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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 t−twwith 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 X∞with

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) [1–5]. 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/),

which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.

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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 [10–17]. 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

t−twwith t>twnot via t−twonly. It is important that decays for these two-time quantities as

functions of time of observation t−tware 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 [1–3] 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

[28–30].

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 ∼τ−zν,where

τ=(T−Tc)/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(t−tw)and R(t,tw)=Req

(t−tw),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 X∞for 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 01−2/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 0≈0.40a

3D XY model [39]MC 00.43(4)

aThe value X∞≈0.40 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, X∞can 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 X∞and 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(t−tw)a+1−d/z(t/tw)θ−1fC(tw/t),

R(t,tw)=AR(t−tw)a−d/z(t/tw)θfR(tw/t), (6)

where fC(tw/t)and fR(tw/t)are ﬁnite for tw→0,a=(2−η−z)/z,θ=θ−(2−z−η)/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 X∞follows as an amplitude ratio X∞=AR/[(1−θ)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 ∼

m−1/θ+β/zν

0(see, e.g., Fig. 1). The initial rise of magnetization is changed to the well known decay

m(t)∼t−β/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 [26,40–42] (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 t−twtwwith 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(t−tw)and R=R(t−tw). In the limit

tw/(t−tw)→∞, the scaling functions fC(tw/t)and fR(tw/t)in (7) are characterized by power

law dependences fC(tw/t)∼(tw/t)θ−1and fR(tw/t)∼(tw/t)θthat lead to their quasi-equilibrium

behavior. The second aging regime is realized for times t−tw∼tw1, where the correlation and

response functions are derived by relations

C(t,tw)∼t−2β/(νz)

wˆ

FC(t/tw),

R(t,tw)∼t−2β/(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/z−a−1was used and the

scaling functions ˆ

FC(t/tw)=[(t−tw)/tw]a+1−d/z(t/tw)θ−1fC(tw/t)and ˆ

FR(t/tw)=[(t−

tw)/tw]a−d/z(t/tw)θfR(tw/t)were introduced). At long time separations with t−twtw1,

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/z−a−θ=d/z−θis 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 X∞were calculated with the use of the ε-expansion method (ε=4−d) 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

2−1

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(t−tw)a+1−d/z(t/tw)θ−1FC(tw/t,t/tm),

R(t,tw)=aR(t−tw)a−d/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=Bmm−1/κ

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

t−tw,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(t−tw)a+1−d/z(t/tw)¯

θ−1¯

FC(tw/t),

R(t,tw)=¯aR(t−tw)a−d/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

t−tw∼twtm, the correlation and response functions are derived by the relations

C(t,tw)∼t−2β/(νz)

w˜

FC(t/tw),

R(t,tw)∼t−2β/(νz)−1

w˜

FR(t/tw)

(14)

with the scaling functions ˜

FC,R(t/tw), which decay at a long time separation limit with t−tw

twtmas a power law:

˜

FC,R(t/tw)∼(t/tw)−φ,(15)

where the exponent φ=d/z−a+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

5−73

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 Si=±1. 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 t−twfor 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 m0→0. 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

C→0

∂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 t−twfor 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 (t−tw)/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 t−twfor 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/(νz)=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/(t−tw)) to the limit tw/(t−tw)→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 X∞is in excellent agreement with the theoretical ﬁeld value

X∞0.78 [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/(t−tw)for t−twtw.

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 ∼m−1/(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 t−twfor

different spin concentrations p.

m(t)∼t−β/zνfor ttcr. However, in the limit of m0→0, 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 t−twtw1, 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/(t−tw)for t−twtwand 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 t−tw∼twto t−twtw. 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/(t−tw)→0, and then, using the obtained values of X(tw,p), we made an extrapolation

of 1/tw→0to 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/tw→0.

Tabl e 2 . Values of the FDR X∞for 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 (hi=±h)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

C→0

∂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/tw→0. 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 X∞for 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 X∞for 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 X∞demonstrate 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 X∞in 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 [16,71–73]. 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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