Disclination Asymmetry in Deformable Hexatic Membranes and the Kosterlitz-Thouless Transitions

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Disclination Asymmetry in Deformable Hexatic Membranes and the Kosterlitz-Thouless Transitions

Jeong-Man Park, T. Lubensky

To cite this version:

Jeong-Man Park, T. Lubensky. Disclination Asymmetry in Deformable Hexatic Membranes and the Kosterlitz-Thouless Transitions. Journal de Physique I, EDP Sciences, 1996, 6 (4), pp.493-502.

�10.1051/jp1:1996226�. �jpa-00247199�

J.

Phys.

I France 6

(1996)

493-502 _APRIL 1996, PAGE 493

Disclination Asymmetry in Deformable Hexatic Membranes and the Kosterlitz-Thouless Transitions

Jeong-Man ^Park (*)

and T.C.

Lubensky

Laboratory

for Research _on the Structure of

Matter, University

of

Pennsylvania, Philadelphia,

PA 19104, USA

(Received

6 November 1995, received in final form 14 December 1995,

accepted

21 December

1995)

PACS.05.70.Jk Critical point

phenomena

PACS.68.10.-m Fluid surfaces and fluid-fluid interfaces

PACS.87.22.Bt Membrane and subcellular

physics

and structure

Abstract. A dischnation _{m a} hexatic membrane favors the

development

of Gaussian _curva- ture localized _near ils _core. The

resulting global

structure of the membrane has _mean curvature, which is disfavored

by

curvature energy. Thus _a membrane with _an isolated dischnation under- goes a buckling transition from _a flan to _a buckled store _as the ratio

~/KA

of the

bending rigidity

~ to the hexatic

rigidity

KA is decreased. In this _{paper we} calculate the

buckling

transition and the energy of both _a

positive

and _a

negative

disclination. A

negative

dischnation has _a

larger

energy and _a smaller critical value of

~/KA

_at

buckling

than does _a

positive

disclination. We

use our results to obtain _a crude estimate of the Kosterlitz-Thouless transition temperature m a membrane. This estimate is

higher

than the transition temperature

recently

obtained

by

the

authors in _a renormahzation calculation.

1. Introduction

The hexatic

phase

_{[1] is} characterized

by

6-fold orientational but _not translational order. Both three-dimensional hexatic

phases

with true

long-range

order _[2] and two~dimensional

phases

with

power-law

order _[3] have been observed. In flat two-dimensional films

(under tension),

the transition to the

isotropic phase

_occurs via _a Kosterlitz-Thouless

(KT)

disclination

unbinding

transition

[4].

Free membranes with _zero surface tension _can also exhibit _a hexatic

phase

and

a KT transition to a fluid

phase.

Both the hexatic

phase

and the transition to the

isotropic phase [5,6]

_are,

however,

_more

complicated

than

they

_{are in a} flat film because of

thermally

induced

shape

fluctuations. Recent renormalization group calculations

[7,8]

show that

shape

fluctuations shift trie bare hexatic

rigidity KA.

As _a consequence, an increase in trie

amplitude

of

shape

fluctuations

produced by decreasing

trie membrane

bending rigidity

_~ will induce _a

transition from trie hexatic

phase

to trie

isotropic

^fluid

phase.

In flat

membranes,

there _{is a} symmetry ^between

positive (5-fold)

and

negative (7-fold)

discli-

nations,

and

they

both have the _same energy. In free deformable

membranes,

this

symmetry

is broken

[9].

^{If the ratio}

~/KA

of the

bending rigidity

_~ to the hexatic

rigidity KA

_is

sufliciently (*)

Author for

correspondence le-mail: jeong©lubensky.physics.upenn.eau)

©

Les

Éditions

_de

Physique

1996

small,

_a membrane with _a

single

dislcination _can lower its energy

by buckling, thereby

_cre-

ating

_a

nonvanishing

Gaussian

curvature,

which _screens trie disclination

charge.

The buckled states of

positive

and

negative

disclinations bave diiferent

height profiles,

^diiferent

energies,

and diiferent critical values of

~/KA

at which

buckling

_occurs. In this paper, we will _{use a} variational

procedure

to calculate the

energies

of isolated

positive

and

negative

disclinations _on free membranes. Dur variational form for _a

positive

disclination is

essentially

exact. Dur form for

negative

disdinations is exact

only

for

~/KA

_near the critical value of

~/KA

for

buckling.

Corrections _near

buckling

_are,

however,

_very

small,

and _{we argue} that _our form is _a very

good approximation

until

~/KA

becomes _very small. Dur results _{are in}

agreement

^with recent cal- culations

by

Deem and Nelson

[loi.

The latter

authors, however,

in addition to

calculating

the energy of _a

negative

disclination

variationally

also carry ^out a numerical minimization of trie full non-linear energy ^to obtain _a lower value for ibis _energy at small values of

~/KA.

In what

follows,

_we will present our calculations of

positive

and

negative disclinations, respectively

_in

Sections 2 and 3. In Section

4,

_we will discuss _our results and _an estimate of the KT transition line

produced by

them.

The continuum Hamiltonian for hexatic membranes _was derived in reference

iii.

If _we

parametrize

membranes

positions

ⁱⁿ terms of _a two-dimensional

parameter

li

=

(~~,~~)

_as

R(ù),

then

7i "

7i~

+

7ic, Ill

~~~~~

7~~

_{= ~}

d~itfiH~,

2

~

is the curvature energy and

7ic

_"

KA / _{d~itfi(tif Si} j(tif _{Si, (3)}

2

is trie Coulomb _energy. Alternative form for trie hexatic _{energy can} be found in reference

il lj.

In the

above,

_g

=

det(gab)

is trie determinant of the metric tensor gab "

ôaR: ôôR

tif ,

=

27r£~q~ô(ù ù~)/@

is the disclination

density

with _q~

=

+1/6,

and H and S _are,

respectively,

the _mean and Gaussian curvatures of the membrane. We bave

ignored

_a scalar field contribution of

7i,

which _{gives rise} to Liouville _measure factors

[12,13],

^which are irrelevant to trie current discussion. On _a

rigid

flat

membrane,

this Hamiltonian reduces to the Coulomb gas form of the XY-model. In the absence of

dischnations,

hexatic order induces

long-range

Coulombic interaction between Gaussian curvatures _on the membrane [Si. The Coulomb _en- ergy

7ic depends only

on the diiference

tif

_S.

Thus,

the

development

of Gaussian curvature on a free membrane that approximates the disclination

dentsiy

tif _can reduce trie Coulomb

energy. Gaussian _curvature

usually

leads to mean curvature, and the lowest energy ^state of _a free membrane with _a disclination will be determined

by

the

competition

between the Coulomb

energy,

7ic,

which

prefers

S

=

tif,

and the _{curvarure energy,}

7i~,

which

prefers

_{zero curvature.}

If there is _a

single

disclination at the _{origin, one can}

expect

that Gaussian curvature will be localized _near the

origin.

Gaussian curvature localized in _a small

region

will give rise to a

buckled state with _{mean curvature} but _zero Gaussian curvature away from the

origin.

2. Positive Disclinations

A _minimum

strength positive

disclination bas

"charge"

_q

=

1/6.

To reduce trie Coulomb energy associated with this

charge,

trie membrane _can distort into trie

shape

of _a

spherical section,

~vith

nonvanishing

Gaussian curvature, ^localized to trie _core of trie disclination. Outside trie

N°4 DISCLINATION ASYMMETRY IN HEXATIC MEMBRANES 495

Fig.

l. Membrane buckled into _{a cone} with

h(il)

= mr in the

Monge

_gauge where _{m is} the

slope.

core region, trie membrane will seek _a

shape

with _zero Gaussian curvature. A _cone with

slope

m has _zero Gaussian curvature and _can be connected

smoothly

to a

spherical

section

(Fig. i).

Thus,

in trie

Monge

_{gauge, we}

parametrize

trie membrane

shape

outside trie _{core as} ù

=

(r,

_ç§)

and

R(ù)

=

(rcosç§,rsinç§,h(ù))

with

h(ù)

= mr.

Thus,

outside the _core, the

components

^of metric tensor and its determinant _are

ÎÎÎ

"

OÎÎ~~~Î

g

=Î~(1 ^~Îi~).

^~~~

The _mean curvature H and the Gaussian _curvature S _are

~ i

)

~2

1'

^{~ ~} ~~~

Thus the

bending

_energy of the _cone with radius R and the _{core size a} becomes

7i«

₌

~~/d~ufiH~

2

1/~27rrdr ~(

^m

^~

2~a

_r~ ^~~~

_fi

~~~

~~

Î

~~~

The Gaussian _curvature vanishes outside the _core

region.

In trie limit of trie infinitesimal size of trie _core region, S _can be described

by

trie

point

curvature

charge

_s+.

S(ù)

=

2irS+à(ù Û+)/v@. ii)

Since there is _no Gaussian _curvature in the _{cone, we can} choose _{any curve} in the _{cone to} calculate _s+

using

^the Gauss-Bonnet theorem:

/Sda

₊

/ _kgdi

= 27r,

(8)

M c

where

kg

is trie

geodesic

curvature of trie

boundary

_curve G of trie surface M

[14].

^We use trie

boundary

_curve of the _cone:

C =

(Rcos#,

Rsin _ç§,

mR). (9)

The

geodesic

curvature of trie

boundary

_curve of _{a cone} of

slope

_m is

(Rfi)~~,

_{and the}

Gauss-Bonnet theorem becomes

M

~~~ ~

Î~~ ~

_~~

~~~~

Using equation iii,

_we obtain

1

~~

fi'

^~~~~

Hence,

the Coulomb _energy for _a

positive

disclination is

7ic

_"

KA / _{d~ufi(tif Si} j _{(tif Si}

2

=

7rKA (~

+)~

27rGc(0), (12)

6

where s+ = 1

-1/Ùfi

and

Gc (iii

is trie Green's function for trie

Laplacian,

V~

=

ôag°flfiôp ₍₁₃₎

@

On _{a cone,}

V~GC là ù')

=

-à(ù ù') /fi. (14)

To determine

Gc,

_{we assume} that it bas trie form -A

In(r/ro)

where _{ro is a}

length.

Then

Id itfiv~Gc(ù)

=

d~itfiô(ù)/fi=-1

~

Î

=

dsag~~jjôbGc

=

dsrg"A/r

Î Î

=

/dsr(g~~/fi)(A/r), ^ils)

where

dsa

=

ôarrd9

is the "surface" element of _a circle

enclosing

the

orgin.

^{Then from} equa- tion

(4), gw/fi

"

r/fi,

A

=

-Ùfil(27r),

and

Gc (iii

=

§/ ^In

~

ln

~)

,

(16)

7r a a

where _we chose _ro to be

equal

to the disclination _core radius _a and _we added the constant term

In(Rla),

where R is trie radius of trie _cone, to

produce

trie

required divergence

of

Gc (fil

at small _r. Trie Coulomb

self-energy

_is

given

in terms of

27rGc(0)

₌

tfiIn(Rla)

where R is trie radius of trie membrane and _{a is} the _cote size.

The _energy of _a

positive

disclination with _q

=

1/6

_on the _cone with trie

slope

_m becomes

~~~~~ l~~À~

~

~~~ ~Î

^~

ù~~

^~

~i

~~

Î

~

~~~ Î6

~

IA ÎÎ~ ^~°~

^~

^ÎÎB IA ^~Î

^~~

Î

~~~~

This energy is shown _in

Figure

2a for _various values of

~/KA.

For

~/KA

_> 11

/72, E+ (mi

has

N°4 DISCLINATION ASYMMETRY IN HEXATIC MEMBRANES 497

E~(m~ _{E_( m~}

l 2

2

~ 3

4 4

(a) (b)

Fig.

2.

a) Energy

of

positive

dischnations _{as a} function of m~ _{for different values} of _p

=

~/KAI (1) p/pf

=

4/3, (2) p/pf

= 1,

(3) p/pf

=

3/4,

and

(4) p/pf

=

1/2,

where

pf

= 11

/72. b) Energy

of

negative

dischnations _{as a} function of m~ _for different values of p =

~/KAI (1) p/p/

=

4/3, (2)

p/p/

=1,

(3) p/p/

=

3/4,

and

(4) p/p/ =1/2,

where pi

=13/216.

a minimum at

m~

= 0 and the membrane remains flot. For

~/KA

_< 11

/72, however, E+(m)

has _a minimum at

m~

=

(11 /36 2~/KA)/(25 /36

+

~/KA)

and the membrane buckles out to form _{a cone} with the

slope

11 2~

~ ~

ÎÎ Î

~~~~

~

Thus the

buckling

transition _occurs at

~/KA

" 11

/72

for

positive

disclination with _{q =}

1/G.

This result for _m with

equation (17)

_can be found in reference

[15]

^and [1G]. The _energy of _a

positive

disclination _{on a} membrane of radius R with the short-distance cutoif

a is

5

~ 36~ R _~ 11 E

3~~~ _KA

^{~ ~}

_25KA

^~~

_{a' KA}

^~

72

jig)

~ 1 R

~ 11

ù~~~~~ _a'

KA

^~

72'

When

KA

_{- cc, m -}

+titi

_and

s+ -

1/6. Thus,

in this

limit,

the disclination

charge

is

totally

screened

by

the Gaussian curvature, there is _no Coulomb _energy, and the disclination

energy

E+(KA

"

ccl

=

Ill /30)7r~ In(Rla)

_comes

entirely

from curvature of the _cone. This energy of _a

positive

disclination is identical to the result obtained

by

Guitter and Kardar _[6]

using

^{the conformal} _gouge.

Dur

simple height profile

for _a

positive

disclination does not break azimuthal

symmetry,

and there is _no

particular

_reason for this

symmetry

to be broken.

Thus,

_we believe that

h(ù)

= mr

provides

_a

complete discription

of the buckled state and _ouf

description

of the

positive

disclinations is exact. In

particular,

_no

symmetry breaking

terms such _{as mir}

cos2ç§

or

m2rcos4ç§

_are needed in the expansion of

h(ii).

Order

pararneters

^such as mi and _{m2 are}

certainly

_not forced

by

the

development

^of _{non-zero m} because

symmetry

^does not permit

terms linear in mp of the form

m~mp

to appear in the

expansion

^of

E+ (mi.

A KT

melting temperatue T+

for

positive

disclinations _can be introduced in the usual way

by setting

the free _energy of _a

single

disclination

equal

to zero;

E+ T+S

=

0,

^{where S} ₌

In(Rla)~

is the

entropy.

Thus

T+

=

E+/21n(Rla).

This

produces

the

phase diagram

obtained in reference _[6] and shown _as the solid _curve

(ai

in

Figure

3. The thin line indicates the

buckling

transitions at

~/KA

= 11

/72,

and the sohd _curve the disclinations

unbinding

transtion obtained

from

T+.

RK

1.4

(b)

1.2

1 K/K~= ^13/216

0.8

0 6

(a)

K/K~= 11/72

0.2

T/K~

O.Ol 0.02 0.03 0.04 0.05

Fig.

3. Estimated

phase diagrams

in the

(T/~, TIRA) plane showing

the Kosterlitz-Thouless tran- sition hne obtained

by balancing

_energy and entropy ^of

la)

a

single positive

dischnation

(T+

=

E+/21n(Rla))

and

(b)

_a

single negative

dischnation

(T-

=

E-/21n(Rla)) (a)

_is identical to the estimate obtained in reference [6]. ^The

straight

hne

through

the ongm m both _{cases is} the buck-

ling

transition hne. The energy of _a negative disclination is

generally higher

^than that of _a positive

disclination,

and T- > T+.

T+(KA

"

oc)

₌

(11 /60)7r~

_ci 0.575959~ and T-

(KA

"

oc)

_ci 1.37941~.

Fig.

4. Membrane buckled into _a saddle with

h(ù)

= mr cos 2# in the

Monge

_gauge where _m is the

slope.

3.

Negative

Disclinations

We _can

similarly

calculate the energy of

negative

disclinations with q =

-1/6.

Since the

corresponding

core

region

should have _a

negative

curvature

charge

to cancel the

topological charge,

_we

expect

the _cote

region

has _a saddle

shape.

The

simplest

saddle

shape (Fig. 4)

is

h

iii)

= mr cos 2ç§

(20)

We will take this _{as a} vanational function and seek the minimum energy solution for _a

negative

disclination with

respect

to variations _in the parameter m. We thus obtain _an upper bound to the energy of _a

negative

disclination. Inclusion of additional terms in

h(il) proportional

to

cos

2nç§

^for n an

integer

will lead _to lower _energies.

Indeed,

recent numencal calculations

by

Deem and Nelson

[10] yield

_a lower energy than _we obtain when

KA/~

_» 1. We will argue,

N°4 DISCLINATION ASYMMETRY IN HEXATIC MEMBRANES 499

however,

that this

simple

variational form is

essentially

exact _near the

buckling

transition.

The components of the metric _tensor and its determinant associated with

h(ii)

= mr

cos2ç§

are

grr = 1 +

m~ cos~

_{2ç§, goô}

"

r~ il

+

rm~ sin~ _2ç§),

~ gr~ = g~r =

-2rm~

_{cos 2ç§ sin 2ç§, g}

=

r~ il

+

m~(1

+ 3

sin~ _2ç§)).

The _mean curvature H and the Gaussian curvature S _are

~

~Î

~°~

~~

il

₊

ÎÎÎ~ ÎÎn~Îç§))3/2'

~ ~ ~~~~

The

bending

_energy of the saddle with

slope

_{m is}

7i~

"

~tl/d~UjjH~

₂

~

Î~ Î ^~~

~~~ÎÎÎÎ~ÎÎ~Î ÎÎ ^Î)~~~~

~~°~ ~~

~~ (Î~ÎÎÎÎÎ Î ~IÎ~ÎÎ)ÎÎ~21

~~

Î

~~~~

Again,

_in the limit of the infinitesimal _size of the _core

region,

S _can be descnbed

by

the

point

curvature

charge

_s-.

S(ù)

=

27rs-à(ù

ù-

/fi. (24)

The

integrated geodesic

curvature

along

the

boundary

G

=

(R

_{cos ç§,} R sin _ç§, mRcos 2ç§) ^is

/c

~~~~

Î~~ _Il

+ m2 cos2

Î)~/ÎÎ~Î~Îm2

sin~ _2ç§)3/2

^~~~~

Thus the Gauss-Bonnet theorem

gives

~~

7r ~~

Il

+ m2 cos2

ÎÎÎ/ÎÎ~ÎÎm2 _sin~

2ç§)3/~

^{' ~~~~}

and the Coulomb

self-energy

of

negative

disclination _on the saddle becomes

7ic

"

)KA / _{d~i~ôlfif SI (2 lfif SI}

=

7rKA

+ _s-

~

27rG~

(0), (27)

where _s- is given

by equation (26)

and G~

(fil

is the Green's function for the

Laplacian ^V~

_on

the saddle. We _can determine G~

(fil following exactly

the same

procedure

_we used to determine

Gc(ù)

for _a

positive

disclination. We find

Gsilii

=

-Ainirlai lniRlail 1281

where

~

~Î ^~~

il

+

Î~ÎÎÎÎ~Î~~Î)]1/2

~

7r ~

~~~

^~

~~~~

^{~ ~~~~}

Equations (23), (27), (28),

and

(29) completely

determine the _energy of _a

negative

discliantion

as a function of _m within the

approximation h(11)

₌

mrcos2ç§.

We _con locate the

buckling instability

and determine _m

just

above it

by expanding

E-

(mi

in powers of _{m up} to order

m~.

The result is

~~~~°~ ~~~ Î6

~

jÎA

ÎÎ6~ ^~°~

^~

~ÎÎÎÎ ÎÎjÎA~ ^~Î

^~~

Î'

~~~~

This energy is shown in

Figure

2b for various values of

~/KA.

For

~/KA

>

13/216,

E-

(mi

has _a minimum at

m~

=

0;

for

~/KA

<

13/216,

E-

(mi

has _{a minimum} at

m~

=

((13/216)

~/KA)/((2743/10368) (23/4) ~/KA).

The

buckling

transition _occurs at

~ 13

1 _216' ⁽³¹⁾

and the

slope

for

~/KA

< 13

/216

is

~° ~

27431111111~ (iÎÎ~

~/KA

^~~~~

The energy of

negative

disclination around

~/KA

" 13

/216

in _a membrane of radius R with

the short-distance cutoif _a is

16~~ ~ (211111111)

ÎÎÎÎÎÎÎÎÎ~A))

^~~

Î'

iA

^~

^~~~~~~

E-

=

(33)

iKA

_In

~, Ù

>

13/216.

36 _a

KA

As in the _case of

positive disclinations,

the Gaussian curvature will

adjust

to

exactly

_can-

cel the

topological charge

when

KA

_{" cc,}

leaving only

curvature energy.

Setting s-(m)

in

equation (26) equal

to

-1/6,

_we obtain

m~(KA

"

ccl

= 0.350417

(34)

and

E-(KA

"

ccl

= 2.75883~ln ~

(35)

E-

(mi

_can be minimized

numencally

^for 0 _<

~/KA

_< 13

/216.

The results _are

displayed

_{as a}

transition temperature in

Figure

3

(see below).

As discussed in the

introduction,

the

height profile

of _a

negative

disclination breaks _az- imuthal

symmetry,

and _we

expect h(ii)

to have _a Fourier series expansion of the form

h(ii)

=

r

£~

_m~ cos2n9. Dur

approximation keeps only

the first _{term in} this _serres. Near the _tran-

sition, higher

order _{terms con} be calculated

by expanding

E- in _a powers series _in all of the

m~'s.

We hâve

already

calculated the contribution from the dominant term mi % m. One

might expect

that the next most

important

term would be _m2. This parameter is not,

however,

forced to

develop

_{a nonzero} value when _m is _nonzero because E- _is invariant under h _-

-h,

1-e-, under _{m~ - -mn} for _{every n.}

Thus,

there _{is a} contribution _to E- of the form

a2m]

but

no term

proportional

_to

m~m2,

which would force _{a nonzero m2.} Thus _m2 will _{remain zero} until the coefficient _a2

changes sign.

The absence of _an

m~m2

term _means that _our expressions for E-

[Eq. (33)]

and _m

[Eq. (32)]

_are exact to order

[(~/KA) (13/216)]~

because there

N°4 DISCLINATION ASYMMETRY IN HEXATIC MEMBRANES soi

RK

0.8

' ' ,'

,' ,' ,' ,'

TM~

Ô.Ol 0.02 0.03 0.04 0.05

Fig.

5. Phase

diagram

_m

(T/~, TIRA) plane.

The fuit fine is the KT transition fine obtained from the RG calculation of references [7] ^and [8]. The dashed fine _is the estimate of the KT transition fine

from the _average

(T+

+

T-)/2

of the

positive

and

negative melting

temperatures

T+

and T-. The

approximations

used in references [7] ^and [8]

apply

below the dotted fine. The RG transition fine

(fuit fine)

obtained in references [7] ^and [8] ^lies below the

simple

estimate

(dashed fine)

_m the

region (below

the dotted

fine)

where the RG calculation

applies.

This is

expected

since the RG calculation includes contributions to the froc energy

arising

from thermal membrane fluctuations which the

simple

estimate

(T+

+

T-) /2

does not include.

is no correction to the

m~

_term

arising

from

couplings

to m2. The third order term m3 is

proportional

to

m~

because inversion symmetry

permit

a term

proportional

to

m3m3.

_More

generally,

there _are

couplings

of the form

m~P+~m2p+1

for _p

=

1, 2,...

Thus the

height profile

can be

expanded

_as

h(r, 9)

= r

£~~~ m2p+1cos[2(2p

+

1)9].

Dur results for E- and _m agree with those obtained

analytically

and

numerically by

Deem and Nelson

[10].

Their numerical result for

~/KA

= 0 is lower than _our

indicating

that the order

parameters

_m2p+1 for _p > 1

are

important

in this limit.

A KT transition

temperature

for

negative

discliantion _can be introduced

just

as for

positive

disclinations: T-

=

E-/21n(Rla).

This _curve is shown _as the solid _curve

(b)

in

Figure

3.

4. Discussion

Clearly

both

positive

and

negative

disclinations will be

thermally excited,

and neither

T+

_nor

T- is _a

good

estimate of the actual

melting temperature, TM.

A better estimate is that

TM

is

simply

the average

(T+

₊

T-)/2.

This

yields

the dashed _curve in

Figure

5. This estimate describes

qualitatively

features that _are in

agreement

^with

simple physical reasoning:

for

large

~, there should be _a disclination-mediated

melting

to the disordered

phase

_as

temperature

is

increased,

and at fixed

KA,

there should be _a transition to the

crumpled phase

_{as ~} is decreased.

In references

iii

and

[8],

we calculated the

melting temperature

_using the renormahzation group

(RG)

recursion relations for the KT transition _{on a}

fluctuating

membrane

subject

to

the constraint of

charge neutrahty.

The result shown _as the solid _{curve in}

Figure

5 is below

the estimate

(T+

+

T-1/2

in the entire region below the dotted _curve where _we believe that

our RG calculations _are vahd. This is

entirely

reasonable. The estimate of

TM

obtained

by

equating

the energy of disclinations to

temperature

times their

entropy completely ignores

the

entropy

associated with

height fluctuations,

which should lead to _a

depression

of

TM.

Dur RG calculations include

height fluctuations,

whose

major

eifect is to decrease the effective

long-wavelength

dielectric constant.

Acknowledgments

We _are

grateful

to David Nelson for

bringing

to our attention the

asymmetry

ⁱⁿ

positive

and

negative

disclination

energies

and to Michael Deem and David Nelson for

sending

_us their

independent

work _on this

subject prior

to

publication.

This work _was

supported

in part

by

the National Science Foundation under

grant

No. DMR94-23114 and in

part by

the Penn

Laboratory

for Research _on the Structure of Matter under NSF

grant

No. DMR91-20GG8.

J-M-P- would like _to grue a

special

thanks to Prof. M.L. Klein for his kindness.

References

[1]

Halperin

B-I- and Nelson

D.R., Phys.

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Nelson D-R- and

Halperin B-I-, Phys.

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2457.

[2]

Birgeneau

R-J- and Lister

J-D-,

J.

Phys.

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Brock J-D- et

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(198G)

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[3]

Cheng M.,

Ho

J-T-,

Hui S-W- and Pindak

R., Phys.

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D.J.,

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D.J.,

J.

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L.,

^J.

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D.R.,

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D.R.,

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David

F.,

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E.,

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