Solvable model exhibiting a first-order phase transition

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Solvable model exhibiting a first-order phase transition

N. Boccara, R. Mejdani, L. de Seze

To cite this version:

N. Boccara, R. Mejdani, L. de Seze. Solvable model exhibiting a first-order phase transition. Journal

de Physique, 1977, 38 (2), pp.149-151. �10.1051/jphys:01977003802014900�. �jpa-00208573�

149

SOLVABLE MODEL EXHIBITING A FIRST-ORDER PHASE TRANSITION

N.

BOCCARA,

^{R. MEJDANI}

(*)

^{and L. DE SEZE}

C.E.N.S., S.P.S.R.M.,

^{Orme des}

Merisiers,

^BP

2,

⁹¹¹⁹⁰

Gif-sur-Yvette,

^France

(Reçu

^le

23 juin 1976, accepté

^{le 2 novembre}

1976)

Résumé. ²⁰¹⁴ On étudie une assemblée d’ellipsoïdes asymétriques ^en interaction constante, isotrope

et de portée infinie. Ce modèle, qui ^{est relié à un} modèle de Potts peut être résolu exactement. Il

présente, ^en fonction de la forme de l’ellipsoïde, ^une transition du premier ordre de la phase isotrope

à une phase ^{uniaxiale et une} transition du second ordre de cette phase uniaxiale à une phase ^biaxiale.

Pour des valeurs particulières ^des paramètres ^de l’ellipsoïde ^{le modèle} présente ^une transition du second ordre isolée de la phase isotrope ^{à la} phase ^biaxiale.

Abstract. ²⁰¹⁴ An assembly ^of asymmetric ellipsoids coupled by ^{a constant} infinite-range isotropic

interaction is studied. This model, which is related to a Potts model, ^{can be solved} exactly. ^It exhibits,

as a function of the shape ^{of the} ellipsoid, ^a first-order transition from the isotropic phase ^{to a uniaxial}

ordered one and a second-order transition from the uniaxial phase ^{to a biaxial one. For a} particular shape ^{of the} ellipsoid the model exhibits an isolated second-order phase transition from the isotropic phase ^to the biaxial one.

LE JOURNAL DE PHYSIQUE TOME 38, ^FTVIUER 1977,

Classification

Physics ^Abstracts

7.480

An

ellipsoid

^{can be}

represented by

^a second-rank

symmetric

^{tensor. As we are} interested in orientational

phase

transitions the scalar part ^{of the tensor}

(its trace) plays

^{no role and we shall}

only

consider irreducible tensors. Each

ellipsoid being asymmetric

^{will be}

represented

in its proper ^axes

by

^{the tensor}

We shall

study

^a model described

by

^the

following

Hamiltonian :

where

qiaB

^{is the}

aB-component

^of the ith irreducible second-rank tensor

(a, fl

⁼

1, 2, 3).

The interaction

J/N

is scaled with the number N of

ellipsoids

^{in order}

to obtain an extensive energy. Models in which

spins

are

coupled by infinite-range

interactions have been studied

previously [1, 2]

and it has been found that molecular-field

theory

^{is exact in this case.}

The Hamiltonian

(1)

may be rewritten as :

(*) University ^of Tirana, ^Albania.

y y (qiaB)2

^{= 2}

Nq2(3

⁺

S2)

^{is a constant which can}

aB i

be discarded.

The form of the Hamiltonian

(2)

^shows

that qiaB >

does not

depend

^{on i}

and,

ⁱⁿ

appropriate

axes, ^we shall write the

thermally averaged

value of the irreducible tensor as :

(3)

^{defines two order} parameters a ^{and b :}

- if a ⁼ b ⁼ 0 the system ^{is in an}

isotropic

^state,

- if a #

0, b

^{= 0 the} system ^{is in an ordered} uniaxial state,

- if a #

0, b #

^{0 the} system ^{is in a biaxial state.}

Landau

theory

^of

phase

transitions

predicts

^{that the}

transition characterized

by

^the parameter ^{a would be} first order

(existence

^{of a cubic}

term)

^and the transition characterized

by

^{b can} be second order

[3, 4].

For the first transition there is a close

relationship

between our model and the continuous

generalisation

of the Ashkin-Teller-Potts model introduced

by

Golner

[5].

^{There is a} contradiction for this model between Landau

theory predictions

^{and exact and}

numerical results which

give

^a second-order

transition,

at least in two dimensions

[6-8].

Renormalization group studies

[9-11]

^{have not} yet

given

^a

satisfactory

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01977003802014900

150

answer to this

problem.

However, ^{as underlined}

above,

ⁱⁿ the limit of the constant

infinite-range isotropic interaction,

molecular-field treatment is exact and we cannot

hope

^{for a} result different from Landau’s

predictions.

The mean value of JC is a function of a and b

only

and is

given by :

where the constant term is not taken into account.

The

partition

^{function can} be written :

where

W(a, b)

^{is a state}

density

^{and we} shall define

S(a, b) by :

(the

^{Boltzmann constant is}

equal

^to

1).

In the

thermodynamic

^{limit the} entropy

S(a, b)

^is

proportional

^{to N} and it is

equal

^to the maximum of the

integral

taking

^{into account the two}

following

conditions :

The

integrals

^{are taken over all the}

possible

^orienta-

tions of the

ellipsoid

characterized

by

^{the tensor} qaB.

Consequently

where A

and p

^are

Lagrange multipliers

^and

The two conditions

(8)

may be

expressed

^as

From

(4)-(11)

^we obtain the

partition

^{function in}

the form

where

In the

thermodynamic

^limit

(N - oo)

^{we can use}

the

Laplace

^{theorem to find}

Z, minimizing F(a, b)

with respect ^{to a} and b. The free energy per

ellipsoid

^is

then the minimum of

F(a, b).

The reduced temperature

being

^defined

by

^{z =}

T/Jq2

we shall determine the transition lines in the

diagram (i, E).

The transition from the

isotropic phase

^{to the}

uniaxial one is first order and it occurs at a

tempera-

ture determined

by

^the

following equations

The uniaxial state can be oblate

(Ao

>

0)

^or

prolate (Ao 0).

The transition from the uniaxial to the biaxial

phase

FIG. 1. ^- Phase diagram (T, s) ^{of a} system ^of asymmetric ellipsoids.

T is the reduced temperature ^{and c is a measure of the} asymmetry of the ellipsoids. ^The phases ^{indicated are : I} isotropic, ^P prolate,

0 oblate and B biaxial. The point ^{C is a} multicritical point ^of type (2, 2).

151

is second order

[12,13]

^{and it occurs at a} temperature determined

by

Our results for the

phase diagram (T, s)

^{are summa-}

rized in

figure

^{1. In order to run}

through

^all

possible shapes

^without

repetition

^{we choose -}

1/12 q

⁰

and then all cases are encountered when 8 goes from 0 to 3. These results show that the

phase diagram

^of

such a system ^{exhibits a critical}

point

^{C. We shall} say that such a

point

^{is of} type

(2, 2) [5].

^{At this}

point

E = 1 and each

ellipsoid

^{can be}

represented by

i.e. the

prolateness along

^{the third axis is the same as}

the oblateness

along

^{the first one.}

Thus,

the transition from the

isotropic

^to the biaxial

phase

^is second order.

Furthermore there is a one-to-one

correspondence

between the transition lines on each side of B ⁼ 1 :

For 8 ⁼ 0

(E

⁼

3)

^the

ellipsoids

^are

symmetric

^and

prolate (oblate)

and the transition which is first order is characterized

by

^the parameter ^a.

Figure

^{2 shows in}

both cases the variation of a versus the reduced temperature ^i.

FIG. 2. - Order parameter u ^{versus reduced} temperature ^{i for} symmetric ellipsoids (s ⁼ 0, ^{e =} 3).

References

[1] KITTEL, C., SHORE, H., Phys. ^{Rev. 138} (1965) ^{A 1165.}

[2] SHERRINGTON, D., KIRKPATRICK, S., Phys. Rev. Lett. 35

(1975) 1792.

[3] LANDAU, L. D., Collected Papers ^{edited by D. ter Haar (Per-}

gamon Press, Oxford) 1965.

LANDAU, L., LIFSHITZ, E., Statistical Physics (Pergamon, London) 1959.

[4] BOCCARA, N., Symétries ^brisées (Hermann Editeur, Paris) 1976.

[5] ^{GOLNER, R. G.,} Phys. ^{Rev. B 8} (1973) ^3419.

[6] BAXTER, R., ^J. Phys. ^{C 6} (1973) ^{L 445.}

[7] STRALEY, ^{J. P. and} FISHER, M. E., J. Phys. ^{A 6} (1973) ^1310.

[8] STRALEY, ^J. P., J. Phys. ^{A 4} (1974) ^2173.

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4159.

[12] ALBEN, R., Phys. ^{Rev. Lett. 30} (1973) ^778.

[13] VIGMAN, ^P. B., LARKIN, ^A. I., FILIEV, ^V. M., ^JETP (in ^Rus- sian) ⁶⁸ (1975) ^1883.