Physical Review B 107 024103 2023 Finite-dimensional signature of spinodal instability in an athermal hysteretic transition

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Physical Review B 107, 024103 (2023)

Finite-dimensional signature of spinodal instability in an athermal hysteretic

transition

Anurag Banerjee1and Tapas Bar2, ∗

1Department of Physics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel

2ICN2-Institut Catal`a de Nanoci`encia i Nanotecnologia (CERCA-BIST-CSIC),

Campus Universitat Aut`onoma de Barcelona, 08193 Bellaterra, Barcelona, Spain

(Dated: January 16, 2023)

We study the oﬀ-equilibrium critical phenomena across a hysteretic ﬁrst-order transition in dis-

ordered athermal systems. The study focuses on the zero temperature random ﬁeld Ising model

(ZTRFIM) above the critical disorder for spatial dimensions d= 2,3,and 4. We use Monte Carlo

simulations to show that disorder suppresses critical slowing down in phase ordering time for ﬁnite-

dimensional systems. The dynamic hysteresis scaling, the measure of explicit ﬁnite-time scaling, is

used to subsequently quantify the critical slowing down. The scaling exponents in all dimensions

increase with disorder strength and ﬁnally reach a stable value where the transformation is no longer

critical. The associated critical behavior in the mean-ﬁeld limit is very diﬀerent, where the exponent

values for various disorders in all dimensions are similar. The non-mean-ﬁeld exponents asymptot-

ically approach the mean-ﬁeld value (Υ ≈2/3) with increase in dimensions. The results suggest

that the critical features in the hysteretic metastable phase are controlled by inherent mean-ﬁeld

spinodal instability that gets blurred by disorder in low-dimension athermal systems.

I. INTRODUCTION

The critical-like features in abrupt hysteretic tran-

sition have recently been observed in various materi-

als including transition metal oxide [1–4], metal alloys

[5, 6], martensitic transformation [7–9], functional mate-

rials [10], amorphous solids [11–13], microbiology, and

social, economic, climate, and other complex systems

[14, 15]. Such “surprising” [7] behavior is not normal

in terms of typical ﬁrst-order phase transition formal-

ism. Some of such transitions have been explained in

terms of classical spinodal instability, a limiting point

of metastability (Fig. 1), where the system behaves

like a mean-ﬁeld [3, 4, 16, 17]. The stability of the

metastable phase depends on the competition of dis-

order, thermal ﬂuctuation, and activation barriers sep-

arating the two phases [10]. Any ﬂuctuations, linked

with disorder or thermal, in the abrupt transition ini-

tiate nucleations before the extreme limit of metastabil-

ity [18]. In the long-range interacting system, thermal

ﬂuctuations are suppressed [16, 19], and the metastable

phase of the system approaches the spinodal point af-

ter multiple cycling of the materials (training) across the

transition [3, 20]. The divergence of correlation length

and relaxation time scale (spinodal slowing down) signals

the instability in experiments [2, 4, 6]. The mean-ﬁeld

spinodal universality in disorder material might be ex-

plained in terms of training-induced self-organized criti-

cality [21, 22]. However, the critical exponents often vary

widely from mean-ﬁeld predictions [23] [see Table II] and

therefore remains unexplained. In general, the training

cannot tune the quenched disorders such as domain walls,

friction, defects due to an underlying heterogeneous sub-

∗tapas.bar@icn2.cat

strate, pinning defects, and kinetically arrested hetero-

geneity. Therefore, the correlation length of the system

would be bounded by the local disorder points, and het-

erogeneous nucleation sites start to emerge before ap-

proaching the spinodal [24–28]. As a result, a suppressed

spinodal slowing down associated with a mild ﬁnite-size

eﬀect is expected to be observed [6, 13, 29] that may

explain such non-mean-ﬁeld critical exponents. In this

article, we investigate spinodal instability using a ran-

dom ﬁeld Ising model (RFIM) in the presence of quench

disorder and under athermal conditions. The athermal

(zero temperature) model mimics the ﬂuctuationless ki-

netics associated with long-ranged potential, whereas the

short-ranged Ising model only deals with the interplay of

disorder and metastable barrier.

In RFIM, the critical signature in hysteretic tran-

sition has generally been observed in two distinct as-

pects: steady-state (slow-driven or quasistatic) and oﬀ-

equilibrium (highly-driven). The steady-state studies are

limited to the avalanche distribution and can explain the

disorder-induced critical transition near the critical dis-

order [30, 31]. Away from the critical point, the power-

law behavior of avalanche distribution is not adequately

understood [31]. One study attempts to explain such

phenomena at a low disorder regime in the context of

spinodal instability [13]. However, most of the hysteretic

transitions in materials are not single-step processes; in-

stead they show a broad transition accompanied by re-

turn point memory indicating the disorder in the system

is greater than the critical disorder [7, 30, 32]. There-

fore, further investigations are required above the crit-

ical point. On the other hand, the oﬀ-equilibrium as-

pect of critical phenomena such as dynamic hysteresis

scaling and phase ordering dynamics are comparatively

easy to measure in experiments. Not surprisingly, nu-

merous assessments have been reported for diﬀerent ma-

terials [3, 6, 33–50]. In theory, several attempts have also

arXiv:2210.04057v2 [cond-mat.stat-mech] 12 Jan 2023

φφ

i n c r e a s i n g f i e l d

d e c r e a s i n g f i e l d

φφ

O r d e r P a r a m e t e r (φ)

E x t e r n a l F i e l d ( H )

FIG. 1. A schematic diagram for the spinodal transitions.

The order parameter φand corresponding free-energy dia-

grams (f−φcurve) are exhibited as a function of increasing

and decreasing ﬁelds. The f−φcurves in the middle rep-

resent the binodal points where the two minima are equal,

and red f−φdiagrams are the two spinodals points (limit

of metastability) where the double-well free energy switches

to a single well, which is a conventional manifestation of con-

tinuous transitions. The system exhibits spinodal transition

when the thermal ﬂuctuations are insigniﬁcant to cross the

free energy activation barrier between binodal and spinodal

points.

been made in diversiﬁed models, but the results are often

inconsistent with one another (except in the mean-ﬁeld

limit). Such studies are designed to describe speciﬁc ex-

perimental result [51–58]. Therefore, the origin of this

general phenomenon is not properly explored. In this

work, we systematically study the oﬀ-equilibrium criti-

cal phenomena from a general perspective that describes

a large class of the experimentally reported dynamical

critical exponents in various systems.

II. THE MODEL AND SIMULATION

We consider a d-dimensional (d= 2,3,4 ) random ﬁeld

Ising model in which every spin interacts with its nearest

neighbors. A random ﬁeld added to an external ﬁeld

acts as a disorder of the system. The Hamiltonian of the

model read as

H=−JX

hi,ji

sisj−X

[H(t) + hi]si,(1)

where Jis the nearest-neighbor coupling strength of Ising

spins si,si=±1, placed on the d-dimensional hypercu-

bic lattice of system of linear size L. The spin interacts

ferromagnetically with strength J= 1 under the periodic

boundary condition. A time-dependent spatially uniform

external ﬁeld, H(t), and a time-independent but site-

dependent random ﬁeld, hiis applied. The random ﬁeld

hiis taken from a Gaussian distribution, V(h),

V(h) = 1

√2πσ2e−h2/(2σ2),(2)

where the width of the distribution represents the disor-

der strength of a single realization. We present all the

physical quantities after averaging over a suﬃcient num-

ber of independent disorder realizations (∼20 −500).

Since we are interested in the athermal system, the ther-

mal ﬂuctuation in the model can be neglected by per-

forming zero-temperature simulations. Therefore the

spin-ﬂip is completely determined by the sign change of

the local ﬁeld at each site [30],

Ei=JX

sj+hi+H. (3)

The zero temperature random ﬁeld Ising model (ZTR-

FIM) shows an external ﬁeld-dependent hysteretic mag-

netic transition (or switching) for a large range of disor-

der values σ. The transition could be a single or multiple-

step (avalanche) process depending upon the strength of

the disorder. There is a critical disorder, σ=σc, above

which single-step transition never happens. At σ=σc,

the avalanche of all sizes exists that follows a long (sev-

eral decades) power law size distribution connected to a

disorder-induced continuous transition (we will say this

is classical-critical point to avoid ambiguity) [30, 31].

Here, we focus on critical-like ﬁeld-induced transitions

for σ≥σc.

A. Phase ordering dynamics

We perform the phase ordering dynamics of the ZTR-

FIM on a d-dimensional lattice. We start with a system

of fully polarized spins and suddenly tune the magnetic

ﬁeld close to the coercive ﬁeld, the ﬁeld at which the

magnetization reverses. We study the time required to

reach the steady state after the quench. During this in-

terval, the system goes through successive set of spin-ﬂips

and ﬁnally arrives at a steady state. Such phase ordering

(or continuous ordering) is generally measured through

quench-and-hold experiments [3, 6, 59]. We extract the

relaxation time constant from the temporal evolution of

the net magnetization. The details algorithm is presented

below.

1. The spin at every site is either up or down (si= 1

or si=−1) depending upon the sign of the initial

ﬁeld H0.

2. We quench the external magnetic ﬁeld to H=Hf

at time t= 0 and check if the local ﬁeld, deﬁned in

Eq. 3, changes sign on any site.

3. If there is a sign change of the local ﬁeld for at least

one site, we ﬂip the spins on those sites in the next

time step t=t+ 1.

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PhysicalReviewB107,024103(2023)Finite-dimensionalsignatureofspinodalinstabilityinanathermalhysteretictransitionAnuragBanerjee1andTapasBar2,1DepartmentofPhysics,Ben-GurionUniversityoftheNegev,Beer-Sheva84105,Israel2ICN2-InstitutCataladeNanocienciaiNanotecnologia(CERCA-BIST-CSIC),CampusUniversitatA...

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Physical Review B 107 024103 2023 Finite-dimensional signature of spinodal instability in an athermal hysteretic transition

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