# Crossover in growth law and violation of superuniversality in the random-field Ising model.

**ABSTRACT** We study the nonconserved phase-ordering dynamics of the d=2,3 random-field Ising model, quenched to below the critical temperature. Motivated by the puzzling results of previous work in two and three dimensions, reporting a crossover from power-law to logarithmic growth, together with superuniversal behavior of the correlation function, we have undertaken a careful investigation of both the domain growth law and the autocorrelation function. Our main results are as follows: We confirm the crossover to asymptotic logarithmic behavior in the growth law, but, at variance with previous findings, we find the exponent in the preasymptotic power law to be disorder dependent, rather than being that of the pure system. Furthermore, we find that the autocorrelation function does not display superuniversal behavior. This restores consistency with previous results for the d=1 system, and fits nicely into the unifying scaling scheme we have recently proposed in the study of the random-bond Ising model.

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**ABSTRACT:**We report a study of nonequilibrium relaxation in a two-dimensional random field Ising model at a nonzero temperature. We attempt to observe the coarsening from a different perspective with a particular focus on three dynamical quantities that characterize the kinetic coarsening. We provide a simple generalized scaling relation of coarsening supported by numerical results. The excellent data collapse of the dynamical quantities justifies our proposition. The scaling relation corroborates the recent observation that the average linear domain size satisfies different scaling behavior in different time regimes.03/2014; 89(4). - SourceAvailable from: Eugenio Lippiello[Show abstract] [Hide abstract]

**ABSTRACT:**We study numerically the phase-ordering kinetics of the two-dimensional site-diluted Ising model. The data can be interpreted in a framework motivated by renormalization-group concepts. Apart from the usual fixed point of the nondiluted system, there exist two disorder fixed points, characterized by logarithmic and power-law growth of the ordered domains. This structure gives rise to a rich scaling behavior, with an interesting crossover due to the competition between fixed points, and violation of superuniversality.Physical Review E 10/2013; 88(4-1):042129. · 2.31 Impact Factor - SourceAvailable from: export.arxiv.org[Show abstract] [Hide abstract]

**ABSTRACT:**We consider the one-dimensional Glauber dynamics with coupling disorder in terms of bilinear fermion Hamiltonians. Dynamic exponents embodied in the spectrum gap of these latter are evaluated numerically by averaging over both binary and Gaussian disorder realizations. In the first case, these exponents are found to follow the nonuniversal values of those of plain dimerized chains. In the second situation their values are still nonuniversal and subdiffusive below a critical variance above which, however, the relaxation time is suggested to grow as a stretched exponential of the equilibrium correlation length.Physical Review E 06/2013; 87(6-1):062102. · 2.31 Impact Factor

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arXiv:1202.1534v1 [cond-mat.dis-nn] 7 Feb 2012

Crossover in Growth Law and Violation of

Superuniversality in the Random Field Ising

Model

F. Corberi1, E. Lippiello2, A. Mukherjee3, S. Puri3and M. Zannetti1

1Dipartimento di Fisica E.Caianiello and CNISM, Unit` a di Salerno,

Universit` a di Salerno, via Ponte don Melillo, 84084 Fisciano (SA), Italy.

2Dipartimento di Scienze Ambientali, Seconda Universit` a di Napoli,

Via Vivaldi, Caserta, Italy.

3School of Physical Sciences, Jawaharlal Nehru University,

New Delhi–110067, India.

Abstract

We study the nonconserved phase ordering dynamics of the d = 2,3

random field Ising model, quenched to below the critical temperature.

Motivated by the puzzling results of previous work in two and three di-

mensions, reporting a crossover from power-law to logarithmic growth,

together with superuniversal behavior of the correlation function, we

have undertaken a careful investigation of both the domain growth

law and the autocorrelation function. Our main results are as follows:

We confirm the crossover to asymptotic logarithmic behavior in the

growth law, but, at variance with previous findings, the exponent in

the preasymptotic power law is disorder-dependent, rather than being

the one of the pure system. Furthermore, we find that the autocorre-

lation function does not display superuniversal behavior. This restores

consistency with previous results for the d = 1 system, and fits nicely

into the unifying scaling scheme we have recently proposed in the

study of the random bond Ising model.

1

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1Introduction

Much recent interest in statistical physics has focused on understanding out-

of-equilibrium phenomena. In this context, of paramount importance are

slow relaxation phenomena, which primarily occur in glassy systems. An im-

portant hallmark of slow relaxation is the lack of time-translation-invariance,

manifested through aging behavior. A similar phenomenology is also ob-

served in systems without disorder, e.g., ferromagnets quenched below the

critical point. The behavior of these systems is well understood in terms of

the domain growth mechanism of slow relaxation [1, 2].

The key feature of domain growth, or coarsening, is the unbounded

growth of the domain size, which entails scaling due to the existence of a

dominant length scale, and aging as a manifestation of scaling in multiple-

time observables. The simplicity of this structure is very attractive and is

expected to be valid beyond the realm of disorder-free phase-separating sys-

tems, establishing domain growth as a paradigm of slow relaxation. However,

as is well known, the applicability of domain growth concepts to hard prob-

lems (such as spin glasses or structural glasses) still remains a debated issue

[3]. Therefore, it is of considerable interest to study the role of disorder in

systems where its presence does not compete with phase ordering [4].

A class of systems of this type are disordered ferromagnets, where disorder

coexists with the low-temperature ferromagnetic order. There are different

ways to introduce disorder in a ferromagnet without inducing frustration.

This can be achieved through bond or site dilution, by randomizing the ex-

change interaction strength while keeping it ferromagnetic, or by introducing

a random external field. These disordered systems have been an active area

of research for quite some time now. The unifying theme of investigation

has been disorder-induced changes in the properties of the underlying pure

systems, with primary interest in the growth law, in the equal-time corre-

lation function and, more recently, in the two-time autocorrelation function

and the related response function [5, 6, 7, 8, 9]. However, despite the many

experimental and theoretical studies [4], a number of issues are still open.

Among these, of primary importance are (i) the nature of the asymptotic

growth law (power-law vs. logarithmic) and (ii) the existence of superuniver-

sal behavior of the correlation and response functions. This is the idea that

scaling functions are robust with respect to disorder, which is expected not

to change the low temperature properties of the system [10]. The lack of a

general framework to understand this complex phenomenology has proven a

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Page 3

major obstacle to development. Recently, in the context of the random bond

Ising model (RBIM) [9], we have shown that the renormalization group (RG)

picture of crossover phenomena may well serve the purpose.

In this paper, we extend the RG conceptual framework to the ordering

dynamics of the random field Ising model (RFIM) [11]. In this system, the

deep asymptotic regime turns out to be numerically accessible, allowing us to

make precise statements regarding the growth law and the superuniversality

(SU) issue (see subsection 2.1). Our principal findings are (a) the existence

of a crossover from power law domain growth (with a disorder-dependent

exponent) to logarithmic growth, and (b) the absence of SU. Both results

fit nicely into an RG picture where disorder acts as a relevant perturbation

with respect to the pure fixed point. This confirms the robustness and the

general applicability of the approach proposed in Ref. [9].

This paper is organized as follows. In Sec. 2, we provide an overview of

domain growth laws and of our scaling framework for phase ordering dynam-

ics in disordered systems. In Sec. 3, we present detailed numerical results for

ordering in the d = 2 RFIM1. These results are interpreted using the scaling

framework of Sec. 2. Sec. 4 is devoted to the presentation of numerical results

in the d = 3 case. Finally, in Sec. 5, we conclude this paper with a summary

and discussion.

2 Domain Growth Laws in Disordered Sys-

tems

Dynamical scaling is the most important characteristic of phase ordering

systems [1, 2]. Let us summarise the concept in the simplest case of pure

systems. As time increases, the typical domain size L(t) grows and becomes

the dominant length in the problem. Then, all other lengths can be rescaled

with respect to L(t). For instance, the two-time order-parameter correlation

function C(r,t,tw), with tw≤ t, scales as [12, 13]

C(r,t,tw) = G(r/L,L/Lw)(1)

where L and Lwstand for L(t) and L(tw), respectively. This contains, as a

special case, the usual scaling of the equal-time (t = tw) correlation function

1As it is explained in subsection 3.1, the spin updating rule we use is equivalent to a

quench to T = 0. Hence, the d = 2 RFIM phase orders even if d = 2 is the lower critical

dimensionality.

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Page 4

C(r,t) = G1(r/L). Further, for r = 0, we have the aging form of the auto-

correlation function C(t,tw) = G2(L/Lw). The validity of scaling is, by now,

a well-established fact. A complete picture of an ordering problem requires

the understanding of the growth law (i.e. how L(t) depends on t) and of the

scaling function G(x,y).

A systematic study of the growth law has been undertaken by Lai et al.

(LMV) [14, 15], who identified four different universality classes of growth

kinetics. LMV considered the role of several factors in the ordering dy-

namics, e.g., temperature, conservation laws, dimensionality, order param-

eter symmetry, lattice structure and disorder. An important distinction is

made between systems that do not freeze (i.e., without free-energy barriers)

and those that do freeze (i.e., with barriers) when the quench is made to

T = 0. To the first category belong pure systems with non-conserved dy-

namics, whose growth follows the power law L(t) = Dt1/z, where z = 2.

LMV designate these as Class 1 systems. To the second category belong

systems whose growth requires thermal activation. This includes pure sys-

tems with conserved order parameter and systems (both conserved and non-

conserved) with quenched disorder. This category is further subdivided into

three classes. In Class 2 systems the freezing involves only local defects, with

activation energy EBindependent of the domain size. In this case, growth

is still power law: L(t) = Dt1/zwith z = 2 and 3 for the non-conserved and

conserved cases, respectively. Furthermore, the prefactor D has a strong tem-

perature dependence, D ∼ e−EB/(zT). Finally, in Class 3 and Class 4 systems,

the freezing involves a collective behavior which depends on the domain size

L. If the corresponding activation energy scales with L like EB(L) ∼ ǫLϕ,

where ǫ measures the disorder strength, the asymptotic growth law is loga-

rithmic

L(t) ∼ (T/ǫ)1/ϕ[ln(t/τ)]1/ϕ

(2)

with τ ∼ T/(ϕǫ). For Class 3 systems, we have ϕ = 1, and for Class 4

systems, we have ϕ ?= 1.

Ferromagnets (with or without disorder) offer examples of the classes

listed above. For simplicity, let us consider systems with non-conserved order

parameter. The pure ferromagnetic Ising model with Glauber kinetics is a

well-known Class 1 system [1]. The d = 1 ferromagnetic RBIM [7] is an

example of a Class 2 system. The d = 1 RFIM [16] belongs to Class 4, with

ϕ = 1/2. The RFIM in higher dimensions, d = 2 [17, 18] and d = 3 [19, 20],

shows logarithmic growth, although it is not easy to unambiguously establish

4

Page 5

the value of ϕ. Recently, we have presented evidence [9] for logarithmic

growth in the d = 2 RBIM, but have not established whether it is a Class 3 or

Class 4 system. This is a particularly interesting system, because its growth

law was previously [21] believed to be power law with a disorder-dependent

exponent. If so, this would have shown the existence of a new universality

class, say Class 5, in addition to the four listed by LMV. We should stress

that a huge numerical effort is involved in accessing the logarithmic growth

regime of the d = 2 RBIM, and our understanding of this system remains

incomplete.

In Ref. [9], we have proposed to unify this wide variety of behaviors for

disordered domain growth into a scaling framework for the growth law itself.

In all the cases we consider in this paper, disorder (h0) and temperature

(T) enter through their ratio h0/T (see Sec. 3.1 below). This will be de-

noted by ǫ and, for short, will be termed as disorder. Let us begin with the

straightforward crossover set-up, where the growth law is assumed to scale

as

L(t,ǫ) = t1/zF(ǫ/tφ), (3)

z = 2 is the growth exponent for non-conserved dynamics in a pure ferro-

magnet and φ is the crossover exponent. With the additional assumption

that the scaling function behaves as

F(x) ∼

?const., for x → 0,

x1/(φz)ℓ(x−1/φ), for x → ∞,

(4)

where x = ǫ/tφ, Eq. (3) describes the crossover from the power law L(t) ∼ t1/z

to the asymptotic form L(t) ∼ ℓ?t/ǫ1/φ?, if φ < 0, and vice-versa if φ > 0.

Alternatively, disorder is asymptotically relevant when φ < 0, and irrelevant

when φ > 0. The key quantity in the analysis of crossover is the effective

growth exponent

1

zeff(t,ǫ)=∂ lnL(t,ǫ)

∂ lnt

=1

z− φ∂ lnF(x)

∂ lnx

,(5)

which depends on t and ǫ through x.

In the following discussion, it will be useful to use the above relations in

the inverted form:

t = Lzg(L/λ), (6)

where

λ = ǫ1/(φz)

(7)

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Page 6

is a length scale associated with disorder. The scaling functions appearing

in Eqs. (3) and (6) are related by

g(y) = F−z(x) (8)

and y = L/λ is related to x by

y = x−1/(φz)F(x). (9)

Then, from Eq. (4) and φ < 0, it follows that

g(y) ∼

?const., for y ≪ 1,

y−zℓ−1(y), for y ≫ 1,

(10)

where ℓ−1stands for the inverse function of ℓ. The opposite behavior holds

for φ > 0

?y−zℓ−1(y), for y ≪ 1,

const., for y ≫ 1.

g(y) ∼

(11)

Finally, the effective exponent as a function of y is obtained from Eq. (6)

zeff(y) = z +∂ lng(y)

∂ lny

. (12)

Therefore, for φ < 0 (disorder relevant), Eq. (10) yields

zeff(y) =

?z, for y ≪ 1,

∂ lnℓ−1(y)/∂ lny, for y ≫ 1,

(13)

and for φ > 0 (disorder irrelevant), we obtain from Eq. (11)

zeff(y) =

?∂ lnℓ−1(y)/∂ lny, for y ≪ 1,

z, for y ≫ 1.

(14)

2.1Superuniversality

One would expect that the above crossover scenario, which is well estab-

lished for the growth law, would extend also to the other observables. How-

ever, this expectation is in conflict with the SU statement that all disorder

dependence in observables other than the growth law can be eliminated by

6

Page 7

reparametrization of time through L(t,ǫ) [10]. Thus, according to SU, for

the autocorrelation function one should have

C(t,tw,ǫ) = G2(L(tw,ǫ)/L(tw,ǫ)) (15)

where G2 is the scaling function of the pure case. The validity of SU is

controversial, since the d = 1 results [7, 16] clearly demonstrate the absence

of SU, while from the study of the correlation function for d ≥ 2, there is

evidence both in favour [19, 20, 21, 22] and against [9] SU validity. Recently,

the validity of SU has been extended to the geometrical properties of domain

structures [23].

In the next sections we present comprehensive numerical results from

large scale simulations of ordering dynamics in the RFIM in d = 2,3. We

will analyze numerical results within the above scaling framework, producing

evidence against SU validity.

3 Numerical Results for d = 2

3.1Simulation Details

We consider an RFIM on a two-dimensional square lattice, with the Hamil-

tonian

N

?

i=1

H = −J

?

?ij?

σiσj−

hiσi,σ = ±1,(16)

where ?ij? denotes a nearest-neighbour pair, and J > 0 is the ferromagnetic

exchange coupling. The random field hi= ±h0is an uncorrelated quenched

variable with a bimodal distribution

P(hi) =1

2[δ(hi− h0) + δ(hi+ h0)]. (17)

The system evolves according to the Glauber kinetics, which models non-

conserved dynamics [2], with spin flip transition rates given by

w(σi→ −σi) =1

2

?1 − σitanh?(HW

i + hi)/T??

(18)

where HW

the limit T → 0 (J/T → ∞), while keeping the ratio ǫ = h0/T finite. In this

i

is the local Weiss field. All results in this paper correspond to

7

Page 8

limit the system undergoes phase ordering in any dimension, down to d = 1

[16]. The transition rates take the form

w(σi→ −σi) =

1 for HW

0 for HW

1

2(1 − sign(σihi)tanh(ǫ)) for HW

iσi< 0,

iσi> 0,

i

= 0

(19)

which shows that disorder affects the evolution through the ratio ǫ = h0/T,

as anticipated in Sec. 2. Moreover, Eq. (19) allows for an accelerated up-

dating rule, with a considerable increase in the speed of computation [24],

by restricting updates to the sites with HW

in time as 1/L(t). The gain in the speed of computation becomes more

important the longer the simulation.

All statistical quantities presented here have been obtained as an average

over Nrun = 10 independent runs. For each run, the system has different

initial condition and random field configuration. We have considered the

values of disorder amplitude ǫ = 0,0.25,0.5,1,1.5,2,2.5and we have carefully

checked that no finite size effects are present up to the final simulation time

when N = 80002spins. In the pure case, since coarsening is more rapid, we

have taken N = 120002.

Numerical results for the growth law and the autocorrelation function are

presented in the following subsections.

iσi≤ 0, whose number decreases

3.2 Growth Law

We have obtained the characteristic L(t) from the inverse density of defects.

This is measured by dividing the number of sites with at least one oppositely-

aligned neighbor by the total number of sites2. The plot of L(t,ǫ) vs. t,

in Fig. 1, shows the existence of at least two time-regimes, separated by

a microscopic time t0 of order 1. In the early-time regime (for t < t0),

there is no dependence on disorder and growth is fast. This is the regime

where the defects seeded by the random initial condition execute rapid motion

toward the nearby local minima. For ǫ > 0 and t > t0, there is a strong

dependence on disorder producing slower growth and deviation from the

power law behavior of the pure case (top cicles line in Fig. 1).

2We have checked that the same results are obtained measuring L(t) from the equal

time corrlation function.

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Page 9

100

102

104

106

t

100

101

102

103

L(t)

ε=0

ε=0.2

ε=0.35

ε=0.5

ε=0.65

ε=0.8

t0

Figure 1: (Color online) Growth law in d = 2. The dashed line is the t1/2

growth law.

9

Page 10

100

101

102

103

t

104

105

106

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

zeff

ε=0

ε=0.2

ε=0.35

ε=0.5

ε=0.65

ε=0.8

t0

Figure 2: (Color online) Effective exponent zeffvs. t in d = 2. The horizontal

dashed lines indicate z, the plateau values of zeff.

In Fig. 2, we show the time-dependence of the effective exponent zeff(t,ǫ),

defined by Eq. (5). For t > t0, this plot shows the existence of an intermediate

power-law regime, characterized by a plateau where zeff is approximately

constant.This is followed by the late regime where zeff is clearly time-

dependent. The disorder dependent values of zeff on the plateaus, denoted

by z, are listed in Table 1 and plotted in Fig. 3. We encountered a similar

crossover in our study of the d = 2 RBIM [9], i.e., a preasymptotic power

law regime with a disorder dependent exponent, followed by an asymptotic

regime where the growth law deviates from a power law.

The appearance of a disorder dependent exponent z in the intermediate

regime suggests to upgrade the crossover picture, presented in Sec. 2, by

replacing the pure growth exponent z by z in all the scaling formulae. Then,

from Eq. (12) it follows that zeff−z ought to depend only on y = L/λ. Indeed,

as Fig. 4 shows, it is possible to determine numerically the quantity λ such

that the plots of (zeff−z) vs. L/λ, for different disorder values, collapse on a

single master curve. The ǫ-dependence of λ is displayed in Fig. 5 and is well

10

Page 11

00.20.4

ε

0.6

0.8

2

2.5

3

z

Figure 3: (Color online) z (taken from Fig. 2) vs. ǫ. The red line is the best

fit z = 2.0 + 1.4ǫ1.35.

ǫ

0

z

2

0.2

0.35

0.5

0.65

0.8

2.20

2.31

2.52

2.77

3.05

Table 1: Plateau exponent z for various disorder strengths.

11

Page 12

02040

60

80

L/λ

-2

-1

0

1

2

3

4

5

zeff-z

ε=0

ε=0.2

ε=0.35

ε=0.5

ε=0.65

ε=0.8

Figure 4: (Color online) Subtracted effective exponent (zeff− z) vs. L/λ.

The dashed line is the best fit zeff− z = 0.0055(L/λ)1.5.

fitted by

λ ∼ ǫ−2. (20)

Comparing this with Eq. (7), the negative exponent implies φ < 0 and,

therefore, that disorder acts like a relevant scaling field. This is also confirmed

by the behavior of zeff(y) in Fig. 4, which is consistent with Eq. (13) but not

with Eq. (14).

Fitting the data of Fig. 4 to the power law zeff− z = byϕ, we find b ≃

0.0055 and ϕ ≃ 1.5. Hence, from the definition of zeffin Eq. (5) follows

∂ lnt

∂L

= z + byϕ

(21)

which, after integrating with respect to L, yields

t = K(ǫ)Lzg(L/λ) (22)

12

Page 13

10-2

10-1

ε

100

101

102

103

104

λ

ε−2

Figure 5: (Color online) Plot of λ vs. ǫ.

ǫK

0.5

1.0

1.5

2.0

0.39

0.13

0.039

0.015

Table 2: Prefactor K(ǫ) for various disorder strengths.

where K(ǫ) is an ǫ-dependent prefactor. Indeed, replotting in Fig. 6 the data

of Fig. 1, as tL−z/K(ǫ) vs. y, an excellent data collapse on the master curve

g(y) ∼ exp

?b

ϕyϕ

?

(23)

is obtained, with the values of K(ǫ) listed in Table 2.

The plot of the scaling function g(y) illustrates quite effectively (i) the

existence of the crossover, and (ii) that our numerical data reach deep into

the asymptotic regime. The flat part of the curve, where g(y) lies on the

horizontal dashed line at g(y) = 1, corresponds to the preasymptotic power-

law regime [cf. Eq. (10)]. The sharp and fast increase of g(y), for large y,

13

Page 14

corresponds to the crossover to the asymptotic growth law

L

λ≃

?ϕ

bln?t/λz??1/ϕ

(24)

which corresponds to the Class 4 form of Eq. (2).

Summarising, our main findings for the growth law, in the d = 2, case

are as follows:

1. Disorder is a relevant perturbation with respect to pure-like behavior.

2. The corresponding growth law shows a clear crossover from power-law

to logarithmic behavior:

L(t,ǫ) ∼

?t1/z, if L ≪ Lcr,

(lnt)1/ϕ, if L ≫ Lcr.

(25)

This differs from previously found results, since the preasymptotic

power law is not pure-like, due to the ǫ-dependence of the exponent

z. This feature, also observed in the d = 2 RBIM [9], means that dis-

order although globally relevant acts like a marginal operator in the

neighborhood of the pure fixed point [25].

Finally, we remark on the considerable numerical advanyage in using the

effective exponent as a probe for the crossover. In fact, while the switch from

preasymptotic to asymptotic behaviors in zefftakes place at about Lcr≃ λ,

from Eqs. (22) and (23), it follows that the condition byϕ/ϕ = 1 puts the

crossover, when looking at the domain size, at the much greater value Lcr≃

50λ, as it is evident from Fig. 6.

3.3Autocorrelation Function and SU Violation

The results presented above show that disorder affects the growth law as

an asymptotically relevant parameter. Therefore, one would expect this to

apply also to other observables. However, as explained in Sec. 2.1, such an

expectation would be in conflict with claims of SU validity.

In this section, we study the autocorrelation function, defined by

C(t,tw,ǫ) = ?σi(t)σi(tw)?(26)

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10-1

100

101

102

L(t)/λ

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

(t L(t)-z)/K(ε)

ε=0.2

ε=0.35

ε=0.5

ε=0.65

ε=0.8

Figure 6: (Color online) Plot of tL−z/K(ǫ) vs. L/λ with various disorder

values. The master curve obeys the exponential form of Eq. (23).

15

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