Physica 128A (1984) 529-534 North-Holland, Amsterdam

S C A L I N G A N A L Y S I S O F T H E d = 3 R A N D O M F I E L D I S I N G M O D E L

Laurence JACOBS ~' and Michael NAUENBERG*

Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA

Received 4 June 1984

We present a Monte Carlo simulation of the d = 3 Ising model in a random magnetic field with values -+H on each lattice site. Calculations have been performed on cubic lattices with linear dimension L = 6, 8, 10 and 12, and several values of H. A finite-size scaling analysis of the data for H = 0.5, 0.75 and 1.0 is consistent with hyperscaling, and a correlation exponent ~ = 0.7-0.1. For H = 1.50, 1.75 and 2.0 some of the L = 12 data show a discontinuity as a function of the inverse temperature ft. The implications of these results are discussed.

R e c e n t e x p e r i m e n t a l resul ts 1-3) have led to a r e ne w e d in te res t in the de te r -

mina t ion of the lower cri t ical d imens iona l i ty , de, of the Ising mode l in a

r a n d o m magne t i c field. Since the p ionee r ing work of Imry and Man), who

a rgued that dc = 2, the re have been cont inu ing con t rove r s i e s abou t the va lue of

d~. These con t rove r s i e s s t a r t ed af ter it was shown that the e - e x p a n s i o n 5) and

s u p e r s y m m e t r y 6) imply d~ = 3, and the d i s a g r e e m e n t r ema ins un reso lved up to

the p resen t t ime. D i s a g r e e m e n t s have occu r red also in e x p e r i m e n t s in dis-

o r d e r e d magne t i c sys tems c o r r e s p o n d i n g to an Ising a n t i f e r r o m a g n e t with

r a n d o m impur i t i e s in a cons tan t magne t i c field, which has been p r o p o s e d as a

r e l a t ed modelT). W h i l e b i re f r ingency m e a s u r e m e n t s 1) give ev idence for a phase

t rans i t ion for the th ree d imens iona l mode l , some neu t ron sca t te r ing da t a 2) and

magne t i za t i on and specific hea t measu remen t3 ) ' i nd ica t e that long range o r d e r is

d e s t r o y e d by the r a n d o m fieldS). F ina l ly there have been M o n t e Car lo s imula-

t ions on the d = 3 Ising mode l in a r a n d o m magne t i c fieldg'~°). H o w e v e r , these

s tudies have not t aken into account the expec t ed c rossover effects for finite

lat t ices be tween the ze ro field and the r a n d o m field as a funct ion of the

s t rength of the field and the size of the lat t ice. In par t i cu la r , the scal ing

a rgumen t s p r e s e n t e d here indica te that for finite lat t ices and small r a n d o m

fields this mode l is expec t ed to exhibi t behav io r s imi lar to the pure Ising model .

In this p a p e r we discuss a M o n t e Car lo s imula t ion of the d = 3 f e r roma gne t i c

Ising mode l with n e a r e s t - n e i g h b o r coupl ing J = 1 in a q u e n c h e d r a n d o m

magne t i c field having equa l ly p r o b a b l e values ___H at each la t t ice site. W e have

~'Permanent address: Bell Laboratories, H03M522, Holmdel, NJ 07733, USA. *Permanent address: Physics Department, University of California, Santa Cruz, CA 95064, USA.

0378-4371/84/$03.00 O El sev ie r Science Publ i shers B.V.

( N o r t h - H o l l a n d Physics Publ i sh ing Divis ion)

5 ~ L. J A C O B S A N D M. N A U E N B E R G

carried out a finite size scaling analysis of our data for cubic lattices of linear

d imensions L = 6, 8, 10 and 12, and for several values of H. Due to the small

lattice sizes we accepted only constra ined r andom field configurat ions with

vanishing total magnet ic field. In this analysis we have not averaged over

quenchings but compared results of different r andom field configurat ions for

the four lattices. For H = 0.5, 0.75 and 1.0 we find what appears to be a

second-order phase transition with a value for the correlat ion exponent ,

P = 0.7 -+ 0.1, consistent with that of the H = 0 Ising model. As an example, we

show in fig. 1 the spon taneous magnet izat ion M for H = 0.5 and L = 12. This was obta ined by starting from an ordered configurat ion at /3 = 0.3 and taking averages of 4000 iterations/spin at successively lower values of /3, after 2000

iterations at each /3 to reach equilibrium. Our conclusion, which will be

substant iated here, is that for these small r andom fields and relatively small lattice sizes one sees only the expected effects of the pure Ising model fixed

point. We want to point out that this result applies also to the simulations in

refs. 10 and 11, and cannot be claimed as evidence for the existence of a second-order phase transition in

1.75 and 2.0 and L = 12, some of the occurrence of a discontinuity

the r andom field Ising model . For H = 1.5, the data for a quenched r a n d o m field shows

as funct ion of /3. Howeve r it will be shown

that this is an artifact of the finite lattice, and it is not evidence for a first-order transition as was indicated in ref. 10, or as was found in mean field theory'3).

Accord ing to finite-size scaling theory the singular part of the free energy F

for the r andom field Ising model has the form

F = L-df( tL '/', H2L~/~), (1)

- I . 0 . i i i i i t i i i i ~ i i ! i i i i

Z X T I

0.5 ¢ -

M o [[III[[I • -05 L = I 2

H=0.5

" 1 . 0 I I I I I i I I I I I I I I I I I I

0.15 0.20 O.Z5 0.50 0.35

Fig. 1. Spontaneous magnetizat ion M as a function of/3.

SCALING ANALYSIS OF RANDOM FIELD ISING MODEL 531

where d is the dimensionali ty if hyperscal ing is valid, u is the correlat ion

exponent , & is a crossover exponent , and t = ( T - T ~ ) / T ~ , where T c is the critical t empera tu re of the pure Ising model . For the existence of an infinite lattice limit we require that for x ~ ~, y ~ ~ and z = x/y 1/~ fixed

f (x, y) = xd"s(z) . (2)

If there exists a crossover to a second-order r andom field phase transition, we expect that s(z ) has a power law singularity 7) at z = z~ of the form

s ( z ) = t z - z J • (3)

Here 07 is the new specific heat exponent associated with the r andom field which occurs at the critical t empera ture

Tc = L - zcT¢ H2/*. (4)

It follows that, in order to see this phase transit ion in Monte Carlo simulations on a finite size lattice, it is necessary that

HZL ~/~ ,> 1. (5)

A simple test would be to find evidence for a crossover in the value of critical exponents as a function of the pa ramete r HE/L ~/v.

The correlat ion exponent ~ for various quenched r andom fields H was

obta ined f rom the Monte Carlo simulation by evaluat ing the effective Ising coupling constant n'~2) gL for a cubic lattice of linear d imension L,

(S 4) gt. = (s2)---5- 3 , (6)

where s" = Z s~', and ( . . . ) denotes the thermal average. Accord ing to finite- size scaling theory,

gL = L- 'g( ' [L UO) , (7)

where i-= ( T - T c ) / T o and to is an exponent which determines the deviation f rom hyperscaling. An exciting consequence of some of the recent experi- ments 1'8) is the possibility of a large hyperscal ing violation. A t the critical point t = 0, eq. (7) implies

532 L. J A C O B S A N D M. N A U E N B E R G

_ ~o = in ln(L/L'). (8) v \ a t / ~ t / ~

For H = 0.5, 0.75 and 1.0 we obtain consistent solutions with to = 0 and = 0.7 - 0.1. As an example , we show in fig. 2 our da ta for gL vs./3 for H = 0.5

and L = 6 and L = 12. O v e r the range 0.21 < / 3 < 0.23, gL is approx ima te ly l inear

in /3 and can be fitted by straight lines which intersect at/3c = 0.225. This value of /3c also cor responds to the onset of spon t aneous magnet iza t ion (see fig. 1), and there fore implies to ~-0.

For H = 1.5, 1.75 and 2.0 we found evidence for discontinuit ies as a funct ion of/3. O u r result for (s 2) in a lattice of size L = 12 and H = 1.5 are shown in fig.

3. Start ing f rom an o rde red configurat ion with all spins up and decreas ing /3 f rom an initial value /3i = 0.32, we found a discontinuity in (s 2) at /3 = 0.28. Start ing f rom a r a n d o m configurat ion and increasing 13 f rom a small initial va lue /3 exhibits the hysteresis effect shown in fig. 3. Howeve r , s tar t ing f rom a configurat ion with all spins down and decreas ing fl gives results similar to the r a n d o m configurat ion run. H e n c e this discontinuity is due to an up-down a s y m m e t r y in the r a n d o m field configurat ion which obviously cannot exist in the infinite lattice limit. O the r r a n d o m field configurat ions give somewha t different results for the discontinui ty in (s2), and an average over quenchings would wash out this discontinuity, giving rise to a region where the values of (s 2) are spread over a band. While the occur rence of a f i rs t-order transit ion is found in the mean field approximations13), the behav io r we obse rved is not in accordance with this result. In this region the behav io r of the effective Ising coupling, gL, is also marked ly different f rom what we obse rved for small H.

0.5

- I .0

g

-I .5

I I I I I I I I I

o1_-'6 H=O.5 \

I I I I | t I I I

-2"00.21 0.22 0 . ~

Fig. 2. T h e Is ing c o u p l i n g c o n s t a n t g as a f u n c t i o n o f / 3 for two la t t ice sizes, L = 6 a n d L = 12.

SCALING ANALYSIS OF RANDOM FIELD ISING MODEL 533

0.8

0 .6

0.4 <$2> -

0.2

0 ' 0.26

l l f l l i ~ t l I

I

I 1

&

I I I

0.28

!

I &

I I I I

[ !

L=I2 H=I.5

I I J I I I I I

0.30 0.32

B

Fig. 3. The average of s 2 vs./3 for two cases: starting from an ordered configuration and decreasing /30) and starting from a random spin configuration and increasing/3(A).

Whereas there gL is approximate ly linear over a relatively large range of

t empera tures (see fig. 2), for H ~> 1.5, gL has discont inuous changes and large

fluctuations fo r /3 > 0.28 which do not allow a scaling analysis of our data. In summary , we have found that for values of the r andom field H = 0.5, 0.75

and 1.0 and lattice sizes L = 6, 8, 10 and 12, a finite size scaling analysis of our Monte Carlo data is consistent with the correlat ion exponent v for the H = 0

Ising Model. There is no evidence for a crossover to a new critical behavior characterist ic of the r andom field for these small lattices. For H = 1.5, 1.75 and 2.0 and L = 12 the data exhibits discontinuities as a function of /3 which do not

permit a scaling analysis. These discontinuities are due to up-down asym- metries in a finite size quenched r andom field, and cannot be claimed to be evidence 9) for a first-order transition as expected from mean field theoryl3). To clarify the issues discussed in the introduct ion of this paper, fur ther studies at larger lattice sizes are needed.

Acknowledgements

We would like to thank K. Binder for useful comment s and suggestions. One of us (M.N.) is also indebted to J. Cardy, G. Grinstein, B. Derrida, V. Jaccarino, R. Pearson and A.P. Y o u n g for st imulating discussions.

534 L. JACOBS AND M. NAUENBERG

Note added in proof

Recently it has been shown theoretically by K. Binder, M. Nauenberg, V. Privman and A.P. Young that the exponent to in eq. (7) must vanish. A.P. Young (private communicat ion) has carried out calculations similar to ours on a d = 3 L = 32 lattice which gives evidence for a crossover to a new regime. Simulations have also been carried out for lattices of size L = 150 which, however, show nonequilibrium effects. D. Stauffer, C. Hartzstein, K. Binder and A. Aharony, Z. Phys. B55 (1984) 325.

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