Controlling the Effect of Griffiths' Singularities in Random Ferromagnets

Brazilian Journal of Physics, vol. 23, no. 4, December, 1993

Controlling the Effect of Griffiths' Singularities in Random Ferromagnets: Smoothness of t he Magnetization J. Fernando Perez Instituto de Fisica, Universidade de São Paulo P. O. Box 20516, 01498-970 São Paulo, SP, Brasil

Received October 28, 1993 The magnetization of random ferromagnets with site or bond disorder is shown t o be a differentiable function of the externa1 magnetic field a t sufficiently high temperatures. This is shown to happen even in a region of the parameters where this function is not analytic as a consequence of the so called Griffiths' singularities. The result is proven through the use of correlation inequalities specific of ferromagnetic systems and with the weakest possible assumptions on the probability distribution of the random couplings. With the same methods one can actually prove infinite differentiability in the same region of parameters. We also discuss the analogous problem for ferromagnets in (d -+ 1) dimensions with d-dimensional disorder. In this case translation invariance in the extra direction produces slower decay rates in the presence of Griffiths' singularities. The results for the latter systems are oE importance in the analysis of quantum disordered ferromagnetic models.

the thermal expectations at inverse temperature P of an

I. Introduction In the Statistical Mechanics of Disordered Sys-

observable A(a) are defined in the usual way:

t e m a special role is played by the family of random ferromagnets.

A typical representative of this

class is the Ising model in z d , with configurations

a = {a, = f1,x E

zd) and

1

( A ) , (J) =- exp [-BHA (u, J)] . ZA ( J > ,,

(I.:!)

with energy function for

the system in a finite volume h C

zdgiven by:

The so called quenched expectations are defined as averages over the random parameters J and will be de-

-

noted by (A),. We will be primarily interested in these quantities considered in the thermodynarnical limit:

(xy) E where J = {JXy,

zd}is a family of identically

distributed random variables satisfying: Jry2 0, where the notation (xy) means that Ix - yl = 1, i.e. the

(A) (J) = lim (A)^(J) and A- Zd

(A) =

lim

A-+

(A)^ .

Zd

summation is taken only over pairs of nearest neigh-

zd. The model will be called bond disordered if the random variables J = { JZy , (XY) E zd}are

We shall be interested mainly in the case where the ex-

taken as independent, whereas if JXy=JxJy J where J =

will have to take the limit h

bor sites in

{J, >_ 0, x E

zd)are independent random variables the

model will be called site disordered. For a given realization of the random parameters J = {JZy, (xy) E

zd)

terna1 magnetic field h is zero, but in many cases we thermodynamic limit A

-t

-+ O

only after taking the

zd.

In 1969 R. ~ r i f f i t h d l ]considered the site diliite model, with

J. Fernando Perez S C Ird to be singular when P J , > KC(d)for a11 bonds

( 1 with probabilityp '( O with probability (1- p) '

(ij) in S . Then, even if the inverse temperature is such (1.3)

and called the attention to the fact that the quenched

that the system is not ordered (so that singular regions are mostly finite or even small) as a whole, there exist with probability one, arbitrarily large singular regions,

magnetization

i.e. finite regions inside which the system is strongly correlated. Another remarkable consequence of these singularviewed as a function of z = e-ph displayed a non

ities is that for those models, with either site or bond

analytic behavior at z = O even at values of the in-

disorder, where the couplings Jxymay assume arbitrar-

verse temperature ,O for which the system has neither

ily large values (even if with very small but non-zero

spontaneous magnetization nor long range order, pro-

probability) the usual high temperature expansions do

vided only ,hJ

>

K,(d), the critica1 value for a homo-

not converge for any values of

/3. An example of this

geneous d-dimensional deterministic system with cou-

situation is the bond disordered model where J,,

pling JXy = J for a11 bonds (xy). This phenomenon

probability p distribution like a one-sided gaussian:

has a

goes nowadays under the name of Griffiths' singulari-

O for J

ties; its existence has been rigorously proved[2] and it

< O2 x

exp -,o,

is now recogriized as a regular feature in the Statistical Mechanics of disordered ~ ~ s t e m s [ The ~ ] . physical ori-

with arbitrary variance o2

gin of this behavior may be better understood in the

p ( J ) > O for a11 J

,

with the property that

.

following situation which dramatizes the phenomenon.

The next question to be asked concerns the nature

Consider the site dilute model given by (1.3) and sup-

of this singularity. The first rigorous result control-

pose p < pc(d), where pc(d) is the critica1 value for the

ling the effect of Griffiths' singularities was obtained by

occupation probability of a site in the site percolation

~ ] discussed the Olivieri, Perez and Goulart ~ o s a [ who

i

problem in Ir . We are therefore in a situation where,

bond disordered ferromagnetic model and showed ex-

with probability one, only finite clusters of sites which

ponential decay of correlation functions in the presence

,are coupled (i e. J,, = J ) with their nearest neighbors,

of Griffiths' singularities. Their results implied also,

so that the system decomposes into a collection of fi-

although not explicitly stated in the paper, infinite dif-

nite independmt subsystems. In this case we may con-

ferentiability of the quenched magnetization (see dis-

clude that wit4 probability one, there is no spontaneous

cussion below). Their techniques however, could only

magnetization nor long-range order for a11 values of the

be applied to the specific situation of bond disorder and

temperature. However, as a consequence of the law of

for Ising systems, i.e. r, = &

large numbers, also with probabilty one, there are arbi-

average of the coupling Jxy .

/í.

It required also finite

trarily large d-dimensional boxes inside which, sites are

More general results, concerning exponential decay

coupled with 1 heir nearest neighbors with strength J.

of truncated correlation functions for not necessarily

Now, if /3J

> .r(,(d) there will be arbitrady large (but

ferromagnetic models, where obtained by ~ e r r e t t i [ ~ ]

finite!) boxes inside which the system is strongly cor-

with strong restrictions on the probability distribution

related, i.e. below the critica1 temperature, thus gener-

of the random parameters and subsequently by Fkohlich

ating the singb lar behavior .

and 1mbrie161through an intricate resummation of high-

The above simplified situation suggests the gen-

temperature or low-activity expansions. More recently

eral mechanisni for the phenomenon. Define a region

Dreifus, Klein and perezI7] produced a very general and

358

Brazilian Journal of Physics, vol. 23, no. 4, December, 1993

simple proof of infinite differentiability of tlie magneti-

tion. Our assumptions on the probability distribution

zation at sufficiently high temperatures, with no as-

of the random variables (Jcy),apart from ferromag-

sumption on the probabiblity distribution of the ran-

netism are the weakest possible. In the case of bounded

dom parameters.

spins (lo,

1 5 c, for some positive constant

c) JXy may

The purpose of this note is to show how the use of

even take the value +m if this happens with sufficiently

correlation inequalities specific of purely ferromagnetic

small prohability. This is to be compared with tlie

systems drastically simplifies the analysis. We are go-

methods of ref.[4], which crucially requires a, = f 1

ing to sliow exponential decay of the two point func-

and

tions and differentiability of the quenched magnetiza-

ity distribution of the coupling JXywe can also elimi-

tion as a function of the externa1 magnetic field, in the

nate the restriction on the values of the spin variable

presence of Griffiths' singularities at sufficiently high-

'Jx .

< m. With some restrictions on the probabil-

temperatures. Actually our methods can be sharpened,

Another interesting class of random ferromagnetic

along the lines discussed in ref. [7] to prove exponen-

spin sytems are the so called (d+ 1)-dimensional systeni

tia1 decay of a11 truncated correlation functions and in-

with d-dimensional disorder. A typical representative

finite differentiability of the qnenched magnetization,

of these models is the Ising model in

but this would go beyond the scope of this contribu-

function given by

where

J

=

( J ( X , ~ ) , < X ~ > E Z and ~)

K ={K (x) , x E Z) are two families of independent,

transverse field in

zd [9],

zdflwith energy

[lO] with Hamiltonian foi-

mally given by:

and within each family identically distributed random variables with J (x, y)

2O

and I( (x)

> O. The

spin variables a (x, t)are taken to be f 1, the symbol

< uv >denotes that u and v are nearest neighbor points

zdand T C Z are finite

where ui (2) , i = 1 , 2 , 3 are usual Pauli spin operators.

subsets. The characteristic feature of these systems is

These models appear also in the study of contact pro-

the fact that both "horizontal" J (x, y) and "vertical"

cesses in random e n ~ i r o n r n e n t s [ ~ ~ ~ ~ ~ ] .

in Zd or Z i.e.(u- v( = I ; A C

li'(x) couplings do not depend on the "time" variable

In this paper we briefly revisit the problem of tlie phase diagram of such models and provide some in-

t E Z. these models

sight into the nature of the effects of Griffiths' sing;u-

when realized in Zd x (;Z), that is with lattice spacing

larities for these systerns. It is intuitively clear that

($)hthe "time" direction and with the replacements

their effect should be even more serious than those .for

Apart from its own interest (see

[$I),

the standard random ferromagnets in z d + l : as a conJ (x, Y) J (x, Y) IO .

# y we have only the clusters with n > lx - yl

give a non trivial contribution:

so that

(c(x, t) u (y,s))

< e-'lx-yl

m

,-(t-sl

(2dp)"-'

exp -(2lx-y(K)exp -(2nK)

n=l

This shows that (õ (x, t )

u (y, s)S

c(K, 1 )

5 e-IIX-~I

-L

[It - slexp- (2 lx - yl I 2 K we have exponential decay in the space direction and polynomial decay in the time direction.

This work has been partially supported by CNPq and FAPESP.

The upper bound (3.6) can be used to prove differentiability of the quenched magnetization provided

&

References

is sufficiently large so that

1. R. B. Griffiths, Phys. Rev. Lett. 23, 19 (1969). whereas the lower bound precludes exponential decay of

2. A. Suto, J. Phys. A15, L749 (1982). 3. J. Frohlich, Mathematical Aspects of the Physics

the quenched correlation functions. The lower bound is

of Disordered Systems in "Critica1 Phenomena,

Brazilian Journal of Physics, vol. 23, no. 4, December, 1993

362

Random Systems and Gauge Theories" edited by

K. Osterwalder and R. Stora (Elsevier, Amsterdam, 1986). 4. E. Olivieri, J . F. Perez and S. G. Rosa Jr., Phys. Lett. 94A, 309 (1983).

5. A. Berretti, J. Stat. Phys. 38, 485 (1985). 6. J . Frohlich and J. Imbrie, Commun. Math. Phys. 96, 145 (1984). 7. H. von Dreifus, A. Klein and J. Fernando Perez, Taming Grifiths Singularities: Infinite Diflerentiability of Correlation Functions. To be published. 8. M. Campanino and A. Klein, Commun. Math.

Phys. 135, 483 (1991). 9. M. Campanino, A. Klein and J. Fernando Perez, Commun. Math. Phys. 135, 499 (1991). 10. A. Klein, Quantum Spin Systems in a Random Enuironment. Mathematical Physics X, edited by K. Schmunden (Springer, Berlin, 1992) p. 384. 11. A. Klein, Extinction of Contact and Percolation Process in a Random Enuironment, to appear in Annals of Probability (1993). 12. M. Fisher, Phys. Rev. 1 6 2 , 480 (1967). 13. H. von Dreifus and A. Klein, Commun. Math. Phys. 124, 285 (1989).