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.
arXiv:1202.1534v1 [cond-mat.dis-nn] 7 Feb 2012
Crossover in Growth Law and Violation of
Superuniversality in the Random Field Ising
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.
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.
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
. Therefore, it is of considerable interest to study the role of disorder in
systems where its presence does not compete with phase ordering .
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 , 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 . The lack of a
general framework to understand this complex phenomenology has proven a
major obstacle to development. Recently, in the context of the random bond
Ising model (RBIM) , 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) . 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. .
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
2 Domain Growth Laws in Disordered Sys-
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
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-
L(t) ∼ (T/ǫ)1/ϕ[ln(t/τ)]1/ϕ
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 . The d = 1 ferromagnetic RBIM  is an
example of a Class 2 system. The d = 1 RFIM  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
the value of ϕ. Recently, we have presented evidence  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  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
In Ref. , 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
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
?const., for x → 0,
x1/(φz)ℓ(x−1/φ), for x → ∞,
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
z− φ∂ lnF(x)
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)
λ = ǫ1/(φz)
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
?const., for y ≪ 1,
y−zℓ−1(y), for y ≫ 1,
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.
Finally, the effective exponent as a function of y is obtained from Eq. (6)
zeff(y) = z +∂ lng(y)
Therefore, for φ < 0 (disorder relevant), Eq. (10) yields
?z, for y ≪ 1,
∂ lnℓ−1(y)/∂ lny, for y ≫ 1,
and for φ > 0 (disorder irrelevant), we obtain from Eq. (11)
?∂ lnℓ−1(y)/∂ lny, for y ≪ 1,
z, for y ≫ 1.
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
reparametrization of time through L(t,ǫ) . 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  SU validity. Recently,
the validity of SU has been extended to the geometrical properties of domain
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
We consider an RFIM on a two-dimensional square lattice, with the Hamil-
H = −J
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
2[δ(hi− h0) + δ(hi+ h0)]. (17)
The system evolves according to the Glauber kinetics, which models non-
conserved dynamics , with spin flip transition rates given by
w(σi→ −σi) =1
?1 − σitanh?(HW
i + hi)/T??
the limit T → 0 (J/T → ∞), while keeping the ratio ǫ = h0/T finite. In this
is the local Weiss field. All results in this paper correspond to
limit the system undergoes phase ordering in any dimension, down to d = 1
. The transition rates take the form
w(σi→ −σi) =
1 for HW
0 for HW
2(1 − sign(σihi)tanh(ǫ)) for HW
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 ,
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.
Figure 1: (Color online) Growth law in d = 2. The dashed line is the t1/2
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 , 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
Figure 3: (Color online) z (taken from Fig. 2) vs. ǫ. The red line is the best
fit z = 2.0 + 1.4ǫ1.35.
Table 1: Plateau exponent z for various disorder strengths.
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.
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
= z + byϕ
which, after integrating with respect to L, yields
t = K(ǫ)Lzg(L/λ) (22)
Figure 5: (Color online) Plot of λ vs. ǫ.
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
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,
corresponds to the crossover to the asymptotic growth law
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:
?t1/z, if L ≪ Lcr,
(lnt)1/ϕ, if L ≫ Lcr.
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 , means that dis-
order although globally relevant acts like a marginal operator in the
neighborhood of the pure fixed point .
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)
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).