Remarks on the multicritical topologies in the n=∞ limit

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Remarks on the multicritical topologies in the n= ∞ limit

Jacques Prost, Jérôme Pommier

To cite this version:

Jacques Prost, Jérôme Pommier. Remarks on the multicritical topologies in the n= ∞ limit. Journal

de Physique I, EDP Sciences, 1991, 1 (3), pp.383-393. �10.1051/jp1:1991140�. �jpa-00246338�

Classification

Physics

Abstracts

64.70 64.70M

Remarks

_on

the multicritical topologies in the

_n

= m

limit

Jacques

Prost

(I)

and Jkr6me Pommier

()

(') Groupe

de

Physico-Chimie Thdorique,

ESPCI, URA 1382 du CNRS, 10 _rue

Vauquelin,

F-75231 Pads Cedex 05, France

~)

Fachbereich

Physik,

Freie Universitit Berlin, Amimallee 14, D-1000 Berlin 33,

Germany (Received

4

July

1990,

accepted

in

final form

5 November

1990)

Rksumk. Nous montrons que Ies

diagrammes

de

phase

mettant en

jeu

des condensations de

paramdtres

vectoriels I la Iimite _n

= cc

(n

= nombre de composantes des vecteurs), rdvdlent _une richesse inattendue de

possibiIit6s topologiquement

distinctes.

Abstract. We show that

phase diagrams

involving the condensation of two vectorial order parameters in the limit _n

= cc (n

=

number of components ^{of the} vectors)

coupled only

to

quartic

order exhibit _an

unexpectedly large variety

of

topologically

distinct cases.

I. Introduction.

Recent

experiments

in

liquid crystalline _systems

raise

questions

_on the

understanding

of the exact

topologies

^of

phase

boundaries in the

vicinity

of multicritical

points.

This is the _case of the NAC

point (I.e.

intersection of the nematic-smectic

A,

nematic-smectic

C,

and smectic A- smectic C

phase boundaries)

which exhibits _a universal

topology,

but not the tetracritical

point (involving

a new biaxial

nematic) theoretically expected [1-3].

This is also the _case of the

SAd-SAT-N point,

for which

general

_arguments also

predict

_a tetracritical

point (with

_an extra incommensurate

phase) [4],

^{and which} turns out to involve both _a tricritical and _a critical end

point

in the

topology

of

figure

5

[5, 6].

More

generally,

a number of mixed

phases

have been

predicted,

which have not yet been observed

[4].

^This situation is somewhat

puzzling

and deserves to be clarified. What

happens

to two second order

phase

boundaries

separating

out one

high

_symmetry and two

(non related)

condensed

phases (called respectively HT,

I and 2 in the

following),

when

they

meet? Renormalization _group and

scaling arguments predict only

two

possible behaviors,

if the lines _are second order up ^to their intersection

point (7) (a jl,

^v

jl

^{critical exponents for the}

^specific

^{heat and} correlation

length)

_:

2 2

if "~

+

fl

~

0,

_one expects the bicritical

topology

of

figure16

_;

vi v~

if

fl

+

fl

<

0,

_one

expects

^the tetracritical

topology

of

figure

2b. The latter

implies

the

vi v~

existence of _a mixed

phase

in which

(I)

and

(2)

_are condensed.

j j

j ⁱ

j HT I _HT

i 1

~

i i /

__

'

~ 2

ia ib

j

/

~

j'

HT

j HT j

i / i '

/

j

/,1'

,~

,/ IT

,/ j

fi,2)

~ ^,

~'

o,2)

~'

~

j

/

Fig.

I. Bicritical

topology (dotted

lines 2nd order transition, solid lines first order _; the omitted

axes

correspond

to temperature ^and pressure, for

instance) a)

mean field _case;

b) according

to E

expansion.

Fig.

2. Tetracritical

topology a)

mean field _case

b) according

to _E

expansion,

The very fact that in

liquid crystals

_many transitions _are KY

like,

has led to the

proposal

that _new mixed

phases

should exist

(4).

A

difficulty

arises when the

(1)-(2)

transition is first order far from the tetracritical

point (for

instance _as

predicted by

_mean

field).

One way ^to connect the first order

(1)-(2)

line to the tetracritical

point

has been

proposed

in reference

[3]

(Fig. 3).

It

keeps

the existence of the mixed

phase.

On the other

hand,

if _one starts from the first order

(1-2)

line and asks how _can it end and

eventually yield

the two second order

(HT- l), (HT-2) lines,

it _seems that the most

generic topology

is that of

figure

4 it involves two critical end

(or Landau) points,

and _a

(fluctuation induced)

isolated critical

point

in the HT

phase.

Such _a

topology

has been

predicted

_on the basis of _a fluctuation

corrected,

dislocation

unbinding

model

[8]. Limiting

_cases of this

generic

situation involves either _one tricritical and

one critical end

point (Fig. 5)

_{or one}

(ordinary) triple

and two tricritical

points (Fig. 6).

As

already stated,

the

topology

of

figure

5 is that observed _at the

NS~~ SAj point.

That of

figure

6 has been

predicted

in _{an E}

expansion

in the _case when the HT-I and HT-2 transitions _are

Ising like,

and for _some values of the

microscopic _parameters [9].

The

topology

of

figure

5 has also

been obtained in _{an E}

expansion,

in the

vicinity

of _a critical

point

of fourth order

[10].

In view of the

large

number of

possibilities,

of the

puzzling

absence of mixed

phases,

and of the

novelty

of the fluctuation induced critical

point,

it _seems worth

investigating

_an

exactly

soluble model in which _some of the fluctuation effects _are retained. Because of its

simplicity,

the _n

= co model _seems well suited for that purpose.

We consider the

simple following

hamiltonian _:

lf

"

d~X(r1 ~/

₊

(~~l)~

_{+ ~2}

~(

₊

(~~2)~

₊

( _~'

+

~(

_{+ "12}

~) ~() (l)

/

_HT

j

,,

, HT

I

j,,"

','

2

2

3 ₄

,

I

,/

j

HT

,' /

i , j HT

,' /

i , i

/

2

2

ad ,,wnitm ,,i»#imp I ml 2 5

Fig, ^3. Possible

phase diagram according

to reference [3]. Note the existence of _one tetracritical

point,

two tricritical

points

and _one

triple point.

Fig. 4. Alternative

topology suggested

by dislocation mediated theory (8), involving _an isolated critical

point

in the

high

temperature

phase

and two critical end

points.

This

topology

is also obtained in the _n

= cc limit _as described in the text.

Fig. 5. Limit of the

topology

of

figure

4, when the isolated critical

point

_merges with the

phase

boundary to

yield

_a tricritical

point.

Fig.

6.

Topology

with two tricritical

points,

as obtained in the _E expansion.

n

in which _n denotes _{the dimension} of each of the fields

#j, #~

_;

WI

₌

jj ^#(

,

(I

=

1,

2

)

and

v=1

WI =

(#/)~,

_{ri and}

r~ ^are

temperature

^and pressure

dependent.

The

gradient

terms

correspond

to short _range

isotropic

interactions. The u; are

positive

definite

coefficients,

called

coupling

constants in the rest of the _paper. This model is

generic

in that

(I)

is the

relevant hamiltonian for the above

quoted physical problems

and _a few others such _as

displacive

transitions. Extensions of the calculations to the frustrated smectics will be

reported

elsewhere

[11].

A

physical

_system described

by

H is

O(n) symmetric

in each of the fields

#j,

#2

^taken

separately. Consequently,

two

independent

^continuous transitions

describing

condensation of fields

#j (Phase I)

and

#~ (Phase 2)

exist in _some

region

of

phase

_space. This is

emphasized by

the _mean field

phase diagrams

sketched in

figures

la and 2a. When the

inequality u)~<uju~

is

satisfied,

the

phase

with both fields

simultaneously

condensed

(Phase (1-2))

is stable for _some values of _ri and r~.

T(rj

= r~ =

0)

in

figure

2a which is the

meeting point

of four second order transition lines is _a tetracritical

point. Otherwise,

_a first order transition line separates

phase

I and

phase

2

(Fig. la)

and

B(rj

= r~ =

0)

is _a bicritical

point.

The purpose of this paper is to illustrate how fluctuations may

modify

these

phase

diagrams.

In section

2,

_we calculate the HT-I and HT-2

spinodals.

The _n

= co

model, belongs

to the

case a <

0,

and when all lines _are second

order,

_{one expects} to find _a tetracritical

point.

This

is indeed the _case when suitable

inequalities

_among

coupling

constants _are satisfied.

However,

when

they

_{are not,} the

spinodals

intersect in such _a way that

necessarily

first order

phase

boundaries _come in. We

investigate

the

resulting topologies

in section

3,

with the direct calculation of the free

energies

of

phases HT,

1, ² and

(1,2).

The calculation _can be

performed

in any dimension between two and four. We find the _cases of

figure 6,

_near 4

dimensions,

that of

figure

6 _or 5 in dimension

3,

in the fluctuation

govemed regime.

Below dimension

3,

_new

possibilities

arise which contain the

generic diagram

of

figure

4. There _are in fact at least three

topologically

distinct other variants

(Figs. 7-9).

The

meaning

of these

results,

and their

expe,rimental

relevance _are discussed in section four.

They

all _are consistent with the work of Golubovib and Kostib _on the related

problem

of

' I

i HT

I cp

I

~i

cP

i

_---~~

l

2

' '

'

' HT

I i ~p

i

cP

2

, I

I I

I ' ~p

HT

i cp

2

Fig.

7. For suitable values of the

coupling

constants

(as

described in the

text)

the

high

temperature first order line

splits

and

gives

rise to two isolated critical

points.

Fig.

8. Variant of

figure

7, when _an

ordinary triple point

separates

phase (I)

with two distinct HT

phases.

Fig. 9. Variant of figure 7, with _an ordinary

triple point separating phases (1),

(2) and HT.

the existence of _a novel

phase

in which symmetry ^of fluctuations is

broken,

without condensation of the

primary

order

parameters [12].

2.

Spherical

limit.

It is well known that

letting

the order

parameter

^dimension n to

infinity yields

the

exactly

soluble

spherical

model

[13]

of Berlin and Kac

[14],

in which fluctuations _may be

integrated explicitly

for _any

spatial

dimension between two and four. This

integration

_may be

performed [15]

with the introduction of

auxiliary

scalar fields

(two

in _our

case),

and the saddle

point technique,

_as described

by

Brbzin _et al.

[16].

It allows to validate the Hartree formula for the inverse

susceptibilities

_{ii, t~ as} functions of the basic parameters :

tj _{= rj} +

~"

~ uj

~d~p

+

~~

"

~ uj~

~

d~p

t~ = r~ +

~~

~~

u~

~

~d"p

+

~~

~ uj~

~d~p

~~~

" 2 + P " + P

Note that uj must be of order to ensure convergence when _n

- co. In the

following,

_we will

n

use the notation u; for the

product

_nu,. In the cut off

independent regime (I.e.

A~ "la

»

I),

_one

_gets

d- 2 d-2

tj =

Pi fir

tj ^~ Rj~ t~ ^~

d- 2 d- 2 ~~~

t~ =

f~

fi~ t~ ^~

fij~

tj ^~

where _we have introduced the reduced variables

~d-

²

~

_~"

~"fi ~"'

~ "~~~

(4)

aTU,

~i

~d

~ _~"

i ~~ ~~

where

(2 ar)" K~

is the surface of _a unit

sphere

in d dimensions.

Note that for d-

4,

_one

always

_gets out of the cut off

independent regime

in

exactly

d

=

4, (3)

is

replaced by

A~+

_t~

A~

_{+ tj}

tj ₌

Pi

fij~t~ In Rj tj ^In

~2 ~l

A~

_{+ t~}

A~

_{+ tj}

(5)

t~ ₌

f~

_fi~t~ ^In Rj~ tj ^In

~2 ~l

For all dimensions _: 2

< d<

4,

the free energy reads

~~= ~"~~(ln(tj+q~)+In(t~+q~))-"~ lj

~

~"~~)~-

n

(2

_ar

)

² _{(t~ + q}

(2

_ar

U2 i

ddq ~

~

i

ddq

i

ddq

~ (t2

+

~~) (~

_"

)"

^~~ _{tl +}

~~ (~

_"

)"

_{t2 +}

~~ (~

_"

)"

~~~

the relevant part ^{of which} may be

expressed

_as

~~li~~ ^{~~ ~~~~}

^~

^~" ~

~~~

~~ ~~~

^~

~

~j~ ^~~

~~~

~

~

~ ~~~ ~

~~)

d d

~r

fit tf~~

_{+ fi~}

t(~~ _fij~ ~j~ ~j~

tj~ t~~

cos

"

(d-3)j

^{~ ~ ~ ~~ ~~ ~ ~ ~ ~}

2

The

spinodal lines,

which _are of _course also the transition lines if all

phase

boundaries _are second

order,

are

easily

obtained

by setting

_t,

=

0

(I

=

1, 2),

in

(3)

2

f~ ^d-2

g~

tj =

0

- ~ ⁼ 12 _~

ij

"12 "12

(7)

2

f~

^d-2 _g~

t2 ₌ 0

- ~ ⁼

ij ~12

"12 "12

(7)

differs from the _mean field

straight

lines in two ways

(setting

aside the translation of rj and

r~

_:

the lines _are

curved,

which _can lead to reentrances

already

discussed

[17],

for

u)~~uj

_u~, the

spinodals

intersect twice

(at

the

origin O(fj

=

f~

=

0),

and at

point

P

(Fig. 10).

1

0 ad 0.6 O-B 0 1

7~ ?~

;i) hi

Fig.

10.

Spinodals

of the

HT-(I)

and HT-(2)

phase

transitions

a)

_u)~

~ u u~,

b)

_{u)~ > u~ u~} note the intersection before the

origin.

This second feature necessitates the existence of first order

phase

boundaries.

Indeed,

if the

point

P _was in the

physically

accessible

domain,

this would _mean that the

high

temperature

phase

could

simultaneously

be in two distinct states,

namely: (diverging pi

+ finite

#~)

fluctuations

together

with

(finite pi

₊

diverging #~

fluctuations. The

uniqueness

of the

high

_symmetry

phase

forbids

this,

and first order transitions _{must occur} before the P

point

is

reached

(note

that P cannot be _a tetracritical

point

since fluctuations do not

diverge

simultaneously

for

pi

^and

#~.

There _{are two} types ^{of first order} transitions

the first _one separates ^the

high

_symmetry

phase

from _a condensed

phase (either

~l

#

0, #2

"

0

#j

₌

0, #2

# 0

#j

#

0, #2

#

0)

the second

separates

^{in the}

high

temperature

phase, regions

of

phase

_space, where fluctuations _are

predominantly

of

type

one, from

regions,

where

they

_are of type ^{two. Since} the evolution _can also be

progressive (I.e.

without _any

singularity),

the existence of such _a first order

line,

necessitates that of _an isolated critical

point

like the

liquid

_{vapor one.} In the

symmetrical

_case, it

corresponds

to the onset of the

partially

ordered state described

by

Golubovib and Kostib

[12].

The discussion of the first order lines of the first

type requires

the

description

of the condensed

phases.

This will be done in the next section. On the other

hand,

_a line of the second type, ^{if it}

exists,

is

entirely

contained in the

high

_symmetry

phase,

and thus included in

equation (6).

It is not

difficult, although

somewhat cumbersome in the

general

_case, to locate the critical

point,

and the local

slope

of the first order line. Below dimension

3,

the critical

point

_occurs in the cut off

independent regime,

_so that

equation (3)

_can be used. One _can

expand

^{the inverse}

susceptibilities

tj, t~ around their critical values in the

vicinity

of that

point

tj = tj

~(l

+

at)

_t~

=

t~~(I

₊

a~).

The variations _{at, a~ are} the _responses to variations

3rj, 3r~

of _{ri, r~ away} from their critical values rj

~, r~

~.

The existence of the critical

point

is

signaled by

the

divergence

of the response.

Linearizing (3) yields (

E'

= d 2

)

_:

"1(tl

c

+ E' _"

I

t/~i~)

₊

"2 E' "12

ti~12

~

3~1

~~~

at E' _u

j~

t[)~/2

₊

a~(t~

~ + E' u~

t()~/2)

=

3r~

The

divergence

occurs when

(tic

+ E' "1

t(~i~)(t2c

_{+ E' "2}

ti~i~)

"

E~

"/2 ~(~~ti~14 (9)

(9)

has _a solution

only

if

u(~

_~ u

j u~. The direction of the first order line in the

vicinity

of the critical

point

is

given by

that of the

eigenvector,

with _zero

eigenvalue

in

(8). Equations (9)

and

(3)

_are not

enough

to

specify

_tj

~,

t~~, ri

~, r~~. One still has to express in the

expansion

of the critical combination _~J of

a j, ^a ~, that the second order term has _a

vanishing

coefficient. One is thus led to _a standard

equation

of the

type

~J 3r _cc

~J

~

(10)

which is

typical

of _{a mean} field critical

point (3r being

some

appropriate

linear combination of

3rj, 3r~).

The _mean field character of this critical

point

is _a

special

feature of the _n

= co limit

[18, 12].

The

symmetrical

_{case uj}

= u~ = u leads _{to a}

simple algebra, keeping

the

physics [12].

Diagonalization

is obtained

directly by choosing

tj ₌

t~(I

_{+ ~J +}

f )

_t~

=

t~(I

_{~J +}

f ).

f,

_~J are the _non

critical,

and critical variables which

obey

_:

f

= a

3r~- p~J~+ O(3r), ~J( 3r~J~) (11)

'i"

3~s

_~ ^~

3~a/E'(E'/~ l)("12 ") ~l'~~+ (li (E'/2 ~)/3 !)

7~

~ +

O(7~i

_7~^~

Bra)

" =

(tc

+

E'(U

₊

_U12) ti'~12)

~j

~~

P

₌

(U

+

U12) (d

2

(d

4

ti"~

_a

/8

3rj

+

3r~ 3rj 3r~

~~~ 2 ^'

~~~

2

Note that _{a more}

satisfactory

definition for the order

parameter

would be _~J

=

(#) _WI).

However,

it is

equivalent

to _~J for all

practical

_purposes in the _n

= co limit. The

divergence

of the

susceptibility

defines the location of the critical

point

2

(d

2 fi

~~

~~~~ ~~

2

d-2

(13)

rj _C " ~2C ~ ~C ⁺

(~

+

fi12)

_~C ^~

The

equation

for

~J shows _:

the

conjugate

field is 3r~=

(3rj-3r~)/2,

and the

susceptibility diverges

like

3rj (mean

field

behavior)

the coexistence line is the

diagonal

3r~

=

0, along

which the

equation

for _~J reads _:

For dimensions

larger

than

three,

the stable solution is

always

_~J

=

0. For 2 _w d<

3,

and

"

~ 2

d/3 (4 d),

_~J

= 0 for 3r _m 0 and _{~J a}

(- 3r~)~/~

for

3r~

_< 0. _~J # 0

corresponds

_to the

"12

partially

ordered state of Golubovib and Kostib. In the

asymmetric

_case, the e~istence of this state is

preserved

and leads to the

topologies

of

figure

4. For ^"

=

~

~ ^a tricritical

"12

~(4 )

point

is reached when

3r~

₌ ^{3r~= 0. For "} _< ^{~ the}

diagonal

3r~=

0,

reaches

uj~ 3

(4 d)

discontinuously

the coexistence _{curve at an}

ordinary triple point.

This

implies

the existence of at least _a

pair

of critical

points

_away from the symmetry ^axis.

Again

this

possibility

survives in the

general

_case for small

enough anisotropy.

This opens the way ^to the

topologies

of

figures

7

to 9. To _assess which _are the relevant _cases, needs the

investigation

of the condensed

phases

this is done in the next section.

3. Ordered

phases.

We describe the condensed

phase (say

of

#j type) by writing

_:

i~j

₌

(Pi

+

fimj, _{i~i ) (15)}

(Pi) =°; 14~ii)

=°.

The

averaged

order

parameter Mj

called

magnetization

in the

following

is choosen in the direction

I,

and

#~

takes into account the

(n I) remaining

_components of the field.

Expression (15)

is

plugged

into

(I)

and with _a

procedure

similar to the _one

previously used,

we

get

the

following expression

for the inverse

susceptibilities

J~ _{~~ ~} ~12

~"P

~

~l

~"P

~~

fif2

~~

(2 ar)" _T~+ p~ ₍₂ ar)d _Tj~ +p~

T~ ₌ r~ +

"~~

~

~"~

~+

"~

~

~"~

~

+ u j~

M)

~~~~

(~") l~li+P (~") 1~2+P

Because the

(n I) perpendicular

directions

correspond

to

hydrodynamic modes, necessarily Tj

~ ⁼

0. This

provides

_an

equation

for

Mj

minimization of the free energy with respect to

Mj gives

the _same result.

The

expression

of the free _energy is _now

~ ~ l~ T~

~ q

~ (~~)d

~

^~

In

d 2Fj ~

Pi

_fij~ ^{"j~j ~~ fit ~}

Pi

fij~

"j~j

fi~

~ ~

i _~'~ fij fit _~ d ~

4 g~ fij

~ 4

~~

_~~~~

The

same

holds of

ourse for

the

#~

type

phase,

with

The procedure _for the

i'l~ (~lll+~fifj>i'li)>

i'2~

The

^inverse ptibilities

are

now

given

by

:

j~=rj+ujm)+uj~mj+

"~~ ~j ~"~

(2ar) ^{(2ar) j~+p}

T~ ~ = r~

(~")

l~li+P

(2") 1~2i+P

With ^gain the _conditions

Fj~ _is

n

+

2

1 _~il~~ in q

2 - ~"

+ "i+

~

hichgives the

cut

independent

egime

(for the ^levant

:

~

₁₂

_

~2'l

~

^12'2

~ _{~l '2}

^~ i12'l

~l

~2'l

~

fi2

~

l~ ~ ~ ~ ~

~

~ ~ ~ ~ ~ ~

~

"2

_~2

_~

12

~l

"12

"1 "2

"12

~l

2 l

~ k~2

~ kl

_k2 ~ k(~

-

kj

k~ k(~ -

kj

k~

~ ~

The

ifferent

diagrams ome from

the

4. Phase

diagrams.

For

u)~

_{< uj u~,} the situation is

quite simple

and in total agreement ^with renormalization group

expectations.

The

diagram

of

figure

2b is relevant. The

phase

boundaries _are all second order and _can be

easily expressed analytically.

The

high HT~(I)

_or

HT~(2)

lines _are

given by (7)

in the cut off

independent regime. Similarly,

the

(1,2)~(l)

and

(1,2)~(2)

lines _are

given by

the

vanishing

of

M~

and

Mj respectively (in

the mixed

phase).

This

yields simply

f~

₌ ^~~~Pi Pi =

~~

f~. (22)

~l

_~12

The _case

u)~

_{~uj u~} is

considerably

richer and

provides

all the other

topologies already

described.

In the cut off

independent regime

for 3 _w d _w

4, topologies (5)

and

(6)

_are obtained. The location of the tricritical

points

_can be

expressed analytically

_on the tj ₌ ⁰ line from

(8)

~~

-~ ~(2 ~l

~2 ^~

2 d 2

~~ ~~

kj kj2

^{~~~ 2}

2

~2 ~l _TC ^{~ ~}

~2TC " t ~l _TC + ₁

"12 "12

(the

tricritical

point

on the t~ =0 line follows of _course similar

expressions

with the

permutation

of I and

2).

The

topology

of

figure

5 is obtained rather than that of

figure 6,

when _one of the would be tricritical

points

is _on the _non

physical part

of the

spinodals (I.e.

_on

the OP

branches).

In dimension three the observation of the

topology

of

figure

5

requires

uj ~ u j~. In the

vicinity

of dimension

four,

the _two tricritical

points

_are

always

in the

physically

accessible domain. As

(4

_{d~ goes} to zero,

they

_are

pushed

into the cut~off

govemed regime,

but the

global topology

is

preserved.

Mean field like

diagrams

_are recovered above four dimensions at an other dimension

depending

_on bare parameters.

For 2 _w d

<

3,

the critical

point

discussed in the second section _comes into the

physically

accessible domain. We checked

numerically

in d

= 2 +

2/3

dimensions that

all'topologies

from

figures

7 to 9

plus

that of

figure

4 _are

possible, depending

_on bare parameters values.

Discussion.

The _n

= co limit of the

simple

Hamiltonian discussed in this paper, reveals many of the

possibilities proposed

in the dislocation mediated model of reference

[8].

It is

interesting

in that it

provides

_a reliable

example

of alternative scenarios to the tetracritical

topology.

Whether all these

possibilities

survive for small _{n or} not, remains to be _seen, but since the first order lines _we discuss in this paper are due to

fluctuations,

_we believe that

they

will. The isolated critical

point

should be

Ising

like _as

already pointed

out in reference

[12].

The _mere fact that the

topology

of

figure

5 has

already

been observed

experimentally

suggests ^{that this} is indeed the _case

[5, 6].

This

implies

that the isolated critical

point

in the

high

_symmetry

phase

should be observed in _some

system [19].

We

suggest

that the _« central

peak

observed in

systems exhibiting displacive

transitions _{many years ago,} and _never

satisfactorily

interpreted,

is

precisely

the

signature

of this isolated critical

point. Indeed,

because of symmetry, the system ^is

naturally

such that _rj

= r~, uj = u~, and _{one can} hit the critical

point

^if it

exists, merely by changing

temperature.

Similarly,

the absence of mixed

phases,

such _as the

biaxial nematic at the NAC

point,

_can be rationalized. If _we

accept

a disorder

parameter

^for the NA transition such _as introduced

by

Toner

[3] together

with _an order

parameter (tilt

_as introduced

by

de Gennes

[20])

for the A~C

transition,

the NAC

topology might correspond

to

figure

5

(with

_a tricritical

point

_very close to the critical end

point [21]).

The observed universal curvatures of the

phase

boundaries would

correspond

to _crossover

exponents,

as

discussed in

[17]. Finally,

it is worth

stressing

that these results do not contradict renormalization _{group ones:} whenever

only

second order transitions _are

observed,

the tetracritical

topology prevails,

but

they

reintroduce _an

interesting sensitivity

to the micro~

scopic

details of the interactions.

AcknowledgTnents.

It is _a

pleasure

to

acknowledge illuminating

discussions with E.

Brezin,

T. C.

Lubensky

and J.

Toner. We _are also indebted _to D. Mukamel for

drawing

_our attention _on reference

[10],

and C.

Garland,

B. R. Ratna and R. Shashidhar for

exposing

_us all the beauties of their

experimental

results.

References

[I]

BRISBIN D., JOHNSON D. L., FELLNER H., NEUBERT M. E.,

Phys.

Rev. Lett. 50

(1983)

178.

[2]

SyASHIDHAR

R., RATNA B. R., KRISHNA PRASAD S.,

Phys.

Rev. Lent. 53

(1984)

2I4I.

[3] ^GRINSTEIN G., TONER J.,

Phys.

Rev. Lent. 51

(1983)

2386.

[4] ^GRINSTEIN G., LUBENSKY T. C., TONER J., Phys. Rev. B 33

(1986)

3306. The arguJnents

developed

in the above reference should

apply

to the S~~ SA~ N _case.

[5] RAJA V. N., SHASHIDHAR R., ^RATNA B. R., HEPPKE G., BAHR Ch., Phys. Rev. A 37 (1988).

[6] ^EMA K., NOUNESIS G., GARLAND C. W., SHASHIDHAR R., Phys. Rev A 39, 2599

(1989).

[7] KOSTERLITz J. M., NELSON D. R., FISHER M. E.,

Phys.

Rev. B is

(1977)

5432

AHARONY A., Phase transitions and critical

phenomena,

URG. Domb C., Green M. S. Eds.

(Academic Press)

1976.

[8] PROST J., TONER J.,

Phys.

Rev. A 36

(1987)

5008.

[9] ^RUDNICK J., Phys. Rev. B 18

(1978)

1406.

[10] KERSzBERG M., MUKAMEL D.,

Phys.

Rev. B 23

(I98I)

3943.

[I

Ii

POMMIER J., PROST J., in

preparation.

[12]

GOLUBOVIt L., KOSTIt D.,

Phys.

Rev. B 38 (1988) 2622.

[13]

STANLEY H. E.,

Phys.

Rev. A 76

(1968)

718.

[14]

BERLIN T. H., KAC M.,

Phys.

Rev. 86

(1952)

821.

[15] POMMIER J., Thesis Bordeaux _n 274

(1989).

[16] BRtzIN E., LE GUILLOU J. C., ZINNJUSTIN J., _« Phase transitions and critical

phenomena,

C.

Domb and M. S. Green Eds.

(Academic

Press, New York,

1976)

Vol. 6.

[17]

PROST J., From crystalline to

amorphous,

C. Godrdche Ed. (Les Editions de

Physique)

1988 after LUBENSKY T.

C., unpublished.

[18] SARBACH S., FISHER M. E., J.

Appl. Phys.

49

(1978)

1350.

[19] More recent

experiments

suggest ^{that the isolated critical}

point

does exist in the

N S~~ SA~ ^{case SHASHIDHAR} R., GARLAND C.,

private

communication.

[20] DE GENNES P. G., Mol. Cryst. Liq. Cryst. 21

(1973)

49.

[21] As pointed out by _one of the referees _one would still have _to understand why the tricritical point is

systematically

close to the critical end

point.