Random-field Ising model: Insight from zero-temperature simulations
We enlighten some critical aspects of the three-dimensional (d=3) random-field Ising model (RFIM) from simulations performed at zero temperature. We consider two different, in terms of the field distribution, versions of model, namely a Gaussian RFIM and an equal-weight trimodal RFIM. By implementin...
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irk-123456789-1535342019-06-15T01:27:50Z Random-field Ising model: Insight from zero-temperature simulations Theodorakis, P.E. Fytas, N.G. We enlighten some critical aspects of the three-dimensional (d=3) random-field Ising model (RFIM) from simulations performed at zero temperature. We consider two different, in terms of the field distribution, versions of model, namely a Gaussian RFIM and an equal-weight trimodal RFIM. By implementing a computational approach that maps the ground-state of the system to the maximum-flow optimization problem of a network, we employ the most up-to-date version of the push-relabel algorithm and simulate large ensembles of disorder realizations of both models for a broad range of random-field values and systems sizes V=LxLxL, where L denotes linear lattice size and Lmax=156. Using as finite-size measures the sample-to-sample fluctuations of various quantities of physical and technical origin, and the primitive operations of the push-relabel algorithm, we propose, for both types of distributions, estimates of the critical field hmax and the critical exponent ν of the correlation length, the latter clearly suggesting that both models share the same universality class. Additional simulations of the Gaussian RFIM at the best-known value of the critical field provide the magnetic exponent ratio β/ν with high accuracy and clear out the controversial issue of the critical exponent α of the specific heat. Finally, we discuss the infinite-limit size extrapolation of energy- and order-parameter-based noise to signal ratios related to the self-averaging properties of the model, as well as the critical slowing down aspects of the algorithm. Застосовуючи комп’ютернi симуляцiї при нульовiй температурi, ми висвiтлюємо деякi аспекти критичної поведiнки тривимiрної (d = 3) моделi Iзiнга у випадковому полi. Ми розглядаємо двi версiї моделi, що вiдрiзняються розподiлом випадкового поля, а саме, гаусову та тримодову моделi Iзiнга у випадковому полi з однаковими вагами. Застосовуючи обчислювальний пiдхiд, що ставить у вiдповiднiсть основному стану системи проблему оптимiзацiї максимуму потоку на мережi, ми використовуємо найсучаснiшу версiю алгоритму проштовхування потоку i моделюємо великi ансамблi випадкових реалiзацiй моделей для широкої областi значень випадкового поля i розмiрiв системи V = L ×L ×L, де L позначає лiнiйний розмiр гратки i Lmax = 156. Використовуючи в якостi скiнчено-вимiрних мiр флуктуацiї рiзних величин фiзичного i технiчного походження, вимiряних для рiзних зразкiв, i примiтивнi операцiї алгоритму проштовхування потоку, ми пропонуємо для обох типiв розподiлу оцiнки критичного поля hc i критичного показника кореляцiйної довжини ν. Отримане значення цього показника чiтко вказує на те, що обидвi моделi належать до одного класу унiверсальностi. Додатковi симуляцiї гаусової моделi Iзiнга у випадковому полi при добре вiдомому значеннi критичного поля забезпечують вiдношення магнiтних iндексiв β/ν з високою точнiстю i прояснюють контроверсiйну проблему критичного iндекса α питомої теплоємностi. Накiнець, ми обговорюємо нескiнченнорозмiрну екстраполяцiю енергiї i базованого на параметрi порядку шуму до сигнальних коефiцiєнтiв, пов’язаних з властивостями самоусереднення моделi, а також аспекти критичного сповiльнення алгоритму. 2014 Article Random-field Ising model: Insight from zero-temperature simulations / P.E.Theodorakis, N.G. Fytas // Condensed Matter Physics. — 2014. — Т. 17, № 4. — С. 43003: 1–14. — Бібліогр.: 81 назв. — англ. 1607-324X arXiv:1501.02338 DOI:10.5488/CMP.17.43003 PACS: 05.50.+q, 75.10.Hk, 64.60.Cn, 75.10.Nr http://dspace.nbuv.gov.ua/handle/123456789/153534 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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We enlighten some critical aspects of the three-dimensional (d=3) random-field Ising model (RFIM) from simulations performed at zero temperature. We consider two different, in terms of the field distribution, versions of model, namely a Gaussian RFIM and an equal-weight trimodal RFIM. By implementing a computational approach that maps the ground-state of the system to the maximum-flow optimization problem of a network, we employ the most up-to-date version of the push-relabel algorithm and simulate large ensembles of disorder realizations of both models for a broad range of random-field values and systems sizes V=LxLxL, where L denotes linear lattice size and Lmax=156. Using as finite-size measures the sample-to-sample fluctuations of various quantities of physical and technical origin, and the primitive operations of the push-relabel algorithm, we propose, for both types of distributions, estimates of the critical field hmax and the critical exponent ν of the correlation length, the latter clearly suggesting that both models share the same universality class. Additional simulations of the Gaussian RFIM at the best-known value of the critical field provide the magnetic exponent ratio β/ν with high accuracy and clear out the controversial issue of the critical exponent α of the specific heat. Finally, we discuss the infinite-limit size extrapolation of energy- and order-parameter-based noise to signal ratios related to the self-averaging properties of the model, as well as the critical slowing down aspects of the algorithm. |
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Theodorakis, P.E. Fytas, N.G. |
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Theodorakis, P.E. Fytas, N.G. Random-field Ising model: Insight from zero-temperature simulations Condensed Matter Physics |
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Theodorakis, P.E. Fytas, N.G. |
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Theodorakis, P.E. |
| title |
Random-field Ising model: Insight from zero-temperature simulations |
| title_short |
Random-field Ising model: Insight from zero-temperature simulations |
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Random-field Ising model: Insight from zero-temperature simulations |
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Random-field Ising model: Insight from zero-temperature simulations |
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Random-field Ising model: Insight from zero-temperature simulations |
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random-field ising model: insight from zero-temperature simulations |
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Інститут фізики конденсованих систем НАН України |
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| citation_txt |
Random-field Ising model: Insight from zero-temperature simulations / P.E.Theodorakis, N.G. Fytas // Condensed Matter Physics. — 2014. — Т. 17, № 4. — С. 43003: 1–14. — Бібліогр.: 81 назв. — англ. |
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Condensed Matter Physics |
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Condensed Matter Physics, 2014, Vol. 17, No 4, 43003: 1–14
DOI: 10.5488/CMP.17.43003
http://www.icmp.lviv.ua/journal
Random-field Ising model: Insight from
zero-temperature simulations
P.E. Theodorakis1 , N.G. Fytas2
1 Department of Chemical Engineering, Imperial College London, SW7 2AZ, London, United Kingdom
2 Applied Mathematics Research Centre, Coventry University, Coventry, CV1 5FB, United Kingdom
Received October 2, 2014
We enlighten some critical aspects of the three-dimensional (d = 3) random-field Ising model from simulations
performed at zero temperature. We consider two different, in terms of the field distribution, versions of model,
namely a Gaussian random-field Ising model and an equal-weight trimodal random-field Ising model. By imple-
menting a computational approach that maps the ground-state of the system to the maximum-flow optimization
problem of a network, we employ the most up-to-date version of the push-relabel algorithm and simulate large
ensembles of disorder realizations of both models for a broad range of random-field values and systems sizes
V = L ×L ×L, where L denotes linear lattice size and Lmax = 156. Using as finite-size measures the sample-
to-sample fluctuations of various quantities of physical and technical origin, and the primitive operations of the
push-relabel algorithm, we propose, for both types of distributions, estimates of the critical field hc and the
critical exponent ν of the correlation length, the latter clearly suggesting that both models share the same uni-
versality class. Additional simulations of the Gaussian random-field Ising model at the best-known value of the
critical field provide the magnetic exponent ratio β/ν with high accuracy and clear out the controversial issue
of the critical exponent α of the specific heat. Finally, we discuss the infinite-limit size extrapolation of energy-
and order-parameter-based noise to signal ratios related to the self-averaging properties of the model, as well
as the critical slowing down aspects of the algorithm.
Key words: random-field Ising model, finite-size scaling, graph theory
PACS: 05.50.+q, 75.10.Hk, 64.60.Cn, 75.10.Nr
1. Introduction
The random-field Ising model (RFIM) is one of the archetypal disordered systems [1–3], extensively
studied due to its theoretical interest, as well as its close connection to experiments in hard [4, 5] and
soft condensed matter systems [6]. Its beauty is that the mixture of random fields and the standard Ising
model creates rich physics and leaves many still unanswered problems. The Hamiltonian describing the
model is
H =−J
∑
〈i , j 〉
σiσ j −
∑
i
hiσi , (1)
where σi =±1 are Ising spins, J > 0 is the nearest-neighbor’s ferromagnetic interaction, and hi are inde-
pendent quenched random fields.
The existence of an ordered ferromagnetic phase for the RFIM, at low temperature andweak disorder,
followed from the seminal discussion of Imry and Ma [1], when the space dimension is greater than two
(d > 2) [7–11]. This has provided us with a general qualitative agreement on the sketch of the phase
boundary, separating the ordered ferromagnetic phase from the high-temperature paramagnetic one.
The phase-diagram line separates the two phases of the model and intersects the randomness axis at
the critical value of the disorder strength hc, as shown in figure 1. Such qualitative sketching has been
commonly used for the RFIM [12–14] and closed form quantitative expressions are also known from the
early mean-field calculations [15–17]. However, it is generally true that the quantitative aspects of phase
diagrams produced by mean-field treatments provide rather poor approximations.
© P.E. Theodorakis, N.G. Fytas, 2014 43003-1
http://dx.doi.org/10.5488/CMP.17.43003
http://www.icmp.lviv.ua/journal
P.E. Theodorakis, N.G. Fytas
R P
F
h / J
0
h
c
T / J
Figure 1. Schematic phase diagram and renormalization-group flow of the RFIM. The solid line separates
the ferromagnetic (F) and paramagnetic (P) phases. The black arrow shows the flow to the random fixed
point (R) at T = 0 and h = hc, as marked by an asterisk.
The criteria for determining the order of the low-temperature phase transition and its dependence
on the form of the field distribution have been discussed throughout the years [15–19]. Although the
view that the phase transition of the RFIM is nowadays considered to be of second order [20–25], the
extremely small value of the exponent β continues to cast some doubts. Moreover, a rather strong debate
with regard to the role of disorder, i.e., the dependence, or not, of the critical exponents on the particular
choice of the distribution for the random fields and the value of the disorder strength, analogously to the
mean-field theory predictions [15–17], was only recently put on a different basis [26]. Currently, even the
well-known correspondence among the RFIM and its experimental analogue, the diluted antifferomagnet
in a field (DAFF), has been severely questioned by extensive simulations performed on both models at
positive- and zero-temperature [27]. In any case, the whole issue of the model’s critical behavior is still
under intense investigation [20–25, 28–43].
Already from the work of Houghton et al. [44], the importance of the form of the distribution function
in the determination of the critical properties of the RFIM has been emphasized. In fact, different results
have been proposed for different field distributions, like the existence of a tricritical point at the strong
disorder regime of the system, present only in the bimodal case [15–17, 44]. Following the results of
Houghton et al. [44], Mattis [45] reexamined the RFIM introducing a new type of a trimodal distribution
P
(trimodal)(hi ) = pδ(hi )+
(
1−p
2
)
[δ(hi −h)+δ(hi +h)], (2)
where h defines the disorder (field) strength and p ∈ (0,1). Clearly, for p = 1 one switches to the pure
Ising model, whereas for p = 0 the well-known bimodal distribution is recovered. In general terms, the
trimodal distribution (2) permits a physical interpretation as a diluted bimodal distribution, in which a
fraction p of the spins are not exposed to the external field. Thus, it mimics the salient feature of the
Gaussian distribution
P
(Gaussian)(hi ) =
1
p
2πh2
exp
(
−
h2
i
2h2
)
, (3)
for which a significant fraction of the spins are in weak external fields. Mattis suggested that for a par-
ticular case, p = 1/3, equation (2) may be considered to a good approximation as the Gaussian distribu-
tion [45]. This in turn indicated that the two models should be in the same universality class. Further
studies along these lines, using mean-field and renormalization-group approaches, provided contradict-
ing evidence for the critical aspects of the p = 1/3 model and also proposed several approximations of its
phase diagram for a range of values of p [46–48]. However, none of these predictions has been confirmed
by numerical simulations up till now, thus remaining ambiguous, due to the approximate nature of the
mean-field-type of the methods used.
The scope of the present work is to shed some light towards this direction by examining several criti-
cal features of the phase diagram of the RFIM at d = 3, using both distributions described above in equa-
43003-2
Random-field Ising model: Insight from zero-temperature simulations
tions (2) and (3). In particular, in the first part of our study we provide numerical evidence that clarify
the matching between the trimodal (p = 1/3) and Gaussian models and we give estimates for the critical
field hc and critical exponent ν that compare very well to the most accurate ones in the corresponding
literature of the RFIM. In the second part of our study we concentrate on the most studied case of the
Gaussian RFIM, for which we present a scaling analysis of critical data for the order parameter and the
specific heat, i.e., data obtained at the best known estimate of the critical field hc. Our analysis points to
a very small, but non-zero, value for the magnetic exponent ratio β/ν, and a critical exponent α→ 0−, in
good agreement with experimental predictions [49, 50]. We also discuss the infinite-limit size extrapola-
tion of energy- and order-parameter-based noise to signal ratios related to the self-averaging properties
of the model, as well as some technical aspects of the implemented numerical method.
Our attempt benefits from: (i) the existence of robust computational methods of graph theory at zero
temperature (T = 0) that allow us to simulate very large system sizes and disorder ensembles, necessary
for an accurate investigation of the delicate properties discussed above, (ii) classical finite-size scaling
(FSS) techniques, and (iii) a new scaling approach that involves the analysis of the sample-to-sample fluc-
tuations of various well-defined quantities. In particular, sample-to-sample fluctuations and the relative
issue of self-averaging have attracted much interest in the study of disordered systems [51]. Although
it has been known for many years now that for (spin and regular) glasses there is no self-averaging in
the ordered phase [52], for random ferromagnets such a behavior was first observed for the RFIM by
Dayan et al. [53] and some years later for the random versions of the Ising and Ashkin-Teller models by
Wiseman and Domany [54]. These latter authors suggested a FSS ansatz describing the absence of self-
averaging and the universal fluctuations of random systems near critical points that was refined on a
more rigorous basis by Aharony and Harris [55]. Ever since, the subject of breakdown of self-averaging is
an important aspect of several theoretical and numerical investigations of disordered spin systems [56–
68]. In fact, Efrat and Schwartz [69] showed that the property of lack of self-averaging may be turned into
a useful tool that can provide an independent measure to distinguish the ordered and disordered phases
of the system. In view of this observation, we discuss here another useful application of the fluctuation
properties of several quantities of the system to obtain information on the ground-state criticality of the
RFIM.
The rest of the paper is organized as follows: In the next section we describe the general framework
behind the mapping of the RFIM to the corresponding network, outline the numerical approach, and
provide all the necessary details of our implementation. The relevant FSS analysis that shows the equiva-
lence of both distributions under study in terms of the critical exponent ν of the correlation length, using
an approach based on the sample-to-sample fluctuations of the model, is presented in section 3. Then, in
section 4 we focus our attention on the most studied case of the Gaussianmodel andwe provide estimates
for the magnetic exponent ratio β/ν and the critical exponent α of the specific heat, via the scaling of the
order parameter and bond energy, respectively, at the best known estimate of the critical field value.
We also investigate the self-averaging properties of the model at criticality, using properly defined noise
to signal ratios and we provide an estimate for the exponent z that describes the critical slowing of the
algorithm used. Finally, we synopsize our findings in section 5.
2. Simulation protocol
As already discussed extensively in the literature (see reference [70] and references therein), the RFIM
captures essential features of models in statistical physics that are controlled by disorder and have frus-
tration. Such systems show complex energy landscapes due to the presence of large barriers that separate
several meta-stable states. When such models are studied using simulations mimicking the local dynam-
ics of physical processes, it takes an extremely long time to encounter the exact ground state. However,
there are cases where efficient methods for finding the ground state can be utilized and, fortunately, the
RFIM is one such case. These methods escape from the typical direct physical representation of the sys-
tem, in a way that extra degrees of freedom are introduced and an expanded problem is finally solved.
By expanding the configuration space and choosing proper dynamics, the algorithm practically avoids
the need of overcoming large barriers that exist in the original physical configuration space. An attractor
state in the expended space is found in time polynomial in the size of the system and when the algo-
43003-3
P.E. Theodorakis, N.G. Fytas
rithm terminates, the relevant auxiliary fields can be projected onto a physical configuration, which is
the guaranteed ground state.
The randomfield is a relevant perturbation at the pure fixed-point, and the random-field fixed-point is
at T = 0 [7–10]. Hence, the critical behavior is the same everywhere along the phase boundary of figure 1,
and we can predict it simply by staying at T = 0 and crossing the phase boundary at h = hc. This is a
convenient approach, because we can determine the ground states of the system exactly using efficient
optimization algorithms [20, 21, 25, 65, 66, 71–76] through an existing mapping of the ground state to the
maximum-flow optimization problem [77]. A clear advantage of this approach is the ability to simulate
large system sizes and disorder ensembles in rather moderate computational times. We should underline
here that, even the most efficient T > 0 Monte Carlo schemes exhibit extremely slow dynamics in the low-
temperature phase of these systems and are upper bounded by linear sizes of the order of Lmax É 32 [70].
Further advantages of the T = 0 approach are the absence of statistical errors and equilibration problems,
which, on the contrary, are the two major drawbacks encountered in the T > 0 simulation of systems with
rough free-energy landscapes [5].
A short direct sketching of how this mapping may in principle occur through some simple consid-
erations is as follows: Let G = (V ,E ) be a directed, weighted graph consisting of a set V of nodes and a
set E of edges, each of the latter connecting two nodes. In a directed graph, for each edge a direction is
specified. The property of being weighted means that to each edge from node i to node j a capacity ci j is
assigned. Let the number of nodes be n +2. We enumerate the nodes V = {0,1,2, . . . ,n,n +1} and define
the first node 0 as source s and the last node n +1 as the sink t . The remaining nodes will be associated
to the lattice sites of the RFIM. We call a directed, weighted graph G with source s, sink t , and capacities
c, as network N = (G,c, s, t). Now, in a network N = (G,c, s, t), an (s, t )-cut (S,S) is defined as a partition
of the set of nodes V into two disjoint sets S and S (S ∩S =∅ and S ∪S =V ) with s ∈ S and t ∈ S. In other
words, one can imagine a cut as a partition that divides the network into two parts, one part belonging to
the source and the other to the sink. Generally, there are many different possible cuts in a network. We
can assign to each of them a capacity C (S,S), defined as the sum of the capacities of the edges that the cut
crosses
C (S,S) =
∑
i∈S, j∈S
ci j . (4)
Note that edges are directed, that is why only edges that start at the source side of the cut and end at the
sink side contribute to the capacity of the cut.
Now, the central idea that allows us to map the RFIM into a network defined above, consists of de-
scribing a cut by a vector X with the property: Xi = 1 if i ∈ S and Xi = 0 otherwise, i.e., if i ∈ S. Then,
by definition, X0 = 1 and Xn+1 = 0. Using this representation, the formula for the cut capacity may be
written in the following form
C (S,S) =
n+1
∑
i=0
n+1
∑
j=0
ci j Xi (1−X j ). (5)
An expansion of equation (5) leads to
C (S,S) =−
∑
i , j
ci j Xi X j +
∑
i
(
∑
j
ci j
)
Xi , (6)
and already a structural similarity to the fundamental Hamiltonian definition of the RFIM [equation (1)]
is clearly seen. Further information on this structural similarities, including a detailed algebra, may be
found for the interested reader in the relevant literature (see for instance reference [70] and references
therein).
The application of maximum-flow algorithms to the RFIM is nowadays well established [75]. Themost
efficient network flow algorithm used to solve the RFIM is the push-relabel (PR) algorithm of Tarjan and
Goldberg [78]. For the interested reader, general proofs and theorems on the PR algorithm can be found in
standard textbooks [77]. The version of the algorithm implemented in our study involves a modification
proposed byMiddleton et al. [21, 79, 80] that removes the source and sink nodes, reducing memory usage
and also clarifying the physical connection [79, 80].
43003-4
Random-field Ising model: Insight from zero-temperature simulations
The algorithm starts by assigning an excess xi to each lattice site i , with xi = hi . Residual capacity
variables ri j between neighboring sites are initially set to J . A height variable ui is then assigned to each
node via a global update step. In this global update, the value of ui at each site in the set T =
{
j |x j < 0
}
of negative excess sites is set to zero. Sites with xi Ê 0 have ui set to the length of the shortest path,
via edges with positive capacity, from i to T . The ground state is found by successively rearranging the
excesses xi , via push operations, and updating the heights, via relabel operations. The order in which sites
are considered is given by a queue. In this paper, we consider a first-in-first-out (FIFO) queue. The FIFO
structure executes a PR step for the site i at the front of a list. If any neighboring site is made active by
the PR step, it is added to the end of the list. If i is still active after the PR step, it is also added to the end
of the list. This structure maintains the cycles through the set of active sites.
When no more pushes or relabels are possible, a final global update determines the ground state,
so that sites which are path connected by bonds with ri j > 0 to T have σi = −1, while those which
are disconnected from T have σi = 1. A push operation moves excess from a site i to a lower height
neighbor j , if possible, that is, whenever xi > 0, ri j > 0, and u j = ui −1. In a push, the working variables
are modified according to xi → xi −δ, x j → x j +δ, ri j → ri j −δ, and r j i → r j i +δ, with δ= min(xi ,ri j ).
Push operations tend to move the positive excess towards sites in T . When xi > 0 and no push is possible,
the site is relabelled, with ui increased to 1+min{ j |ri j>0} u j . In addition, if a set of highest sites U become
isolated, with ui > u j +1, for all i ∈U and all j ∉U , the height ui for all i ∈U is increased to its maximum
value, V , as these sites will always be isolated from the negative excess nodes.
Periodic global updates are often crucial to the practical speed of the algorithm [79, 80]. Following
the suggestions of references [21, 79, 80], we have also applied global updates here every V relabels, a
practice found to be computationally optimum [25, 76, 79, 80].
Using this scheme we performed large-scale simulations of the RFIM with both type of distributions
discussed above in section 1. Let us note here that prior to the commencement of these large-scale simu-
lations, a set of preliminary runs with smaller system sizes revealed the critical h-regime that we should
work on. In particular, for the trimodal (p = 1/3) RFIM simulations have been performed for lattice sizes
L ∈ {24,32,48,64,96,128} and disorder strengths h ∈ [2.7−3.3]. For the Gaussian model lattice sizes in the
range L ∈ {Lmin−Lmax}, where Lmin = 24 and Lmax = 156, were used and disorder strengths h ∈ [2.0−3.0].
In both cases a disorder-strength step of δh = 0.02 was used. Regarding the disorder averaging procedure,
which is of paramount importance in the study of the RFIM, for each pair (L, h) an extensive averaging
over Ns = 50×103 independent random-field realizations has been undertaken, much larger than in pre-
vious relevant studies of the model [21, 65, 66, 72, 73]. Additionally, for the Gaussian RFIM we performed
some further and even more extensive simulations, at the best known estimate of the critical field hc,
using in this case an ensemble of Ns = 200×103 random realizations.
3. Universality aspects
As the outcome of the PR algorithm is the spin configuration of the ground state, we can calculate
for a given sample of a lattice with linear size L the magnetization via m = V
−1 ∑
i σi . Taking the aver-
age over different disorder configurations we may define the order parameter of the system M = [|m|],
where [· · · ] denotes disorder averaging. Another physical parameter of interest is the bond energy per
spin that corresponds to the first term of the Hamiltonian (1), i.e. eJ = −V
−1 ∑
〈i , j 〉σiσ j , and its disorder
average, defined hereafter as EJ = [eJ]. Our analysis in the sequel will be mainly based on these three
thermodynamic quantities, as well as a relevant algorithmic quantity, namely the number of primitive
operations of the PR algorithm, that is the number of relabels per spin R.
At this point, let us start the presentation of our FSS approach with figures 2 (a) and (b), where we plot
the sample-to-sample fluctuations over disorder of two quantities, of physical and technical origin, for the
case of the trimodal RFIM. In particular, we plot the fluctuations of the bond energy EJ [figure 2 (a)] and
the number of primitive operations of the PR algorithm [figure 2 (b)]. All these fluctuations are plotted as
a function of the disorder strength h for the complete lattice size-range L = 32−128. It is clear that for
every lattice size L, these fluctuations appear to have a maximum value at a certain value of h, denoted
hereafter as h∗
L , that may be considered in the following as a suitable pseudo-critical disorder strength.
By fitting the data points around the maximum first to a Gaussian, and subsequently to a fourth-order
43003-5
P.E. Theodorakis, N.G. Fytas
Figure 2. (Color online) (a) Sample-to-sample fluctuations of the bond energy VEJ
of the trimodal RFIM
as a function of the disorder strength for various lattice sizes in the range L = 32−128. Lines are simple
guides to the eye. (b) Same as in panel (a) but now the sample-to-sample fluctuations of the number
of primitive operations of the PR algorithm, VR, are shown. (c) Simultaneous fitting of the form (7) of
the pseudo-critical disorder strengths h∗
L
, obtained from the peak positions of the fluctuations shown in
panels (a) and (b). The shared parameters of the three data sets of the fit are the critical strength hc and
the correlation length’s exponent ν.
polynomial, we have extracted the values of the peak-locations (h∗
L ) by taking the mean value via the two
fitting functions, as well as the corresponding error bars. Using now these values for h∗
L we consider in
the panel (c) of figure 2 a simultaneous power-law fitting attempt of the form
h∗
L = hc +bL−1/ν, (7)
simultaneous meaning that the values of hc and ν for all data sets in the fitting procedure are shared
during the fit. The quality of the fit is fair enough, with a value of χ2/dof of the order of 0.6, where
dof refers to the degrees of freedom, and produces the estimates hc = 2.745(7) and ν = 1.37(2) for the
critical disorder strength and the correlation length’s exponent, in agreement with recent estimates in
the literature [76].
We now turn our discussion on the Gaussian RFIM. For this case we show in figure 3 (a) the number of
relabels per spin R as a function of the disorder strength for various lattice sizes in the range L = 24−156.
Again, we observe that for every lattice size L, R has amaximum at a certain value of h, denoted as before
with h∗
L , that may be considered now as a relevant pseudo-critical disorder strength. Following a similar
procedure, we extracted the values of the peak-locations (h∗
L ) as well as the corresponding error bars,
whose shift-behavior is now plotted in panel (b) of figure 3. The straight line is power-law fitting attempt
of the same form (7) and the outcome for hc and ν is 2.274(4) and 1.37(1), respectively. The quality of the
fit is also in this case good, with a value of χ2/dof of the order of 0.4.
A few comments on the scaling analysis are now in order: Having simulated more than five lattice
size-points in each case, we also tried to perform the above analysis including higher-order scaling cor-
rections of the form (1+ b′L−ω), where ω is the well-known correction-to-scaling exponent, obtained
very recently to be ω = 0.52(11) for this model [26], using the scaling behavior of universal quantities.
However, no improvement has been observed in the quality of the fit of our data. On the contrary, the
43003-6
Random-field Ising model: Insight from zero-temperature simulations
Figure 3. (Color online) (a) The number of relabels per spin R of the Gaussian RFIM as a function of the
disorder strength for various lattice sizes in the range L = 24−156. Lines are simple guides to the eye. (b)
Fitting of the form (7) of the pseudo-critical disorder strengths h∗
L
, obtained from the peak positions of
panel (a).
corrected scaling assumption resulted in an unstable fitting procedure with significantly large errors in
the values of the exponent ν, the coefficient b′, as well as the exponent ω.
Our suggestion of choosing these newly defined pseudo-critical disorder strengths h∗
L as a proper
measure for performing FSS, closely follows the analogous considerations of Hartmann and Young [72]
and Dukovski and Machta [73], also for the Gaussian RFIM. The first authors [72] considered pseudo-
critical disorder strengths at the values of h atwhich a specific-heat-like quantity obtained by numerically
differentiating the bond energy with respect to h attains its maximum. On the other hand, the authors
of reference [73] identified the pseudo-critical points as those in the H − h plane (with H a uniform
external field), where three degenerate ground states of the system show the largest discontinuities in
the magnetization.
Respectively, Middleton and Fisher [21] using similar reasoning on the Gaussian RFIM, characterized
the distribution of the order parameter by the average over samples of the square of the magnetization
per spin and the root-mean-square sample-to-sample variations of the square of the magnetization. They
identified a similar behavior to that of figures 2 (a) and (b), i.e., with increasing L, the peak magnitude of
this quantity moved its location to smaller values of h, defining another relevant pseudo-critical disorder
strength. However, in reference [21] the authors were only interested in the scaling behavior of the height
of these peaks. The practice followed in the current paper, employing the FSS behavior of the peaks of
the sample-to-sample fluctuations of several quantities of physical (M and EJ) and technical (R) origin,
was inspired by the intriguing analysis of Efrat and Schwartz [69]. These authors, studying also the d = 3
RFIM, showed that the behavior of the sample-to-sample fluctuations in a disordered system may be
turned into a useful tool that can provide an independent measure to distinguish between the ordered
and disordered phases of the system. The analysis of figures 2 (a) and (b) above verifies their prediction,
and the accuracy in the estimation of relevant phase diagram features, like the critical field hc and the
critical exponent ν, turns out to be a clear test in favor of the overall scheme.
Let us make at this point a small comment concerning the errors inherent in these types of approx-
imations. The errors induced in the scheme based on the sample-to-sample fluctuations of figures 2 (a),
2 (b), or the primitive operations of the PR algorithm shown in figure 3 (a), have their origin in the applica-
tion of some polynomial, or peak-like, function in order to extract the relevant position of the maximum
in the h-axis. On the contrary, in similar definitions of pseudo-critical points, such as through the use of
some properly defined specific-heat-like quantity at T = 0 [72], one should numerically differentiate the
data of the bond energy EJ, and then consider a smoothing function to locate the position of the maxi-
mum. This scheme is subjected to two successive fitting approximations, thus increasing the errors in the
estimation of the pseudo-critical points.
To summarize, in this section, we have investigated the matching between the trimodal, p = 1/3,
RFIM and the Gaussian RFIM. Clearly enough, our estimates for the critical exponent ν of both models
43003-7
P.E. Theodorakis, N.G. Fytas
indicate an equivalence among both distributions within error bars, justifying the original prediction of
Mattis [45]. Furthermore, we have suggested the values for the critical field hc which compare very well
to the most accurate estimations of the literature. For instance, the best known estimate for the Gaussian
RFIM is hc = 2.27205 [26], very close to the value 2.274(4) of the present work. This is also true for the
reported values of the correlation-length’s exponent, as for the Gaussian RFIM, previous high-accuracy
estimates suggest a value of ν = 1.37 [21, 26, 72]. An interesting aspect of our analysis that led to the
above results was the illustration that quantities related to the sample-to-sample fluctuations of several
quantities of the system or simply the, originally technical, number of primitive operations of the PR
algorithm, constitute a useful alternative to investigate criticality.
4. Gaussian RFIM
In this last part of our work, we concentrate on the Gaussian distribution, which is the most studied
case in the literature of the RFIM, and present further results on important aspects of its critical behavior.
As already mentioned above, we have performed additional runs at the best-known value of the critical
field, that is the value hc = 2.27205 [26]. Thus, the data and analysis of this section are based on extensive
simulations performed at this value of the critical field.
In principal, we are interested in the extraction of an accurate estimate for the magnetic exponent
ratio β/ν, whose small value casts some doubts on the order of the transition of the RFIM. The route we
follow here is via the scaling of the order parameter M at the critical field. This is shown in figure 4, and
the solid line is a power-law fitting of the form M (h=hc ) ∼ L−β/ν. The resulting estimate of the magnetic
exponent ratio, given also in the figure, is β/ν = 0.0131(3), a rather small, but non-zero value, also in
agreement with some of the most accurate estimations in the literature [21].
The next part of our FSS analysis concerns the controversial issue of the specific heat of the RFIM. The
specific heat of the RFIM can be experimentally measured [49, 50] and is, for sure, of great theoretical
importance. Yet, it is well known that it is one of the most intricate thermodynamic quantities to deal
with in numerical simulations, even when it comes to pure systems. For the RFIM, Monte Carlo methods
at T > 0 have been used to estimate the value of its critical exponentα, but were restricted to rather small
systems sizes and have also revealed many serious problems, i.e., severe violations of selfaveraging [61,
64]. A better picture emerged throughout the years from T = 0 computations, proposing estimates of
α ≈ 0. However, even by using the same numerical techniques, but different scaling approaches, some
inconsistencies were recorded in the literature. The most prominent was that of reference [72], where
a strongly negative value of the critical exponent α was estimated. On the other hand, experiments on
random field and diluted antiferromagnetic systems suggest a clear logarithmic divergence of the specific
heat [49, 50].
In general, one expects that the finite-temperature definition of the specific heat C can be extended
to T = 0, with the second derivative of 〈E〉 with respect to temperature being replaced by the second
Figure 4. FSS of the order parameter at the critical field hc.
43003-8
Random-field Ising model: Insight from zero-temperature simulations
Figure 5. FSS behavior of the bond part of the energy density at the critical field hc. The line is a fitting of
the form (8).
derivative of the ground-state energy density Egs with respect to the random field h [21, 72]. The first
derivative ∂Egs/∂J is the bond energy EJ, already defined above. The general FSS form assumed is that
the singular part of the specific heat Cs behaves as Cs ∼ Lα/νC̃
[
(h−hc)L1/ν
]
. Thus, one may estimate
α by studying the behavior of EJ at h = hc [21]. The computation from the behavior of EJ is based on
integrating the above scaling equation up to hc, which gives a dependence
E
(h=hc)
J
= c1 +c2L(α−1)/ν, (8)
with ci constants. Alternatively, following the prescription of [72], one may calculate the second deriva-
tive by finite differences of EJ(h) for values of h near hc and determine α by fitting to the maximum of
the peaks in Cs, which occur at h∗
L −hc ≈ L−1/ν. However, as already noted in [21], this latter approach
may be more strongly affected by finite-size corrections, since the peaks in Cs found by numerical dif-
ferentiation are somewhat above hc, and furthermore it is computationally more demanding, since one
must have the values of EJ in a wide and very dense range of h-values.
In the present case, where the critical value hc is known with good accuracy, the first approach seems
to be more suitable to follow. The numerical data of the critical bond energy and the relevant scaling
analysis are presented in figure 5. The solid line is a power-law fitting of the form (8) and the estimate for
the exponent ratio (α−1)/ν is −0.799(28), as also given in the figure. Using now our estimate ν= 1.37(1),
we calculate the critical exponent α of the specific heat, resulting in an estimate α=−0.095(37), which is
fairly compatible to the experimental scenario of a logarithmic divergence (α= 0) [49, 50].
Following the discussion above in section 1, our numerical studies of disordered systems are carried
out near their critical points using finite samples; each sample is a particular random realization of the
quenched disorder. A measurement of a thermodynamic property, say Z , yields a different value for
every sample. In an ensemble of disordered samples of linear size L, the values of Z are distributed ac-
cording to a probability distribution. The behavior of this distribution is directly related to the issue of
self-averaging. In particular, by studying the behavior of the width of this distribution, one may qualita-
tively address the issue of self-averaging, as has already been stressed by previous authors [54, 55, 58]. In
general, we characterize the distribution by its average [Z ] and also by the relative variance
RZ =
VZ
[Z ]2
=
[Z 2]− [Z ]2
[Z ]2
, (9)
that we employ here to investigate the self-averaging properties of the RFIM.
In particular, we study the behavior of the ratio RZ , in the framework of the two main quantities
typically used, the order parameter M and the bond energy EJ of the model. In figure 6 we plot the ratio
RZ , estimated at the critical field hc, for both quantities, as a function of the inverse linear size. The solid
lines are simple linear extrapolations to the infinite-limit size L →∞. As it is straightforward from the
extrapolations, the order-parameter that carries the effect of the disorder — we remind here that the
43003-9
P.E. Theodorakis, N.G. Fytas
Figure 6. (Color online) Illustration of the self-averaging properties of the model in terms of the magneti-
zation and bond energy. The lines are linear extrapolations to the infinite-limit size.
random field couples to the local spins — is a strongly non-self-averaging quantity. On the other hand, as
expected, the bond energy restores self-averaging in the thermodynamic limit.
Closing, we present some computational aspects of the implemented PR algorithm and its perfor-
mance on the study of the Gaussian RFIM. Although its generic implementation has a polynomial time
bound, its actual performance depends on the order in which operations are performed and which
heuristics are used to maintain auxiliary fields for the algorithm. Even within this polynomial time
bound, there is a power-law critical slowing down of the PR algorithm at the zero-temperature transi-
tion [21, 71]. This critical slowing down is certainly reminiscent of the slowing down seen in local al-
gorithms for statistical mechanics at finite temperature, such as Metropolis, and even for cluster algo-
rithms [81]. In fact, Ogielski was the first to note that the PR algorithm takes more time to find the ground
state near the transition in three dimensions from the ferromagnetic to paramagnetic phase [71]. This
has already been qualitatively seen in figure 3 (a), where, indeed, the number of primitive operations R
of the PR algorithm is maximized in the suitably defined pseudo-critical fields h∗
L . Assuming the standard
scaling of the form R ≈ Lz w
[
(h−hc)−1/νL
]
, where the dynamic exponent z describes the divergence in
the running time at h = hc, and w(x) ∼ x−z at large x and w(x) ∼ |x|−z ln |x| as x →−∞, to be consistent
with convergence to constant R for h > hc and R ∼ ln L for small h. Our fitting attempt of this scaling
form is plotted in figure 7 and the obtained estimate for the dynamic critical exponent z is 0.487(7).
Figure 7. Estimation of the critical slowing down exponent z of the PR algorithm via the FSS behavior of
the number of relabels per spin at the critical field hc.
43003-10
Random-field Ising model: Insight from zero-temperature simulations
Figure 8. Sample-to-sample fluctuations of the order parameter of an L = 48 trimodal RFIM with varying
probability p as a function of the external random field h. A clear shift behavior is observed.
5. Summary and outlook
To summarize, we have investigated the ground-state criticality of the d = 3 RFIM with two types
of the random-field distribution, namely a uniform trimodal and a Gaussian distribution. In particular,
we have estimated for both cases the critical disorder strength hc and the critical exponent ν of the
correlation length. These values, compare well enough to the most accurate estimates of the literature,
with the values of ν placing the trimodal (p = 1/3) RFIM into the universality class of the Gaussian model,
thus verifying a scenario suggested many years ago byMattis [45]. Technically, our effort became feasible
through the implementation of a modified version of the PR algorithm that enabled us to simulate very
large system sizes, up to 1563 spins, and disorder ensembles of the order of up to 200×103 , for several
values of the random-field strength.
On physical grounds, we have implemented a FSS approach based on the sample-to-sample fluctu-
ations of various quantities of physical and technical origin and the primitive operations of the PR al-
gorithm. The outcome of this analysis indicated that the fluctuations of the system may be used as an
alternative successful approach to criticality, paving the way to even more sophisticated studies of dis-
ordered systems under this perspective. Furthermore, we have provided high-accuracy estimates for the
controversial issues of the magnetic-exponent ratio of the order parameter β/ν and the critical exponent
α of the specific heat. In particular, the magnetic exponent ratio β/ν was found to be very small, but
clearly non zero, ruling out the possibility of a first-order phase transition, whereas the exponent α was
found to be compatible with zero, in agreement with the experiments [49, 50]. Particular interest has
been paid to the self-averaging properties of the model, by studying the infinite-limit size extrapolation
of energy- and order-parameter-based noise to signal ratios, as well as the critical slowing down aspects
of the PR algorithm.
A future challenge emerging out of the current work, is the production of the phase diagram of the
trimodal RFIM in the hc−p plane and the verification, or challenge, of the originally proposed for p = 1/3
universal behavior of the trimodal and Gaussian models in higher dimensions, below the upper critical
dimension of the RFIM du. Preliminary simulations for various values of the probability p in the spectrum
0.1−0.5 of the trimodal RFIM indicate a smooth scaling behavior of the sample-to-sample fluctuations of
the order parameter, as illustrated in figure 8 for a system size of L = 48 and ensembles of the order of
Ns = 5×103 realizations. We expect this task and analysis to bring forward new results on the RFIM that
will be useful to the community of disordered systems.
43003-11
P.E. Theodorakis, N.G. Fytas
References
1. Imry Y., Ma S.-K., Phys. Rev. Lett., 1975, 35, 1399; doi:10.1103/PhysRevLett.35.1399.
2. Aharony A., Imry Y., Ma S.-K., Phys. Rev. Lett., 1976, 37, 1364; doi:10.1103/PhysRevLett.37.1364.
3. Parisi G., Sourlas N., Phys. Rev. Lett., 1979, 43, 744; doi:10.1103/PhysRevLett.43.744.
4. Belanger D.P., Young A.P., J. Magn. Magn. Mater., 1991, 100, 272; doi:10.1016/0304-8853(91)90825-U.
5. Rieger H., In: Annual Reviews of Computational Physics II, Stauffer D. (Ed.), World Scientific, Singapore, 1995,
295–341.
6. Vink R.L.C., Binder K., Löwen H., Phys. Rev. Lett., 2006, 97, 230603; doi:10.1103/PhysRevLett.97.230603.
7. Villain J., Phys. Rev. Lett., 1984, 52, 1543; doi:10.1103/PhysRevLett.52.1543.
8. Bray A.J., Moore M.A., J. Phys. C: Solid State Phys., 1985, 18, L927; doi:10.1088/0022-3719/18/28/006.
9. Fisher D.S., Phys. Rev. Lett., 1986, 56, 416; doi:10.1103/PhysRevLett.56.416.
10. Berker A.N., McKay S.R., Phys. Rev. B, 1986, 33, 4712; doi:10.1103/PhysRevB.33.4712.
11. Bricmont J., Kupiainen A., Phys. Rev. Lett., 1987, 59, 1829; doi:10.1103/PhysRevLett.59.1829.
12. Newman M.E.J., Roberts B.W., Barkema G.T., Sethna J.P., Phys. Rev. B, 1993, 48, 16533;
doi:10.1103/PhysRevB.48.16533.
13. Machta J., Newman M.E.J., Chayes L.B., Phys. Rev. E, 2000, 62, 8782; doi:10.1103/PhysRevE.62.8782.
14. Newman M.E.J., Barkema G.T., Phys. Rev. E, 1996, 53, 393; doi:10.1103/PhysRevE.53.393.
15. Aharony A., Phys. Rev. B, 1978, 18, 3318; doi:10.1103/PhysRevB.18.3318.
16. Aharony A., Phys. Rev. B, 1978, 18, 3328; doi:10.1103/PhysRevB.18.3328.
17. Andelman D., Phys. Rev. B, 1983, 27, 3079; doi:10.1103/PhysRevB.27.3079.
18. Galam S., Birman J.L., Phys. Rev. B, 1983, 28, 5322; doi:10.1103/PhysRevB.28.5322.
19. Saxena V.K., Phys. Rev. B, 1984, 30, 4034; doi:10.1103/PhysRevB.30.4034.
20. Hartmann A.K., Nowak U., Eur. Phys. J. B, 1999, 7, 105; doi:10.1007/s100510050593.
21. Middleton A.A., Fisher D.S., Phys. Rev. B, 2002, 65, 134411; doi:10.1103/PhysRevB.65.134411.
22. Vink R.L.C., Fischer T., Binder K., Phys. Rev. E, 2010, 82, 051134; doi:10.1103/PhysRevE.82.051134.
23. Fernández L.A., Martín-Mayor V., Yllanes D., Phys. Rev. B, 2011, 84, 100408(R); doi:10.1103/PhysRevB.84.100408.
24. Fytas N.G., Malakis A., Eftaxias K., J. Stat. Mech.: Theory Exp., 2008, P03015;
doi:10.1088/1742-5468/2008/03/P03015.
25. Theodorakis P.E., Georgiou I., Fytas N.G., Phys. Rev. E, 2013, 87, 032119; doi:10.1103/PhysRevE.87.032119.
26. Fytas N.G., Martín-Mayor V., Phys. Rev. Lett., 2013, 110, 227201; doi:10.1103/PhysRevLett.110.227201.
27. Ahrens B., Xiao J., Hartmann A.K., Katzgraber H.G., Phys. Rev. B, 2013, 88, 174408;
doi:10.1103/PhysRevB.88.174408.
28. Rieger H., Young A.P., J. Phys. A: Math. Gen., 1993, 26, 5279; doi:10.1088/0305-4470/26/20/014.
29. Rieger H., Phys. Rev. B, 1995, 52, 6659; doi:10.1103/PhysRevB.52.6659.
30. Falicov A., Berker A.N., McKay S.R., Phys. Rev. B, 1995, 51, 8266; doi:10.1103/PhysRevB.51.8266.
31. Swift M.R., Bray A.J., Martian A., Cieplak M., Banavar J.R., Europhys. Lett., 1997, 38, 273;
doi:10.1209/epl/i1997-00237-5.
32. Anglés d’Auriac J.-C., Sourlas N., Europhys. Lett., 1997, 39, 473; doi:10.1209/epl/i1997-00379-x.
33. Sourlas N., Comput. Phys. Commun., 1999, 121, 183; doi:10.1016/S0010-4655(99)00308-2.
34. Nowak U., Usadel K.D., Esser J., Physica A, 1998, 250, 1; doi:10.1016/S0378-4371(97)00580-3.
35. Duxbury P.M., Meinke J.H., Phys. Rev. E, 2001, 64, 036112; doi:10.1103/PhysRevE.64.036112.
36. Hernández L., Ceva H., Physica A, 2008, 387, 2793; doi:10.1016/j.physa.2008.01.021.
37. Crokidakis N., Nobre F.D., J. Phys.: Condens. Matter, 2008, 20, 145211; doi:10.1088/0953-8984/20/14/145211.
38. Salmon O.R., Crokidakis N., Nobre F.D., J. Phys.: Condens. Matter, 2009, 21, 056005;
doi:10.1088/0953-8984/21/5/056005.
39. Hadjiagapiou I.A., Physica A, 2011, 390, 2229; doi:10.1016/j.physa.2011.02.029.
40. Hadjiagapiou I.A., Physica A, 2011, 390, 3204; doi:10.1016/j.physa.2011.05.012.
41. Hadjiagapiou I.A., Physica A, 2012, 391, 3541; doi:10.1016/j.physa.2012.02.007.
42. Akinci Ü., Yüksel Y., Polat H., Phys. Rev. E, 2011, 83, 061103; doi:10.1103/PhysRevE.83.061103.
43. Tissier M., Tarjus G., Phys. Rev. Lett., 2011, 107, 041601; doi:10.1103/PhysRevLett.107.041601.
44. Houghton A., Khurana A., Seco F.J., Phys. Rev. Lett., 1985, 55, 856; doi:10.1103/PhysRevLett.55.856.
45. Mattis D.C., Phys. Rev. Lett., 1985, 55, 3009; doi:10.1103/PhysRevLett.55.3009.
46. Kaufman M., Klunzinger P.E., Khurana A., Phys. Rev. B, 1986, 34, 4766; doi:10.1103/PhysRevB.34.4766.
47. Sebastianes R.M., Saxena V.K., Phys. Rev. B, 1987, 35, 2058; doi:10.1103/PhysRevB.35.2058.
48. De Arruda A.S., Figueiredo W., Sebastianes R.M., Saxena V.K., Phys. Rev. B, 1989, 39, 4409;
doi:10.1103/PhysRevB.39.4409.
49. Belanger D.P., King A.R., Jaccarino V., Cardy J.L., Phys. Rev. B, 1983, 28, 2522; doi:10.1103/PhysRevB.28.2522.
43003-12
http://dx.doi.org/10.1103/PhysRevLett.35.1399
http://dx.doi.org/10.1103/PhysRevLett.37.1364
http://dx.doi.org/10.1103/PhysRevLett.43.744
http://dx.doi.org/10.1016/0304-8853(91)90825-U
http://dx.doi.org/10.1103/PhysRevLett.97.230603
http://dx.doi.org/10.1103/PhysRevLett.52.1543
http://dx.doi.org/10.1088/0022-3719/18/28/006
http://dx.doi.org/10.1103/PhysRevLett.56.416
http://dx.doi.org/10.1103/PhysRevB.33.4712
http://dx.doi.org/10.1103/PhysRevLett.59.1829
http://dx.doi.org/10.1103/PhysRevB.48.16533
http://dx.doi.org/10.1103/PhysRevE.62.8782
http://dx.doi.org/10.1103/PhysRevE.53.393
http://dx.doi.org/10.1103/PhysRevB.18.3318
http://dx.doi.org/10.1103/PhysRevB.18.3328
http://dx.doi.org/10.1103/PhysRevB.27.3079
http://dx.doi.org/10.1103/PhysRevB.28.5322
http://dx.doi.org/10.1103/PhysRevB.30.4034
http://dx.doi.org/10.1007/s100510050593
http://dx.doi.org/10.1103/PhysRevB.65.134411
http://dx.doi.org/10.1103/PhysRevE.82.051134
http://dx.doi.org/10.1103/PhysRevB.84.100408
http://dx.doi.org/10.1088/1742-5468/2008/03/P03015
http://dx.doi.org/10.1103/PhysRevE.87.032119
http://dx.doi.org/10.1103/PhysRevLett.110.227201
http://dx.doi.org/10.1103/PhysRevB.88.174408
http://dx.doi.org/10.1088/0305-4470/26/20/014
http://dx.doi.org/10.1103/PhysRevB.52.6659
http://dx.doi.org/10.1103/PhysRevB.51.8266
http://dx.doi.org/10.1209/epl/i1997-00237-5
http://dx.doi.org/10.1209/epl/i1997-00379-x
http://dx.doi.org/10.1016/S0010-4655(99)00308-2
http://dx.doi.org/10.1016/S0378-4371(97)00580-3
http://dx.doi.org/10.1103/PhysRevE.64.036112
http://dx.doi.org/10.1016/j.physa.2008.01.021
http://dx.doi.org/10.1088/0953-8984/20/14/145211
http://dx.doi.org/10.1088/0953-8984/21/5/056005
http://dx.doi.org/10.1016/j.physa.2011.02.029
http://dx.doi.org/10.1016/j.physa.2011.05.012
http://dx.doi.org/10.1016/j.physa.2012.02.007
http://dx.doi.org/10.1103/PhysRevE.83.061103
http://dx.doi.org/10.1103/PhysRevLett.107.041601
http://dx.doi.org/10.1103/PhysRevLett.55.856
http://dx.doi.org/10.1103/PhysRevLett.55.3009
http://dx.doi.org/10.1103/PhysRevB.34.4766
http://dx.doi.org/10.1103/PhysRevB.35.2058
http://dx.doi.org/10.1103/PhysRevB.39.4409
http://dx.doi.org/10.1103/PhysRevB.28.2522
Random-field Ising model: Insight from zero-temperature simulations
50. Slanič Z., Belanger D.P., J. Magn. Magn. Mater., 1998, 186, 65; doi:10.1016/S0304-8853(98)00065-1.
51. Brout R., Phys. Rev., 1959, 115, 824; doi:10.1103/PhysRev.115.824.
52. Binder K., Young A.P., Rev. Mod. Phys., 1986, 58, 837; doi:10.1103/RevModPhys.58.801.
53. Dayan I., Schwartz M., Young A.P., J. Phys. A: Math. Gen., 1993, 26, 3093; doi:10.1088/0305-4470/26/13/014.
54. Wiseman S., Domany E., Phys. Rev. E, 1995, 52, 3469; doi:10.1103/PhysRevE.52.3469.
55. Aharony A., Harris A.B., Phys. Rev. Lett., 1996, 77, 3700; doi:10.1103/PhysRevLett.77.3700.
56. Eichhorn K., Binder K., J. Phys.: Condens. Matter, 1996, 8, 5209; doi:10.1088/0953-8984/8/28/005.
57. Pázmándi F., Scalettar R., Zimányi G.T., Phys. Rev. Lett., 1997, 79, 5130; doi:10.1103/PhysRevLett.79.5130.
58. Wiseman S., Domany E., Phys. Rev. Lett., 1998, 81, 22; doi:10.1103/PhysRevLett.81.22.
59. Ballesteros H.G., Fernández L.A., Martín-Mayor V., Muñoz Sudupe A., Parisi G., Ruiz-Lorenzo J.J., Phys. Rev. B,
1998, 58, 2740; doi:10.1103/PhysRevB.58.2740.
60. Tomita Y., Okabe Y., Phys. Rev. E, 2001, 64, 036114; doi:10.1103/PhysRevE.64.036114.
61. Parisi G., Sourlas N., Phys. Rev. Lett., 2002, 89, 257204; doi:10.1103/PhysRevLett.89.257204.
62. Berche P.E., Chatelain C., Berche B., Janke W., Eur. Phys. J. B, 2004, 38, 463; doi:10.1140/epjb/e2004-00141-x.
63. Monthus C., Garel T., Eur. Phys. J. B, 2005, 48, 393; doi:10.1140/epjb/e2005-00417-7.
64. Malakis A., Fytas N.G., Phys. Rev. E, 2006, 73, 016109; doi:10.1103/PhysRevE.73.016109.
65. Wu Y., Machta J., Phys. Rev. Lett., 2005, 95 137208; doi:10.1103/PhysRevLett.95.137208.
66. Wu Y., Machta J., Phys. Rev. B, 2006, 74, 064418; doi:10.1103/PhysRevB.74.064418.
67. Gordillo-Guerrero A., Ruiz-Lorenzo J.J., J. Stat. Mech.: Theory Exp., 2007, P0601;
doi:10.1088/1742-5468/2007/06/P06014.
68. Fytas N.G., Malakis A., Phys. Rev. E, 2010, 81, 041109; doi:10.1103/PhysRevE.81.041109.
69. Efrat A., Schwartz M., Preprint arXiv:cond-mat/0608435, 2006.
70. Hartmann A.K., Rieger H., Optimization Algorithms in Physics, Wiley-VCH, Berlin, 2004.
71. Ogielski A.T., Phys. Rev. Lett., 1986, 57, 1251; doi:10.1103/PhysRevLett.57.1251.
72. Hartmann A.K., Young A.P., Phys. Rev. B, 2001, 64, 214419; doi:10.1103/PhysRevB.64.214419.
73. Dukovski I., Machta J., Phys. Rev. B, 2003, 67, 014413; doi:10.1103/PhysRevB.67.014413.
74. Seppälä E.T., Alava M.J., Phys. Rev. E, 2001, 63, 066109; doi:10.1103/PhysRevE.63.066109.
75. Alava M.J., Duxbury P.M., Moukarzel C.F., Rieger H., In: Phase Transitions and Critical Phenomena, Vol. 18,
Domb C., Lebowitz J.L. (Eds.), Academic Press, San Diego, 2001, 143–317.
76. Fytas N.G., Theodorakis P.E., Georgiou I., Eur. Phys. J. B, 2012, 85, 349; doi:10.1140/epjb/e2012-30731-8.
77. Papadimitriou C.H., Computational Complexity, Addison-Wesley, Reading, 1994.
78. Goldberg A.V., Tarjan R.E., J. ACM, 1988, 35, 921; doi:10.1145/48014.61051.
79. Middleton A.A., Phys. Rev. Lett., 2002, 88, 017202; doi:10.1103/PhysRevLett.88.017202.
80. Meinke J.H., Middleton A.A., Preprint arXiv:cond-mat/0502471, 2005.
81. Landau D.P., Binder K., A Guide to Monte Carlo Simulations in Statistical Physics, Cambridge University Press,
Cambridge, 2000.
43003-13
http://dx.doi.org/10.1016/S0304-8853(98)00065-1
http://dx.doi.org/10.1103/PhysRev.115.824
http://dx.doi.org/10.1103/RevModPhys.58.801
http://dx.doi.org/10.1088/0305-4470/26/13/014
http://dx.doi.org/10.1103/PhysRevE.52.3469
http://dx.doi.org/10.1103/PhysRevLett.77.3700
http://dx.doi.org/10.1088/0953-8984/8/28/005
http://dx.doi.org/10.1103/PhysRevLett.79.5130
http://dx.doi.org/10.1103/PhysRevLett.81.22
http://dx.doi.org/10.1103/PhysRevB.58.2740
http://dx.doi.org/10.1103/PhysRevE.64.036114
http://dx.doi.org/10.1103/PhysRevLett.89.257204
http://dx.doi.org/10.1140/epjb/e2004-00141-x
http://dx.doi.org/10.1140/epjb/e2005-00417-7
http://dx.doi.org/10.1103/PhysRevE.73.016109
http://dx.doi.org/10.1103/PhysRevLett.95.137208
http://dx.doi.org/10.1103/PhysRevB.74.064418
http://dx.doi.org/10.1088/1742-5468/2007/06/P06014
http://dx.doi.org/10.1103/PhysRevE.81.041109
http://arxiv.org/abs/cond-mat/0608435
http://dx.doi.org/10.1103/PhysRevLett.57.1251
http://dx.doi.org/10.1103/PhysRevB.64.214419
http://dx.doi.org/10.1103/PhysRevB.67.014413
http://dx.doi.org/10.1103/PhysRevE.63.066109
http://dx.doi.org/10.1140/epjb/e2012-30731-8
http://dx.doi.org/10.1145/48014.61051
http://dx.doi.org/10.1103/PhysRevLett.88.017202
http://arxiv.org/abs/cond-mat/0502471
P.E. Theodorakis, N.G. Fytas
Модель Iзiнга у випадковому полi: моделювання при
нульовiй температурi
П.Е. Теодоракiс1, Н.Г. Фiтас2
1 Вiддiл хiмiчної iнженерiї, Емпiрiал Коледж Лондон, SW7 2AZ, Лондон, Великобританiя
2 Центр прикладних математичних дослiджень, Унiверситет м. Ковентрi, Ковентрi, CV1 5FB,
Великобританiя
Застосовуючи комп’ютернi симуляцiї при нульовiй температурi, ми висвiтлюємо деякi аспекти критичної
поведiнки тривимiрної (d = 3) моделi Iзiнга у випадковому полi. Ми розглядаємо двi версiї моделi, що
вiдрiзняються розподiлом випадкового поля, а саме, гаусову та тримодову моделi Iзiнга у випадковому
полi з однаковими вагами. Застосовуючи обчислювальний пiдхiд, що ставить у вiдповiднiсть основно-
му стану системи проблему оптимiзацiї максимуму потоку на мережi, ми використовуємо найсучаснiшу
версiю алгоритму проштовхування потоку i моделюємо великi ансамблi випадкових реалiзацiй моделей
для широкої областi значень випадкового поля i розмiрiв системи V = L ×L ×L, де L позначає лiнiйний
розмiр гратки i Lmax = 156. Використовуючи в якостi скiнчено-вимiрних мiр флуктуацiї рiзних величин
фiзичного i технiчного походження, вимiряних для рiзних зразкiв, i примiтивнi операцiї алгоритму про-
штовхування потоку, ми пропонуємо для обох типiв розподiлу оцiнки критичного поля hc i критичного
показника кореляцiйної довжини ν. Отримане значення цього показника чiтко вказує на те, що обидвi
моделi належать до одного класу унiверсальностi. Додатковi симуляцiї гаусової моделi Iзiнга у випадко-
вому полi при добре вiдомому значеннi критичного поля забезпечують вiдношення магнiтних iндексiв
β/ν з високою точнiстю i прояснюють контроверсiйну проблему критичного iндекса α питомої теплоєм-
ностi. Накiнець, ми обговорюємо нескiнченнорозмiрну екстраполяцiю енергiї i базованого на параметрi
порядку шуму до сигнальних коефiцiєнтiв, пов’язаних з властивостями самоусереднення моделi, а також
аспекти критичного сповiльнення алгоритму.
Ключовi слова: модель Iзiнга у випадковому полi, скiнченнорозмiрний скейлiнг, теорiя графiв
43003-14
Introduction
Simulation protocol
Universality aspects
Gaussian RFIM
Summary and outlook
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