RECENT PROGRESS IN NUMERICAL SIMULATION OF SPIN GLASSES

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RECENT PROGRESS IN NUMERICAL SIMULATION OF SPIN GLASSES

K. Binder

To cite this version:

K. Binder. RECENT PROGRESS IN NUMERICAL SIMULATION OF SPIN GLASSES. Journal de

Physique Colloques, 1978, 39 (C6), pp.C6-1527-C6-1534. �10.1051/jphyscol:19786595�. �jpa-00218088�

RECENT PROGRESS I N NUMERICAL SIMULATION OF S P I N GLASSES

K. Binder

f i s t . fiir F e s t k i j r p e r f o r s c h u n g d e r KFA J i i l i c h , 0-5170 J i i l i c h , W-Germany

R6sum6.- On p a s s e e n revue des c a l c u l s p a r l a m6thode de Monte-Carlo appliqu6e 5 d i v e r s modPles : mo- d P l e s d t I s i n g avec de proches v o i s i n s a l d a t o i r e s ou une p a r t i e d ' i n t e r a c t i o n a l 6 a t o i r e i n f i n i e ; mo- d P l e s dlHeisenberg avec de proches v o i s i n s a l d a t o i r e s ou une i n t e r a c t i o n RKKY. On compare l e s suscep- t i b i l i t d s , l e s c h a l e u r s s p d c i f i q u e s e t l e s d i s t r i b u t i o n s de champ i n t e r n e aux expdriences e t aux t h e o r i e s a p p r o p r i 6 e s . On d 6 c r i t l e s p r o p r i d t 6 s B tempdrature n u l l e s e l o n l e s concepts de l a n a t u r e de l ' o r d r e des s p i n s dans un v e r r e de s p i n s . On p r 6 s e n t e l e s p r o p r i d t c s d6pendant du temps pour un v e r r e de s p i n s d f I s i n g B proches v o i s i n s . On s u i t l a r e l a x a t i o n anormalement l e n t e dans 1 ' 6 t a t "gel6"

depuis l a d6g6ne'rescence 61evbe de 1 ' 6 t a t fondamental e t on en d i s c u t e l e s consdquences pour l ' e x i s - tence de t r a n s i t i o n s de phases.

A b s t r a c t . - Monte C a r l o (MC) c a l c u l a t i o n s on v a r i o u s models a r e reviewed : I s i n g models with random n e a r e s t neighbor a s w e l l a s random i n f i n i t e range exchange, Heisenberg models w i t h random n e a r e s t neighbor a s w e l l a s RKKY exchange. S u s c e p t i b i l i t i e s , s p e c i f i c h e a t s and i n t e r n a l f i e l d d i s t r i b u t i o n s a r e compared t o r e l e v a n t experiments and t h e o r i e s . Zero-temperature p r o p e r t i e s a r e d e s c r i b e d i n t h e c o n t e x t of concepts on the n a t u r e of s p i n o r d e r i n g i n s p i n g l a s s e s . Time-dependent p r o p e r t i e s a r e p r e s e n t e d f o r n e a r e s t neighbor I s i n g s p i n g l a s s e s . The anomalous slow r e l a x a t i o n i n t h e "frozen" s t a - t e i s t r a c e d back t o t h e high degeneracy of t h e ground s t a t e , and i m p l i c a t i o n s f o r t h e e x i s t e n c e of phase t r a n s i t i o n s a r e d i s c u s s e d .

1 . INTRODUCTION.- MC computer experiments s i m u l a t e thermal f l u c t u a t i o n s of many-body systems / 1 , 2 / . They can be used t o check ( i ) t h e v a l i d i t y of theo- r i e s f o r a model h a m i l t o n i a n

]l,

( i i ) t h e e x t e n t t o which a model approximates a r e a l system. Microsco- p i c i n f o r m a t i o n both i n space and time i s a v a i l a b l e t o o b t a i n ( i i i ) r e l e v a n t q u a l i t a t i v e i n s i g h t a s w e l l a s ( i v ) q u a n t i t i e s i n a c c e s s i b l e by experiment ( l i k e s t a g g e r e d n o n p e r i o d i c o r d e r parameters, e t c .)

.

^Thus

t h i s method i s very v a l u a b l e f o r a c o n t r o v e r s i a l f i e l d l i k e s p i n g l a s s e s ( f o r experimental reviews s e e / 3 , 4 / and f o r t h e o r e t i c a l / 5 , 6 / ) . However, t h e l i m i t a t i o n s of the method must be k e p t i n mind. By MC one numerically r e a l i z e s a master e q u a t i o n f o r t h e p r o b a b i l i t y P ( x , t ) of +

+ + -+

x = (Sl

,...,

S N ) f i ,

d + + + +

P ( x , t ) =

-

CW(~+~')P(~,~)+CW(:'+:)P(:' , t ) (1) The t r a n s i t i o n p r o b a b i l i t y W c o n t a i n s an undetermi- ned t i m e s c a l e -r and depends on t h e e n e r & c o s t 6%

-kO+

of the change e x '

.

Thus MC sampling i s "time" ave- r a g i n g { b u t t i n e q u a t i o n ( I ) need not d i r e c t l y c o r r e s p o n d ' t o t h e time of t h e p h y s i c a l system evol- v i n g v i a d e t e r m i n i s t i c e q u a t i o n s of motion / 1 , 2 / ) . One average; systems of f i n i t e s i z e N over f i n i t e times { t y p i c a l l y l o 3 MC s t e p s l s p i n (MCS)}. Using

model N = 500 1181 and f o r RKKY exchange N = 96 119, 201 {RKKY ground s t a t e s t u d i e s use N = 324 1211, N = 1000 1221). Thus f i n i t e s i z e and p e r i o d i c boun- dary c o n d i t i o n s i n f l u e n c e t h e r e s u l t s . S t r i c t l y s p e a k i n g , f i n i t e systems do n o t have phase t r a n s i - t i o n s 1231 ; t h u s s t a t e s w i t h nonzero o r d e r para- meter a t b e s t a r e m e t a s t a b l e d u r i n g some l i m i t e d o b s e r v a t i o n time. This s m a l l s i z e - m e t a s t a b i l i t y i s h a r d t o d i s t i n g u i s h from the m e t a s t a b i l i t y of macro- s c o p i c systems ( t h e f r o z e n s t a t e of r e a l s p i n glas- s e s could be m e t a s t a b l e r a t h e r t h a n t r u l y s t a b l e / 3 , 4 , 2 4 / ) . The MC method has l i t t l e t o say about t h i s d i s t i n c t i o n , which i s t h e o r e t i c a l l y i n t e r e s - t i n g b u t makes l i t t l e p r a c t i c a l d i f f e r e n c e (even diamond i s only a m e t a s t a b l e phase of g r a p h i t e ) . Another problem i s t h e a p p r o p r i a t e i n t e r p r e t a t i o n of e q u a t i o n (1) f o r r e a l systems. The dynamics of i n d i v i d u a l magnetic i o n s i n t e r a c t i n g v i a RKKY i s n o t w e l l r e p r e s e n t e d by e q u a t i o n ( I ) , which y i e l d s r e l a x a t i o n only and no s p i n waves. But one.may in- t e r p r e t / 6 , 9 / Si a s an e f f e c t i v e moment of a +

" g

t e r " r a t h e r than'a s i n g l e s p i n . The c l u s t e r m a y be d e f i -

-

ned e i t h e r v i a chemical c l u s t e r i n g 141 o r by r e a u i - r i n g t h a t a l l s p i n s b e l o n g i n g t o t h e c l u s t e r a r e locked t o g e t h e r by i n t e r a c t i o n s J>>kBTf (Tf=free- z i n g temperature) /6,9,24,25/. There may be f e r r o - , random n e a r e s t neighbor (n.n-) exchange N t y p i c a l l y a n t i f e r r o m a g n e t i c o r random s p i n o r d e r i n g w i t h i n i s l o 3 t o lo4 16-171, w h i l e f o r an " i n f i n i t e range" t h e c l u s t e r , depending on t h e p a r t i c u l a r system.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19786595

C6-1528 ^{JOURNAL DE PHYSIQUE}

Since the cluster magnetic moment may exceed gp S, B a high sensitivity to the magnetic field H results /6,9/. Then T means the typical time for thermal- ly activated reorientation of the cluster : clear- ly, for T+O and large clusters T will be large.

0

In fact, superparamagnetism 1261 is a special ca- se of equation (I), where one neglects che inter- actions among clusters. Including these interac- tions, however, one again ends up with an Edwards- Anderson (EA) 1271 model but with rescaled para- meters /6,9/. Thus the simulations using n.n. ex-

change 16-1

11

may well agree with experimental data.

2. SUSCEPTIBILITY

x

AND SPECIFIC HEAT C OF SPIN GLASSES.- Results for Ising spins with n.n. gaus- sian excPange (of width AJ) are given in figure I

0 ^{1 1 I}

1 2 w 3

X A J AJ

0 25-*-•

~"i/.~ ^{, , *}

0.5 10 l S k _ a ~ 2 0 , A J deal paramagnet

v\

0'

is

i b 1 ' 5 ~ 2:o

-

AJ

Fig. 1 : A) X, C plotted vs. T (d=2 Ising case,

~ 1 3 4 ~ to ~ = 8 0 ~ ) at various H (from /7,9/)

B) X r C plotted vs. T for d=3 with N = I ~ ~ at H=O.

for dimensionalities d=2,3. Typically, C and

x

are computed as c = < { < ~ ~ > - & ~ I > / { N ( ~ ~ T ) ~ I , X=<{<~Z>- RI>')>/(N~~T) = magnetization, inner bracket

< * * * > is a thermal (i.e., time-) average, outer one on average with P(J)~. The peak of C occurs at higher T than that of X, and is much broader (the

"cusp" of

x

is always rounded due to finite size and finite time effects). This behavior resembles experiment /3,4/ much closer than the EA mean field theory 1271. The peak of

x

is strongly rounded and depressed for H # 0. While it is no surprise that

time effects influence

x

near Tf (see /lo/), note that

x

^and

xderiv

5 a<M>/aH disagree strongly for T<<Tf. We will come back to this effect below.

For T>T both methods yield the Curie law

xa

IIT, f

as expected. Figure 2 A shows that Ising spins with n.n.

+

J exchange behave similarly, apart from, a

1/T-divergence of

x

for T-+O due to "loose" Spins.

The spin glass (figure 2 B) where arbitrary Pairs

Fig. 2 : A) X, C plotted vs. T(d==2, ~=80', +J mo- del, from /13/)

B) X, C plotted vs. T for the infinite range model (N=500, from 1181).

cusp both in

x

and in C in contrast to experiment.

But this simulation is very valuable for theory : the "replica" solution 1281 is shown to be presu- mably correct for TLTf but incorrect for T<T f' while Thouless et al. 1291 (TAP) give the correct behavior for T+O.

Classical d=3 Heisenberg systems with n.n.

gaussian exchange are shown in figure 3 A. While

Fig. 3 : A) X, C plotted vs. T for d=3 Heisenberg spin glasses (from /9,12/)

some data 1121 again were interpreted as a phase transition, it was argued /6,9/ that the smallness of "Tfn, the nonequilibrium effects there (C and a a > / a ~ disagree strongly) and the very gradual freeze-in of the EA order parameter

t

q(t) = &

(i

Si(tT)dtl~t)2/~ are better consistent

1

with Tf = 0. This result would agree with real spa- ce renormalization 1301 and gauge theories 1311 suggesting that d=3 is a "lower critical dimensio- nality" d for a phase transition (i.e. T >O for

f d>d ) . The RKKY results are even less conclusive near Tf since N=96, figure 3 B. But one gets a peak of

x

roughly at the experimental T /32/ (the

f

magnitude of

x

to T- was adjusted but the strength A of RKKY exchange had its known value /19/). Whi-

le C agrees with the coefficient of the 1/T-law 1331 for T-, C+1 for T-Y) (as expected for a clas- sical system). One gets more realistic results from a numerical spin wave analysis 1211 based on the ground state spin configuration. The resulting C 1211 agrees well with experiment 1341 (insert of figure 3 B)

.

EXPERMNT

@Mn itat % I Soa

T IKI- 0 6

OL

0 2

04 ₀ b I ₁

.

_{2 3 4 5}

O3KsTlA- tO1 kaT (A-

Fig. 3 : B) X, C plotted vs. T for d=3 Heisenberg RKKY model with N=96(from 1191, insert showing C for small T, from 1211).

is a key quantity for mean field theories I35

-

371 not using replica methods /27/

k . g .

"mean ran- dom fieldl'method (MRF) /36g. Figure 4 A shows

Fig. 4 finite (lower

: A) P(Heff) plotted vs. Heff at TPO for in- range / 18/ (upper part) and RKKY models 1221 part)

the MRF to be wrong at T = 0 in the infinite range model : P(Heff=O)=O rather than being maximal the- re. Since reasonable mean field methods should be- come correct for infinite range this failure is ve- ry serious. While this theory 1371 could be fitted 1221 to RKKY results (figure 4 A, lower part), it was shown later 1211 that the predicted field Ho is incorrect, the maximum of P(Heff) occurs at a larger value. For n.n. Heisenberg models the form 1361 P . ( H ~ ~ ~ ) ~ H ~ ~ ~ exp{-Hzff/P(A~) 'zq

(TU

1 ^=co- ordination numbed implies a rapid variation with T which is not found (figure 4 B). In the Ising

2U

-

^T.3

C6-1530 JOURNAL DE PHYSIQUE

!!&!

A I

Fig. 4 : B) P(Heff) plotted vs. k f f at various T for the n.n. Heisenberg spin glass 191.

case P(Hef

ex^{-^^

eff/p(~~)2~q(~fl 1 1361 fails similarly, and due to correlation effects similar to the long range cases P(Heff) is minimal at Heff = 0 rather than maximal for low T (figure 5).

symbol

-.

Fig. 5 : P(Heff) plotted vs. Heff at various T for the nearest neighbor d=2 (A) and d=3 (B) Ising spin glass 191.

4. TIME-DEPENDENT PROPERTIES.- Figure 6 shows the autocorrelations of the n.n. gaussian d=2 Ising spin glass 171. The decay with time is not a sim-

Fig. 6 Autocorrelation function of d=2 Ising spin glass plotted vs. time in two ways. ~ = 2 4 ~ . From17 1.

ple exponential. It was argued / 7 / that these da- ta would be as well consistent with a gradual free- ze-in rather than a phase transition. Below Tf hysteresis occurs (figure 7 A) and the remanence

(M>

Fig. 7 : A) Magnetization process of a d=2 Ising spin glass with ~ = 8 0 ~ at kgT/AJ=0.7. From

/B/.

7 B). A detailed analysis /I !I of the irreversible

C

Fig. 7 : B) Decay of remanent magnetization. From/7/.

behavior brings out a great similarity to real experiments /3,4,24./ : hence there is no need to

invoke a Ne6l model I261 to explain them, similar irreversibility follows from the EA model 1271 as well. The

+

^J n.n. Ising system does not yield this slow nonex~onential decay, however 1131, and in the infinite range model one finds only t- lh

laws below Tf 1181.

The slow decay near and below Tf must lead to pronounced time-effects on X, as noted in figu- re I and in 171. Bray et al. 116,171 took these effects as an evidence against a phase transition.

The peak of

x

would then be a nonequilibrium ef- fect as in the d=l Ising spin glass (figure 8 A),

Fig. 8 : A) Susceptibility of

+

^J Ising spin glas- ses plotted vs. T at various t for d=l,d=3.From/l7/.

however : for d=l the peak position very distinct- ly decreases with t, the slope of

x

in the linear region below the peak increases. For d=3, however, the data are consistent with a unique peak posi- tion, and the slope of

x

decreases with t, as in experiments (figure 8 B) where

x

"saturates" af-

6 10 1L 18 T(K1 22

Fig. 8 : B) Magnetization of AuFe plotted vs. T at various time during which a field H=lO gauss was applied. From 1381.

ter macroscopic times at a limit far below the Curie law. Due to the finite size of the simulated systems and the much smaller time intervals acces- sible, MC work neither proves nor disproves the uniqueness of Tf (for a detailed discussion see

1101).

5. GROUNJI STATE PROPERTIES OF ISING SPIN GLASSES.- The spin glass ordering can be analyzed /6,7/ by the vector

6 ' ' )

=

((:'I. . .

,mN )/A, the "phase factor"

' ; 4 1

being the direction of spin S. in the L'th ground state. The order parameter $

='b*;b(')>

can then also be found from MC /9,10/ (figure 9 A).

JOURNAL DE PHYSIQUE

Pig. 9 : A) Order parameters J,

,

q plotted vs. T for d=2,3 n.n. Ising spin glasses.

J, measures the alignment of the spins to a (parti- cular) ground state configuration which is used as initial state for the averaging. As expected, J, vanishes at Tf. Figure 9 A includes q(t) which relaxes slower and hence is less well suited to locate Tf. At T=O a finite field Hc is necessary to destroy the.ordering (figure 9 B). The magneti-

~ i g :B) ~ . $, M plotted vs. H at T=O for d=2,3.

From /9,10/.

zation process at T=O is also shown. The resulting x's at T=O are consistent with figure 1.

While typically the projection of two ground states onto each other is of order cos4

am= -;b(L)*;b')

= const /&, also ground states exist with nearly arbitraty

Ices$

[ < I /lo/. Thus one may continu-

Em -

ously "rotate" the system from one ground state to the other. Hence the order parameter symmetry is quasi-continuous (i.e., a continuous symmetry exists only globally and not locally). Thus one expects the boundary energy (per boundary site) E between two phases with different -+

4

to vanish

b

for N- /lo/, as indeed found by Reed et al. /39/, figure 10 A. Since Eb vanishes also in the 3d Hei-

Fig. 10: A) Boundary energy (per boundary site) plotted vs. Y/AJ {~=J-P(J)J~J}. From 1391.

senberg-ferromagnet of -Mattis spin glass /40/, this fact does not necessarily imply unstable or- dering at T>O as was concluded /39/. In the Hei- senberg magnet the total boundary energy is

ad-2'da

which exceeds the total energy fluctuation (a~")

for d>4 only -thus the change of behavior at d=4 found by Reed et al. 1391 need not support the re- sult d =4 obtained from series /41/. But these studies /39/ and the average length L of strings ,between frustrated /42/ squares (for d=2) allow

to locate the ferromagnet-spin glass-boundary (fi- gure 10)

.

0 03 0.2 0.3 0.L 0.5

X

Fig. 10 ^: B) Average length of strings plotted vs fraction of wrong bonds in the

+

J model (d=2).

From 1131.

glass with continuous symmetry the order parameter q(t) relaxes to zero for all T>O in a finite sys- tem since the order parameter direction "diffuses"

there / l o / . The same effect may be responsible for the slow relaxation of q(t) seen in EA Ising spin glasses (figure 1 1 B). Physically this "diffu- sion" of the order parameter means a rearrangement of large clusters of correlated spins with no cost in energy. An analytic treatment of such fluctua- tions should provide the basis to both describe the irreversible behavior of spin glasses and to clarify the questions about dc.

symbol

TIT= N ' ~

0

10.9651

12

Fig. 1 1 : A) q(t) plotted vs. t for various N,T in the d=3 Heisenberg Mattis spin glass.

0.8

t

size 502 ( 1 run 1 252 (6runs) 1!j2 (single runs)

lo2

( 5 runs 1

~ i 1 1 ~ .B) . q(t) plotted vs. t for various N,T in the d=2 fsing EA spin glass. (From /lo/).

JOURNAL D E PHYSIQUE

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21

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35

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B14

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