Randomly branched polymers and conformal invariance

(1)

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Randomly branched polymers and conformal invariance

Jeffrey Miller, Keith De’Bell

To cite this version:

Jeffrey Miller, Keith De’Bell. Randomly branched polymers and conformal invariance. Journal de

Physique I, EDP Sciences, 1993, 3 (8), pp.1717-1728. �10.1051/jp1:1993211�. �jpa-00246827�

(2)

Classification Physics Abstracts

5.40 5.70J 36.20C

Randomly branched polymers and conformal invariance

Jeffrey

D. Miller

(I)

and Keith De'Bell

(~)

(1)

Service de Physique Thdorique de Saclay, 91191 Gif-sur-Yvette Cedex, France (~) Trent University, Department of Physics, Peterborough, Ontario K9J 788 Canada

(Received

I December 1992, rev~sed 3 March 1993, accepted 27 April

1993)

Abstract We show that the field theory that describes randomly branched polymers does

not have the structure _one expects ^of _a two-dimensional conformal field theory. In particular,

we show that the lowest dimension operator in the theory cannot be primary. We show

moreover that the free field theory obtained by setting the potential equal to _zero in the branched polymer theory is not _even classically conformally invariant. Finally, _we present numerical data for the exponent

0(a),

defined by

TN(«)

+~

l~N~~l"J+~,

_where

_TN(«)

_is _number _of _distinct

configurations of _a branched polymer rooted _near the _apex of _a _cone with apex angle _a. The data indicate that

0(a)

is not linear in

lla,

providing further evidence that correlation functions which generate randomly branched polymers do not transform simply under conformal

transformations.

1 Introduction.

It is

widely

believed that statistical mechanical systems at the critical point ^of a second order

phase

transition _are

conformally

invariant _on scales much

larger

than _any

microscopic

distance

iii (~).

In

two-dimensions,

where the conformal

algebra

is infinite

dimensional,

conformal invariance

places

_strong constraints _on the fixed

point

correlation functions. The

analysis

of these constraints,

starting

with the _paper of

Belavin, Polyakov,

and Zamolodchikov [2], ^has lead to _a rather

complete understanding

of the

large

distance behavior of twc-dimensional

critical theories [3].

As is

well-known,

the

large

^distance behavior of certain random

geometrical

systems exhibit critical

fluctuations,

and _can be described

by

continuum field theories. The conformal _proper- ties in two dimensions of most of these

systems-linear polymers,

branched

polymers

with fixed

topology,

theta

polymers, percolation,

dense

polymers,

etc. _are

fairly

well understood [4].

(1)

More precisely, translational, rotational, and scale invariance, in systems with only short range

interactions, _are thought to imply conformal invariance.

(3)

1718 JOUIIIiAL DE PHYSIQUE I N°8

Randomly

branched

polymers

_[5] are a notable

exception (~).

It is not known which conformal field

theory

^describes them, or even whether

they

_are ^described

by

_a conformal field

theory

at all.

Numerical work indicates that

randomly

branched

polymers

behave

anomalously

under _con~

formal transformations. The rates of

decay

of correlation functions in _a

strip,

for systems ^that

are

conformally

invariant in the

bulk,

_are

simply

^related to the

scaling

dimensions of the operators ^that appear in the correlation functions [6].

Extremely

accurate transfer matrix cal- culations of lattice animals

(which

_are believed to be in the _same

universality

class _as branched

polymers)

in the

strip

_[7]

yield

correlation

lengths

which do not seem to be

proportional

to any of the known exponents in the branched

polymer problem. Also,

numerical enumerations of

randomly

branched

polymers

in the

wedge

_[8] indicate that the exponent

0w(a),

defined

by TJ~(a)

+~

A~N~°(")+I,

_where

_TJ~(a)

_is _number of distinct

configuratations

^of _a ^branched

polymer

rooted _near the _apex of _a _cone with _apex

angle

_a, is not linear in

lla

^unlike what

one finds in

conformally

invariant systems, like linear

polymers

[9].

Previous theoretical

investigations

of branched

polymers

have also encountered

problems

in

regard

to the

question

^of ^conformal invariance.

Duplantier, Kostov,

and Saleur have solved the branched

polymer problem

_on _a random lattice [10]. ^These ^authors

calculate,

_on _a random lattice, ^universal

properties

of

polymer partition

functions. Unlike the _case of other

(conformally invariant) geometrical

systems ^that can be solved _on _a random lattice

ill], however,

these

partition

functions cannot

always

be

interpreted

in terms of correlation functions of

scaling

operators.

Furthermore,

unlike the _case of linear

polymers

or

percolation,

it is not clear in this

case how to relate the

quantities

calculated _on the random lattice to the

corresponding

_quan~

tities in the

plane.

^This

correspondence

should exist in theories that _are

conformally

invariant [12]. ^Saleur [13] ^has ^also

pointed

out that the violation of naive

hyperscaling

in the branched

polymer

field

theory (discussed below), together

with finite size

scaling

in _a

strip

of width

L,

leads to an L

dependence

for the

singular

part ^of ^the ^branched

polymer

free _energy that is at

variance with what _one expects ⁱⁿ a conformal field

theory.

In this _paper _we address the

question

^of conformal invariance of branched

polymers by studying

the structure of the continuum field

theory

that describes their

scaling

behavior.

This field

theory

_was first studied within the _context of the

epsilon expansion

around the upper critical dimension of branched

polymers,

d~ ₌

8, by Lubensky

and Isaacson [5].

Later,

Parisi and Sourlas [14] ^showed ^that ^this

theory

is, _up to

perturbatively

irrelevant operators,

supersymmetric.

^A consequence of the supersymmetry ^is ^that ^certain correlation functions of the branched

polymer

field

theory

in d dimensions _are

equal

_to correlation functions in the field

theory

that describe the

Yang~Lee edge singularity

_[15] _near its upper critical dimension of six in d 2 dimensions

(~). Although

the

original

_paper of Parisi and Sourlas

only

demonstrated the supersymmetry _near the _upper critical dimension of the branched

polymer problem,

and then

only perturbatively,

there is

good

_reason to believe that the supersymmetry and its consequence dimensional reduction _to the

Yang-Lee edge problem persist

all the _way down to d

= 2.

First,

the

predictions

of dimensional reduction in two and three dimensions for the exponent

0,

which _governs the number of branched

polymer configurations,

the

prediction

in three dimensions for the exponent u which _governs the radius of

gyration

[14], ^and ^the

prediction

for the

scaling

form of the

pair

correlation function also in three dimensions

ii?]

_are

all well confirmed

by

numerical work.

Second,

there exists _a

supersymmetric

lattice

model,

(~) ^A randomly branched polymer is _a tree

(no loops)

_on _a regular lattice. Each _tree with

a given number of bonds is taken be equally probable _[5].

(~) ^We emphasize that _one _can _prove non-perturbatively that

a Parisi-Sourlas supersyrr~metry implies dimensional reduction [16] the only question is whether the theory in question is really _super~

symmetric.

(4)

due to

Shapir

[18], ^which ^holds ⁱⁿ ^all

dimensions,

and which has _as its continuum limit the Parisi-Sourlas field

theory, again

_up to operators which _are

perturbatively

irrelevant.

We therefore take this field

theory

_as _our

starting point.

Our

assumptions

_are,

explicitly,

that the supersymmetry ^and ^the operator content of the Parisi-Sourlas field

theory persist

down to dimension d

= 2. In

light

of the above

remarks,

this

assumption

_appears to be _a

good

one. The

question

that _we would then like to _answer is whether these _two

assumptions

are

compatible

with conformal invariance.

By

^conformal

invariance,

_we _mean here

having

the structure of _a conformal field

theory,

with the operators of the

theory arranged

in conformal families

(highest weight representations

of the Virasoro

algebra),

the lowest dimension operator in each

family being

_a

primary

operator [2].

In order that the branched

polymer

field

theory

has this structure, ^it is clear that the lowest dimension operator in the

theory

must be

primary.

For if this operator _were not

primary, (and

if the

theory

had the usual _structure of _a conformal field

theory),

it would have to be the descendent of another operator ^with ^lower

scaling dimension,

and

by assumption

there _is _no lower dimension operator.

The

precise question

that _we want to ask then is whether the lowest dimension operator in the branched

polymer theory,

the operator ^which appears in all of the branched

polymer

generating functions,

is

primary.

We find that it is not, ^and ^thus ^conclude ^that this

theory

does _not have the _structure that _one expects ^of a conformal field

theory.

To show

this,

_we _assume that the lowest dimension operator in the

theory

is

primary

and

show, by conformally mapping

the

plane

to the _cone, that _one then obtains inconsistent predictions for the exponent

0(a)

defined

by TN(a)

_+~

A~N~°(")+I,

where

TN(a)

is number of distinct

configurations

of _a branched

polymer

angle

£X.

To further

investigate

the

question

of conformal invariance in this

model,

_we also consider the free field

theory

obtained

by setting

the

potential equal

to _zero in the branched

polymer

field

theory.

We show that this

theory

is not _even

classically conformally

invariant. The _reason for this is that the free Parisi~sourlas

theory

is

essentially

_a

higher

derivative Gaussian

theory,

and it is known

that,

in

higher

derivative

theories,

scale invariance does _not

imply

^conformal

invariance [19]. ^{It is}

important

to

emphasize

that the free

theory

_no

longer

describes branched

polymers,

_even

non-self-avoiding

ones. That the free

theory

is _not _even

classically conformally

invariant

is,

in

light

of the

problems

that arise in the

interacting theory, however, suggestive.

Finally,

_we present ^numerical enumerations of

randomly

branched

polymers

in the _cone. A

cone is _a

wedge

with

opposite

sides

identified,

and is

conformally

related to the

punctured plane.

If the lowest dimension operator in the branched

polymer problem

_were

primary,

the exponent

0(a)

in the _cone would be

simply

related to the exponent ⁰ in the

plane

and _to the apex

angle

_o. More

precisely, 0(a)

would be be linear in I

la.

Our enumerations indicate that this linear relation does _not hold.

The

plan

of this paper is _as follows. In section 2 _we discuss the field

theory

that describes

randomly

branched

polymers,

in section 3 _we show that the lowest dimension operator in this

theory

is not

primary,

in section 4 _we show that the free Parisi-Sourlas

theory

is not

conformally

invariant, in section 5 _we present our results for the exponent

0(a)

obtained

by

exact ennumerations of branched

polymers

in the cone, and in section 6 _we _restate _our

conclusions.

(5)

1720 JOURNAL DE PHYSIQUE I N°8

2. Structure of Parisi~sourlas field

theory.

The effective Hamiltonian of the field

theory

that describes the universal

properties

of

randomly

branched

polymers

_near their _upper critical dimension _[5] _can be written [14]

HB.p

₌

/ _d~rlw(-V~#

+

V'(#))

₊

W~

+

~l(-V~

₊

V"(#))lbl. (i)

where

V(#)

=

)r#~

₊

i)#~. #

and _w _are

commuting scalars;

~b and 1i anti-commute. HB.P. is invariant under the supersymmetry transformations

b#

=

-aepxH~b,

bw

=

2aep6H~b,

b~b = 0, and

bli

=

a(epxHw 2ep6H#)

where _a is _an

anti-commuting

number and ep an

arbitrary

vector

[20].

^We ^take ^this ^field

theory

_as _our

starting point,

and in

particular

assume, for the _reasons

given

in the

introduction,

and _as is usual

(~),

^that ^the operator content of the

theory

in two

dimensions is that of the field

theory

^defined

by equation (I).

The supersymmetry

imposes

relations between the correlation functions of the

theory.

For

our purposes, the

following

identities _are useful

(<(r)w(o))

₌

(~b(r)I(o))

=

4$1<(r)<(o)). ⁽²⁾

FYom these

equations

it follows that the the

scaling

^dimensions _~~, _xw, _x~, and x~ ^{of the} ^fields

#, _w, _~b, and are related

by

x~=xw-2=x~-I=x~-1. ⁽³⁾

#

is therefore the lowest dimension field in HB.P:

In accordance with the discussion in the

introduction,

the field

#

must be

primary

if the field

theory

described

by equation (I)

is to have the usual _structure of _a conformal field

theory.

Given that

#

is

primary,

it follows from

equation (2)

that

w =

kV~#

₊ _Q

₍₄₎

where _&l is a sum of

primary

operators, ^and ^the constant k

= This is because in

x~

a conformal field

theory

^the two

point

function of operators

belonging

to different conformal families is

always

_zero; since

(#w)

is not zero, w must have _a _term _that

belongs

to the conformal

family

of

#.

Since the

scaling

dimension of _w is x~ ⁺ 2, this term must be _a descendent of

#

at level two, ^and ^the

only

descendent of

#

at level two which is also _a scalar is

V~#. Likewise,

&l

must be the _sum of

primary

operators, since if iD has _a _term which is not

primary,

that term must be the descendent of another operator ^with

scaling

dimension _xw _n where _n is _a

positive integer.

But

again

^there is _no such operator in the

theory (other

than

#;

_&l is _a

commuting

variable and therefore cannot be _a first

generation

^descendent of _~b

or1i).

It is

important

to note that conformal invariance

together

with

equation (2) imply

that _w

=

kV~#

₊ _iD holds _as

an operator

identity,

and not

just

_as _an

equality

in the

plane.

(~) For example, the lowest dimension fields, # and #~

(the

spin and energy

operators),

of the scalar #~ theory ^that describes the Ising model in sufficiently high dimensions correspond to primary

operators ⁱⁿ two dimensions; ^the ^lowest field dimension field, #, in the #~ theory that describes the

Yang-Lee theory [15] corresponds to a primary operator [21]; ^and ^the ^lowest ^dimension "operator" in the _n _- 0

O(n)

model, which generates linear polymers, corresponds to _a primary operator.

(6)

3. Lowest dimension field not

primary.

We _now show that

#

cannot be

primary.

The correlation function

(#(r)w(0))

is the

generating

function for the number

wN(r)

of

randomly

branched

polymer configurations

with N bonds

containing

the sites 0 and _r

[17,18]

(#(r)Ld(0))

+~

/~ dNe~"~KfWN(r). ₍₅₎

Here K is _a

fugacity

for the number of bonds in the

polymer,

K~ is the

(non-universal)

value of the

fugacity

at which the _average

polymer

size in the fixed

fugacity

ensemble

diverges,

and

e measures the deviations of K from K~: _e

=

~

Equation (5)

holds for _e

small,

and

+~

means that both sides of the

equation

have the

sane leading singular

behavior. In the

scaling

region

we expect

(#(r)#(0))

=

~/,

^where ^the correlation

length (

+~ e~" FYom

equation

r 4

(2), (#(r)w(0))

=

~j~~)j. Using

this

scaling

form and

integrating

both sides of

equation IS)

with respect to r

a~~d~hen taking

the inverse

Laplace

transform with respect to _e _we have

~nunrooted

~

~N

~f-@

~

)N ~fu(d-2x~-2)-3 (~)

N j~

c

where T@~~°°~~~ ^is the number of unrooted branched

polymers

^of size N in the

plane,

and

£~ wN(r)

=

N~T]°°~. Setting

d

= 2 and

using

the

approximate

value of _u

= 0.64 [7] and

the exact value of 0

= [14], we find x~ = -I

Iv

R -1.5. The field

#

therefore has

negative scaling

dimension.

Under the

assumption

^that

#

is

primary

_we can also calculate the exponent

0(a),

defined

by

TN

(a)

+~

l~N~°(")+I,

where

TN(a)

is the the total number of

randomly

branched

polymer configurations

angle

_a (~). ^To ^calculate

0(o)

_we need

j

to know

(#w)

in the _cone. The transformation _z _-

(

₌ z2« maps the

plane

onto the _cone.

Since, by assumption, #

is

primary,

_we have at

criticality

(<(Zl)<(22))Plane

"

i)(Zl)ii)(22)i(<(<1)<(<2))cone ⁽⁷⁾

so

that,

(§~((1)§~((2))cone "

())~'(l' ^~~'(2(

^~~~~ _m _m

~

(~~

'(l" (2"

~~

where _we have normalized

#

_so that

(#(r)#(0))plane

_"

fi.

FYom equation

(4)

it then follows

r

that

(~((l)~J((2))cone

"

(§~((1)(ki7~~

+

ld))cone

_"

k~)2 ^(~((l

)§~((2))cone

(9)

where _we used the

orthogonality

of

#

and _&l to obtain the second

equality. Combining equations (8)

and

(9)

then

gives

(§~((1)~J((2))cone (2 (

^~~~~^~~~~~~ ~~~~~~~~~ ~ ~~~~

j(~ _~f

(2x~

~~~~

(~) A _cone _can be thought of _as _a wedge in the plane, with opposite sides identified. By _apex angle

We mean the angle in the plane between the two sides of the wedge.

(7)

The correlation function

(#((i)w((2))cone

is the the

generating

function for the total number of branched

polymers

in the _cone. This statement is true

regardless

of whether _we put

#

in

the bulk and _w _near the apex or w near the apex and

#

in the bulk. On the other

hand,

_we

see from

equation (10)

that the exponent

0(a) depends

_on whether

#

_or _w is _near the _apex.

For if _we put w in the

bulk,

that

is,

if _we

integrate

_over

(2, holding (i

fixed and _near the apex of the _cone, _we find

0(a)

= 2

~'~

(II

a

while if

integrate

_over

(i holding (2

^fixed we find

something ^different, ^namely 0(a)

₌ 2

~'~

2u

(12)

for a < 27r. Since _we obtain different _answers for

0(o) depending

_on whether _we put

#

_or _w

near the apex,

equation (10)

cannot be correct: the correlation function

(#w)

in the _cone is not

equal

to the conformal transformation of

(#w)

in the

plane,

under the assumption that

#

transforms _as _a

primary

operator. There _are in fact _two additional

problems

with

equations (II)

and

(12):

₁₎ the the coefficient of

27rla

is

negative, implying

that the smaller the _apex

angle,

the greater ^the ^number ^of ^rooted

polymer configurations,

which is

impossible,

and

2)

the exponent

0(a)

in

equation (12) jumps discontinuously

from _one at _a ₌ 27r to about

1/3

for _a

just

smaller than 27r. Such _a

jump

_seems

unlikely,

and in any case is in the _wrong direction

(0(a)

should increase _as _a

decreases).

A final

point

must be

considered, namely

that the _cone _is _not

conformally

related to the

plane,

but to the

punctured plane.

In most

theories,

the distinction between the

plane

and the

punctured plane

is irrelevant. A puncture in the continuum

theory corresponds

to _a finite size hole in the lattice

theory.

The presence of the hole

changes

the interactions between sites

variables,

and _so

corresponds

to the _presence of _an _energy like operator. The _energy operator

on the lattice _can be written _as _a _sum of operators in the continuum

theory.

In

general,

one

expects _every operator ⁱⁿ the continuum

theory

to appear in this _sum unless it is forbidden to do _so

by

_some symmetry. The lowest dimension operator ⁱⁿ ^the sum is the _most

relevant,

and dominates the

large

distance

physics.

In most theories the lowest dimension operator ^with the

sane

_symmetry

as the _energy operator is the

identity

operator, _so _a ^defect in the lattice does _not

change

the

large

distance behavior of correlation functions. But in the

theory

_we

are

considering (and

in the

Yang-Lee theory)

the operator

#

^has ^the same symmetry as the energy operator and has _a smaller

scaling

dimension than the

identity

operator,

If, therefore,

#

_appears in the _sum

(and

there is _no symmetry which forbids it from

doing so)

the

large

distance behavior of correlation functions _on _a lattice with _a

single

defect would be different from the behavior of correlation functions _on the

perfect

lattice. In the continuum this _means that correlation functions in the

plane

would not be the _same _as those in the

punctured plane.

The exponent 0 in the

punctured plane,

_or _on _a lattice with _a finite size

hole,

would therefore be different from 0 in the

plane.

Since correlation functions in the _cone _are

conformally

related to those in the

punctured plane, they

would also be

modified,

_as would

predictions

for

0(a).

This

proposal,

while

raising interesting questions,

has _a number of

problems. First,

it is difficult to _see how the _presence of _a finite size hole _on the lattice could alter the exponent 0.

To check

this,

_we have enumerated all branched

polymer configurations (rooted

_near the

origin)

with twelve and fewer bonds _on _a _square lattice with the site at the

origin

removed. Branched

polymer configurations containing

the

origin

_were disallowed. The ratio of the number of

polymer configurations

in the _presence of the puncture to the number in the

plane

_appears to converge ^to a constant _near 0A _as the number of bonds in the

polymer

increases. This indicates that the defect does not

change

0.

Second,

_we expect that the _presence of the puncture should

(8)

not affect the

boundary

^conditions at

infinity.

Thus the state, at

infinity

should still be the

SL(2, C)

invariant _vacuum, I.e. the state

corresponding

to the

identity

operator. In the Hilbert space

formulation,

the state

corresponding

to the operator

#

is _an energy

eigenstate,

and is

orthogonal

to the

SL(2, C)

invariant _vacuum. Hence the

partition

function Z should not be affected

by

the puncture. Correlation

functions, however,

will _now have _an extra

#(0)

inserted in them. Since _we _are interested in two

point functions,

the correlation functions with the extra insertion of

#(0)

_are three point ^functions ^and are

computable. Performing

^these computations,

one can show

explicitly

that the correlation function

(#w)

in the

punctured plane

still

depends

on whether

#

_or _w is put at the

origin.

4. Free Parisi-Sourlas

theory

not

conformally

invariant.

In order to

gain insight

into what

might

go wrong with conformal invariance in the branched

polymer theory,

_we consider here the

simplest theory

with _a Parisi-Sourlas supersymmetry.

This is obtained

by setting

the

potential V(#)

in equation

(I) equal

to _zero. The

commuting

fields

#

and _w then

decouple

from the

anti~commuting

fields

~b and _@.

Moreover,

the fermion part ^of ^the ^free Hamiltonian HF has the form of _an

ordinary

Gaussian

model,

and is

conformally

invariant with _~b and 1i

transforming

_as

primary

fields with conformal dimension _zero. The

remaining

part ^{of the} ^free

theory

is

HF

₌

/ d~x[w(-V~#)

+ ~w~].

(13)

2

or,

integrating

out the field _w,

HF _"

d~r(V~#)~ (14)

a

higher

derivative Gaussian

theory.

For

HF

to be scale

invariant, #

must have

scaling

dimen~

sion -I, or conformal dimension

-1/2.

Let _us _assume, _as _we did for the branched

polymers,

that

#

transforms under conformal transformations _as _a

primary

operator.

Then,

under the

transformation _z _- _z ₊

e(z),

4(~, 2)

_~

(i ))~~/~(i ))~~/~#(~'~l' ^(is)

Taking

into account the transformation of the _measure d~r and the derivatives

6~,

_we ^find ^the

variation in HF

~

~~~

/ ^~~i~

~~~~~~~~~ ~ ~'~'~' ~~~~

If

e(z)

is constant,

corresponding

to _a

translation,

_or linear _{in z,}

corresponding

to _a dilatation

or

rotation, bHF

vanishes

identically,

while if

e(z)

is

quadratic

ⁱⁿ _z, ^the

integrand

is _a total

derivative,

and _so bHF vanishes in this _case _as well. These three transformations generate ^the

special

^conformal _group,

SL(2, C).

^On ^the other

hand,

the

integrand

is not _a total derivative for _more

general

conformal transformations.

Thus, assuming

that

#

is

primary,

the action in

equation (14)

is invariant under the _group

SL(2,C),

transformations

mapping

^the

plane

to

itself,

but not invariant under the full conformal _group in _two dimensions.

Furthermore,

there is _no way ^to transform

#

_so that

bHF

" 0. To _see

this,

_suppose that under _z _- _z +

e(z)

<(z z)

_-

(i t)-~/~(i t)~/~<(z z)

+

b<(z z). (17)

JOURNAL DE PHYSIQUE J T 3,N. 8, AUGUST 1993 62

(9)

with

b#

linear in

#

and _e. Then

bHF

=

/ _d~rl( _~j)i~ _(3#)(63#)

+

2(33#)(63b#))

₊ _C-Cl

(18)

In order for

bHF

to

vanish,

_we need to choose

b#

in such _a _way that the

integrand

is _a total derivative.

Considering

the first _term in

equation (18)

_we _see that this is

only possible

if

b#

is of the form

ae(z)6#

+

b6e(z)#.

But _we cannot add _a term of this kind _to the transformation law for

#,

because this has

already

been fixed

by

translational and scale invariance

(that

is to say, if _we add term of this form _to transformation law for

#,

HF will not be invariant under the

special

conformal

group).

Thus there is _no way ^to transform

#

_so that

bHF

" 0 under

general

conformal transformations. Hence the

simplest theory

^with a Parisi-Sourlas supersymmetry is not

conformally

invariant _even at the classical level. It _seems

likely

that this is related _to the

problems

with conformal invariance encountered in the _more

complicated interacting theory

that describes branched

polymers.

5 Numerical results.

In

light

of the discussion in section

three,

it is of interest _to know the

dependence

of the exponent

0(a)

_on the the

reciprocal

of the _cone

angle

_a for branched

polymers

confined to _a

cone. For linear

polymers,

_one finds [9], as a

simple

consequence of the fact that the

generating

function for _a linear

polymer

is the correlation function of two

primary

operators, an exponent that is linear in I

lo.

In the _case at

hand,

_we do not know what to expect,

because,

_as _we

have shown in section 3, the

generating

function for

randomly

branched

polymers

is not the correlation function of

primaries,

_or ^of _a

primary

and _a descendent. A

previous analysis

of

exact enumeration data _[8] for branched

polymers

confined _to _a

wedge

found _a non-linear

relation between the exponent and the inverse _cone

angle.

It

is, however,

worth

studying

branched

polymers

in the _cone, because the _cone, unlike the

wedge,

is

conformally

related to the

(punctured) plane.

We have

analysed

exact enumeration data for lattice _trees confined to _a _cone. The _cone is formed

by applying cyclic boundary

conditions to _a

wedge,

cut from

a square

lattice,

in such _a way that

corresponding

lattice sites _on the two boundaries of the

wedge

become identified _as

a

single

site _on the surface of the _cone. The number of

(weak) embeddings

of _trees rooted _at the _apex of the _cone _were enumerated for trees with up ^to 16 vertices.

We _assume that the number of trees _on the surface of the _cone

diverges

_as

TN(o)

+~

l'~N~°(")+~ (19)

1 is the

growth

constant for lattice _trees _on the square lattice and is the _same for trees confined to a

wedge

or cone as it is for _trees in the bulk lattice [22]. ^The

yth

moment of the

generating

function for

TN(o)

will therefore have critical behaviour described

by

G~Y)(x)

=

£TNX'~NY

+~ (x~

x)~+Y~° (20)

N

From

1/~

_" °.~~~~~

(21)

-0.00002

(10)

To obtain estimates of

0(a)

_a Baker-Hunter confluent

singularity analysis

_[24] _was

applied

to

the _exact enumeration data at each of the _cone

angles

considered. This type of

analysis

_was

used since _we expect ^the presence of confluences in the

generating

function for lattice trees [25].

The initial

analysis

used the central estimate of _x~

given

above

and,

since

0(o)

is

bigger

than

unity

for the smaller

angles,

the

first,

second and third moments of the

generating

^function

were

analysed

_[8]. The results _are shown in table I. As this type ^of

analysis

_may be sensative to the value of _x~

assumed,

_we also

performed

_an

analysis

which eliminates the need _to

specify

_x~.

To do this the ratio of the number of

embeddings

in the bulk lattice TN and the

corresponding

number of

embeddings

in _a _cone with

wedge angle

_a,

TN (a),

_was formed

TN "

TN/TN(a)

+~

N°(")~° (22)

Table I. Estimates of

0(o)

for various _cone

angles

_o. Columns labelled 1, 2 and 3 tabulate results obtained from the

first,

second and third _moment of the

generating

function

respectively.

The column labelled 4 tabulates the results obtained

by taking

the ratio of the number of _trees in the bulk lattice to the number _on the _cone.

Angle

1 2 3 4

90° _2.091 _2.134 2.126 2.l14

+0.017 +0.025 +0.017 +0.014

12i° 1.98 1.96 1.85 1.852

+0.ll +0.09 +0.03 +0.007

143° 1.85 1.83 1.78 1.73

+o.06 +o.os +o.io +o.09

180° _1.55 1.551 1.550 1.553

+o.02 +o.oos

±((((

_+o.oio

233° 1.418 1.4080 1.39 1.39

+0.010 +0.0007 +0.09 +0.04

270° 1.2539 1.250 1.252 1.236

+0.0012 +0.006 +0.004 +0.014

The

generating

^function ^for ^these ^ratios ^has a critical behavior

given by

~

(~) £

~

~N

~ (~ ~)l+@(a)-@

(~~)

r N

N

This has the

advantage

^that the critical

point (x~

₌

I)

is known

exactly

and it is not necessary to use

higher

moments as the exponent is

always

greater than I. The

resulting

estimates of

0(a)

_are shown in column 4 of table 1.

(11)

In most cases the

spread

in the central estimates of

0(a)

in the columns of table

I,

for _a

given

value of _a, is

comparable

with the _error estimate in _a

single

column obtained

by considering

the variation in the estimate from different Pad4

approximants

to the Baker-Hunter

auxillary

function. In order to make _a

comparison

with the

expected

linear relation _we take the overall estimate of

0(a)

to have _a central value

given by

_an _average, _over the columns of table

I,

of the central estimates and _error bounds such that all of the central estimates fall within these

bounds. These values _are

plotted against

I

la

in

figure

I. Even

allowing

for _some

degree

of

subjectivity

in the value of the _error bounds the estimates of

0(a)

_are

clearly

not consistent with _a linear

relationship (although

_we cannot

altogether

rule out short series effects _as the

reason for this deviation from _a linear

relation).

2.2

~

4l

1.8

§

Sin)

^~~ o

lA @

1.2 ~

l 8

0.00i 0.004 0 006 0.008 0.01 0 012

1In

Fig. I. Overall estimates of @(a) plotted against I

la.

The bulk value of

= I has been included

as the value for _a

= 360°.

6. Conclusions.

In this _paper _we have considered whether

randomly

branched

polymers

_are

conformally

invariant, We have shown that the usual structure of conformal field

theories,

in which the lowest dimension operator in the

theory

is

primary,

does not hold. This rules _out the most

straightforward

formulation of conformal invariance for branched

polymers.

We also showed that the free Parisi-Sourlas Hamiltonian is not _even

classically conformally invariant,

because it is

essentially

_a

higher

derivative

theory.

This is

interesting,

because this

higher

derivative

theory

is the

simplest

local scale invariant

theory

that _one _can write down which is not also

conformally

invariant. It _seems

likely

that the lack of conformal invariance in the free

theory

is related to the

problems

that arise in the

interacting

branched

polymer theory. Finally,

from numerical ennumerations, _we showed that

0(a)

is not linear in the inverse _cone

angle

_a, as is the _case when the

generating

function is _a correlation function of primaries, or of _a

primary

and _a descendent.

We would like to conclude with several remarks.

First,

_our results rest _on the

assumption

that the supersymmetry and operator content of the Parisi-Sourlas field

theory,

valid _near

eight dimensions, persist

^down to two dimensions. As noted in the

introduction,

the

predictions

of the

supersymmetric

formulation _agree

extremely

well with numerical work in two and three di-

mensions,

_so ^that ^these

assumptions

seem to be

good

_ones.

They

_are nevertheless

assumptions.

Second,

it would be nice _to have _a better

understanding

^{of the} ^role ^of

boundary

conditions in

theories,

like this _one, that have

negative

^dimension operators. ^As we discussed in section

4,

the

(12)

existence of

negative

^dimension operators ^makes ^the ^relation

between,

for

example,

the

plane

and the

punctured plane

rather

ambiguous. Finally,

the _reason for

problems

with conformal invariance may be related to

non-unitarity

of the model. Since the

theory

is

non~unitary,

the Hamiltonian in the Hilbert _space formulation of the

problem

is not

Hermitian,

and therefore need _not be

diagonalizable.

^If H is not

diagonalizable,

the usual arrangement ^of operators in

conformal field theories in

highest weight representaions

of the Virasoro

algebra

_[2] would be

impossible.

Acknowledgements.

J.M. thanks John

Cardy

^for

suggesting

^this

problem

to

him,

and for _many

helpful

_conversa~

tions. He also thanks Mark Goulian for

discussions,

and Bertrand

Duplantier,

Ivan Kostov and Hubert Saleur for

explaining

their

unpublished

work _to him. This work _was

supported,

in part,

by

NSF Grant PHY 86-14185 and the Natural Sciences and

Engineering

Research Council of

Canada.

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