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Submitted on 1 Jan 1978
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REMANENT MAGNETIZATION IN SPIN GLASSES
-A MONTE C-ARLO INVESTIG-ATION
W. Kinzel
To cite this version:
W. Kinzel.
REMANENT MAGNETIZATION IN SPIN GLASSES - A MONTE CARLO
JOURNAL DE PHYSIQUE Colloque C6, supplément au n" 8, Tome 39, août 1978, page C6-905
REMANENT MAGNETIZATION IN SPIN GLASSES - A MONTE CARLO INVESTIGATION
W. Kinzel
Institut fur Festkdperforschung der KFA Jiiliah, D-5170 Jiiliah, Germany
Résumé.- Utilisant des simulations de type Monte Carlo pour un verre de spins Ising à deux dimen-sions, on calcule différentes sortes d'aimantation rémanente en fonction d'un champ appliqué et de la température. L'aimantation rémanente isotherme (IRM), l'aimantation thermorémanente (TRM) et les susceptibilités réversibles et irréversibles sont en accord qualitatif avec les résultats expéri-mentaux.
Abstract.- Using Monte Carlo simulations for a two dimensional Ising spin glass, different kinds of remanent magnetization are calculated as a function of applied field and temperature. Isothermal
(IRM), fieldcooled (TRM) remanence and reversible and irreversible susceptibility are all in quali-tative agreement with experimental data.
In spin glasses, there are many magnetic moments coupled by an interaction much stronger
than the freezing temperature T. /I,3/. Thus, at T,, such spins may be considered as belonging to rigid clusters with an effective moment S. and an rela-xation time x.. The interaction between the
clus-1
ters may be approximated by a random interaction of the order of k„T,. There are several ways of under-standing spin glasses by a model of isolated clus-ters- with a distribution of moments S. and relaxa-tion times T. /4,6/. However, as the interacrelaxa-tion is of the order of k„T,, the neglect of it can be only
D r
a rough approximation. In the present work, we study effects of the collective random interaction alone. We use the Ising model on a square lattice with a Gaussian distribution of nearest neighbour inter-actions and the dynamics are simulated by a master equation /2,3/. Monte Carlo simulations (MCS) have demonstrated many properties observed in experiments
(see K. Binder, this conference). Here we use MCS to study long-time relaxation.
When an external field is applied to spin glasses and then switched off, one observes remanent magnetization which decays non-exponentially with a
time scale of minutes or hours /4-6/. Such a beha-viour has also been found in MCS HI. We have, simu-lated a 50 x 50 spin glass with runs of about 4000 Monte Carlo steps per spin (M/s) and have investi-gated three kinds of thermoremanent (TRM) and isothermal remanent (IRM) magnetization.
1) TRM. The spin glass is suddenly cooled from infinite temperatures (random spin configuration)
to T = 3T,, then over 2000 M/s cooled in an external field B to T = Tf/4. After a further 400 M/s, the field is switched off.
2) IEM (sc = slowly cooled). As in TRM, but without external field. During the last 20 M/s a field is applied and then switched off.
3) IRM (fc = fast cooled). The spin glass is
suddenly cooled from T = » to T = T./4 ; after 20 M/s a field is applied for 20 M/s and then switched off.
After a short relaxation of 20 M/s the remanence has been calculated by averaging the magnetization over 1400 M/s. Figure 1 shows the remanence as a function of the applied field energy - BS.. AJ (= k T. II/) is the width of the
distri-l a t bution of couplings.
03
-M
/
^
^
°
/ / / '
T R M 01 •/ 7 / " I R M (fc)' \ / J x I RM (sc)
0 1 2 3 B/AJ 4Fig. 1 : Remanent magnetization as a function of
applied f i e l d . The bar shows M(B = «)
The remanence states may be characterized by
dependence of the irreversible and reversible magne-
their internal energy and magnetization. Figure 2
tization in a field of B
=0.2 AJ. The dots are the
shows the same TRM and IRM states as in figure 1.
magnetization
M(B)in field B obtained for the field-
cooled spin glass, the crosses represent the diffe-
rence M(B)-TRM(B)
between magnetization and rema-
Fig.
.?.
:Energy and magnetization of remanence
states of figure 1. The stars represents
states produced by an external field
B
= 1.1AJ
The state with zero magnetization and lowest energy
is presumably one of the ground states. The stars
show three states produced by the same field
B
= 1.1AJ. It is interesting that this field ofthe
order of kBTf helps the fast-cooled spin glass to
reach lower energies and creates the highest TRM.
For long times, all states tend to one of the ground
states, and so the whole region between the three
lines represents metastable states. Of course we
cannot exclude that metastable states also occur in
the outside region. Figure 3 shows the temperature
nence. In the limit
B
+0, the curves correspond to
irreversible
(x.
)and reversible
(xrev)
suscepti-
1rr
bility. These points are in agreement with computer
data of
xirr
obtained by lowering the field from in-
finity to zero and
x
obtained by calculating the
rev
fluctuations of magnetization in thermal equilibrium
171. At kBT
=
AJ, the irreversible susceptibility
become constant for lower temperatures, whereas the
reversible susceptibility shows
apeak. The same be-
haviour of
x
and
xirr
can be observed in experi-
rev
ments/4-6/. Note that in MCS the transition is roun-
ded due to the nonzero field and finite size effects.
We conclude that many irreversible and rever-
sible processes in spin glasses can be described
qualitatively by an Ising model with short range
random interactions.
References
/I/ Mydosh, J.A., AIP Conf. Proceedings
26
(1975)
131
/2/ Binder, K., Advances in Solid State Physics,
Vol.
XVII(1977), p. 55
/3/ Fischer,
K.H., Physica 86-88B (1977) 813
141 Tholence, J.L. and Tournier, R., J. Physique
Colloq.
35
(1974) C4-229
/ 5 /
