Disordered ferromagnetic materials are the subject of intense theoretical and experimental research due to their physical characteristics associated with the domain structure [1–3]. Recent research surveys [4–7] elucidate that these materials, especially low dimensional samples and ferromagnetic-antiferromagnetic heterostructures, are considered promising materials for modern technology applications. Disordered ferromagnets possess intricate domain structure leading to the hysteresis behavior [8,9] in the external magnetic field; see also recent review [10] and references therein. Hysteresis behavior is vital for various applications of magnetic materials [11]. Therefore more attention is devoted to predicting and optimizing hysteresis properties using different numerical tools [12]. During the reversal processes by slow ramping of the external field over the hysteresis loop, the moving domain walls interact with the structural and magnetic disorder [13], leading to stochastic changes of the magnetization. The magnetization fluctuations are experimentally measured as Barkhausen noise (BHN), for example, in bulk materials [14–17] and thin films [18–21], as well as in samples with varied thickness [22,23]. Considering the Ising model, one of central pillars of statistical physics [24], theoretical investigations focusing to the dynamic phenomena on hysteresis use the field-driven spin models with random bond [25] or random field disorder [26–28]. These investigations revealed that the collective nature of the magnetization changes with avalanche-like behavior on the hysteresis are associated with a disorder-induced critical point [29–32].

This out-of-equilibrium critical dynamics is characterized by long-range temporal correlations and multifractal features of the BHN [33]. Among different types of models of disordered systems, the Ising model with random fields [34–37] appeared as the most attractive for theoretical investigations of the in- and out-of-equilibrium criticality. It was also recognized as an appropriate model for weakly disordered antiferromagnetic materials in an external field, where structural disorder induces local random fields [38–40]. For more details regarding recent studies on antiferromagnetic systems, see [41–44]. It should be noted that the 100 years of the Ising model, celebrated this year, have shown its relevance to many different phenomena in complex systems [45]. However, certain limitations of the model in describing the magnetism of solid materials are evident, for example, taking into account more complex domain walls in domain structures that are associated with hysteresis phenomena, distinguishing hysteresis properties along different easy axes and the influence of other types of disorder that do not break the local rotational symmetry. In addition to the theoretical interpretation of the behavior of disordered ferromagnetic systems, using the random-field Ising model in interpreting experimental Barkhausen noise proved crucial for some systems [17,46]. In a more general context, statistical physics and random field theory were recently applied for spatial data modelling, as described in the book [47].

The model treats a system of Ising spins located at lattice sites and quenched impurities interpreted by randomly distributed magnetic fields. Among the nearest neighboring spins the ferromagnetic interaction occurs, in addition to the interaction with the external magnetic and a quenched random field. The zero-temperature variant of the model proposed in [48] captures the essential features of avalanching dynamics while neglecting thermal effects. Triggered by the external magnetic field changes, the system relaxes in spin-flipping avalanches, reflected by the magnetization jumps. The stochastic process of magnetization reversal occurs along the hysteresis loop, characterizing the collective response of the system. The occurrence of the out-of-equilibrium critical dynamics on the hysteresis loop together with its dependence on pertinent physical parameters, system’s spatial dimensionality and shape, represent essential challenges for controlling and predicting the reversal processes in these materials.

Recently, it has been recognized that besides hysteresis-loop criticality in disordered ferromagnets, numerous other complex systems exhibit a similar avalanche-like response to the external driving forces. Further examples of such systems span from earthquakes [49–53] and propagation of interfaces in various kinds of random media [54–58] to brain networks and neuronal activity [59–62], dislocations in crystal structures [63–66], invading imbibition fronts in porous media [67,68], response of the mechanically pressured wooden materials [69] to financial booms [70] and epidemics [71], all having in common that the underlying systems evolve through the metastable states due to an avalanche type of relaxation. Despite being different at a microscopic level, these systems have some universal features of their critical states in common. This universality implies the possibility of extending the findings and methodologies developed on the systems that are tractable for experimental realizations, such as disordered ferromagnets, to characterize such phenomena in systems that are elusive to controlled experiments.

Here, we provide a comprehensive review of numerical approaches to model the criticality of the hysteresis loop in disordered ferromagnets based on extended nonequilibrium (NEQ) Random Field Ising Model (RFIM) with different physical parameters and driving regimes inspired by experimentally achievable conditions. The presented overview of the results shows how these modeling approaches, supported by advanced computational techniques using the inherent scale invariance of the dynamics, adequately describe the experimentally observed hysteresis behavior and provide new physical insights into the dynamic critical phenomena of disordered ferromagnetic systems. The developed computational techniques can be adjusted to analyze nonequilibrium critical dynamics of ferromagnetic/antiferromagnetic bilayers [72,73] and various quantum [74–76] and classical complex systems, in particular, nanonetworks [77] and higher-order self-assembled nanostructures [78–82] that are increasingly interesting to modern technology.

Challenges in Numerical Modeling of Disordered Ferromagnets Hysteresis Behaviour

Occurrence of extreme events such as infinite or system-spanning avalanches underpins the system criticality, characterized by the scale invariance of the created events whose features are power-law distributed and diverging correlation length [34]. Over time, several models were developed to describe this complex phenomenon and answer whether the model under study exhibits a critical behavior [83]. Unraveling this is not an easy task due to the multitude of mutually intertwined factors to which the underlying systems are sensitive, such as the degree of disorder or the type and rate of driving, presence of the demagnetizing fields and thermal effects, sample geometry, which is of particular relevance for applications, and other. In the following, we briefly describe these factors that represent challenges for computational modeling of the hysteresis-loop critical phenomena.

As the extensive studies [28] have shown, the critical behavior of the zero-temperature (ZT) RFIM depends on the dimension d of the studied system [31,84]. In the range of dimensions 2≤d<6, ZT NEQ RFIM displays a nontrivial critical behavior [31], while for d≥6, the behavior of the system is described by mean-field approximation [35,84,85]. In the nonequilibrium field-driven version, the system evolves through metastable states that represent local, and not global energy minima, in contrast to the equilibrium critical behaviour studied by the RFIM [86–88].

Criticality of the equilibrium and nonequilibrium version of the model for systems with dimension d≥3, displayed in the matching of many avalanche properties, lead to the presumption that they share the same universality class [37,87,88]. However, that is not the case for the 2D model since the ferromagnetic phase, whose existence is demonstrated in the case of nonequilibrium model [89,90], is not possible in the thermodynamic limit of the equilibrium version [91]. In an attempt to describe this nontrivial critical behavior, both perturbative [92–94] and nonperturbative renormalization group approaches [95,96] were used. Recently some advances have been made on understanding the principles of universality [36,97], dimensional reduction [98] and supersymmetry [99] in the equilibrium version of the model. On the other hand, dimension and geometry influence of underlying lattice are shown to have different impact on the nonequilibrium critical behavior [100–103].

Particular attention in the recent studies of nonequilibrium systems is devoted to the extreme, catastrophic events in which most of the system’s constituents change their state [104,105]. Understanding the particular conditions under which these events occur would reveal essential details regarding the mechanism of some natural phenomena such as earthquakes, snow avalanches, cracks in materials, etc. [27,106–108], that are of immense importance due to the possible significant consequences they might cause.

In recent years, due to the increased interest in the practical applications of thin ferromagnetic materials, studies of nonequilateral systems with different geometry aspects [109] and thin systems [110–114] emerged, with some aspects in resemblance to the criticality of spin systems situated on a complex network topology [115–118]. Disordered ferromagnetics are mainly used as memory materials in the form of thin films [119–122] and nanowires [123–126], putting forward the experimental investigations of critical dynamics of BHN [127–129]. The ZT NEQ version of the RFIM was very suitable for interpreting actual experiments since the thermal fluctuations are negligible in most of them, and the underlying dynamics is closer to the one in externally driven ferromagnets. Driving these systems by the external magnetic field at finite rates [130] provides a more realistic framework for the numerical interpretation of experimental measurements conducted on ferromagnetic strips and ribbons [21,131,132] and has been of increasing interest in recent experimental studies [17,22,114,133].

Besides dimensionality, one may ask to which extent the model describes the influence of a variation of the geometry of the underlying lattice [109], including the reconsideration of the universality classes within the RFIM [100,134]. Recently, a competitive conjecture was put forward that the universality classes in the NEQ model are determined by the topology of the lattice emphasizing the role of its coordination number in addition to its spatial dimensionality [102,134–136]. This conjecture was to some extent confirmed by contrasting the criticality of the model on triangular [100], quadratic [89] and hexagonal (i.e., honeycomb) [103] 2D lattices. The absence of critical behavior for the latter lattice introduced doubt in the presumed existance of a single universality class for all periodic 2D lattices in the case of ZT NEQ RFIM.

Another significant aspect, particularly for experimental studies, is the presence of unwanted external noise in the recorded data requiring the inevitable procedure of using the discrimination threshold. In the experimentally obtained signals, avalanches are identified as signal parts outside the imposed threshold region (i.e., the region between lower and upper threshold levels). Recent experimental [137] and theoretical [138–141] studies delivered some relevant results regarding the consequences of the implementation of the finite detection threshold when analyzing the original signal, which leads to correlated bursts of activity by separating the avalanche events into subavalanches. Effects of thresholding have also been considered in some other types of signals, e.g., in the context of fracture [142–144], and argued to be of importance in seismicity [145,146].

Driving mode represents one of the essential factors influencing the avalanche-like response. Theoretically, the three types of driving are recognized: adiabatic, in which the external magnetic field is increased for the exact amount needed to trigger the least stable spin and kept unchanged until the avalanche stops; quasistatic, in which the incrementing of the external magnetic field is performed in fixed steps until the conditions for the initiation of the avalanche are met, from which moment is kept constant as long as the avalanche propagates; and a finite rate driving during which, throughout the whole simulation time, the external magnetic field is increased at a constant rate. According to the value of the driving rate, the regimes of slow, intermediate and fast driving are identified [130]. The underlying dynamics in the finite driving regime is profoundly influenced by the interplay of the system’s disorder and the driving rate at which the magnetic field is incremented. In this type of driving, multiple avalanches may simultaneously propagate, making the analysis and interpretation of the obtained data of the avalanching dynamics far more complex.

The different ways of driving have a significant impact on the system’s behavior, with one of the most noticeable effects being the time/space profile of avalanche evolution. Adiabatic driving, during which only one avalanche is active at a time, is conducted under precisely defined conditions realizable in the numerical simulations but not in the actual experiments. A little bit closer to realistic is the quasistatic driving type, in which the regimes with small (adiabatic-like) and large field increments can be identified, promoting the temporal and occasionally spatial merging of avalanches [147]. This amalgamation of avalanches is expressed the most at the fast regime of finite rate driving protocol, comprising an overall system-spreading activity [148] without the possibility of distinguishing the contribution of individual avalanches [130,147–149]. This type of driving is even more realistic and comparable to the experimental situations. In one of the previous studies [150], it has been shown that due to the merging of avalanches (swelling) the overall duration of avalanches can be extended; at the same time, the merged avalanche will appear in the distribution while the merging avalanches vanish (merging absorption). Similarly, for the avalanches overlapped in time (and spatially separated), the overlapped avalanche appears, whereas the overlapping avalanches disappear from the distribution (temporal absorption). Until now, many experiments have been conducted on disordered ferromagnetic materials using the finite driving rate protocol for example, in [14–16,133]. Works in [26,150–153] attempt to describe theoretically the observed critical dynamics phenomena with the finite driving rate and its impact on the hysteresis loop and coercive field was considered in [18,19,154–156].

The study of the effects that the type of driving inflicts on the system is further augmented by employing the newly proposed stochastic driving, realized by stochastic increments of the external magnetic field in each time step of the system’s evolution, regardless of the current state of the ongoing system’s activity. This type of driving opens the possibility to extend the findings and perform the more realistic interpretation of seismic activity [50], owing to its importance due to the unwanted ramifications. In some recently conducted experiments on the propagation of a crack line in a random environment [57,142,157,158], a seismic-like behavior has been demonstrated, also found in the studies of compressional fracture [159]. Stochastic driving represents so far the most realistic scenario, allowing much closer benchmarking with the results of the experimental studies and the possibility to extend to the numerous models developed in an attempt to explain and interpret this complex avalanche dynamics [143,160–163].

Another aspect concerns the thermal effects and the presence of the demagnetization fields in the system. These physical factors bring challenges, especially in understanding the intricate interplay of various parameters, interpreting and implementing the solution numerically, and comparing the numerical results with experiments. As shown in [164], the extended form of RFIM can be an excellent alternative for the micromagnetic modeling, which is particularly relevant for applications. The stochastic nature of the thermally triggered spin flips tends to impede accurate avalanche detection by contaminating the signal, producing an excess of minor activity events, and limiting the spread of large avalanches. Nonetheless, the demagnetizing field limits the amount of underlying spin activity events, and the maximum length of magnetization jumps by counteracting the external field and introducing non-local interactions. In a recent study [164], an appropriate algorithm to study these combined effects was developed. Notably, it demonstrates that the non-zero temperature modifies the remanent magnetization and coercive field, leading to the shrinkage of the hysteresis loop and the amplification of minor activity events. By introducing extended linear segments, the higher demagnetizing coefficient modifies the loop shape and changes the multifractal nature of the magnetization fluctuations. Similar to the analogous ZT dynamics, the statistics of well-identified intermediate-range activity events are controlled by the same scaling exponents.

In the next section, we describe the most general version of the NEQ RFIM to study the critical behavior on the hysteresis loop of disordered ferromagnetic systems, assuming different driving conditions and the aforementioned physical factors, followed by the section presenting the appropriate computational techniques. The rest of the review is structured so that each section presents the results obtained in numerical studies of particular hysteresis phenomena, emphasizing the influence of specific physical factors.

The Random Field Ising Model describes systems of Ns mutually interacting classical Ising spins si=±1 exposed to external magnetic field H and a quenched local random magnetic field hi. The spins are situated at the sites i of some underlying finite lattice with space dimensionality d, shape and size specified in each model realization. Mostly employed are the two-dimensional (2D), and three-dimensional (3D) lattices, however lattices of other dimensions are also used (for d>3, see, e.g., [31]). Frequently utilized are finite lattices cut out of a corresponding infinite lattice with translation symmetry like the quadratic lattice of size (Lx,Ly) in 2D, cubic lattice of size (Lx,Ly,Lz) in 3D, and analogously hypercubic lattice for d>3. In all three of the foregoing lattice examples the number znn of nearest neighbors for each site equals 2d, however, this number can be different for other types of lattice elementary cell, like znn=6 for the triangular [100], and znn=3 for the honeycomb [103] 2D lattices. Finally, the model can be situated on a lattice without translation symmetry, such as Bethe lattice, and/or translation symmetry may be absent due to the presence of irregularly distributed vacancies (lattice sites not ‘occupied’ by a spin), and/or irregular interfaces consisting of spins with fixed value.

In general, the interaction of each spin si→ with the external magnetic field is accomplished through the coupling −μ0mi→⋅Hi→(t) between the magnetic dipole moment mi→=γi(s)ℏsi→ associated with si→, and the external magnetic field Hi→(t) at the spin’s site i at the current moment (of time) t; here, γi(s) is the gyromagnetic ratio for the spin si→, usually taken to be the same for all spins in the model, γi(s)=γ(s). Because the RFIM is not a vector spin model, one can assume here that some (arbitrary) direction is chosen in space such that for all spins the projection s→i⋅u→0 of si→ onto the unit vector u→0 of this direction equals si enabling in the RFIM the incorporation of the coupling between spins and the external magnetic field as −Hi(t)si; here, Hi(t)=H→i(t)⋅u→0 is the projection of H→ onto u→0, and the units are appropriately rescaled for simplicity (which is allowed in models).

Analogous form −hisi is taken for the coupling between the spin si and the random field hi at the spin’s site i. This local random field {hi}i=1Ns is postulated as quenched (meaning that its values are fixed in time), and could be considered to originate from various types of real systems’ imperfections not specified in the RFIM. Besides, this field is random, meaning that its values hi at different lattice sites are chosen out of some type of zero-mean random distribution ρ(h) with finite standard deviation R measuring disorder in the RFIM systems, which could be site-dependent. Here, the common approach is to use Gaussian distribution, ρ(h)=1R2πexp⁡(−h2/2R2), with the same disorder R for all sites. The RFIM studies, performed with another choice of the random filed distribution ρ(h), exist but are relatively rare-see, e.g., the case of parabolic and uniform distributions in [87], Laplace and double Gaussian distribution in [97], and the case of Gaussian distribution with R modulated by the presence of crystal grains [165].

Independent on the choice of the distribution ρ(h) is the question whether there is some correlation between the values of random field at different lattice sites. When the values are uncorrelated, then ⟨hihj⟩=Ri2δij, where δij is the Kronecker delta function and Ri is the value of disorder at the site i. This is the case in majority of past RFIM studies although the model itself doesn’t exclude possible correlations.

Besides the interaction of spins with magnetic fields (external and random), the spins also mutually interact. The strongest type of interaction between the spins is the (electrostatic in nature) exchange interaction between pairs of spins, giving the contribution −Jijsisj from each pair si and sj, to the overall system Hamiltonian through the coupling constant Jij. When Jij>0 the coupling is ferromagnetic, for Jij<0 the coupling is antiferromagnetic, while for Jij=0 the spins si and sj are exchange decoupled. The exchange interaction is typically considered as short-ranged, meaning that it is the most prominent for the nearest neighbor spins and significantly weaker for further neighbors (e.g., exponentially decaying with the inter-site distance). On this ground, frequently used is the nearest-neighbor approximation in which all pairs of spins except nearest neighbors are considered as exchange decoupled.

Another type of spin-spin interaction is the dipole-dipole interaction between the magnetic dipole moments m→i and m→j associated with pair of spins s→i and s→j. In general, the dipole-dipole interaction is of the form −μ0(γ(s))2ℏ24πrij5[3(si→⋅rij→)(sj→⋅rij→)−si→⋅sj→rij2],where rij→=ri→−rj→ is the position vector of site i relative to site j. In the RFIM this expression is simplified to −Jdipole3cos2θij−1rij3sisj,where Jdipole=μ0(γ(s))2ℏ2/4π, and θij is the angle between rij→ and u→0. Being inversely proportional to the cube of inter-site distance rij, the dipole-dipole interaction is not (like the exchange interaction) limited to near spins, but instead, it affects all pairs of spins, i.e., it is long-ranged. Yet, it is typically weaker than the exchange interaction, and therefore often omitted in the model analysis.

Besides, the spins located at the boundaries of finite lattices generate inside the sample a long-range (effective) demagnetizing magnetic field which acts against the external field. In the RFIM, this field is taken as a homogeneous field −JDM and its coupling with individual spins as JDMsi, where M=(∑isi)/N is the actual magnetization of the system and JD≥0 is the demagnetization coefficient (factor). The adopted form of demagnetizing field is by all means approximate, so in simulations and analyses it is most appropriate to assign such values to JD that conform to the values theoretically found for the homogeneously magnetized samples (e.g., JD=1/3 for cubic and spheroidal samples).

Therefore, the most general form of Hamiltonian for the RFIM spin systems in the external magnetic field with a time profile Hi(t) reads (1)ℋ=−∑{ij}(Jij+Jdipole3cos2θij−1rij3)sisj−∑i=1Ns(Hi(t)+hi−JDM)si,where the summation in the first term is performed over all pairs of distinct spins, so that each spin si is under the influence of the effective magnetic field (2)hieff(t)=∑j≠i(Jij+Jdipole3cos2θij−1rij3)sj+Hi(t)+hi−JDM,giving contribution −si(t)hieff(t) to Hamiltonian (1). If si(t)hieff(t)≤0, the spin si(t) is field-unstable at the moment t, and the value of Hamiltonian (1), henceforth simply the system energy, will be reduced by flipping of si (i.e., change of si to the opposite value −si). Otherwise, if si(t)hieff(t)>0, the spin si(t) is field-stable, so its flipping would increase the system energy.

Although absent in Hamiltonian (1), the lattice-spin interactions are indirectly present in the RFIM through the thermal fluctuations of spins manifested when the system temperature Θ>0 in which case the model is called thermal. Otherwise, when Θ=0, the thermal fluctuations of spins are absent and the model is athermal or zero-temperature. This Θ=0 version of RFIM is of twofold importance because in many real-world systems, to which the RFIM can be applied, the thermal fluctuations are negligible in the relevant range of temperatures, and also because the athermal RFIM version is much simpler than the thermal, both for simulations and analyses.

In the RFIM, the state {si}i=1Ns of the spin system changes through the flipping of individual spins. In the nonequilibrium (NEQ) model version, which is the subject of this review article, the individual spins flip due to appropriate change of their effective magnetic field and/or thermally if Θ>0. Thus, in the athermal NEQ RFIM version, only the first cause applies, and this is usually done by parallel updating of orientation of spins, meaning that all spins which are field-stable at the moment t will remain unchanged and the field-unstable spins will be flipped at the next moment t+Δt (in simulations, both t and its increment Δt are discrete, usually with Δt=1). Due to such local dynamical rule and parallel updating, the system evolves deterministically, traversing (almost exclusively) through the nonequilibrium states in a tendency to reduce its energy. Other (and less used) way, employed in the athermal model version, is to perform sequential field-stability testing and flipping of spins one by one, in which case the system’s evolution depends on the random order in which the spins are tested.

Flipping of spins, together with a possible change of the external magnetic field, modifies the effective magnetic field hieff of the system’s spins. Thus, when JD>0, the magnetization M is (typically) altered due to spin flipping which causes a change in hieff for all spins through the term −JDM in (2). Also, the dipole-dipole interaction term in (2) becomes different for all (especially near) spins, provided that this type of interaction is included in the model. However, the contribution coming from the change in the exchange-coupling term in (2) is the most prominent, or the only one when JD=0, Jdipole=0, and constant H at the next moment t+Δt.

In thermal (i.e., Θ>0) model, together with the preceding field-stability testing, the spins are also checked for thermal flipping. Thus, in [166], the changes ΔH of the external magnetic field are sandwiched between a preselected number Nth of Monte-Carlo sweeps sequentially applied, so that in each sweep a spin si is randomly chosen and flipped with the probability min(1,e−ΔEi/Θ), where Θ>0 is the temperature, and ΔEi is the change in energy proposed by flipping of si. By this rule, the selected spin si is certainly flipped if it is field-unstable (ΔEi<0), however, if it is field-stable (ΔEi>0), it may be also flipped with probability e−ΔEi/Θ. As a consequence, the central role is played by the time-scale of thermal flipping, set by the number Nth regardless of the system size, while the time-scale for the field changes is introduced indirectly, relative to the thermal flipping time-scale.

To check the stability at the current moment for all (and not only selected) spins, a different approach is proposed in [164]. There, the thermal flipping is tested on a set of spins, randomly selected at each moment t and containing a preselected fraction c=ΔNs/Ns of spins. At the next moment t+Δt, each spin outside this set will be flipped if it is field-unstable, while each spin si from the set will be flipped with the (Boltzmann-type) probability of thermal flipping probability (3)pi(th)=exp⁡(−sihieff/Tr)exp⁡(−sihieff/Tr)+exp⁡(sihieff/Tr),where the parameter Tr=Θ/Tc stands for the temperature relative to some temperature Tc that is characteristic in the model for the underlying system (e.g., Tc=J for the system of purely ferromagnetic spins with Jij=J). Hence, for sihieff<0 (i.e., if the selected spin si is field-unstable) it is likely (but not certain) that it will be flipped, whereas for sihieff>0, the selected spin si has a small nonzero chance for flipping.

As stated, the external magnetic field varies in time in the NEQ RFIM. This is realized either in some deterministic variation pattern (e.g., adiabatic, quasistatic, finite driving rate), or by stochastic external field increments (see, e.g., Reference [149]). In the adiabatic and quasistatic driving protocols, employed in the athermal model versions, the external field is changed only if all spins in the system are field-stable. In the adiabatic case the field is changed in the exact amount that destabilizes the least stable spin (and flips it at the next moment), while in the quasistatic driving the field increment is constant, ΔH=const, possibly causing flipping of one or more spins, especially for the large systems. Flipping of spin(s) may destabilize some spins (most often the nearest neighbors of the flipped spin); their flipping may further destabilize other spins, and so on, leading to a time-series of flipping in the form of an avalanche. In the case of adiabatic models with JD=0, Jdipole=0, and the exchange coupling limited only to the nearest neighbors, each avalanche is nucleated by flipping of a single spin and afterward spread over a cluster of (connected) spins. In other adiabatic cases, or when the driving is quasistatic, multiple avalanche nucleations at spatially distant locations may happen. This leads to the onset of avalanches that simultaneously propagate over several spin clusters initially space-separated and later possibly merging in space. Whether due to such, or to individual avalanches, the system response is expressed in terms of the numbers n+(t) and n−(t) of spins that flip up and down at the moment t, giving the response signal of the system, V(t)=n+(t)−n−(t), and the corresponding change in magnetization ΔM(t)=2V(t) during one time-step at the moment t. Each subsequence of the V(t) sequence, consisting of consecutive terms such that the underlying n+(t) and n−(t) satisfy n+(t)+n+(t)>0, represents a system activity event, which is a generalization of an individual avalanche.

In addition to adiabatic and quasistatic driving (which tends to adiabatic when ΔH→0 for any finite system), there is the finite rate driving protocol in which the external magnetic field, regardless of the system activity, increases/decreases along the rising/falling part of the magnetization curve at some constant driving rate Ω=ΔH/Δt (given by the field increment/decrement ΔH when Δt=1 and also tending to adiabatic when Ω→0). This type of driving, enhancing spin flipping and thus facilitating activity propagation due to perpetual modifications of spins effective magnetic field, is of considerable importance because it is encountered in a large number of experiments performed on magnetic systems.

Compared to athermal model, the system response is essentially different in the thermal model regarding events duration. Thus, while all events in athermal model are of finite duration this is almost never the case in the thermal model due to the thermal flipping of spins. For this reason, unless some threshold is introduced, the decomposition of system response into activity events separated in time is impossible, making the adiabatic and quasistatic driving unrealistic. So, in [164], the external magnetic field is incremented in each step, whereas in [166] the field is incremented after Nth (elemental) time-steps, while meanwhile the thermal flipping of spins is performed. Note that in pure ferromagnetic case, due to thermal flipping and other factors influencing the effective magnetic field, some spins may be back-flipped, i.e., changed from +1 to −1 on the rising part of magnetization curve (and opposite on the falling part). Such back-flipping is possible when JD>0 even in athermal model and impossible when JD=0 and Jdipole=0.

The analysis of RFIM systems evolution is commonly performed on their response signal decomposed into events; the exception is the analysis of power spectra P(f) [167] giving the variation with frequency f of the released power frequency density which is performed without any decomposition on the entire selected part or on the whole response signal. In the athermal version of ferromagnetic systems each activity event gives a part of consecutive non-zero values in the response signal, V(t)>0 along the rising part of the magnetization curve (and V(t)<0 along the falling part). These parts are taken as events which are the subject of further statistical analysis. In other cases, such simple decomposition is impossible, and the decomposition of the response signal into events is accomplished with the aid of some (suitably chosen) discrimination threshold Vth. Provided that Vth≥0 is selected, the response signal is decomposed into longest subsequences {V(ti),V(ti+1),…,V(te)} consisting of signal values |V(t)|>Vth, all positive (or all negative), and registered in consecutive moments between the initial moment ti and ending moment te of the subsequence in question. Each such subsequence is considered as an event, and for any moment t that is outside all events |V(t)|≤Vth. Each event can be characterized by several parameters like its duration T=te−ti+1, size S=∑te≤t≤tiV(t)Δt, energy E=∑te≤t≤tiV2(t)Δt, and amplitude A=max{|V(ti)|,|V(ti+1)|,…,|V(te)|}, and for each such parameter X the corresponding distribution DX(X) is collected. These distributions can be either integrated (i.e., collected from the entire signal), or windowed (i.e., collected in a selected window (Hmin,Hmax) of the external magnetic field). Besides, the distributions can be classified according to the event’s type (e.g., nonspanning and different types of spanning events). Among possible distribution shapes, of particular importance are those of the type (4)DX(X)=X−a𝒟X(X),representing a power-law with cutoff specified by the power-law exponent a and the so-called cutoff function 𝒟X(X) which (beside X) depends on some parameters for simplicity not shown in notation. The analytic form (4) describes a pure power-law, DX(X)∝X−a, only in some limits of the cutoff function parameters leading to 𝒟X(X)→const. Outside such limits the cutoff function rapidly tends to zero for large X (and possibly for very small X), while in between, a region of moderate X-values may exist such that 𝒟X(X)≈const. If so, this region is called the scaling region, and it could be used for determination of the value of the power-law exponent as a equals the gradient (i.e., slope) of the scaling region data presented in a log-log plot; other (and preferred) method is by the collapsing of data which can be applied when the cutoff function has universal scaling properties as will be demonstrated in the Results sections. For the later convenience, here we quote the common RFIM exponents: τ, α, ε, and μ for the windowed distributions of event’s size, duration, energy, and amplitude, respectively; additional NEQ RFIM exponents will be introduced in the Results sections.

Besides, other quantifiers of response signal can be defined, like the magnetization as a function of the external magnetic field, M=M(H), and analogously the susceptibility dM/dH, the correlation functions, average event shape, and the distributions of various types of waiting time, like the external waiting time Text measuring the separation in time of consecutive events.

Integral part of model is the specification of boundary conditions which are either closed or open on each system boundary. Closed conditions mean that the spin values are periodic along the corresponding direction, e.g., s(ix,iy,iz)=s(ix+Lx,iy,iz) in the case of (Lx,Ly,Lz) 3D cubic lattice and closed boundary condition along x− direction. Closed boundary conditions are important because they enable faster convergence towards the infinite systems. On the other hand, they are less realistic than the open boundary conditions for which there are no spins outside considered lattice.

When studying system evolution, initial conditions have to be specified that include the initial state for all spins and the external magnetic field. If suitable, terminal conditions are specified as well.

Finally, let us explain the role of averaging. Averaging is performed in order to collect more reliable statistics of system response. However, for fully deterministic variants of the model repeating the simulation under identical conditions is useless because it gives exactly the same results. In such cases, the so-called quenched averaging is performed in which the simulation is repeated using different configurations of the random field corresponding to the same choice of disorder and the same values of the remaining model parameters. Otherwise, the simulations may be repeated using the same configuration of the random field but with randomized other thermal ingredients, e.g., the different sets of spins tested for thermal flipping.

The theoretical analyses of the NEQ RFIM was focused so far on the possible critical behavior of this model. Thus, for the adiabatically driven ZT model situated on the (hyper)cubic lattices, the renormalization group (RG) analyses [28,37,84] have revealed a mean-field behavior for dimensions d≥6 and a nontrivial critical behavior for 3≤d<6, but without the final RG conclusions about the 2D case.

Having explained the main features of various versions of the NEQ RFIM, let us briefly state the main characteristics of the equilibrium model. In this RFIM version at each external (commonly homogeneous) magnetic field of interest is only the ground spin state (or states in the case of frustration when more than one state have minimal energy) at Θ=0, whereas for Θ>0, the corresponding thermal distribution of spin states is analyzed. Hence, the temporal evolution of the underlying spin system remains out of the scope together with all associated features (events, power-spectrum, waiting times, etc.). For this reason, the equilibrium version is less informative than the nonequilibrium version, especially for the analyses of response of the real-world systems. Nevertheless, the equilibrium version is very important in theoretical studies revealing many conclusions that are not attainable within the nonequilibrium version.

As mentioned in the Introduction, the focus of this review is on extensive computational techniques and simulations of the field-driven spin reversal dynamics, avalanches statistics, finite-size scaling, correlations and multifractal analysis of Barkhausen noise signals, that are developed to study the nonequilibrium criticality of the hysteresis loop. They are detailed throughout the results presented in different sections. More precisely, in the following section, we give the fundamental aspects of the simulations, which are then adapted and elaborated in each section to analyze different driving modes, dimensionality and sample shapes, athermal and thermal fluctuations and demagnetizing effects. See also the program flow in Section 6.1, which refers to the model with demagnetizing fields. In addition, a systematic comparison of the analysis of simulated data and data collected in the experiment with the nanocrystalline sample in [168] is presented; see Section 7.1. It demonstrates the advantages and disadvantages of presented numerical experiments compared to laboratory experimental data.

Even in the simplest model version, the RFIM simulations are computationally very challenging. Realization of the finite-size scaling analysis, and avoiding the finite-size effects, demand simulations of very big systems (e.g., with 109 spins) which require large amounts of computer memory and long running time even for a single run, let alone for repeated simulations with different realizations of the random magnetic field necessary for quenched averaging. Efficient algorithms are therefore a must regarding both memory (RAM and storage) space and execution time.

An essential breakthrough in simulations is achieved by the sorted-list algorithm and bit-per-spin algorithm detailed in [169] for the adiabatically driven athermal NEQ RFIM (with Jij=J>0 for nearest neighbors and zero otherwise, Jdipole=JD=0, and homogeneous H). In this case, the external magnetic field H is to be increased only after all spins become stable and that by the amount triggering the least stable spin. To find it, the easiest tactic is to check the values of the effective magnetic field hieff at all lattice sites i, find the site i0 with the least negative value of hieff, increase H by ΔH=−hi0eff, and flip the spin si0. Then, iteratively until the system becomes stable, for all spins flipped at the previous moment, flip at the next moment their nearest neighbors that became unstable. Despite being simple, the preceding algorithm, named the brute-force method, is extremely slow because its total running time scales as O(Ns2); therefore it is not used in simulations of larger systems (with, e.g., more than 105 spins) even by modern (sequential) computers.

Traversing the whole lattice in search for the least stable spin is avoided in the sorted-list algorithm. To this end, the array {hi}i=1Ns of the random field values, generated before the start of the simulation, are sorted in descending order, {hπ(1)>hπ(2)>…>hπ(Ns)}, where π:{1,…,Ns}→{1,…,Ns} is the permutation such that π(k) points to the index of the element in the original array occupying k-th position in the sorted array h∘π. Also is introduced a pointer (integer) array {Pn↑}n↑=0znn and all its elements initially set to 1. After the system becomes stable at the current value of H, each element Pn↑ of this (nondecreasing) array is updated (increased) so to point to the smallest index k in the sorted array, i.e., to the largest element in the sorted array h∘π, such that sπ(k) would flip if it had n↑ upward oriented nearest neighbors, and the external field for that n↑ set to Hnew(n↑)>H equal to Hnew(n↑)=−[(2n↑−znn)J+hπ(k)]. Thereupon, the external magnetic field is updated to the new value Hnew=min{Hnew(n↑):n↑=0,1,…,znn}, and the new avalanche nucleated by flipping the least stable spin caused by H→Hnew.

Having the total running time that scales as O(Nslog⁡Ns), the sorted-list algorithm is, so far as is known, the fastest algorithm used for simulations of the adiabatically driven athermal NEQ RFIM. However, the algorithm is memory-hungry because, besides the arrays used for storing the spins and the random magnetic field, it requires an additional integer array {πi}i=1Ns keeping track about the permutation used in sorting the original array {hi}i=1Ns, requiring for storage at least 64 bits of memory per each element.

The bit-per-spin algorithm greatly reduces memory demands because it eliminates the need for the array {hi}i=1Ns. Besides this, memory is additionally saved by storing each spin in only one bit of memory, also possible in any other algorithm regarding the storage of Ising spins. Although the algorithm’s historical name refers to bit-per-spin memory storage, the main aspect of its optimization is generating the random field values on the fly, resulting in at least 96 times reduced memory demands, but at the cost of approximately two-fold increase in the execution time. Thus, the array {hi}i=1Ns is not generated before the start of the simulation, nor the memory for its storage is reserved. Instead, provided that the system is stable at the external field Hold, a new value of the external field, Hnew is found. To this end, using the complementary error function erfc=(2/π)∫x∞exp⁡(−t2)dt, one firstly calculates the probabilities (5)P↓(n↑,H)=12erfc(H+(2n↑−znn)JR2),that at the value H of the external field a spin with n↑ up neighbors is pointing down, and consequently the probability (6)Pnone(Hold;Hnew)=∏n↑=0znn[P↓(n↑,Hnew)P↓(n↑,Hold)]Nn↑,that no spin has flipped between Hold and Hnew. After that, the equation (7)rn=Pnone(Hold;Hnew),is numerically solved for a chosen random number rn (uniformly distributed between zero and one), using in this solving the recipe from [169], for example. In the next step, the rates for spin flipping for each n↑ at the Hnew are calculated according to (8)Γ(n↑,Hnew)=Nn↑ρ(Hnr(n↑,Hnew))P↓(n↑,Hnew),where Nn↑ is the number of spins with n↑ up neighbors (taken before H is set to Hnew), and ρ(Hnr(n↑,Hnew)) stands for the Gaussian distribution ρ(h)=exp⁡(−h2/2R2)/R2 of the random field at h=Hnr(n↑,Hnew), where Hnr(n↑,Hnew)=Hnew+(2n↑−znn)J. For these rates Γ(n↑,Hnew), the total rate (9)Γ(Hnew)=∑n↑=0znnΓ(n↑,Hnew)is calculated and n↑ selected using another random number uniformly distributed between zero and Γ(Hnew). Finally, randomly searching the lattice, a down spin with n↑ up neighbors is found and flipped. This way a new avalanche is nucleated and its propagation at Hnew is followed like in the brute force method until the ongoing new avalanche dies. The whole preceding procedure (for finding Hnew and propagating the emerged avalanche) is iteratively repeated, until all spins become stable in the up state.

The bit-per-spin algorithm enabled large-scale simulations in the last decade of the XX century standing as a breakthrough in the numerical studies of the model [31,170]. Together with the sorted-list algorithm, it was later extended to be applicable for the quasistatic [147], finite-rate [130,148,171,172], and stochastic driving [149] along the entire hysteresis loop. Nevertheless, their extension was impossible for the RFIM versions modelling the systems at finite temperatures and/or systems having a more complex inter-spin interaction (e.g., nonhomogenous exchange interaction, combined ferromagnetic-antiferromagnetic layers, vacancies, interfaces, etc.). In these cases, the single run execution time is possible to be reduced by converting the brute-force method from the sequential to parallel code, which is particularly convenient for the shared-memory systems using Message Passing Interface (MPI)–a standard designed to function on parallel computing architectures. Note, however, that the sequential code provides the shortest run-time per thread, and that the run-time acceleration usually rapidly saturates with the number of involved parallel threads. This, together with the repetition of simulations with different random field configurations requested for the quenched averaging, means that the number of involved parallel threads has to be compromised to achieve the least overall execution time of the whole set of simulations. An example of the flowchart of a parallel algorithm is given in Section 6.1.

Besides sophisticated simulation algorithms, the RFIM studies also have computational challenges regarding the extraction of the relevant statistics and their subsequent analysis. As an illustrative example, let us describe the algorithm proposed in [89] for the collapsing of the suitably scaled distributions. These distributions are of the power-law type and are represented by their histograms collected in the logarithmic bins which reduces the random scattering at the side of large events. Thus, for a given set of discrete histograms, one firstly finds the union set X of all their ordinates, next for each x in X the (weighted) mean of the interpolated curves, and then the width around this mean curve, resulting in a merit function (i.e., width w). Various types of interpolation and weighted widths can be used, which is important for noisy data having different uncertainties. The simplest choice is the linear interpolation, no weights, and width w=D2/Np, where D2 is the sum of squared distances from the mean curve, and Np is the number of degrees of freedom (crudely equal to the number of points in X).