CENTRAL RESEARCH INSTITUTE FOR PHYSICSBUDAPEST

KFKI-1982-83

A . S Ü T Ö T . Y A L C I N C . G R U B E R

A PROBABILISTIC APPROACH TO THE MODELS OF SPIN GLASSES

‘H u n g arian ‘Academ y o f S c ie n c e s

CENTRAL RESEARCH

INSTITUTE FOR PHYSICS

BUDAPEST

A PROBABILISTIC APPROACH TO THE MODELS OF SPIN GLASSES

A. Sütő, T. Yalcin*, C. Gruber*

Central Research Institute for Physics H-1525 Budapest 114, P.O.B. 49, Hungary

*Institut de Physique Théorique Ecole Polytechnique Fédérale de Lausanne PHB - Ecublens - С Н - Ю 1 5 Lausanne, Switzerland

Submitted, to J. Stat. Phys.

HU ISSN 0368 5330 ISBN 963 371 974 7

Introducing the notions of quenched and annealed probability measures, a systematic study of some problems in the description of spin glasses is attempted. Inequalities and variational principles for the free energies are derived. The absence of spontaneous breakdown of the gauge symmetry is dis­

cussed and some high temperature properties are studied. Examples of an­

nealed models with more than one phase transition are shown.

АННОТАЦИЯ

Для изучения моделей спиновых стекол нами введены понятия мер вероятное т и , относящихся к случаям быстрого и медленного охлаждения. Выведены неравен ства и вариационные принципы для соответствующих свободных энергий. Обсужде­

но отсутствие спонтанного нарушения калибровочной симметрии, и изучены высо­

котемпературные свойства. Приведены примеры для моделей с медленным охлажде- ним, имеющих несколько фазовых переходов.

K I V O N A T

A spinüveg-modellek leirására bevezetjük a "gyorshütött" és "hőtempe­

rált" esetekhez rendelt valószinüségi mértékeket. Egyenlőtlenségeket és va­

riációs elvet vezetünk le a megfelelő szabadenergiákra. Tárgyaljuk a gauge- szimmetria megőrzését, magas hőmérsékleti tulajdonságokat. Példát adunk olyan hőtemperált modellekre, melyekben egynél több fázisátmenet történik.

A comm o n l y accepted way to describe d i s o r d e r e d systems is to represent them by statistical ensembles in which the degrees of freedom are cou p l e d by r a ndom interactions. This additional randomness may be treated in different ways: R andom i n t eractions may be c o n s i d e r e d as new degrees of f r e e d o m and in extreme

cases they may be in thermal e q u i l i b r i u m with the rest of the system (annealed state) or completely frozen in some r andom position (quenched state). Many years after the pioneering

work of Brout'*' the quenched state of c e r t a i n spin models b e c a m e the center of interest of the theoretical research on spin

glasses. Edwards and Ande r s o n 2 p ointed out that these systems can p r o p e r l y be d e s c r i b e d by the q u e n c h e d state of randomly

interacting spin m o d e l s on regular lattices. Meanwhile, one encounters two mai n difficulties: the first is to calculate, in a respectable approximation, the q u e n c h e d free energy and the second is to give a reliable proof that there exists a phase transition - in the static sense - between the high t e m ­ perature p a r a m a g n e t i c and the low t e m p e r a t u r e spin glass state.

Neither of these prob l e m s has got so far a reassuring solution, excepted probably in the case of the so c a l l e d S h e r r i n g t o n - K i r kpatrick model .3

While not pre t e n d i n g to contribute to the solution of these great questions, in the present paper we attempt a s y s tematic study of what were c a l l e d the "annealed" and "quenched" states.

To this end we d efine in sec 2 the a n n e a l e d and quenched p r o bability m e a s u r e s which pl a y the same role as the Gibbs measure in e q u i l i b r i u m systems. In Se c t i o n 3 we derive some

inequalities for the quenched free e n e r g y and establish in

sec 4 a variational p r i n ciple c h a r a c t e r i s i n g both the a n n e a l e d and quenched free energies in the space of the joint p r o b a b i l i t y distributions for spins and bonds. In Section 5 we raise the

question of the un i q u e n e s s of the q u e n c h e d state and show that the gauge invariance cannot be broken by different choices

for the b o u n d a r y condition. This suggests that the "order p a r a ­ meter" of the spin glass state must be the e x p e c t a t i o n value of a gauge invariant and n o n - local observable. The o r d e r parameter, propo s e d by Edwards and A n d e r s o n 2 has indeed these properties, as we p o i n t out in Section 6 . A discussion of high temperature p r o perties is also given t h ere and the functional r e l ationship between o r d e r p a r a meter and free energy is established. Finally, in Section 7 we return to the study of annealed m o d e l s and

give e x a m p l e s for one, two and three cons e c u t i v e p h ase tr a n ­ sitions in such models.

2. Definition of the system

Let L be a lattice; at each site i of L is ass o c i a t e d a single spin space S w h e r e S is a subset of . The spin c o n f i g u r a t i o n s are defined by o' : L — ► S and the formal h a m iltonian of the

system is given as

H(J.^ = -2L 1ь Фи(сг)

b c L b

Here the ф ^ S are bounded, real v a l u e d functions d e p e n d i n g on

< 4 = [ Л ; i f e b } and the 's are real random v a r i a b l e s with probability d i s t r i b u t i o n the m e a n value of w h ich is finite.

The finite par t i a l sums of (2.1) are well defined for any <T with ^ - p r o b a b i l i t y 1. In particular

З ь ф и Со-)

b c v b

exists for all finite V c L.

On S is given an a priori f inite m e a s u r e d.^i0 not n e c e s s a r i l y n o r m alised and díAví0*) denotes TT d u (G i ) while dj (J) =

-j-r ' U v ' o Jv

l' d5,(J, )• In the following d i s c u s s i o n we consider on l y bcv D D

finite volumes a n d we omit the label V.

For given interactions J, the free e n e r g y F (J) and the e q u ilibrium state at inverse t e m p e r a t u r e are defined as usual by:

and by the Gibbs prob a b i l i t y mea s u r e

where

d G 3 ( < 0 = ■ j a . e O d l f U G ' )

(2.4)

(2.4) can ill so be written as

F ( J ) = H U , 0 + | j ^ 3 (j,o-)

(2.5)

which yields (since the Gibb m e a s u r e is normalised)

(2 .6 )

= E U ) - S O )

The quen c h e d f ree energy F and the q u e n c hed state are def i n e d by

F - U Q ( 3 /(T')[Ht3,<r) + t In^CJ/r)] - E - | S

We shall impose that the d i s t r i b u t i o n d ^ (x) falls off s u f ficiently r a p i d l y so that F is well defined.

The anne a l e d free e nergy F and the anne a l e d state are def i n e d --- —--- an ---

by the p r e s c r i p t i o n that the average over the interactions has to be perfo r m e d in the p a r t i t i o n function, i.e.

Far, - - X W 3 >at W ) e ~(iH<I/ni

and by the q u e n c h e d p r o b a b i l i t y m e a s u r e

dQ (3,СГ

-

(2.8)

which yields

(2.10)

and by the a n n e a l e d p r o b a b i l i t y m e a s u r e

(2.1 2) (2.11 )

which yields

Fao~ ^dA U ,(T)lН О /г) + 1 £ пЦ 1 /гХ | = Ean- i San (2.13)

Introducing the space (& of joint p r o b a b i l i t y d i stributions

< B =

^ d SA/»i

and the free e n e r g y functional F l U de f i n e d on (£> by

F t ? b ^ d jC iH iiUr)((J,<F)tH (3(o + te n (t3 ,F ')]

we can express F and F r e s p e c t i v e l y as

F = F Q g ] F an = F

Finally for any subset В = (b^, . .., b^) of b's in V we introduce J = \ J. \ „ and

В L b ) beB F U 6 ) =

b £ B which implies in particular

F (JB> = F (J ) if В II er er о <

F < V = F if в = ф

Let us note that for any

J B “ I,Jb ^ b £ B ' F (JB ) r e P resents the quenched free energy with respect to the new me a s u r e d ^ ( J ) where

d S' C n = T V « W - d ^ T T < r C 3 b - 3 b ) d 3 , •

t>CV \o€,Eb

b ^ 6

(2.14)

(2.15)

(2.16)

(2.17)

3. Inequalities for the q u e n c h e d free energy

In this section we first col l e c t i n e qualities for F (J ); we В then discuss the d e p endence of F on the d i s t r i b u t i o n dy.

Propo s i t i o n 3.1

i) F (J_) ^ F ( J _ , ) for any В CB' (3.1)

D О

where ^ dj^ (x) X

i i ) F < F <. F (3) (3.2)

an ^

The proof of i) follows from Jensen's i n e q uality using the known fact that F(J) is a c oncave function of eac h J, 's and

b thus F(J„) is also a concave function of each J, 's , b£,B.

В b

The proof of ii) follows from Jensen's i n e q uality using the fact that (- = - n Q is a c o n v e x functional of the p a r t ition

function Q.

4

This p r o p o s i t i o n was e arlier p u b l i s h e d by Rosa ; as men- tionned in Sec. 2 the i n e qualities (3.1) can be regar d e d as the c o m p a r i s o n of two d i f f erent averages of the same function F(J); we thus have

^ F 4 ) d s a F 0 .3 )

where the p r o b a b i l i t y mea s u r e d ^ is sharper than d£^.

The ques t i o n n a t u r a l l y arises whe t h e r (3.3) is g e n e r a l l y true, i.e. whe t h e r a "sharpening" of the d i s t r i b u t i o n causes the quenched free e nergy to increase. The inverse p r o b l e m is also of interest: does a b r o a d e n i n g decr e a s e F? At first we show that the b r o a d e n i n g p r o b l e m can always be solved.

Lemma 3.1

Let f = {R — * R be concave and let ^ a n d У be two p r o b a b i l i t y meas u r e s on IR such that

i i ) ^ dvix") x = 0

exists

and

exists for all у in IK.

the n ^ d(<£*v) ( * ) |( x ) 4

where < А Ц* ^ 0 denotes the convolution of the measures,

(3.4)

P r o o f :

Using Jensen's inequality together w i t h ^dvCx") X ■= 0 , implies

J d ^ C x ) >y ^ d ^ ( x ) ^ d v ( ^ Í (x + ^ ) — ^ d(<j*v) ( f

This lemma can be immediately e x t e n d e d to con c a v e functions of several v a r i ables and implies the f o l l owing result.

Prop o s i t i o n 3.2

Let b cV, be a set of p r o b a b i l i t y m e a s u r e s with zero

means and such that e xists for all J

then \ d ( S * v ) ( З Ж З ) 4 ^ < 4 ( 3 ) F U ) = F (3.5)

Remarks

1. If E denotes the mean value and A the mea n square d e v i ation

then E ^ y = E ^ + E v and . Hence in proposition

"X. дХ

3.2 we have - E у and A ^ v >/ A y , so that £ * V is indeed b r o a dened in comparison to ^ and does decrease F.

2. Replacing

d S b bY d ( ?v>*v ^

b 4 b "b r b

d i s t r i b u t e d a c c ordin^to dU>b .

3. The inequalities (3.1) follow from this th e o r e m if in the latter we replace by <£(х-ТГь) for b 6 B 7 and take

for b ^ B* a n d b t В

Vb(X^ - + for b € b / 6>

In vi e w of the second remark, the s h a r pening p r o b l e m can be f o r ­ mulated in this way: given a r a n d o m variable we have to find a non trivial decompo s i t i o n of into the sum of two ind e p e n ­ dant random variables, one of t h e m having zero mean. This p r o b l e m can not be g e n e r a l l y solved. However if the J ’s are

D г

Gaussian R a n d o m variables then such a d e c o m p o s i t i o n is possible and we have the following result.

P r o position 3.3

Consider the quenched free e n e r g y ass o c i a t e d with two different set of G a u s s i a n distributions i C i and such that

EL^U>= and Д о ( Ц >/ Д\> U ) for all b -

then ^ ( 3 ) F ( J ) ^ ^ < ^ U > 0 ) F ( 3 ) (3.6)

P r o o f :

Let be the gaussian measure w i t h zero m e a n and m e a n square deviation A - Л у ti) . Th e n ^ ° = V b and the statement follows from p r o p o s i t i o n 3.2.

There is a nother way to ge n e r a l i z e the inequality (3.1) which shows that the approximation of a given d i s t r i b u t i o n with a

suitably chosen discrete distribution results in the increase of the quenched free energy. The better the discrete d i s t r i ­ bution approaches the original one the smaller is the upper b o u n d .

Proposition 3.4

Let dj*b (x) = dj^(-x) for all b and let

+ + w h e r e = \ < Ц к ( . х М М

then: — r _

F i J lv(3) F (3) Í F U ) .

P r o o f :

We m a y fix the interactions wit h the e x c e p t i o n of the single J, and it is suf f i c i e n t to show that for the concave function

b

F (J, ) the inequalities b

) F(*)asucx

á L(F(lTblUFC-l

b

) if(o)

- OÖ

hold. Here the s econd inequality comes from the def i n i t i o n of a concave function. Now 2 dl^ is a p r o b a b i l i t y measure on bot h

[R+ and [Rf and hence

i ^ F O O d ^ c * ) « F ( a ^ * < * f k0 o )

— OO

2 ^ F U ) i ? t O O $

О

But

2 ^ x J L . O O = - 2 - '

0 J b Л

which concludes the proof.

The g e n e r a l i s a t i o n of this propo s i t i o n for non-even d i stributions

and for mo r e d e t a i l e d p a r t itions of the support of ^ is easy and we leave it to the reader.

4. V a r iational principles

The e x i s t e n c e of a variational p r i n c i p l e for the free energy is a m a n i f e s t a t i o n of equilibrium. Since the quen c h e d s ystem is not in e q u i l i b r i u m we cannot e x p e c t that a variational principle c o m p l e t e l y a n a l ogous to that of e q u i l i b r i u m systems will also hold for F. In mean field c a l c u l a t i o n (Edwards and Ande r s o n , 2 S h e rrington and Kirkpatrick^) the free energy of the quenched state at low t e m perature is above the c o n t i n u a t i o n from high temperature» this c o n t i n u a t i o n being esse n t i a l l y the annealed free e n e r g y (see P r o position 3.1). In this section we give some insight on this mea n field result by showing that the quenched free e nergy s a t i sfies a v a r iational principle on a subspace (0>o of the space <2.14); on the other hand the a n n e a l e d free energy satisfies a variational p r i n c i p l e on the whole space (5.

t

Le us r e call that for given i n t e r a c t i o n J, the v a r iational principle for the f i nite volume free energy is e x p r e s s e d by the ine q u a l i t y

F L 3; О S f(c*) L H(

+^en((G*)l (

.

)

Indeed the inequality (4.1) is a consequence of the inequality

(

.

)

where the free e n e r g y functional is defined by

on the space of d i s t r i b u t i o n functions f = f (O') , and

= g ( 3 , 0 is t h e ^ i b b s function (2.4).

For random systems we consider the free e n e r g y functional (2.15) d efined on (& and we also i n t r o d u c e d the subspace of <2> defined

о b y :

Proposition 4.1

1) F = F [ h } = min F [ f аП f & d

2) For a n ^ f G 5 )

F [ ( l l - F ( I ) > / 0

and the equa l i t y holds for f = g (J,C*)

3) F = F [ g ] = min F Í f ] f€(B0

Proof: 1

(4.4)

(4.5)

(4.6)

1) Using the defi n i t i o n s (2.12), (2.13), (2.15), we have

2)

Ft { 3-Га = - -i- ( (t 3 6*) tn ^ V [I у о

M 5 r [ ( 1 , 0 '

Ft{] -JayCJ)a^(tf)((j/a*)F(3) =

- - ~ ^ d ^ ( 3 ) 3,(3*) Cr\ ^(-3,0*) Уу О 3) Using (4.5) we have for any f € (^

Í U / )

F t f 1 ^ F

which concludes the proof.

Another way to formulate the above result is the following: in the space (£> of the joint p r o bability d i s t r i b u t i o n s the m i n i m i ­

zation of F £ f yields the anne a l e d free energy, while the q u e n c h e d free e n e r g y is o b t a i n e d by m i n i m i z i n g the difference

(4.5) .

To conclude this section we remark that the q u e n c h e d en t r o p y S and the annealed entropy S are not n e c e s s a r i l y positive for the scales of the entropies d epend on the norm c h o s e n for the

a priori measure JXo . Indeed changing to CjiD will change S(J)C2-0 to S (J ) + IVI ln C. If the nor m a l i s a t i o n of yxQ is c hosen so

that S ( J ) ^ о (e.g. if g ( J , C f ) ^ . d - for all (J,6*) ) then the quenched entropy S will be non-negative; h owever the annealed entropy may still be negative, in p a r t i c u l a r for large values of ^ . As an ex a m p l e we m e n t i o n that for the Ising spin b models, the choice

du.(ti') = c O’-l') ♦ i (Cf-tl')} dc" (4.6)

dSbU1 = d?b(-x)

will give

f an — - — L W l + 2 .

tv\

* ~\

U v J

and

This shows that goes to |V|ln 2 if goes to 0, changes sign wit h increasing ^ and goes to - ®° with going to infinity . Therefore one can imagine that u n d e r e s t i m a t i n g F w i t h an i m p r o ­ per c h o i c e of f in ß may r esult that F [ f ] app r o a c h e s F s u f f i ­ ciently c losely that a n e g a t i v e entropy will be o b t a i n e d for large v alues of p> . A similar mi s t a k e really o c c u r r e d in the original paper w r i t t e n on the S h e r r i n g t o n - K i r k p a t r i c k model'*.

* At least for di s t r i b u t i o n s d ^ C x ) - Jx. , with C l.

5„ S t a bility of the Gauge Symmetry under B o u n d a r y Perturbation

The boundary c o n d itions p l a y a predo m i n a n t role in the d e f i n i ­ tion of infinite volume e q u i l i b r i u m states. In par t i c u l a r if the h a m i l t o n i a n has some internal symmetries the i n t e r a c t i o n with the fixed b o u n d a r y spins breaks the symm e t r y and may lead to a

"spontaneous breakdown" of this symmetry in the thermodynamic l i m i t .

The random H a m iltonian (2.1) has also internal symmetries. H o w ­ ev e r we shall sh o w in this section that in "pure" models of

spin glasses the gauge symm e t r y cannot be b r o k e n by the boundary conditions. This implies that a spin glass c annot gener a l l y be c a r a c t e r i z e d by a local order parameter; in fact the order para- m e t e r prop o s e d by Edwards and Ande r s o n 2 is the e x p e c t a t i o n value of a non-local quan t i t y taken with the m e a s u r e (2 .8 ) (see 6 .2 ).

This order p a r a m e t e r can be c o n s i d e r e d as a local observable o n l y if one introduces the so-called replicas; in this case how e v e r one looses the c lear descr i p t i o n of the quenc h e d states as p r o b a b i l i t y measures. This explains some w h a t the d i f ficulty to prove the e x i s tence of phase trans i t i o n s in quen c h e d models.

In this section we c o n s i d e r h a m iltonian of the form

H(3,cr) = - Z Í _ \¥ = _ 2.

ь °<ri «еь ^w<4>

w h e r e (Г1. is the tk c o m p o n e n t of the spin at site i, and the l ,<*

^ ' s are independent r a n d o m variables.

We assume that the i n t eractions have finite range, i.e. there exists some R>0 such that ^ is stri c t l y zero for any b w i t h diam ( b ) > R . F u r thermore we cons i d e r o n l y "p u r e " models of Spin glasses, i.e. m o d e l s with even d i s t r i b u t i o n for each interactions

1 b .o<.

(x.) - d<

F i n a l l y we assume that the a priori m e a s u r e ^ on S c R i s even in all components

d|A.0 С Ь!б\,. . . . • /Sy a"v ') -

(с^,..../CJW ) (5.2b)

bet C| = j Ь : {b,/04^ 2 S.> = S 1 ] у for апУ s in

^ we introduce the a u t o m o r p h i s m ^ ^ d e f i n e d on the algebra of local o b s ervables by

U . o = f ( b 3 , scr)

w h e r e :

^ S J

i€b

(ьо-).,ч - Si,

<r

(5.3)

T h e s e transformations are internal s y m metries of the system, since they leave the H a m iltonian and the m e a s u r e invariant ^i.e.

and d ^ W ) = d^Ctr) = dj(3)

Let us note that ^ is a g r oup of gauge t r a n s f o r m a t i o n s since it is generated by t ransformations involving on l y a finite

n u m b e r of lattice sites. We shall not cons i d e r other symmetries the system might also have.

We are interested in local observables of the form

(5.4)

w here n^ ^ and nr are non-n e g a t i v e integers which are d i f f e ­ rent from zero only for a finite number of b's and i's. We introduce the notation

supp f ü b U U l C L

b:V ! to

For any f of the form 5.4 we have

Nf (s) -- 2 * v * |v K( i ) n t l + 2. w.-.jfc I V(v(s)n {0

v,* fs) - { J* L ;

and therefore f is gauge breaking, i.e. И f, if N f (s.) is odd for some s in G in which case

о Ó

Zi<t = -t

From the invariance property of H y ancl d Q v we have immediately the following result.

Proposition 5.1

Let f be any g a u g e - b r e a k i n g observable of the form (5.4).

Then

^ f a q v = °

if V D •

(A similar result was obtained by Avron et al^ in the case where f is a pure product of Ising spins).

To show the absence of spontaneous breakdown of this gauge symmetry we consider quenched states with boundary conditions dQ^*’0^ . Such states are defined through (2.8) by the

hamiltonian

H v = - b

b n v

f(f>

together with the boundary conditions ( С П - Í ; if

T (5.5)

?u CO if b^v/

t 3 bjoc

for "fixed" b o u n d a r y c o n d i t i o n s - b ( X - J b/4^ •

P r o p o s i t i o n 5.2

Let f be any g a u g e - b r e a k i n g local o b s e r v a b l e of the form (5.4).

Th e n for any b o u n d a r y c o n d i t i o n s (5.5)

< f > = 5 i 4 k c ) = °

if dist ( / V C ) > ^ P r o o f :

Let us take S0 in C such that and s . = 1 for

Q ( b e )

i o u t s i d e supp f. The m e a s u r e dQ^ * is th e n invariant under this t r a n s f o r m a t i o n and y ields < f > = - < f > = 0 .

We n o t i c e that this t h e o r e m does not e x c l u d e the exist e n c e of a phase tra n s i t i o n in the sense that d i f f e r e n t weak limits of the quen c h e d m e a s u r e s d Q v ^ ' C ^ can be obtained, i.e. the p o s s i b i l i t y still stands for the n o n - u n i q u e n e s s of the e x p e c ­ t a t i o n value of some local g a u g e - i n v a r i a n t quantity. However, as far as we know, such a h y p o t h e s i s has n e v e r a p p e a r e d in the the literature and can be q u a l i f i e d as "unphysical". An e v e n t u a l s p i n -glass t r a n s i t i o n is e x p e c t e d to be c h a r a c t e r i s e d by a s i n g u l a r i t y in the t h e r m o d y n a m i c f u n c t i o n s and in a n o n ­ local order parameter. Th i s we discuss in the f o l l o w i n g section

6 . The E d w a r d s - A n d e r s o n Order Parameter

The o r der p a r a meter p r o p o s e d by Edwards a n d A n d e r s o n 2 to d e s c r i b e the spin glass t r a n s i t i o n is d e f i n e d as <l<0'o>l >

w h e r e the first average is the thermal one a n d the second is t a k e n over the i n teractions. In o r der to o b t a i n a non zero value

one has to c hoose some b o u n d a r y conditions. We take for the d i s t r i b u t i o n of i n t e r a c t i o n s J, across the b o u n d a r y the

D f

same d i s t r i b u t i o n as the one d e f i n i n g the s ystem and

j ь /«x. /\ £

for the external spins some c o n f i g u r a t i o n СГ ; we denote the quen c h e d state a s s o c i a t e d with this b o u n d a r y condition.

W i t h our notation the o r d e r p a r a m e t e r is the t h e r m o d y n a m i c limit of

£ Л ( 6 -1

= < U < r . > ”

)l >

* Л

p r o v i d e d this limit exists. In (6.1) 9 ^ ( 3 , o') is t W e

p r o b a b i l i t y d e n s i t y (2.4) a s s o c i a t e d wit h the b o u n d a r y condi- л

tion (Г* .

Л £

We note that °* has two p a r t i c u l a r features: F i r s t l y

is a non-local o b s e r v a b l e wit h respect to the q u e n c h e d measure.

Indeed

C = o'. <o-.>ft3)

and

O'.

(3)

S e c o n d l y is the e x p e c t a t i o n value of an o b s e r v a b l e w h ich is invariant with respect to § v = l s > i; = 1

I ndeed for any s in ^ we have

■'s

■&s!X,4 <0'ол >Г (3)3

( htre f o r e is i n v a riant u n d e r any gauge t r a n s f o r m a t i o n s in G (see also A v r o n et a l 6 ) .

B ecause of this g a u g e - i n v a r i a n c e c h a r a c t e r will not d i s t i n g u i s h b e t w e e n different gauge b r e a k i n g phases. It is h o w e v e r an order

p a r a m e t e r in the sense that it is zero at hig h temper a t u r e s (Proposition 6 .1)and is perhaps non zero at low temperatures if the d i m e nsion of the l attice exceeds some finite value. As we shall see b e low (Proposition 6.2) Q is i n d e p e n d e n t of the

1

b o u n d a r y conditions cr'. This b e h a viour of Cj implies a new type of low t e m perature p h ase - the spin glass - for the local m o m e n t < < (J'p'^vanishes at all t e m p e r a t u r e s in any model s a t i s f y i n g (5.2) ( P roposition 6.3).

P r o p o s i t i o n 6.1 (

Let us consider Ising m o d e l s ( 0"^ = ±1) w i t h finite range even

★

i n t e r a c t i o n s and eve n d i s t r i b u t i o n of s p ins and bonds (5.2).

Sup p o s e the i n t e r a c t i o n s are i n d ependent r a n d o m v a r i a b l e s and the number of d i f f e r e n t d i s t r i b u t i o n s is finite. Th e n the o r d e r p a r a m e t e r Cj vanishes for s u f f i c i e n t l y h i g h temperature.

P r o o f :

7

Using G r i ffith's i n e q u a l i t y and | 0*o | ^ 1 we find that

4 * = < | < с - . > " ' ( л ) Г > í < о * 0 >

w h e r e |J| = |j J | ^ and + means that the b o u n d a r y spins are positive. N o w let R be the range of the i n t e r a c t i o n and

Л

Я

c o r r e l a t i o n s using self-avoiding w a lks g ives

« о : о з о $

Here the prime indicates that (b^o . . . . о b n )П V = { 0 ^ and

there is no n o n - empty subset i b V •' bc ] C { b t/. • • ,ЬЛ ^ such that bijO-'-’pbj^ = 0 (А о B = А \ в О Л О В ) . U n der re a s o n a b l e a s ­ s u m p t i o n on the lattice and the set of ^.b] there e xists a c o n s ­ tan t c, depending on L. and on the i n t e r a c t i n g sets, such

i.e. J h = 0 if I b I is odd.

that the number of sets £ ь А , bn ^ in (6.3) is smaller than C n . Moreover,

0 ^ W A j b l x l d ^ U O .< í Cfb) ^ 1 - oO

w here £(|J) goes to zero with ^ g o i n g to zero.

Therefore, for small e nough (b , C . £ (p) < 1 and

X t c . K ^ " í

-Vr

< < < r . \

C m ) > í Z - ' v ■ . <6-4)

V r v ^ A / R i - c . u p

If V increases then

A

In the f o l l owing part of this s e c t i o n we di s c u s s some formulas o b t a i n e d for Ising m odels by g a u g e fixing.

P r o p o s i t i o n 6.2

Consi d e r an Ising m o d e l s ( 0^ = + 1) with even d i s t r i b u t i o n (5.2) and let f (J , 0*) be a func t i o n of 0" -s for i £ V and of J. ’s for

1 b

b О V ф 0. Then

i) If f is invariant under

^ d Q v £ = 2 ^ cif (.J) J * ( 2,+ ) £ ( G" J / + ) (6.5)

/4 xs A

where in <T 3 , O' is e x t e n d e d to V w i t h the v a l u e CJ'c = 1 for i £ V and + denotes the c o n f i g u r a t i o n (Tj, = + 1.

ii) If f is gauge invariant

= alvl^a5C 3 ) ^ ( 3 (+)(C3/+) (6.6)

P r o o f :

Using the invariance p r o p e r t y (5.2) and ‘ft.f = f for all s £ C

S

*

we have:

U Q v | 1 - í <Ц 1 з) < y u r ) ^ ( 3 ,«') fia,«1)

■ U ? C J ) d r « r ) < £ ( 3 / + ) f ( J , + )

- ^dj(.3)dyA(.tr) 3 ^{+(. J,} + ) f ($3,+ )

- W 3 ' 3 Í

w h i c h c o n c l u d e s the proof of (i) and (ii).

Let us note that accor d i n g to (6 .6 ) the e x p e c t a t i o n v alue of any gauge-invar iant. g u a n t i t y is i n d e p e n d e n t of the b o u n d a r y c o n d i t i o n and can be o b t a i n e d by "fixing the gauge " at

G"\

14/1 .

= 2 cj* ( 3 , + ) c i ^ C 3 ) (6 .7)

P r o p o s i t i o n 6.3

For Ising m o d e l s ( (Tj= -1) with eve n d i s t r i b u t i o n s (5.2)

> Г (6.8)

= ^ > g t

w h ere A and В are subsets of V and CT* denotes Т Т Л .

A L£A

P r o o f :

For A = В (6 .8 ) follows immedi a t e l y from (6 .6 ) since

A

Z b ( C a « 4 ^ ( 3 ) ) = < С А >Т ( 1 ) ( í 3 ) =■ <(TA > ; ( ? )

and

For A ^ В we can write

« ° Á > C < < ^ > v > =

u s i n g the fact that

flA < ° A > v ( J > = « v > f (a>3>

and /n A

3vr (3,<r) = 9£ < « , ♦ )

we find

■ < « r A > f < c r 6 > f > = ^ / » v C<r)C>A fl,e ^ d , W ) 5 J u /t ) « lfA > V , , l

= 2 ,vl i A/6 [ $ < * j l * > g í U , 0 ( 3 ) ]

w h i c h concludes the proof.

C o n s e q u e n c e s of P r o p o s i t i o n 6.3

1. For A = В = ^ 0 ^ we find

q f V = < « o ; v <6Л0)

/\

w h i c h shows that C| is i n d ependent of the b o u n d a r y c o n d i t i o n s (Г .

2. For A = ^ i^ and В =

j

« ^ « t i ^ y - 0 (6-u >

This h e u r i s t i c a l l y obv i o u s result was used e a r l i e r (see e.g.

F i s c h e r )to conclude that the q u e n c h e d s u s c e p t i b i l i t y X is p r o p o r t i o n a l to 1-^j . i ndeed

x v ~ i Z ( ««'io, j>> - <«u><°j>>)

iv| vjev

Г ~ Z _ ( 1 - < < 0'1 >г > ) +

/V/ i€v

+ X Z _ ( « G \ G j » - « 0*1 > < О > )

Using P r o p o s i t i o n 6.3 it follows that b o t h terms in the s e c o n d

summation v a n i s h w h ile « 0*i> >t ends to C| in the t h e r m o d y n a m i c , limit, at least for a t r a n s l a t i o n a l l y invariant state.

« 3. For any А <<С(Г ^ ~ 0 i n d ependent of the b o u n d a r y c o n d i -

6

tions (see a l s o Prop. 5.2).

The i n e quality

< ( < < ^ > $ У > = \ а 3; и х < г А > Ч з ^ > °

suggests that dg^(J) favors the f e r r o m a g n e t i c interactions.■f This is cor r e c t in the f o l l owing sense.

P r o p o s i t i o n 6.4

For Ising m o d e l s w i t h even d i s t r i b u t i o n s (5.2),

^ ° (6.12)

holds for all interactions.

P r o o f :

It is sufficient to show that O x ) is an i n c r e a s i n g function

of J . Indeed } ^

-p. ^ 0 , 0 - P 3 t ( J ' + ) < < r b > v •

C o r o l l a r y

The avera g e d e n e r g y of any bon d is n o n-positive. Indeed, the a v e r a g e d e nergy of the b o n d b is

- \ ^ V v t 3 ) é ° , <6-i3)

a c c o rding to U - O and (6 .12).

To conc l u d e this section we e s t a b l i s h the c o n n e c t i o n b e t w e e n the order p a r a m e t e r q and the d e r i v a t i v e of the c[u e n c h e d free energy, w i t h re s p e c t to an e x t e r n a l f i eld h. Let F (J) be the

AV

free e nergy in v o l u m e V with b o u n d a r y condit i o n e r . S i nce F^(J) is gauge invariant, (6 .6 ) implies

(6.14)

However

< < = S J т Л ) ir,L«) a S ( 3 )

= ^ F +v C 3 ) d j O ) - T v

(6.15)

T h e r efore the q u e n c h e d free e n e r g y is i n d e p e n d e n t of the boun d a r y c o n d i t i o n and

Fv = (-3 ^ v < ¥ 3)

Let now F^ (J, h) be the free e n e r g y d e f i n e d by the e q u a t i o n

e x p ( - f F j c 3 , V . O = 2 L м р ( Р,5 г Л к/ ь

dV.UV bOVff

w h ere = 3 - 1 v. €-V . Let m o r e o v e r

? V U ) = l ' V ‘ ^ F v+ ( 3 , W ) 3 M 3 , * ) d ? ( 0

We should e m p h a s i z e that, in the above d e f i n i t i o n g* does not depend on h. Therefore, F^(h) is not e q ual to the q u e n c h e d free energy in the p r e s e n c e of a n o n - r a n d o m e x t e r n a l field, though F (0) = F . The r eason for the i n t r o d u c t i o n of F (h) through

v v v

Eq. (6.17) is that it is cou p l e d to the a v e r a g e d order p a r a m e t e r

Q „ = - i - 2 . q (i) (6.18)

v IVI f c v V

^ I <*

where < ^ (0 - 2 U { ( T ) 3 + ( X + l

4 just in the same w a y as in n o n - r a n d o m m o d e l s the free e n e r g y is

c oupled to the a v e r a g e magne t i z a t i o n : the c o m p a r i s o n of Eqs. ( 6 18) and (6.17) yields

" 1 ц ( * Г ? ''о Л ) = <3''

Therefore, in a t r a n s l a t i o n a l l y i n v a r i a n t phase the o r der p a r a ­ meter Q can be o b t a i n e d as the t h e r m o d y n a m i c limit of the l.h.s.

of Eq. (6.19) - p r o v i d e d that this limit exists.

7. A n n e a l e d m o d e l s w i t h one, two and t hree phase t r a n s i t i o n s

The annealed m o d e l s (see Section 2) are u s u a l l y c o n s i d e r e d to be trivial and h e n c e of no further interest. This o p i n i o n c o mes from the fact that an anne a l e d m o del w i t h even d i s t r i b u t i o n s for the bonds is in fact equ i v a l e n t to a m o d e l w i t h o u t any interaction (see (4.7) and (7.2)).

Non trivial results can be obta i n e d e i t h e r by intr o d u c i n g i n t e r ­ actions among the bonds or by destroying the symmetry of their a priory d istributions. As an exa m p l e to the former p o s s i b i l i t y we me n t i o n the A s h k i n - T e l l e r model 10 in w h i c h two c o n s e c u t i v e phase transitions were c o n j e c t u r e d by W e g n e r ^ and p r o v e d r i g orously by Pfi s t e r12

Here we exhibit s imple examples of a n n e a l e d models wit h a s y m m e t r i c

»

b o n d distribution, in which one, two or t h ree p hase transitions take p l ace as the t e m p e r a t u r e changes, d e p e n d i n g on the choice of the la t t i c e and of c ertain parameters.

We shall cons i d e r o n l y Ising models, i.e. (T{, = ±1; our d i s c u s s i o n is based on the f o l l o w i n g simple observ a t i o n :

Let f = f (O') be a local obse r v a b l e ; then

=• j j A a o f c o = ^ ! ь (тк ) е ? Ть°*и

But <?, — ТГ CT^ ~ + 1_ implies

,СЬ c rt 2-К.<Г.

< £ > -

o f w i e ; _ _ _ _ _ _ (7. D

where K W = = IW «’ -2)

t

*

Equa t i o n (7.1) shows that the a n n e a l e d s y s t e m is e q u i v a l e n t to a s p i n ^ - s y s t e m with the same l a t t i c e and b o n d s t r u c t u r e with fixed intera c t i o n s ^ (|b) . T h e r e f o r e the p o s s i b l e phase t r a n s i t i o n s of the a n n e a l e d s y s t e m can be i n v e s t i g a t e d using the known phase tra n s i t i o n of the s p i n Ц system. It s h o u l d be

stressed that the i n t e r a c t i o n s in the c o r r e s p o n d i n g model are (h d e p e n d e n t and this will lea d to the e x i s t e n c e of several p hase transitions. One should a l s o note that

and sign =

In p a r t i c u l a r

K, (0) = 0 b

Sign ( $<^Cx)sUftx)

for small

sign Kb = s x g n ^ ^ cx;.x ^

To illustrate the p o s s i b l e existence of several p h a s e t r a n s i ­ tions we restrict o u r s e l v e s to the s i m p l e s t a n n e a l e d Ising m o d e l s

H = - Z - Jji (Г: (7 .3a)

wit h nearest n e i g h b o u r i n teractions d i s t r i b u t e d a c c o r d i n g to

$ ( 3 ; р = p £ ( 3 i j - a ) + (7.3b)

and the a priori even me a s u r e jXD

^ Д < Г ) -

S

In this case (7.2) yields

. H - p ) ^ \ ,7.4,

2 Ч р е - Р л г ( 1 - ^ е Р Ь >

T h e r efore for small sign К = sign (a - b)

for large (Ь К rsJ ^ (b.ln (a-b) if a > b -*s ^ . In (b-a) if a <, b h In (7E-) if a = b

>1- -1 - - ж -

I) a > b II) a = b III) a < b

а b

P i) - P > *5 i)--- a > £ b

P a < ^ - £ b

P ii) ---- p < Ц i i ) ---- a < b

P

In c o n c l u s i o n if the spin ^ - s y s t e m has an o r d e r e d p h a s e for 1KI у Kc , then the a n n e a l e d s y s t e m will have at least one phase t r a n s i t i o n in the case I and III; it will have at least one phase t r a n s i t i o n in the case II if *51 In -■£ - I s К ; it will

1-p ' c

have at least t h ree phase t r a n s i t i o n in the case I-ii if К . ^ — К and in the case Ill-i if К > К .

c m a x ' с

P r o p o s i t i o n 7.1

Let us consider the a n n e a l e d Ising model (7.3) w i t h a = b = J on a d - d i m e n s i o n a l simple c u b i c lattice w i t h d^,2.. Let Kc be the c r i t i c a l value of the spin H model w i t h J.. = J > 0.

T h e n :

L) For P > fc -

lfcc

1 £IK, (7.5)

there e x i s t s two f e r r o m a g n e t i c a l l y o r d e r e d phases for

(ь > 6 P = Л Л п М 1 * - * Л 2 7

ii) for p ^ 1-p^ there e x i s t s two a n t i f e r r o m a g n e t i c a l l y o r d e r e d phases for P > ( W

Let us recall that for d = 2 we have sh 2 К = 1 w h i c h yields

1 c

p c NfT * P r o o f :

For a g i v e n Pj К (p) given by (7.4) is p o s i t i v e m o n o t o n i c a l l y increasing if p > >5 (resp. n e g a t i v e m o n o t o n i c a l l y dec r e a s i n g if

р < ^ ) and (7.5) implies that К ( | Ъ ) > К с for (resp.

к (p) < - for p > p i - F ) which c o n c l u d e s the proof.

P r o p o s i t i o n 7.2

Let us consider the a n n e a l e d Ising m o d e l (7.3) w i t h b > a > 0 ; then there exists some p > h such that for p > p

c c

i) on the d-dimensional simple cubic l attice w i t h d 2, there exists 0 < ß © < о© such that for p><

and for there exists a u n i q u e (paramagnetic) «

e q u i l i b r i u m state, for Jb<, <-|b < , there e x i s t s a f e r r o ­ magnetic o r d e r i n g and for [b>^>2 there exists an a n t i ­ ferroma g n e t i c o r d e r i n g .

ii) on the 2 - d i m e n s i o n a l triangular l attice there exists such that for p<Jb0 and for |i > there exists a u n i q u e (paramagnetic) e q u i l i b r i u m state while for ß 0 < £»> < p 3 > there exists a f e r r o m a g n e t i c ordering.

P r o o f :

Let ^ ^ < for f i xed p , к ) Eq. (7.4) is an i n c r e a s i n g function of ^ w h i c h tends to |2»G- as ^ tends to infinity; t h e r e ­ fore for any a m a x (, K ( f b ^ ) К if ^ c . Furt h e r m o r e

^ К ( о

- 0 if |b is the solution of

ch (a + b )

P

г I - b (b

which is u n i q u e l y s p e c i f i e d by (a, b, p); t h e r efore for g i v e n p , К (p) is a c o n c a v e function which shows that there exists

exactly two v alues and such that К (p) = .

N o w on simple c u bic lattices the c r i t i c a l t e m p e r a t u r e s are de t e r m i n e d by К ( ^>л ) = К ( ^ ) = and К (Л-») = -К^. For the triangular lattice К ( = К and there is some so that К (p ) < 0 for p>|3 • However, the a n t i f e r r o m a g n e t i c

model does not u n d e r g o any phase t r a n s i t i o n at (h < oo (Wannier1 3 )

Acknow l e d g e m e n t

One of the authors (A.S.) w i s h e s to t h a n k the Institut de Physique Théorique, U n i v e r s i t é de L a u s a n n e and p a r t i c u l a r l y Profs P. Erdős, J.-J. Loeffel, G. Wanders and F. R o t h e n for their ki n d h o s p i t a l i t y during his stay in Lausanne. He is a l s o indebted to the members of the Institut de Physique Théorique, EPF L a u s a n n e for m a n y useful d i s c u s s i o n s .

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G. P á r i s i and G. Toulouse, J. P h y s i q u e Lettres ££, L36 1 (1980) 4. S.G. Rosa, Physics Letters 78A, 468 (1980)

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У

Szakmai lektor: Dr. Forgács Gábor Nyelvi lektor: D r . Tüttő István Példányszám: 570 Törzsszám: 82-573 Készült a KFKI sokszorosító üzemében Felelős vezető: Nagy Károly

Budapest, 1982. november hó