ii mite _{n -} 0 et la

possibilitd

de brisure de

symdtrie

des

r£pliques.

-Abstract. Directed

Feynman paths

ⁱⁿ I+I dimensions that

acquire

random

phases,

encountered in the

study

of electron and

light propagation

in disordered

media,

_are examined

by

means of the

replica

trick. Two models _are considered _one with

short-range correlations,

the other with very

long-range

correlations. The

large

and small

(finite)

_n behavior of the

resulting 2n-body _quantum

Hamiltonians with

competing

interactions is calculated. the difficulties _that arise in

extrapolating

the results to _{n -} o and the

possibility

of

replica _symmetry breaking

are discussed.

The

importance

of the

forwardscattering

interference mechanism in the

(hopping)

conduc-

tivity

of insulators has been

pointed

out

by

Shklovskii

[ii.

Each

hop actually

consists of the _su-

perposition

of many

scatterings patlls.

^Since the electrons _are

loca1i2ed,

each

path's

contribu- tion

decays exponentially. Consequently,

models

involving only

the shortest _or "directed"

paws

have been considered

[2-7j.

^lb

study

the effect of

disorder,

we consider here a model

(some-

times called

"complex

directed

polymers")

ⁱⁿ which directed walks

acquire

random

phases

with each

step.

^{Thin model} may ^be viewed _{as an} extension of "directed

polymers" [8-13],

_a model in which walks

acquire

random

amplitudes,

and is

appropriate

in the

presence

^of a

strong magnetic

field _or

magnetic impurities.

The transverse fluctuations of directed

polymer

in I+I dimensions with uncorrelated radomness have been found to scale with "time"

(distance

measured

along

the

directed

axh)

as:

(~2(t))~

c~

t2" (where _)R

indicates

averaging

over the

randomness)

with

v =

2/3,

and the

sample-to-sample

fluctuations in the walk's free energy ^{have been found to}

614 JOURNAL DE PHYSIQUE I N°5

scale _as: AF

=

((F ^F)R)~)~~~

_c~ t~ with _w

=

1/3.

We aim to examine similar

quantities

in

R

the case of directed walks with random

phases by

means of the

replica

trick. These models may also be useful in the

study

^of

light propagating

in _a random media

[14, 15].

In two dimensions

(2d),

the wavefunction it

(~t, t)

of _an electron at time t

(position along

the directed

axh)

and

(tranverse) position

_~i is

given

in the continuum limit

by

the sum of all directed walks

connecting (~i, t)

to

(0, 0),

^{which in} the

path integral

formulation _can be written:

(~<>t) _k

4t

(~t, t)

_CK

/

_D~ ~

eXp

~ /

_dt

_(+t)

+ I

/

_{dt B}

_{(~t, t)}

,

(1)

(0, 0)

where the first term

/

_dt

_(ii

)~

provides

the

exponenthl decay

of the localized _wave functions and the second term 8

(~i, t)

is _a random variable

corresponding

to the

phase. it(~, t)

satisfies the

following equation:

~~ _~~

~ ~~ ~~~' _~~ ^~' ~~~

The conductance between the sites

(0, 0)

and

(~, t)

is

proportional

to the

probability

for the electron to

hop ([it[2(~, t)) Analysb _{[2, 3]}

indicates that the

logatithm

of

the

conductance is

given by

the average of the

logarithm

over the different realizations of the

dhorder; consequently, (log (it* ~))~

is of

primary

interest. lb facilitate the

log-averaging,

one can

implement

the

replica trick,

in which _one averages ^the

replicated quantity, ((it*it)" _((~(~)

,

t))~

((it* ^it)" ((~(~))

,

t)

c~

~~~~~~

^{~~ ~~}

D~(~~

exp

(- ( /

_dt

(k(~~)

)

~ (1°1, °) _r=I

x

exp

I

( /

_dt

_er8 ~(~~, t)

,

(3a)

r=I R

+1, ill<r<n;

whereer=

-I, ifn+I<r<2n.

The above

expression

is _{a sum} over the rea1i2ations of 2n directed

paths

which connect

(0, 0)

to

(~(~), t)

with random

phases era (~i, t) acquired along

^each

path.

The first _n

paths

arise from

it;

^the second _n from it*.

Assuming

that 8 is

normally

dbtributed with:

je(~, t)

e

(~', t'))~

=

2c',2 (t t')

_u

(~ ~')

,

(3b)

leads to the

following:

((it*iP)" ((~(~))

,

t)

c~

~~~~~~

^{~~ ~~}

D~(~~

exp

(- ( /

_dt

+(~~)~).

~ (1°1, °) _r=I

x exp

-c'a2 ( ( /

_dt

erer,u

~)~~ )~'~) l. ^(3c)

r=I r'=I

The

averaging

results in interactions among ^the

replica

with

magnitudes proportional

to the vari-

ance [8]. Notice that

oppositely "charged" replica (where

_e; is the

charge)

attract and

similarly

charged replica repel

and that there _are

n2

_such attractions and

n(n I)

such

repulsions

a net of _n attractions. lb obtain this

result,

one

ignores

the

compactness

of the

phase,

wllich

only

seems reasonable when the variance b

quite

small

(a2

<

1)

^The

periodicity

^of the

phases

_may

be crucial but its

study

is

beyond

the

scope

^{of the}

present investigation.

We will denote

log (it*it) by

F

(the analog

of the free

energy).

Information about the

log-

average

(F)R

₌

log (it* it))~

is then obtained

by examining

the cumulant

expansion [8]:

lli~*i~)")R

₌

xP1i§ )

((1°g ^(i~* i~))')~ ⁽⁴⁾

I=i

When

adopting

this

method,

one b faced with the difficulties of

solving

the

resulting (interacting) problem

for

arbitrary

_n.

((it* it)" _)~ (the "propagator")

can be

expanded

in terms of functions

#;

1(i~*i~)")R

₌ GSS

((~(~~), ^1°1; t)

=

£ ~li(1°1)~li ((~)~~))

^exP

^{I-Eitl, (5)}

where

#;

are

eigenfunctions

of the Hamiltonian:

2n _fi2

255

H(n)

₌

~

fi + 2C

~j

_eye;

U

(~;

~;

,

(6)

I=I

~~i

I<I

(with

k

= I and _{c =}

c'a2/2

_and

_E;

_{are the}

corresponding eigenenergies.

Since the

ground

state

(#o> Eo)

dominates the

long-time behavior,

_{we can} concentrate on it

[9].

Assume that the

ground-state

_energy has the form

Eo(n)

_{= ein,-}

e2nP.

Then

fl

is related to the

exponents

w and _v

through [7, 8, _16]:

2v-1=w= ^~

(7)

fl

The first

scaling

relation has been confirmed for the random

amplitude

case

(directed polymers),

but it is less established for the random

phase

case.

We concentrate on two

types

^of correlations:

(I)

disorder with

short-range

correlations

u(~)

_c~

b(~)

for which

contradictory

results have been claimed

(see below);

and

(2)

dborder with

long-

range correlations

u(~)

_c~

~2,

which has been studied in the directed

polymer

case

only.

1.

Shod-range

correlations.

First _we will consider the version of the

problem

with uncorrelated

randomness, I.e., u(~)

+w

b(~).

In 2d random directed

polymer

^it leads to _an

exactly

solvable Hamiltonian with _a

ground-state

energy:

Eo(n)

₌ c2

(n _n~) /12 (for

all

n).

Kardar _[8] has

exploited

this result and the

scaling

relations above to obtain

fl

₌

3,

_w

=

1/3,

and _{v =}

2/3.

Parisi

[10, 12]

^{and M6zard}

[12, 13]

^have re-examined the

problem

and found _some evidence for _a weak

replica symmetry breaking,

which effects no

change

in the

previous scaling

results.

Unlike the random

amplitude problem just mentioned,

the random

phase problem

^leads to a Hamiltonian which is not

exactly

solvable

(I.e. integrable),

as evidenced

by

the fact that the

Yang-Baxter

conditions _are not satisfied. Directed

Feynman paths

^with random

phases

have been

simulated; however,

there remains _a

discrepancy

among the simulations. Medina _et al [4] ^have

616 JOURNAL DE PHYSIQUE I N°5

performed

_a simulation _{on a} square ^{lattice in} which _an

independent

random

phase (with

a unTorm

distribution,

0 < 9 _<

2~)

was associated with each bond

belonging

to the walk

They

have found that the

quantities ([~])~)

^and

_[~2]~~)

^have

asymptotic scaling ^t2"

with _v

= 0.68 +

0.05;

while We difference

(( _(~2]

_~~)

([z])~))

^{~~~ scales} as

tl/2

_{On the other}

_hand, Zhang

_[5j has found in his

simulation:

([[~]av[)

c~ iv with _{v =} 0.74 + 0.01 and

(( _(~2]

_~~)

_([~])~))

^~~~ _c~

^tl/2.

lb add credence to these

simulations, analytic arguments

^have been

proposed.

Medina et al [4] ^have

argued

that _a bond

belonging

to a walk

arising

form it" _must also

belong

to a walk from

(it*)" otherwise, averaging

over the

phase

would

give

zero

(when

the distribution is uniform and uncorrelated to other

bonds). Hence,

walks fornl

tightly-bound (conjugate) pairs.

When two

pairs

meet,

they

can switch

partners, leading

to a statistical

(exchange)

interaction.

Thus,

one

recovers the attractive .boson

problem (and fl=3)

with

pairs replacing particles

and the

exchange

interaction

replacing

the delta function attraction. Note that thb is

essentially

a

strong

disorder

argument (renormalization

studies

suggest

a flow to

strong

disorder

[4])

^about a

system

on a lattice and that it maintains the

compactness

^of

phase. Zhang

has

presented

_an

argument

^based on

Hartree

theory

_[6] with _a

screening

effect. The _n net attractive interactions _are redistributed among ^{the total}

n(2n 1)

interactions

yielding

once

again

the attractive delta function

problem by

this time with an effective

coupling

constant c"

+w

cn~l/2

_{which leads to}

fl

₌ 2.

Clearly

these

conflicting arguments

and numerical results call for further

investigation

into this model. We have thus

opted

to examine the

large

^{and small} _n behavior of this

intriguing

Hamiltonian via various trial wave-functions. For

large

_n, a lower bound _on

fl

of 5

/3

_can be found

by using

the

following

trial wavefunction:

if _"

ifrep

(Xl> Zni

L) ilrep(Zn +1,

,

z2ni

L), (8)

where

itrep

^{is the} wavefunction for

repulsive

bosons in _a box of

length

L

(with periodic boundary conditions),

where L is used _{as a} variational

parameter [17].

The energy ^h

separated

into two

pieces.

The first h the energy ^{due to the attractive} interactions _cum;ke and h

readily

calculated:

eunuke =

-2cnp,

where p is the

density

of _one

charge (p

_{= n}

IL).

The second is the kinetic and

repulsive potential energies

_eiike and is

equal

to the energy ^{of the ld}

repulsive

^boson

problem, provided by

the Bethe ansatz. In the

high density limit,

Lieb and

Liniger _[18]

and

subsequently

Gaudin

[19]

derived the

following expression:

e,i~~in,p)

₌ ²ⁿ

op

)pi/2c3/2

⁺

¹⁹⁾

16 this must be added _a kinetic energy per

particle

of

~2 /L2,

_to allow for localization within _a

length

L. Then the minimization with

respect

^{to L leads} to

Eo

< eiike + euni~ke c~

-n~/3.

_This

bound

fl

> 5

/3

is the

analog

of the bound

fl

> 7

IS

found

by Dyson [20]

^{for the 3d}

system

^{of boscns}

interacting

via _a Coulomb

potential (u(i,)

_{c~ eie;}

/r)

The details of

calculating

this

bound,

as well as

arguments proposing

that indeed

fl

= 5

/3,

will be

presented

elsewhere

[17j.

If the attractions were

stronger

^{than the}

repulsions (even by

the

slightest amount),

then the p terms in _efike and eun,;ke would not

cancel, resulting

in

fl

= 3. In fact in that case, many ^{trial wave-} functions would

yield fl

= 3

(for example,

_a wavefunction of the Hartree

type) [17j.

The attractions would be

stronger

^{if the}

problem

called for random

amplitudes

in addition to the random

phases.

(On

the other

hand,

if the

repulsions

_were

stronger

^{than the}

attractions,

then similar

arguments give fl

₌

[17j.)

^The

system

^with

equal strengths

is

quite

delicate. This

fl

= 5

/3

solution may

prove

unstable _to additional considerations which _were irrelevant in the

purely

attractive _case, such _as

higher

order interactions.

Moreover, taking

the

periodicity

of the

phase

into account could alter the Hamiltonian in a

significant

_way.

The

fl

₌ 5

/3

result was obtained

using

wavefunctions of the

product-type

with _no additional at- tractive correlations. While such a choice

maybe justifiable

for

large

_n

_[17j,

it h

clearly inadequate

for small _n. The best variational wavefunction _we have found for small _n is of the form:

n n

16 =

exp

-a

£ £ _[~i

~n+; ⁺ b

£ _((~i

_{q +}

_[~n+;

~n+; [)

,

(10)

I j I<I<I<n

which b motivated

by

the

groundstate

of the attractive boson

system.

^The variational calculations of

£o(n)

=

(16[H(n)

[16)ruin for small _n

provide

evidence for _an

exchange

interaction among

pairs.

The

energy

^for a

pair £o(I)

b

-c2/2;

_{while the energy per}

_{pair (£o(n) In)} (found by numerically minimizing (H(nj)

with

respect

to a and under the

normalizability

conditions _{a >} 0 and <

a)

are

-0.609c~, _0.725c2, 0.837c2

_and

0.943c2,

for _n

=

2, 3,

4 and

5, respectively. (See Fig.

I for _a

plot

of

£o(n)

_versus

n.)

s

4.0

3.0

2.0

1-o

0

0 1 2 3 4 S

n

Fig.

1. The values of

Eo(n)

obtained

variationally

for _n

= o 5. The solid line is the best fit _to these

point using

the form:

Eo(n)

=

(earl

+

(0.500 e)n)

^Mith e = 0.122 and fl

= 1.952

(see

the text for other

possibilities.).

We have fit the variational values to the

expression ij(n)

₌

_(enP

+

(0.5t© e)n)

,

where the

exact results

£o(0)

₌

0and£o(1)

=

-0.500(fore

₌

I)

are built

directly

into this from.

(We

assume a "smooth" behavior _{at n}

=

I.)

The linear term follows from the

(undisputed)

linear increase of the average "free

energy" (F)R

with t. The best fit to the

remaining

four

points (n

₌

2, 3, 4, 5)

has _e

= 0.122 and _an effective

exponent fl

₌ 1.952.

(See Fig. I.)

With

only

two

parameters

^to

618 JOURNAL DE PHYSIQUE I N°5

fit,

we can

emphasize

"small" _n

by fitting only

to the values at n =

2,

3

(e

₌ 0.092 and

fl

=

2.127)

or

"larger"

_n

by fitting only

the

points

_{n =}

4,

5

(e

₌ 0.132 and

fl

₌

1.914).

From these values

we see that the data do not conform to such _a

simple expression. Supposing

nevertheless that the

expression captures

the _{n -} 0

asymptotic behavior,

_we

_propose

that the effective

exponent fl

increases with

decreasing

n. It h

plausible

that this trend continues to smaller _n and that

implies fl

_> 2.127 in thb

range,

^{in contrast to the}

proposed fl

₌

5/3

behavior for

large

_n.

2.

Long-range

correlations.

Next we consider _a

toy

^{model with soluble}

quadratic

correlations introduced

by

Parisi in the ran- dom

amplitude

case

[iii.

The

correxpqnding

Hamiltonian

H(n)

^h:

For this

problem

one can derive the "full

propagator' Gn ((~; )

,

(0) t) (see Eq. (5)).

lntroduc-

ing

normal

modes,

one finds a

single plane

wave mode

(yi

the center of

mass)

and

(n I)

harmonic oscillator modes

(y;

with 2 < I <

n)

each with

frequency ^an1/2

The

ground-state energy

^h

given by Eo(n)

=

(n I)nl/2a (for

all _n >

I).

The full

propagator

^{h the}

product

of the

propagators

for the individual

(independent)

modes. Recall that the

propagator

^for a free

particle

is:

G(~, 0; t)

=

(£)

exp

(-~) (12)

~

~

and that for the harmonic oscillator with

frequency

_wo

_Iwo

₌

ani12)

is:

w~ 1/2 _w~

G(~, 0; t)

~

=

(p

_coth

(won)) ^exp (-

~

^coth

^(wet)

^~

⁽¹³⁾

~

Hence,

the full

propagator

^h:

Gn ((~;), 0; t)

₌

N(t) xp1- ~

'l~~~

^coth

(ant/2t) f ⁾⁾

,

(14a)

;=2

where

N(t)

₌

_eXp -) _{log(2~) log(t)}

)a~t~

⁺

~ a~t~

₊

~a~t~

^{+ O}

(n~)

,

(14b)

~

which agrees with

equations (13)-(15)

^{in reference}

[11].

^{Note that the short-time behavior h that} of _a

non-interacting _system

as has been

suggested _[9, lo, _13].

The

n2t4

_{term in}

N(t)

indicates that

w = 2. From thin

propagator

one can calculate

quantities

such _as

(~))

and the _n

- 0 limit _can be taken without any

difficulty;

it

yields:

~~~io~~~~

'~~~

~ ~' ~~~~

as found

by

Parhi

[I ii.

^Notice that the t _{- oo} limit and the _{n -} 0 limit do not commute

[11,

_13]

and that the fluctuations _{are more} than ballhtic.

Despite

the unusual

result,

the

scaling

relation

(Eq. (7~)

^{still holds}

(fl

=

1/2,

_{w =}

2,

_{v =}

3/2).

This

system

can be solved in d dimensions and

yields

the _same result

independent

^of d.

The Hamiltonian

corresponding

to the random

phase

_{case can} also be solved

exactly

when the interactions _are

quadratic.

The Hamiltonian is:

255

fi2

~ ~

H(n)

_"

£

$

^~'

£~i~i _{(~i ~i) (16)}

I=I S I<I

Again,

one can introduce normal modes. This time there are

(2n 1) plane

wave modes

(z;

^with

I < I <

2n-1)

and one harmonic oscillator mode

(z~n)

^with

frequency (2n)1/2a;

the

ground-state

energy ^is

Eo

₌

(2n)

^II

^2a.

The

propagator

associated with this Hamiltonian is:

On ((z;

,

0; t)

=

fl(t)

_exp

(- ~f ^{z) ~~(~~~~}

^coth

(a(2n) ~/~t) ^z(~

_,

(17a)

t ,_~

where

~

~~~~

~~~ ~

~°~~~~~

~ ~°~~~~

'~~~) ~ _'~~~

~ ~ ~~~~ ~~~~~

If one

naively

takes the _n

- 0 limit of

(~)),

_one finds:

(~))

₌

^a2t~

+ t. This is of course

inadmissible

(for large t)

and

provides

a

strong

^{indication that} a more

sophisticated

treatment of the n - 0 limit is

required (e.g. replica syrnmetry breaking).

In

conclusion,

_we have

applied

the

replica approach

to two models of directed walks with _ran- dom

phases:

the first has disorder with

short-range correlations,

and the second has dhorder with

long-range

correlations. Each leads to a

2n-body

Hamiltonian with

competing

interactions. We have studied the

large

and small

(finite)

_n behavior of each model.

In the

short-range

_case, the Halniltonian is _more difficult to handle than its random

ampli-

tude

counterpart,

^which is

exactly

solvable. We have

employed

trial wavefunctions _to

study

the

dependence

on n and have found different behavior for

large

_n and small

(finite)

n. With the

interpolation

between

large

and small _n

uncertain,

a continuation to n - 0 seems

premature.

Moreover,

we have found it to be _a very delicate

system

^the

slightest @nbalance

in the interac- tion

strengths changing

the

large

n behavior

significantly.

This

instability

rakes the issue of the

periodicity

of the

phase,

which _was

ignored

in order _to obtain the Hamiltonian. A proper ^treat- ment of the

phase

could

very

^well

upset

^{this balance. Tl1ese issues should be addressed in the} future.

In the

long-range

_case, we have found _{an exact} solution

using

normal modes for all

integer

n > I.

Nevertheless,

_even in thin

simple

model of _a random

phase system,

^{there have arisen}

complications

in

taking

the _{n -} 0 limit _not encountered in its random

amplitude counterpart.

Recently

weak and

strong replica symmetry breakings (depending

on the correlations of the

disorder)

have been found in the interface

(directed polymer)

case

[12].

^The

complexities

en-

countered in the

n - 0 behavior in the random

phase

models studied here are indicative that

replica symmetry breaking

of some sort also may occur in these

systems.

Acknowledgements.

The research was

supported

in

part by

^{the U.S.}

Department

of

Energy

under

grant

^{No. DE-FG02-} 88ER40425 _at the

University

of Rochester.

620 JOURNAL DE PHYSIQUE I N°5

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