Phase transition in a ferromagnetic fluid

Physica 138A (1986) 220-230 North-Holland, Amsterdam

PHASE TRANSITION IN A FERROMAGNETIC FLUID Christian GRUBER Zmtitut de Physique

Thiorique,

Ecok

and Robert B. GRIFFITHS*

Polytechnique

Fidt+ale,

1015 Lausanne,

Switzerland

The correlation inequalities of Fortuin, Kasteleyn, and Ginibre (FKG) are used in conjunction with those of Griffiths, Kelly, and Sherman (GKS) to demonstrate the existence of a ferromagnetic phase transition in a continuum fluid of classical particles carrying Ising-like spins and having suitable magnetic and non-magnetic interactions. The procedure involves finding a lower bound on the magnetization of the fluid in terms of that of a suitable spin system on a lattice.

1. Introduction The standard methods to prove the existence of phase transitions for lattice models are restricted either to ferromagnetic systems (Peierl’s argument’)), or to systems with a finite number of ground states (Pirogov-Sinai theory2)), or to systems with special properties (reflection positivity3)). They are therefore not applicable to systems with a hard core having an infinite number of ground states. In the case of continuum models, or fluids, the only systems for which the existence of a phase transition has been established are the WidomRowlinson models in which it is assumed that there is a hard core between pairs of particles of different species, but no hard core between identical particles4”), or the Coulomb systems in two dimensions6); thus such methods cannot be applied for general systems of identical particles with a hard core. In the last twenty years, methods based on correlation inequalities have often been used to study phase transitions; those are in particular the Griffiths, Kelly, Sherman7) (GKS) inequalities valid for ferromagnetic lattice systems

(where A, I3 are subsets of Z” and uA = n

ui, cri = 21)

iEA

*Present and permanent 15213, USA.

address: Physics Dept., Carnegie-Mellon

University,

0378-4371/86/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

Pittsburgh,

PA

PHASE TRANSITION IN A FERROMAGNETIC

FLUID

221

and the Fortuin, Kasteleyn, Ginibre’) (FKG) inequalities valid for any lattice systems with probability measure such that P(AUB)P(Af7B)kP(A)P(B)

VA,BCZ”,

which states:

whenever f and g are increasing functions on the lattice, i.e. f(B)af(A)

if

B>A.

We shall use these inequalities to prove the existence of phase transitions for “ferrofluid models” either in the continuum or on a lattice. These models are systems of particles having an internal degree of freedom called “spin” which interact by means of “ferromagnetic” (“spin’‘-dependent) as well as “nonmagnetic” (particle-particle) potentials. Such models have been used to study a variety of physical systems: magnetic ions’), phase separation in 3He-4He mixture”) or ternary mixtures, amorphous ferromagnet”), magnetic gas”), . . . . However, all these investigations were based on the molecular field approximation. Rigorous results concerning these ferrofluid systems were obtained by Griffiths13) and Bricmont14) for the spin 1 model (without hardcore) with nearest neighbour interactions of the form H=-KC

+;-.c,Y;, (ii)

siE{-l,O,+l} i

and by Friihlich and Huckaby”) hard core on a lattice.

for a very special model of ferrofluid with a

2. “Spin f ferrofluids” 2.1. Definition

We consider a system of identical, spin f particles in R’, with a hard core of radius R, interacting with each other through non-magnetic (potential) and ferromagnetic (exchange) two-body interactions. The hamiltonian of the system is given by

222

C. GRUBER AND R.B. GRIF’FITHS

H(X, a) = -

c 1(x, -

Xj)Ujia; +

Iki)

c

f#& - Xi) = H,(X,

a) + G(X)

)

(1)

{i,il

where

J(x) = J(lxl) 3 0 ,

i.e. “ferromagnetic”

interaction,

(2)

and J(x), 4(x) are bounded functions of x which decrease faster than IxI-@‘+“), E > 0, as 1x1 tends to infinity. The Gibbs states of the finite system A C IF!“,at inverse temperature p and activity z = esP, is defined by means of the boundary condition (Y, +) where Y is a simple cubic lattice with side a and ~~=+lforallyEYc={EY;y~A}. This boundary condition is indicated in the expressions below by means of a superscript (+) with the arguments in Y’ omitted. That is to say, the Hamiltonian will contain the interaction of particles inside A with those in the “boundary”. 2.2. Comparison continuum and lattice system Let Z” be some simple cubic lattice with side a such that a>4R.

(3)

With this condition, whatever be the configuration outside a given cell (Yof Z’, it will always be possible to introduce a particle in the cell LYand the volume available to this particle will be larger than u, where ~=(a-4R)“.

(4)

Let us denote by C, D, . . . a family of cells of the lattice Z” and by C, the family of cells occupied by one or more particles in the configuration (X, a). The magnetisation of the system defined by (A, p, z) and the boundary condition (Y, +), where Y is the lattice defined by the centers of Z “, is given by

where A adm= {(x1,.

. .,x,);Ixi-xjl>2R},

m(o)=f

i=l

0;.

PHASE TRANSITION

Introducing

IN A FERROMAGNETIC

223

FlUID

the notation

{dX=T

-$

j- dxi...dx,, Aadm

dXzlXl (MIX)

(M)n =f

IXl=n, 2;;)

e-@+‘(x)

(6)

where

Z,(X)

= c e-Pfk%? (I

(7)

a)

is the magnetisation and the partition function of a spin i lattice system, located on the sites X (with + boundary condition on Y’ = Y fl A”). We thus have (M),

= c

$

(8)

dXrlX1(MIX)~e-a”‘)‘X’.

CCA c,=c

Since the spin 4 lattice system is ferromagnetic,

(MIX)

2

the GKS-inequality

W>‘c,

yields (9)

9

where ( M) Lxis the magnetisation of a spin $ Ising model with sites located at the centers of the cells in C, and nearest-neighbour interaction JO where J-=

O-

J(x) . ,x,~~,

(10)

Therefore

I

p(c)

=

;CdXzlXi c,=c

& ,

C e-pH(+)(X7a)

czAP(C)

(11) = 1.

CT

To compare the magnetisation of the continuum system with that of a “diluted-1sing model” (i.e. each site can be either empty or occupied by a spin 1. rather than a “quenched” type of dilution) 2, note that this is an “annealed” we write:

224

C. GRUBER AND R.B. GRIFFITHS

where P,(C) is the probability that the set C of sites is occupied and Z,_,(A) is the partition function of the diluted Ising model, with interaction Jo: (WA 3 &

W):f(C)MC).

2.3. Existence of a phase transition Since the interaction implies:

Furthermore,

the GKS inequality

it was shown in ref. 16 that Pi(C) satisfies the FKG-condition:

P*(C u D)P,(C Therefore

of the Ising model is ferromagnetic,

t-l D) 2 Pi(C

.

if

f(D)sf(C)

for D> C

we can apply the FKG-inequality

(WA a (c. (W%C))@ C

to conclude

C

f(C)P,(C))

7

i.e.

and the magnetisation of the ferromagnetic fluid is bounded by the one of the diluted Ising model with parameter ( h, Jo). - / Lemma 1. The condition f(D) 2 f(C) if D 3 C is satisfied whenever /Asatisfies the following inequality: (a _ 4R)’ ePr 32es(k%+~~Jo+l19+II) .

PHASE TRANSITION IN A FERROMAGNETIC Proof.

FLUID

225

It follows from eq. (12) that (13)

where

p(D)

f = Cx=D

dX[. . .]

P(C) J c,=c

dX [. . .]

Let D contain one more cell than C, i.e. D = C U {a}. Since the cell (Y is occupied by at least one particle, we obtain a lower bound by considering in the numerator only those configurations with exactly one particle with position x in cr and spin I+= +l or -1. Furthermore H’+‘(XUx,ac+)=H(+)(X,a)+

c

[-J(x-z)ao*+4(x-z)]

ZEXUYC

II 4,

II = s;P

{c

4+(x)]

7

XEX

where the supremum is taken over subsets X of R” such that lxila2Z?,

Ixi-xjls2R

Vxi,xjEX,

i#j

and 4, is the positive part of 4 (i.e. 4 = 4, - +_, +,a

0). Therefore (14)

and by iteration we have

f’(D) > P(C)

-(

zu

e-P1t++II ID=1

1

.

On the other hand for the diluted-Ising

model

226

C. GRUBER AND R.B. GRIFFITHS

which implies:

P,(D)

(

2,a(~+*~Jo)

P,(C) -[

ID\CI

1

*

(15)

Therefore

P(D) ~ P,(D) P(C) P,(C) whenever

the condition

of the lemma is satisfied, which concludes the proof.

Since for J,, > 0 and EL,,+ vJ, > 0 the diluted Ising model has two ground states which are related by spin inversion, then, using the standard Peierls argument, we know that this system exhibits a spontaneous magnetisation at low enough temperature (large j3) and the same will be true for the ferrofluid system (if Y 2 2). We have thus established the following result: Property 1. Any “spin 1 ferrofluid”,

min

J(x)>0

with magnetic interactions

such that

(&>O)

lxl
exhibits at low temperature a spontaneous transition, for p > [[4+[[ (V > 2).

3. Spin f ferrofluid

magnetisation,

and thus a phase

with hard core on a lattice

In the lattice version of the ferrofluid, the system is defined by a lattice 3 = Z”; at each site x of 9 is associated a spin variable s, which can take the value (- 1, 0, + 1); the value 0 corresponds to an empty site, and the value + 1 to an occupied site. The “hard core condition” between a pair of sites (xy) means that the two sites cannot be simultaneously occupied by a particle. Let {hc} denote the family of pair of sites related by the hard core condition. The characteristic function for admissible configurations is

PHASE TRANSITION IN A FERROMAGNETIC

n

x(S)=

FLUID

227

(l-s:+

(xy)E{hc)

and the radius R of the hard core is defined by 2R=

sup Ix-yl. (xy)E(hc)

The hamiltonian

of the system is given by

(16) where we now assume that both magnetic interaction Jxy and the non-magnetic interaction .sxy are non-negative and invariant under translations. Let us note that the diluted Ising model introduced in subsections 2.2 and 2.3 is a special case of (16) ({hc} = 0, cxy = 0, Jxy = J, if Ix - yI = 1). The magnetisation of the finite system with (+) boundary condition (i.e. ax = + 1 on a cubic sublattice with side a > 2R) is given by

with X(X) the characteristic function for admissible occupied lattice sites and fl%= +1. Let (MIX) be the magnetisation of a spin 1 lattice system with sites X and hamiltonian H(“X)

=

-

=

2

I

(,zx

Jijaiaj

-

P

Ix1- (Ijzx &ij;

then

where Z,(X)

exp[-PH(+)(u,)]

.

=x

Let Q C A be a maximal admissible subset of A in the sense that x( Q’) = 1 for

228

C. GRUBER AND R.B. GRIFFITHS

all Q” c Q and X( Q”) = 0 if Q”3 Q, i.e. the sites of Q can all be occupied but it is impossible to add one more particle. Then

Z(X)=

c Q>X

l= number of maximal admissible subsets of A compatible with X .

We thus have

(17) where

G(Q)=

2 (MIX)

XCQ

j&E

and

With these definitions (&I),, is a weighted average of &I(Q) for different and &f(Q) is a weighted average of (MIX). The ferromagnetic condition Jxy 2 0 implies

(MlY)3(MlX)

Q

VY>X

and by definition Z(Y) s Z(X) for Y 3 X. Furthermore since .I_ 2 0 and exy 3 0 then Z,(X) /Z,_,( Q), where Z,_,(Q) = c xCQ Z,(X), satisfies the condition to apply the FKG inequality16). Therefore the FKG inequality can be applied to eq. (17) and we have

zz’&M(Q) =xzQ(MIX) L d-1

z(x)

Z,(X) z,.,(Q)

PHASE TRANSITION IN A FERROMAGNETIC

FLUID

229

which yields

2 (MIX)

G(Q)2

XCQ

;‘(xe’, =4.,(Q) d-1

and

We thus have: Lemma

2. If the diluted Ising model without

MdAQ)

3 m,l4,

then (WA

hard core on Q is such that

2 m&4.

To obtain a bound on Md_r(Q) we consider a cubic sublattice 6p, of 3 with side a larger than 4R (R = hard core radius). Then any Q, a maximal admissible subset of A, contains at least one site in each cell of AR = A fl 5?R and thus, using the GKS inequality Md_I( Q)

5

(M) ~;F’o*eo) ,

(18)

where Jo =

min

Ix-yft2a

JXY,

We have therefore

co =

min

Ix-y1<2.

established

axr .

the following result (see section 2):

Property 2. Any spin 1 ferrofluid on a lattice, with a hard core of radius R such that for some a > 4R, J,, (eq. 18) is strictly positive, exhibits at low temperature a spontaneous magnetisation if p 3 0 (V 3 2).

References 1) R. Peierls, Proc. Cambridge Philos. Sot. 32 (1936) 477. R.B. Griffiths, in: Phase Transitions and Critical Phenomena, vol. 1, C. Domb and M.S. Green, eds. (Academic Press, New York, 1972). 2) Ya.G. Sinai, Theory of Phase Transitions: Rigorous Results (Pergamon, New York, 1982). 3) J. Friihlich, R.B. Israel, E.H. Lieb, B. Simon, J. Stat. Phys. 22 (1980) 297. 4) D. Ruelle, Phys. Rev. Lett. 27 (1971) 1040.

230

C. GRUBER

AND R.B. GRIFFITHS

5) J. Bricmont, K. Kuroda and J.L. Lebowitz, Comm. Math. Phys. 101 (1985) 501. 6) J. Frdhlich, private communication. 7) R.B. Griffiths, J. Math. Phys. 8 (1967) 478, 489. D.G. Kelly and S. Sherman, J. Math. Phys. 9 (1968) 466. 8) C.M. Fortuin, P.W. Kasteleyn and J. Ginibre, Comm. Math. Phys. 22 (1971) 89. 9) H.W. Capel, Physica 32 (1966) 966. 10) M. Blume, V.J. Emery and R.B. Griffiths, Phys. Rev. A 4 (1971) 1071. 11) N.E. Frankel and C.J. Thompson, J. Phys. C8 (1975) 3194. 12) S. Inawashiro, N.E. Frankel and C.J. Thompson, Phys. Rev. B 24 (1981) 6524. 13) R.B. Griffiths, Physica 33 (1967) 689. 14) J. Bricmont and J. Slawny, First order phase transitions and perturbation theory (preprint). 15) J. Frijhlich and D.A. Huckaby, J. Stat. Phys. 38 (1985) 809. 16) J.L. Lebowitz, Comm. Math. Phys. 35 (1974) 87.