Scaling approach to glassy stationary states of spin-glasses under chaos effects
a r X i v :c o n d -m a t /0407459v 2 [c o n d -m a t .d i s -n n ] 3 A u g 2004
Progress of Theoretical Physics Supplement
1
Scaling approach to glassy stationary states of spin-glasses under
chaos e?ects
Hajime Yoshino 1,and Petra E.J ¨o
nsson 2,1Department
of Earth and Space Science,Faculty of Science,Osaka University,
Toyonaka,560-0043Osaka,Japan
2ISSP,University of Tokyo,Kashiwa-no-ha 5-1-5,Kashiwa,Chiba 277-8581,Japan
Dynamics of spin-glasses subjected to slow continuous changes of working enviroment such as slow changes of temperature or interaction bonds are studied based on scaling argu-ments and numerical simulations of continuous bond changes.Such perturbations lead to a glassy stationary state where the age or domain size of the system is pinned to macroscopic but ?nite values due to competition between relaxation and chaos e?ects (rejuvenation).Flutuation-dissipation relation is also pinned to that of a ?nite age.The scenario explains the anomalously weak cooling rate dependece of spin glasses.
Introduction -Dynamics of glassy systems have been actively studied over the past several decades.One general interest has been the aging e?ect,1)i.e.a glassy system ages extremely slowly in a ?xed working environment speci?ed for example by the temperature of the heatbath.Recently responses of such glassy systems to various kinds of perturbations have attracted much attention including restart of aging or rejuvenation of spin-glasses after sudden changes of the temperature of the heatbath,2),3)responses of soft glassy systems to shear.4)
In the present work we study relaxation of spin-glasses subjected to slow changes of the working environment,namely slow changes of temperature or interaction bonds.This study is motivated by the recent experimental observations that spin-glasses exhibit surprisingly weak dependence on the cooling rates.2),3)We propose a scaling ansatz for the e?ective age of the system under continuous changes of the working environment assuming that the competition between aging and rejuvenation due to the chaos e?ects 5)leads to a stationary state.We performed Monte Carlo (MC)simulations of a 4-dimensional Edwards-Anderson (EA)model subjected to continuous changes of bonds and found good agreement with the scenario.
Model -Speci?cally we consider the Edwards-Anderson (EA)spin-glass model with Ising spins S i (i =1...N )on a d -dimensional (hyper)-cubic lattice with N =L d
lattice points.The Hamiltonian is given by H =?
i S i .We use the ±J model where the interaction bonds J ij between nearest-neighbours i,j take random ±J values.The temperature is measured as k B T/J with the Boltzmann’s constant set as k B =1.The magnetic ?eld h is measured as h/J .The relaxational dynamics is modeled by the usual heatbath single spin ?ip MC algorithm.
A continuous bond changes is realized by ?ipping the sign of a fraction p ?1of the bonds chosen randomly in 1Monte Calro step (MCS).This amounts to ?ip a fraction p e?(t )=[1?(1?2p )t ]/2of the original set of bonds J ij (0)to create the set of bonds J ij (t )at t (MCS).For pt ?1p e?(t )=pt and p e?(t )→1/2for pt ?1.
A continuous temperature changes is speci?ed by a cooling or heating rate v T =
2Hajime Yoshino and Petra E.J¨o nsson
δT/δt whereδT is the change of the temperature within an interval of timeδT.Thus the resultant change of the temperature after a time t is?T(t)=v T t.Unfortunately the temperature-chaos e?ect is much weaker than the bond chaos e?ect6)in the present model and we limit ourselves to the bond perturbations in the simulations.
Scaling arguments-Let us consider a spin-glass whose spins are evolving in a?xed working environment W=(J,T)speci?ed by a set of bonds J={J ij} and the heatbath temperature T.If the initial spin con?guration is far from the equilibrium state associated with W,the system is equilibrated only up to a certain length scale L T(t)after a time t.The domain size L T(t)grows slowly with time t by thermally activated droplets excitations as L T(t)=L0[(T/J)log(t/τc(T))]1/ψwhere L0is a unit of length scale,τc(T)is the attempt time of the activated dynamics and ψ>0is the exponent of the free-energy barriers(See7)for the details).
Now we perturb the system by slowly changing W either by continuous tempera-ture changes characterized by?T(t)=v T t or continuous bond changes characterized by?J(t)=J
3
be present in systems with activated dynamics but without the chaos e?ect.
00.20.40.60.8
1
C (τ+t w ,t w )
τ (MCS)
10?2
10?1
100
C (τ+t w ,t w )
τ
(MCS)
Fig.1.Spin autocorrelation function C (τ+t w ,t w )=(1/N )
N
i =1
···
respectively.The lines in the left ?gure are reference data of standard isothermal aging.7)
The system exhibits interrupted aging:waiting time t w dependence saturates.The tail parts of the autocorrelation functions are ?tted to a simple exponential form A exp(?t/τ?(p ))by which the characteristic relaxation time τ?(p )is extracted.In this example τ?=6013(MCS)is obtained.
00.20.40.60.8
110101010C b o n d (τ*)
p
L e n g t h
τ*
(MCS)
Fig.2.The correlation C bond (τ?)between the set of bonds at two times separated by the re-laxation time τ?(p )is shown in the left ?gure.It is de?ned as C bond (t )≡
p e?(t )at the relaxation time
t =τ?
(p )with p =0.002×2?n
with n =0,...,7(from left to right)obtained at T /J =0.8,1.2and 1.6.Here ζ=0.9as obtained in Ref.3)(see also 6))and chosen the prefactor a =0.22.
Numerical analysis -We performed MC simulations of the dynamics of the 4-dimensional EA model (T g ?2.0J )under continuous bond changes starting from random initial con?gurations.In Fig.1we show the data of the spin autocorrelation function C (τ+t w ,t w ).Apparently the system becomes stationary within a ?nite
4Hajime Yoshino and Petra E.J¨o nsson
time scale τ?due to the continuous bond changes.The relaxation time τ?extracted
from the data are examined as shown in Fig.
2.It appears that the actual change of the bonds within the relaxation time τ?of the bonds becomes negligible as the intensity of the driving p becomes smaller.As shown in the right ?gure of Fig.2,the relaxation time τ?follows the expected scaling Eq.(1).
χZ F C
C
Fig.3.The ZFC susceptibility χZFC (τ+
t w ,t w )vs.the spin autocorrelation func-tion C (τ+t w ,t w )with t w ?τ?.In this
example p =2.5×10?4.χZFC is measured as usual 7)with h/J =0.1but under con-tinuous bond changes.The lines represents FDT χZFC =(1?C )/T associated with the
heatbath temperatures T .
Finally let us examine the ?uctua-tion dissipation relation (FDR)as pro-posed by the dynamical mean ?eld the-ory (DMFT).9)As shown in Fig.3the
?uctuation dissipation theorem is satis-?ed at short time scales and violated at
larger time scales in a non-trivial man-ner as predicted by the DMFT.Thus
the stationary state can be considered as a glassy state.The result is very sim-ilar to that of the same model during isothermal aging 7)and driven dynamics induced by asymmetric coupling.10)It
must be noted however that the FDR systematically depends on p as those of isothermal aging systematically depends
on t w so that the FDR in the asymptotic
limit p →0may be di?erent from the
prediction of the DMFT.7)
Discussion-To conclude we studied glassy stationary dynamics of spin-glasses under continuous temperature or bond changes.Let us consider a typical cooling rate v T =O (10?1)(K/sec)used in “qunech”experiments,which is actually as slow as v T =10?15J/MCS since T g =O (10)K and the microscopic time scale is τ0=10?13(sec).The scaling ansatz Eq.(1)implies it yields τ?of only order 1?10(sec)at a target temperature T m =0.7T g in the case of a canonical spin-glass AgMn sample,3)which allows one to observe “clean”aging at T m .Indeed the e?ective age of the system after such slow cooling is found to be only O (10)(sec)experimentally.3)
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