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Aging and non-equilibrium critical phenomena in Monte Carlo simulations of 3D pure and diluted Ising models

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Abstract

We investigate the non-equilibrium critical evolution of statistical systems and describe of its some features, such as aging and violation of the fluctuation-dissipation theorem. We consider some theoretical results of computations for universal quantities that have been obtained in recent years, such as the exponents determining the scaling behavior of dynamic response and correlation functions and the fluctuation-dissipation ratio, associated with the non-equilibrium critical dynamics. We derive our original Monte Carlo simulation results for 3D pure and diluted Ising models with Glauber and Metropolis dynamics. For these models, we analyze the influence of critical fluctuations, different non-equilibrium initial states, and the presence of nonmagnetic impurities on the two-time dependence of correlation and response functions on characteristic time variables, such as waiting time $t_{\rm w}$ and time of observation $t - t_{\rm w}$ with $t > t_{\rm w}$ . We discuss the obtained values of the non-equilibrium exponents for the autocorrelation and response functions and values of the universal long-time limit of the fluctuation-dissipation ratio $X^{\infty }$ . Our simulation results demonstrate that the insertion of disorder leads to new values of $X^{\infty }$ with $X_{\rm diluted}^{\infty } > X_{\rm pure}^{\infty }$ .
Prog. Theor. Exp. Phys. 2015, 053A01 (20 pages)
DOI: 10.1093/ptep/ptv055
Aging and non-equilibrium critical phenomena
in Monte Carlo simulations of 3D pure and
diluted Ising models
Pavel V. Prudnikov, Vladimir V. Prudnikov, Evgeny A. Pospelov, Petr N. Malyarenko, and
Andrey N. Vakilov
Department of Theoretical Physics, Omsk State University, Omsk 644077, Russia
E-mail: prudnikp@univer.omsk.su
Received September 18, 2014; Revised January 26, 2015; Accepted March 19, 2015; Published May 1, 2015
...............................................................................
We investigate the non-equilibrium critical evolution of statistical systems and describe of its
some features, such as aging and violation of the fluctuation-dissipation theorem. We consider
some theoretical results of computations for universal quantities that have been obtained in recent
years, such as the exponents determining the scaling behavior of dynamic response and correla-
tion functions and the fluctuation-dissipation ratio, associated with the non-equilibrium critical
dynamics. We derive our original Monte Carlo simulation results for 3D pure and diluted Ising
models with Glauber and Metropolis dynamics. For these models, we analyze the influence of
critical fluctuations, different non-equilibrium initial states, and the presence of nonmagnetic
impurities on the two-time dependence of correlation and response functions on characteristic
time variables, such as waiting time twand time of observation ttwwith t>tw. We discuss
the obtained values of the non-equilibrium exponents for the autocorrelation and response func-
tions and values of the universal long-time limit of the fluctuation-dissipation ratio X.Our
simulation results demonstrate that the insertion of disorder leads to new values of Xwith
X
diluted >X
pure.
...............................................................................
Subject Index A22, A40, A56
1. Introduction
The collective behavior of statistical systems close to critical points is characterized by an extremely
slow dynamics that, in the thermodynamic limit, does not give them time to relax to an equilibrium
state after a change in some thermodynamic parameters and conditions. The non-equilibrium evolu-
tion displays in this case some of the features typically observed in glassy materials, such as aging
and violation of the fluctuation-dissipation theorem (FDT) [15]. It can be detected through deter-
mination of dynamic susceptibilities and correlation functions of the order parameter, the scaling
behavior of which is characterized by universal exponents, scaling functions, and amplitude ratios.
This universality allows one to calculate these quantities in different statistical models [6,7]and
Monte Carlo methods are a natural approach for this analysis.
We consider here some of the theoretical results of computations that have been obtained in recent
years for universal quantities, such as the fluctuation-dissipation ratio, associated with the non-
equilibrium critical dynamics, with particular focus on our original Monte Carlo simulation results
for the 3D pure and diluted Ising models with Glauber and Metropolis dynamics.
© The Author(s) 2015. Published by Oxford University Press on behalf of the Physical Society of Japan.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/),
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The investigation of the influence of structural disorder on the critical behavior of different statisti-
cal systems remains one of the main problems in condensed-matter physics. The disorder breaks the
translational symmetry of the crystal and thus greatly complicates the theoretical description of the
material. It is known that the critical behavior of a system is characterized by an anomalously large
response even for any weak perturbation; therefore, the presence of even low defect concentration
might lead to drastic changes of this behavior and its characteristics. The description of such systems
requires the development of special analytical and numerical methods [8,9].
Research into the effects produced by weak quenched disorder on critical phenomena has been
carried out for many years [1017]. This work determined the criterion [10] under which the disorder
affects the critical behavior only if the specific heat exponent of the pure system αis positive. In
this case, a new universal critical behavior, with new critical exponents, is observed. Otherwise, for
α<0, the presence of disorder is irrelevant for the critical behavior. Only systems with an effective
Hamiltonian that is isomorphic near the critical point of the Ising model satisfy this Harris criterion,
and a drastic change in the universality class of the critical behavior is realized with its introduction
to a system of whatever concentration of short-range correlated quenched defects [9,18,19].
Studies of the critical behavior of diluted Ising-like magnets have been carried out by
renormalization-group (RG) methods, by numerical Monte Carlo methods, and experimentally, and
the results of these investigations have been presented in a large number of publications (for a review,
see Ref. [20]). In Refs. [21,22], the authors declared interesting ideas about replica symmetry break-
ing in the systems with quenched disorder. A field-theoretic description of the critical behavior of
weakly disordered systems with introduced replica symmetry-breaking potentials, which was car-
ried out in the two-loop approximation [23,24], has shown the stability of the critical behavior of
disordered 3D systems with respect to the replica symmetry-breaking effects. The results of the
investigations confirm the existence of a new universal class of critical behavior for diluted Ising-
like systems. However, it remains unclear whether the values of critical exponents are independent
of the rate of dilution of the system, how the crossover effects change these values, and whether two
or more regimes of the critical behavior exist for weakly and strongly disordered systems. These
questions are of interest to researchers and have been discussed before now [25,26].
This study is devoted to numerical investigation of the non-equilibrium critical dynamics with a
non-conserved order parameter (model A) [27] in 3D pure and site-diluted Ising systems with spin
concentrations p=0.8and 0.6. In the following section, we derive the main concepts and charac-
teristics of non-equilibrium relaxation in statistical systems with slow dynamics and peculiarities in
their non-equilibrium behavior at critical points. In Sect. 2, we also review some theoretical results
of computations in the range of non-equilibrium phenomena that have been obtained in recent years
for some statistical models. In Sects. 3 and 4, we introduce the 3D pure Ising model and 3D Ising
model with quenched point-like defects, and derive the results of our original Monte Carlo inves-
tigations into two-time autocorrelation and response functions and the fluctuation-dissipation ratio
for systems starting from high-temperature disordered and low-temperature ordered initial states.
The critical exponents and the asymptotic values of the universal fluctuation-dissipation ratio X
obtained under these two conditions are compared. The final section contains an analysis of the main
results and our conclusions.
2. Non-equilibrium critical dynamics and its main peculiarities and characteristics
Statistical systems with slow dynamics have recently attracted considerable theoretical and exper-
imental interest, in view of the rich scenario of phenomena they display: extreme slowing down
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of relaxation processes, memory effects, hysteresis, etc. After a perturbation, a system with slow
dynamics does not achieve equilibrium, even after a long time, and its dynamics is not invariant
either under time translations or under time reversal, as it should be in thermal equilibrium. Dur-
ing this never-ending relaxation, aging occurs: two-time quantities such as response and correlation
functions depend on characteristic time variables such as waiting time twand time of observation
ttwwith t>twnot via ttwonly. It is important that decays for these two-time quantities as
functions of time of observation ttware slower for larger waiting times tw.
At variance with one-time quantities like the order parameter converging to asymptotic values in
the long-time limit, two-time quantities clearly demonstrate aging. Aging is known to occur in such
disordered and complex systems as glassy materials [13] and only in the last ten years has attention
been focused on simpler systems such as critical ones, whose universal features can be rather easily
investigated by using different methods, and which might provide insight into more general cases
[2830].
Consider a system with a critical point at temperature Tc, order parameter S(x,t), and prepare
it in some initial configuration that might correspond to an equilibrium state at a given tempera-
ture T0. At time t=0, bring the system into contact with a thermal bath with a temperature Tsnot
equal to T0. The relaxation process is expected to be characterized by some equilibration time teq(Ts)
such that, for tteq(Ts), equilibrium is attained and the dynamics is stationary and invariant under
time reversal, whereas, for 0<tteq(Ts), the evolution depends on the specific initial condition.
Upon approaching the critical point Ts=Tc, the equilibration time diverges as teq τzν,where
τ=(TTc)/Tcis the reduced critical temperature, zis the dynamic critical exponent, νis the
exponent for correlation length, and therefore equilibrium is never achieved. To monitor the time
evolution, we consider the average order parameter M(t)=1
VddxS(x,t), the time-dependent
correlation function of the order parameter
C(t,tw)=1
VddxS(x,t)S(x,tw)S(x,t)S(x,tw),(1)
where < ... > stands for the mean over the stochastic dynamics, and the linear response (suscepti-
bility) Rx(t,tw)to a small external field, applied at time tw, which is defined by the relation
R(t,tw)=1
VddxδS(x,t)
δh(x,tw)h=0
.(2)
Note that causality implies that R(t,tw>t)=0. According to the general picture of the relax-
ation process, one expects that, for t>twteq(Ts),C(t,tw)=Ceq(ttw)and R(t,tw)=Req
(ttw),whereCeq and Req are the corresponding equilibrium quantities, related by the fluctuation-
dissipation theorem (FDT):
Req(t)=−1
Ts
dCeq(t)
dt .(3)
The FDT suggests the definition of the so-called fluctuation-dissipation ratio (FDR):
X(t,tw)=TsR(t,tw)
twC(t,tw)(4)
with t>tw.Fort>twteq (Ts), the FDT yields X(t,tw)=1. This is not generically true in the
aging regime. The asymptotic value of the FDR,
X=lim
tw→∞ lim
t→∞ X(t,tw), (5)
is a very useful quantity in the description of systems with slow dynamics, since X=1when-
ever the aging evolution is interrupted and the system crosses over to equilibrium dynamics, i.e.,
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Tabl e 1. Values of the fluctuation-dissipation ratio Xfor some systems for a high-tem-
perature initial state with m01.
Model T<TcT=TcT>Tc
Free Gaussian field [33] Exact – 1/21
dD spherical model [34] Exact 012/d1
1D Ising model [35] Exact – 1/21
2D Ising model [34]MC 0.26(1)
[36]MC 0.340(5)
[37,38]MC 0.330(5)
2D 3-state Potts model [37,38]MC 0.406(1)
2D 4-state Potts model [37,38]MC 0.459(8)
3D Ising model [34]MC 00.40a
3D XY model [39]MC 00.43(4)
aThe value X0.40 was announced in Ref. [34] as result of preliminary simulations
on the 3D Ising model without demonstration of any obtained data either in Ref. [34]or
later publications.
teqTs<. Conversely, X= 1is a signal of an asymptotic non-equilibrium dynamics. More-
over, Xcan be used to define an effective non-equilibrium temperature Teff =T/X, which might
have some features of the temperature of an equilibrium system, e.g., controlling the direction of heat
flows and acting as a criterion for thermalization [31].
What is known in general about X? As a consequence of the fluctuation-dissipation theorem,
X(T>Tc)=1. On the other hand, on the basis of general scaling arguments for the phase-
ordering regime [32], it has been shown that X(T<Tc)=0. These results are expected to be
actually independent of the specific system and of its microscopic details. For T=Tc,thereareno
general arguments constraining the value of Xand therefore it has to be determined for each spe-
cific model. In Table 1we report some of the values that have been found either by exact solutions
or by means of Monte Carlo (MC) simulations (a more complete table can be found in Ref. [28]).
Clearly, X(T=Tc)depends on the model and on the space dimensionality d. Nevertheless, it
has been argued on the basis of scaling arguments [32,34]thatX(T=Tc)should be a universal
quantity associated with the critical dynamics.
At the present time, it is well known that the two-time dependence for autocorrelation and response
functions for systems starting from a high-temperature initial state with m0=0(or m01) satisfies
the following scaling forms:
C(t,tw)=AC(ttw)a+1d/z(t/tw)θ1fC(tw/t),
R(t,tw)=AR(ttw)ad/z(t/tw)θfR(tw/t), (6)
where fC(tw/t)and fR(tw/t)are finite for tw0,a=(2ηz)/z,θ=θ(2zη)/z,
and θis the initial slip exponent [40]. ARand ACare non-universal amplitudes that are fixed by the
condition fR,C(0)=1. With this normalization, fR,Care universal. From these scaling forms, the
universality of Xfollows as an amplitude ratio X=AR/[(1θ)AC][28,32,34].
A remarkable property of the non-equilibrium relaxation process in this case is the increase
of magnetization m(t)tθfrom a non-zero initial magnetization m01at short times t<tcr
m1/θ+β/zν
0(see, e.g., Fig. 1). The initial rise of magnetization is changed to the well known decay
m(t)tβ/zνfor ttcr. The critical exponents θand θdepend on the dynamic universality class
[27] and have been calculated by the RG method for a number of dynamic models, such as a model
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m(t)
t (MCS/s)
t
tvz
Fig. 1. Time evolution of the magnetization m(t)from the initial state with magnetization m0=0.03 1at
critical temperature Tc=3.499 48 as a result of Monte Carlo simulation of samples with spin concentration
p=0.8and linear size L=128.
with a non-conserved order parameter [26,4042] (model A), a model with an order parameter cou-
pled to a conserved density [43] (model C), and models with reversible mode coupling [44] (models
E, F, G, and J).
Three stages of the non-equilibrium relaxation process can be distinguished. The first quasi-
equilibrium stage is observed for small time separation ttwtwwith tw1, where aging does
not exist and the dynamic evolution of the correlation and response functions exhibits a station-
ary part and does not depend on waiting time with C=C(ttw)and R=R(ttw). In the limit
tw/(ttw)→∞, the scaling functions fC(tw/t)and fR(tw/t)in (7) are characterized by power
law dependences fC(tw/t)(tw/t)θ1and fR(tw/t)(tw/t)θthat lead to their quasi-equilibrium
behavior. The second aging regime is realized for times ttwtw1, where the correlation and
response functions are derived by relations
C(t,tw)t2β/(νz)
wˆ
FC(t/tw),
R(t,tw)t2β/(νz)1
wˆ
FR(t/tw),
(7)
and therefore at different waiting times do not superpose and are characterized by dif-
ferent slopes for each tw(in (7) the relation 2β/(ν z)=d/za1was used and the
scaling functions ˆ
FC(t/tw)=[(ttw)/tw]a+1d/z(t/tw)θ1fC(tw/t)and ˆ
FR(t/tw)=[(t
tw)/tw]ad/z(t/tw)θfR(tw/t)were introduced). At long time separations with ttwtw1,
the scaling functions in (7) decay as power laws:
ˆ
FC(t/tw)(t/tw)ca,ˆ
FR(t/tw)(t/tw)cr,(8)
where the exponent ca=d/zaθ=d/zθis the same; it describes the time dependence of
the autocorrelation function in the short-time regime of non-equilibrium behavior [26,45,46]. At this
short-time dynamics stage, aging effects are not developed either. Scaling analysis of the response
function R(t,tw)behavior in the short-time dynamics regime predicts that cr=ca.
Renormalization-group investigations of non-equilibrium critical behavior in d-dimensional pure
systems with n-component order parameter and a weakly dilute random Ising model (RIM) for purely
dissipative dynamics (model A) were carried out in Refs. [47]and[48]. The asymptotic values of
the FDR Xwere calculated with the use of the ε-expansion method (ε=4d) in the two-loop
approximation for pure systems [47],
(X)1
2=1+n+2
4(n+8)ε+ε2n+2
(n+8)2n+2
8+3(3n+14)
4(n+8)+c+Oε3,(9)
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where c=−0.0415 ··· (its analytic expression is given in Ref. [47]), and in the one-loop approxi-
mation for the diluted Ising model [48],
X=1
21
46ε
53 +O(ε). (10)
The following results were obtained: X
3DIs =0.429(6)for the 3D Ising (ε=1,n=1) model,
X
3DXY =0.416(8)for the XY (ε=1,n=2) model, and X
2DIs =0.30(5)for the 2D Ising (ε=2,
n=1) model, which are in good agreement with the Monte Carlo results for these models, given
in Table 1. For the 3D diluted Ising model (which is the only physically relevant case), the value
X
3DRIM 0.416 was obtained. To this order, it is not clear whether disorder really changes X
in a sensible way or not. In any case, this could not be safely stated from low-order computations
since the ε-expansion is known to be not well behaved at d=3[20,24,49]. The results of Monte
Carlo investigations into the non-equilibrium critical dynamics for the 3D diluted Ising model will
be presented below.
If the initial state of a system is characterized by magnetization m0= 0, the renormalization-group
analysis of non-equilibrium dynamics for model A predicts that the correlation C(t,tw)and response
R(t,tw)functions display the following scaling behaviors after a quench to T=Tc[40,50]:
C(t,tw)=aC(ttw)a+1d/z(t/tw)θ1FC(tw/t,t/tm),
R(t,tw)=aR(ttw)ad/z(t/tw)θFR(tw/t,t/tm), (11)
where modification of these scaling relations in comparison with (6) is connected with a new
timescale tm, set by the initial value of the magnetization m0, and which displays a universal
dependence on it:
tm=Bmm1
0,(12)
where the universal scaling exponent κ>0is given, in terms of static and dynamic equilibrium and
non-equilibrium exponents, by κ=θ+a+β/(νz).
The two-time quantities C(t,tw)and R(t,tw)are homogeneous functions of the three timescales
ttw,tw,andtm. In particular, when tw<ttm, which is always the case with m0=0, the scaling
forms of Cand Rbecome as they are in Eq. (6) with FC,R(x,0)=fC,R(x). In the opposite case, with
large times compared to tm, i.e., tmtw<t, the scaling forms of C(t,tw)and R(t,tw)become [50]
C(t,tw)aC(ttw)a+1d/z(t/tw)¯
θ1¯
FC(tw/t),
R(t,tw)aR(ttw)ad/z(t/tw)¯
θ¯
FR(tw/t), (13)
where the new exponent ¯
θ=−βδ/(νz)=−
1+a+β/(ν z)and ¯
FC,Rare universal scaling func-
tions related to the large-ybehavior of FC,R(x,y). In the aging regime that is realized for times
ttwtwtm, the correlation and response functions are derived by the relations
C(t,tw)t2β/(νz)
w˜
FC(t/tw),
R(t,tw)t2β/(νz)1
w˜
FR(t/tw)
(14)
with the scaling functions ˜
FC,R(t/tw), which decay at a long time separation limit with ttw
twtmas a power law:
˜
FC,R(t/tw)(t/tw)φ,(15)
where the exponent φ=d/za+βδ/(ν z).
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In Ref. [50], the non-equilibrium behavior of the d-dimensional Ising model with purely dissipative
dynamics was investigated during its critical relaxation from a magnetized initial state. The uni-
versal scaling forms of the two-time response and correlation functions were derived within the
field-theoretical approach and the associated scaling functions were computed in first order of the
ε-expansion. It was shown that aging behavior is clearly displayed and the asymptotic universal
fluctuation-dissipation ratio is characterized by relation
X=4
573
600 π2
100ε+Oε2,(16)
which gives X
3DIs 0.78 for the 3D Ising (ε=1,n=1) model and X
2DIs 0.75 for the 2D Ising
(ε=2,n=1) model. The obtained results were confirmed by Monte Carlo simulations of the 2D
Ising model with Glauber dynamics, from which it was found that X
MC =0.73(1).
3. Monte Carlo simulations of 3D pure Ising models with relaxation from
high-temperature and low-temperature initial states
One of the simplest non-trivial models in which aging occurs is the Ising model in d-dimensions
evolving with a purely dissipative dynamics after a quench to the critical point. Its Hamiltonian on a
hypercubic lattice is given by
H=−J
<i,j>
SiSj,(17)
where J>0is the short-range exchange interaction between spins Sifixed at the lattice sites, and
assuming values of Si1. We performed our Monte Carlo simulations using the heat-bath updat-
ing rule [51], simulating a large Ising spin system on a cubic lattice with linear size L=128 with
periodic boundary conditions at Tc=4.5114(1). We compute the magnetization,
M(t)=1
L3
L3
i=1
Si(t),(18)
and the two-time autocorrelation function,
C(t,tw)=1
L3
L3
i=1
Si(t)Si(tw)M(t)M(tw), (19)
where the angle brackets stand for an average over the initial configurations and MC realizations.
The averaging of C(t,tw)is carried out on 3000 MC runs for every tw.
In the case when we simulated the dynamics of systems starting from the high-temperature initial
state with magnetization value m0=0.02, the response function and fluctuation-dissipation ratio
were calculated using the relations [37,38]:
R(t,tw)=1
TL
3
L3
i=1Si(t)Si(tw+1)SW
i(tw+1),(20)
where SW
i=tanh(Jm=iSm/T),and
X(t,tw)=TR(t,tw)
twC(t,tw)=N
i=1Si(t)Si(tw+1)SW
i(tw+1)
N
i=1Si(t)[Si(tw+1)Si(tw)].(21)
The averaging of R(t,tw)and X(t,tw)is carried out on 90 000 MC runs for every tw.
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t
t
t
t
t
t
t
t
tCt
tCt
ttt
(a) (b)
Fig. 2. Time dependence of correlation function C(t,tw)as a function ttwfor different initial non-equilib-
rium states (a)m01;(b)m0=1.
We must note that the non-equilibrium behavior of R(t,tw)is characterized by very large fluctua-
tions at the critical point and these fluctuations drastically increase in the limit m00. Therefore,
for systems starting from the high-temperature initial state, we used m01, but not m0=0,and
realized the calculations of R(t,tw)with greater statistics than calculations of the autocorrelation
function with the same m0.
In the case when we considered the dynamics of systems starting from the low-temperature initial
state with magnetization value m0=1, we calculated the integrated response function [50,52]:
χ(t,tw)=tw
0
dtR(t,t)=1
TcN
N
i=1Si(t)Si(tw)(22)
with response function determined by (2) and the function Si(tw), which is computed during the
simulation from time t=0to twand is defined by the relation
Si(tw)=
tw
s=0Si(s)SW
i(s).(23)
In the large time limit,
Tcχ(C)=C
0
X(q)dq,
and the fluctuation-dissipation ratio can be defined as
X(t,tw)=lim
C0
Tcχ(t,tw)
C(t,tw).(24)
The MC-obtained time dependences of the autocorrelation function C(t,tw)and the response func-
tion R(t,tw)from observation time ttwfor different initial non-equilibrium states and twvalues
are presented in Figs. 2and 3. The curves of C(t,tw)and R(t,tw)demonstrate the aging effects, i.e.,
the slowing down of time correlations and decreasing of response with increasing system age tw.
In the aging regime, the time dependence of correlation and response functions is character-
ized by scaling relations (7)and(8). These scaling relations are rather well displayed by our data,
as shown in Fig. 4, where we used the exponent values β=0.325(1),ν=0.630(1)[53], and
z=2.024(6)[54]. For time intervals with (ttw)/tw1, we determined the values of exponents
ca=1.333(40)and cr=1.357(18), which demonstrate a good agreement with each other and with
value ca=1.362(19), obtained by the short-time dynamics method in Ref. [45].
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tRt
tt
t
t
t
Fig. 3. Time dependence of response function R(t,tw)as a function ttwfor a high-temperature initial state
with m01and for different values of tw.
t
ttCt ttRt
ttt
zz
(a) (b)
Fig. 4. Scaling collapse of correlation C(t,tw)(a)and response R(t,tw)(b)functions for a high-temperature
initial state with m01.
For a completely ordered initial state with m0=1, the short-time dynamics regime for autocorrela-
tion function C(t,tw)(Fig. 5) is characterized by the exponent φ(15) and we obtained the value φ=
2.742(32), which is in good agreement with the theoretical value φ=1+d/z+β/z)=2.737(8).
The obtained data for the fluctuation-dissipation ratio for the case with m1are presented in
Fig. 6. The asymptotic value of the FDR, X=0.380(13), was obtained by linear extrapolation of
data for X(tw/(ttw)) to the limit tw/(ttw)0.
For a completely ordered initial state with m0=1, the asymptotic value of the FDR,
X=0.77(6), was obtained as a realization of the limit dependence Tcχ(C)(Fig. 7), in accor-
dance with relation (24). This value of Xis in excellent agreement with the theoretical field value
X0.78 [50].
It should be noted that the non-equilibrium evolution of a system from a completely ordered initial
state is accompanied by a change in the magnetic domain structure from a single-domain initial
structure to a multidomain fluctuation structure at the critical point. In diluted systems, this process is
additionally characterized by pinning of domain walls on defects. These phenomena have an essential
influence on the two-time dependence of the correlation and response functions and lead to a new
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tt
tt
tt
Ct
z
Fig. 5. Scaling collapse of correlation function C(t,tw)for a low-temperature initial state with m0=1.
tX
ttt
t
Fig. 6. Fluctuation-dissipation ratio X(t,tw)as a function of tw/(ttw)for ttwtw.
t
Tt
t
X
Ct
Fig. 7. Dependence of Tcχ(t,tw)on C(t,tw)for different values of twwith a demonstration of its deviation
from a line with X=1.
asymptotic value of the FDR. At present, our investigations into the influence of disorder on the non-
equilibrium dynamics of the 3D Ising model with evolution from a completely ordered initial state
are in progress.
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4. Monte Carlo simulations of 3D diluted Ising models with relaxation from
a high-temperature initial state
The Hamiltonian of the ferromagnetic Ising model diluted by nonmagnetic impurity atoms is given
by
H=−J
<i,j>
pipjSiSj,(25)
where random occupation numbers pitake the values 0 or 1, and piequals 1 if the site of the lattice
contains spin and 0 otherwise. We considered the cubic lattice with periodic boundary conditions.
Let us denote Ns=pL3as the number of spins in a lattice with linear size Lwhere pis the spin
concentration.
We also performed our Monte Carlo simulations with the use of the heat-bath updating rule for Ising
spin systems with spin concentrations p=0.8and 0.6 on a cubic lattice with L=128. We computed
the magnetization
M(t)= 1
Ns
Ns
i=1
piSi(t)av
,(26)
and the two-time autocorrelation function
C(t,tw)= 1
Ns
Ns
i=1
piSi(t)Si(tw)av
,(27)
where the square brackets [···]av stand for an additional average over disorder configurations. The
averaging of C(t,tw)was carried out on over 1000 samples with different disorder configurations
with 15 MC runs for each sample.
In this part of the investigation, we have simulated the non-equilibrium critical dynamics of the
systems starting from a high-temperature initial state only with magnetization value m0=0.01 for
p=0.8and m0=0.005 for p=0.6. The response function and fluctuation-dissipation ratio were
calculated using the relations
R(t,tw)=1
Tc1
Ns
Ns
i=1Si(t)Si(tw+1)SW
i(tw+1)av
(28)
with SW
i=tanh(Jm=iSm/T),and
X(t,tw)=Ns
i=1Si(t)Si(tw+1)SW
i(tw+1)av
N
i=1Si(t)Si(tw+1)Si(tw)av
.(29)
The averaging of R(t,tw)and X(t,tw)was carried out on over 5000 samples with different disorder
configurations with 15 runs for each sample.
We give in Fig. 8the obtained curves for the time evolution of the magnetization M(t)from the
initial states with magnetization m0=0.02 at Tc=4.5114(1)for the pure Ising model, m0=0.01
at Tc=3.4995(2)for samples with spin concentration p=0.8,andm0=0.005 at Tc=2.4241(1)
for samples with p=0.6[25,26], which are characterized by a power-law dependence in the form
M(t)tθin the initial stage of evolution. The values of the exponent θcalculated with the use
of corrections to scaling are as follows: θ=0.106(4)for p=1,θ=0.127(16)for p=0.8,and
θ=0.186(39)for p=0.6[26,55,56]. The timescale of a critical initial increase of the magneti-
zation is tcr m1/(θ+β/zν)
0. The initial rise of magnetization is changed to the well known decay
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tM
t
p
p
p
Fig. 8. Time dependence of magnetization M(t)for different spin concentrations p.
ttC
ttR
tt tt
t
t
p
p
(a) (b)
Fig. 9. Time dependence of correlation C(t,tw)(a)and response R(t,tw)(b)functions from ttwfor
different spin concentrations p.
m(t)tβ/zνfor ttcr. However, in the limit of m00, the timescale tcr goes to infinity. The
times with t<tcr can be considered as the time intervals of non-equilibrium behavior of the systems
considered.
The curves in Fig. 8demonstrate that time intervals of non-equilibrium behavior increase with
increasing dilution of spin systems. Therefore, if, for investigation of different non-equilibrium
regimes in the pure system with tcr 103MCS/s, we considered the values of waiting time tw<102
MCS/s, then, for diluted systems with tcr 104MCS/s, we can use the values of tw<103MCS/s.
The curves for the time dependence of the autocorrelation C(t,tw)and response R(t,tw)func-
tions are plotted in Fig. 9for different values of waiting time twand spin concentrations p,which
demonstrate that the aging effects are increased with increasing defect concentrations.
To check the scaling predictions for C(t,tw)and R(t,tw)given by relations (7), we plot the depen-
dences of t2β/(νz)
wC(t,tw)and t1+2β/(νz)
wR(t,tw)versus t/twin Fig. 10 with the use of exponent
values: z=2.191(21),2β/ν =1.016(32)for p=0.8[26]andz=2.663(30),2β/ν =0.924(80)
for p=0.6[55,56]. These functions demonstrate the collapse of curves for different twwith fixed
spin concentration pinto a single curve with universal scaling dependence. Note that systems with
different pare characterized by different scaling functions FC,R(t,tw,p).
For an evolution stage with long time separations ttwtw1, the scaling functions
FC,R(t,tw,p)have power-law dependences (8), which are characterized by exponents caand cr.
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ttCt
tt tt
zttRt
z
pp
p
p
p
p
(a) (b)
Fig. 10. Scaling collapse of correlation C(t,tw)(a)and response R(t,tw)(b)functions for different spin
concentrations p.
Xtt
ttt
pt
t
t
t
t
t
t
pp
p
p
p
p
Fig. 11. Dependence of the FDR X(t,tw)on tw/(ttw)for ttwtwand for different p, where
p=1.0,tw=15 (); p=1.0,tw=30 (); p=0.8,tw=30 (); p=0.8,tw=50 (); p=0.6,tw=30 ();
p=0.6,tw=50 (); p=0.6,tw=150 ().
We determined the values of these exponents: ca=1.237(22),cr=1.251(22)for p=0.8and
ca=0.982(30),cr=0.950(8)for p=0.6. They demonstrate very good agreement with each
other for fixed spin concentration p, but differences between values caand crfor different pand
ca=1.333(40)and cr=1.357(18)for the pure Ising model exceed the statistical errors of their
determination. These values of caare in good agreement with ca=1.242(10)for a weakly diluted
system with p=0.8and ca=0.941(21)for a strongly diluted system with p=0.6, obtained in
Refs. [26,55,56] by the short-time dynamics method, but are in poor agreement with ca=1.05(3),
determined in Ref. [57] as a characteristic, independent of p, of the non-equilibrium critical behavior
of the autocorrelation function. The reasons for this discrepancy have been discussed in detail in our
earlier paper [26].
In later investigations, we computed the fluctuation-dissipation ratio in compliance with relation
(29). The data obtained are plotted in Fig. 11 for different spin concentrations p. For the analysis, we
used a time interval from ttwtwto ttwtw. In contrast to the pure Ising model, the data
for X(t,tw)for the diluted Ising model are characterized by an explicit dependence from tw.Atthe
beginning, we calculated the asymptotic values of X(tw,p)for different twfrom the plot in Fig. 11 in
the limit tw/(ttw)0, and then, using the obtained values of X(tw,p), we made an extrapolation
of 1/tw0to gain the asymptotic fluctuation-dissipation ratios X(p). The results are presented
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Xt
p
p
p
t
Fig. 12. Dependence of the FDR X(tw,p)on 1/twfor different p. The asymptotic value of X(p)can be
obtained in the limit 1/tw0.
Tabl e 2 . Values of the FDR Xfor spin concentra-
tions p=1.0,0.8, and 0.6.
X
twp=1.0p=0.8p=0.6
10 0.361(2)
15 0.371(4)
20 0.365(10)0.373(9)
25 0.369(9)
30 0.374(14)0.384(5)0.382(1)
50 0.379(10)0.397(4)0.407(3)
100 0.406(6)0.427(6)
150 0.412(9)0.437(9)
→∞ 0.380(13)0.413(11)0.446(8)
in Fig. 12 and in Table 2. The final values of the asymptotic fluctuation-dissipation ratio for the
systems considered are X(p=1)=0.380(13),X(p=0.8)=0.413(11),andX(p=0.6)=
0.446(8).
The obtained value X(p=0.8)=0.413(11)for the 3D weakly diluted Ising model is in very
good agreement with the result X
3DRIM 0.416 from low-order renormalization-group computa-
tions by the ε-expansion method [48], but the value X(p=0.6)=0.446(8)for the 3D strongly
diluted Ising model does not agree with the above-denoted values in the limits of errors. Also, we
must mention that the obtained value X(p=1)=0.380(13)for the 3D pure Ising model does
not agree with X
3DIs =0.429(6), which has been calculated with the use of the ε-expansion method
in the two-loop approximation for pure systems with a non-conserved order parameter dynamics
[47], but it is characterized by close agreement with X(p=1)0.40, noted in Ref. [34]asthe
result of preliminary simulations on the 3D pure Ising model, without, however, demonstration of
any obtained data in either Ref. [34] or later publications.
Simulations with probing magnetic field
Another way to compute the asymptotic fluctuation-dissipation ratio is the application of random
probing magnetic fields hiwith small amplitude hin the simulation process after tw[58]. For this
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tt
ttt
ttt
ttt
p
p
p
t
tt
t
Fig. 13. The integrated response function time dependences χ(t,tw)for different values of twand different
spin concentrations p.
case, the Hamiltonian for the diluted Ising model, taking into account a local magnetic field hi,is
given by
H=−J
<i,j>
pipjSiSj
i
hipiSi,(30)
and the two-time integrated response function χ(t,tw)is introduced through the relation
χ(t,tw)=t
tw
dtRt,t=
1
h2pL3
pL3
i=1
pihi(tw)Si(t)
av
,(31)
where the line stands for an average over the random field realizations with hi=0and hihj=h2δij.
For simplicity, the random hiwere taken from a bimodal distribution (hih)with small amplitude
h=0.01 in order to avoid nonlinear effects.
In the large time limit,
Tcχ(C)=1
C
X(q)dq,
and the fluctuation-dissipation ratio can be defined as
X(t,tw)=−lim
C0
Tcχ(t,tw)
C(t,tw)(32)
through dependence on the integrated response function χ(t,tw)computed from the autocorrelation
function C(t,tw).
In this part of the investigation, we determined the time dependence of the integrated response
function using the Metropolis algorithm. The simulations were also performed on lattices with linear
size L=128 with spin concentrations p=1.0,0.8,and0.6. The spin systems considered were
quenched in the critical point from high-temperature initial states with m0=0.02 for pure samples,
m0=0.01 for samples with spin concentration p=0.8,andm0=0.005 for samples with p=
0.6. The averages were taken with the use of 5000 samples characterized by different independent
configurations of defects and 10 different realizations of random fields.
The computed time dependences for the integrated response function are plotted in Fig. 13 for
different values of waiting time twand spin concentrations p.
To obtain the FDR on the basis of relation (32), we analyzed the dependences Tcχ(t,tw)from
C(t,tw), found the slopes of curves for different tw, and then made an extrapolation, tw→∞.
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t
t
tCt
tCt
Ttt
Ttt
p
p
p
p
p
Fig. 14. Tcχ(t,tw)versus C(t,tw)for different p. The error bars are smaller than the symbols.
X
t
p
p
p
p
p
Fig. 15. Dependence of the FDR X(tw,p)on 1/twfor p=0.8and 0.6. The asymptotic value of X(p)can
be obtained in the limit 1/tw0. In the inset, the dependence of X(tw)on 1/twis given separately for the
pure Ising model.
Tabl e 3. The values of the FDR Xfor spin concen-
trations p=1.0,0.8, and 0.6computed with applica-
tion of a probing magnetic field.
X
twp=1.0p=0.8p=0.6
10 0.586(24)
25 0.460(22)
50 0.437(26)
250 0.708(16)0.726(13)
500 0.553(17)0.583(14)
1000 0.494(17)0.519(29)
→∞ 0.391(12)0.419(11)0.443(10)
The results for these pure and diluted systems are given in Figs. 14 and 15 and also presented in
Tabl e 3. It should be noted that the values of X(tw), in accordance with relation (32), are calcu-
lated in the limit C(t,tw)0, which corresponds to the stage in the interval ttw1. The inset
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in Fig. 14 highlights the sections of the dependences of Tχ(t,tw)on C(t,tw), which satisfy these
criteria and on which the values of X(tw)were determined.
The comparison shows that the final values of the asymptotic fluctuation-dissipation ratio,
X(p=1.0)=0.391(12),X(p=0.8)=0.419(11),andX(p=0.6)=0.443(10), obtained
with application of a probing magnetic field, are in very good agreement with the values
X(p=1)=0.380(13),X(p=0.8)=0.413(11),andX(p=0.6)=0.446(8), computed
with realization of heat-bath dynamics simulations.
5. Conclusions
In the introduction to this paper, we reviewed some theoretical results of computations in the range of
non-equilibrium phenomena that have been obtained in recent years for universal quantities, such as
the exponents determining the scaling behavior of dynamic response and correlation functions and
the fluctuation-dissipation ratio, associated with the non-equilibrium critical dynamics. It was noted
that the value of the asymptotic fluctuation-dissipation ratio Xfor quenches from the disordered
initial state with m01to the critical temperature Ts=Tcdepends on the universality class of
critical behavior to which the specific statistical system belongs. In the original part, we considered
the results of our MC simulations for 3D pure and diluted Ising models with Glauber and Metropolis
dynamics for quenches to their critical temperatures as functions of spin concentration. The influence
of critical fluctuations, different non-equilibrium initial states, and site-quenched disorder on the two-
time dependence of the correlation and response functions in a non-equilibrium critical regime was
investigated for these models.
Analysis of the time dependences for the autocorrelation and response functions in the aging
regime of evolution showed that the aging effects increase with increasing defect concentration.
The obtained values of the autocorrelation and response function critical exponents ca,crand the
fluctuation-dissipation ratio Xdemonstrate that statistical systems described by 3D pure and
diluted Ising models with a non-conserved order parameter dynamics belong to different classes
of non-equilibrium critical behavior.
Analysis of the simulation results shows that the insertion of disorder leads to new universal FDRs
with X
strong diluted >X
weak diluted >X
pure.
As mentioned above, the values computed for the 3D weakly diluted Ising model with p=0.8,
X=0.413(11)and 0.419(11), are in very good agreement with the result X
3DRIM 0.416
from low-order renormalization-group computations by the ε-expansion method [48], but, for
the 3D pure Ising model, the obtained values, X=0.380(13)and 0.391(12), do not agree with
X
3DIs =0.429(6), which has been calculated with the use of the ε-expansion method in the two-loop
approximation for pure systems with a non-conserved order parameter dynamics [47]. We think that
this difference of results is because the series of ε-expansion obtained in two-loop order (9) is not
an oscillating series. Therefore, it is impossible for summation of this series to apply the more exact
algorithm of the Pade–Borel or Pade–Borel–Leroy summation methods, but instead to apply only the
simplest Pade approximant method accounting for the direct [2,0] and inverse [0,2] approximants (the
[1,1] approximant has a pole in the important range). To obtain more reliable renormalization-group
results, one needs to calculate Xin the next three-loop order by the ε-expansion method or to
apply this more accurate method of field-theoretic renormalization-group description directly to 3D
systems with the use of different resummation methods [24,54,59].
When discussing the influence of disorder on the universality of the critical behavior in the
site-diluted 3D Ising model, we must note that the results of Monte Carlo investigations into this
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model are quite contradictory. The results of some researchers are aimed at proving the concept
that the values of the critical exponents do not depend on the concentration of defects down to
the percolation threshold with ν=0.684(5),β=0.355(3),γ=1.342(10)[60], z=2.62(7)[61],
z=2.35(2)[62], and θ=0.10(2)[63] obtained by a specific procedure for fitting intermediate
values of the effective exponents and amplitudes in the scaling dependence of the calculated thermo-
dynamic characteristics for different spin concentrations, using a fitted exponent of the corrections
to the scaling ω=0.370(63)[60], ω1=0.50(13)[61], and ω2=0.82(8)[62]. The results of other
researchers indicate that there are two universal classes of critical behavior for weakly diluted sys-
tems with ν=0.68(2),β=0.34(2)[64], z=2.38(1)[65], ν=0.682(3),β=0.344(3)[66,67],
ν=0.683(4),β=0.310(3),γ=1.299(3)[68], ν=0.696(3),γ=1.345(4),ω=0.23(13)[25],
z=2.20(7)[14,15], z=2.191(21),ω=0.256(55),θ=0.127(16)[26], and z=2.28(7)[69],
and for strongly diluted systems with ν=0.72(2),β=0.33(2),γ=1.51(3)[64], z=2.53(3)
[65], ν=0.717(7),β=0.313(12)[66,67], ν=0.725(6),β=0.349(4),γ=1.446(4)[68], ν=
0.725(4),γ=1.415(11),ω=0.28(15)[25], z=2.58(9)[14,15], z=2.663(30),ω=0.286(10),
θ=0.167(18)[55,56], and z=2.67(8)[69].
In this discussion, we consider the results obtained in Ref. [70] very important, where the fixed-
point structure of the 3D site- and bond-diluted and ±JIsing models using the numerical domain-
wall renormalization-group method has been studied. It was shown that the observed random fixed
points are characterized by close values. This fact allows us to suggest that there exists a universal
fixed point characterizing the 3D disordered ferromagnetic Ising model, irrespective of the type of
disorder. Unfortunately, this global conclusion was made on the basis of simulation results of systems
with very small linear sizes, L=8and 12. For the 3D site-diluted Ising model, the influence of the
correction to the finite-size scaling on the critical temperature, the values of the critical exponents, and
other critical characteristics is very important, especially for strongly diluted systems. In addition,
we must note that, in the renormalization-group field-theory description of the random Ising model
(RIM), a model with two vertices characterizing the interaction of the order parameter fluctuations is
applied [16,7173]. In this case, the critical behavior of RIM is determined by a stable fixed point for
these two vertices with negative values for both exponents for the eigenvalues of the stability matrix.
In Ref. [70], the author uses a method with a single effective renormalization parameter r, responsible
for the influence of disorder and stability of a fixed point (another parameter tis connected with the
effective temperature of the system). Such a transition from two-parametric phase space to one-
parametric space can be characterized by inaccuracy in the obtained results. However, the method
applied in Ref. [70] is very interesting, and it is desirable to see its development to 3D RIM with the
largest system sizes and with possible two-parametric modification.
Acknowledgements
This work was supported by the Russian Scientific Fund through project No. 14-12-00562. The simula-
tions were supported by the Supercomputing Center of Lomonosov Moscow State University and Joint
Supercomputer Center of the Russian Academy of Sciences.
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