# Study of Quasi-Static Magnetization with the Random-Field Ising Model
## 2 Random-Field Ising Model  
The model space is defined as a set of unit cells associated with the \(S_{\mathrm{zi}}\) spin operators. Spins denoted as \(S_{\mathrm{zi}}\) are surrounded by \(j=2^{D}\) adjacent spins \(S_{\mathrm{zi}}\). The \(S_{\mathrm{zi}}\) spin is coupled with \(S_{\mathrm{zi}}\) through exchange interactions. Spin operators take values from the set of \(S_{\mathrm{zi}}\), \(S_{\mathrm{zi}}=\{\,S_{z}^{-}\,=\,-1,\,S_{z}^{+}\,=\,+1\}\) and change the values to the opposite by spin flip. A set of periodically arranged unit cells creates either a spatial or a flat model lattice with spins placed in all sites of the lattice. The number of spins in the model space depends on the form of the unit cell, the number of space dimensions \(D\) and the lattice length \(L\) for each dimension. The unit cell is here referred to as a site in the lattice. The model space is therefore defined as the \(D\)-dimensional lattice of regular unit cells limited in each direction by its pertinent length \(L_{\mathrm{D}}\), and comprising \(N\) sites with localized spins (Equation (1)). \[N\,=\,\prod_{\mathrm{D}}(L_{\mathrm{D}}) \tag{1}\]  
Other magnetic structures require consideration of both other forms of unit cells and other numbers of localized spins coupled by exchange interactions [20; 27]. The use of regular cells instead of more complicated forms is a typical simplification of the model space. In other cases, when the structure is not a crystal lattice or has a very heterogeneous form, one can quantize the space with regular cells and estimate equivalent parameters. The fundamental Hamiltonian form \(\mathcal{H}\) of the Ising model takes into account both the exchange interactions between the nearest-neighbored spins in the lattice and the applied external field \(H\). The Hamiltonian of the RFIM (Equation (2)) imposes quenched disorder in the form of a random molecular field \(h_{\mathrm{i}}\) assigned to all sites \(i\) of the lattice [26]. \[\mathcal{H}\,=\,-J_{\mathrm{ij}}\sum\
olimits_{\mathrm{i,j}}S_{\mathrm{zi}}S_ {\mathrm{zi}}-\sum\
olimits_{\mathrm{i}}S_{\mathrm{zi}}h_{\mathrm{i}}-\sum \
olimits_{\mathrm{i}}S_{\mathrm{zi}}H \tag{2}\]  
The value of the \(h_{\mathrm{i}}\) field in Equation (2) has an impact only on triggering the spin \(S_{\mathrm{zi}}\) flip induced either by adjacent spins \(S_{\mathrm{zi}}\) or by the external field \(H\)[28]. Parallel polarization of all spins in the lattice occurs under the influence of exchange interactions in the Weiss molecular field [12; 13]. In the RFIM,the subsequent values of the \(h_{\rm i}\) field are random variables and depend on the parameters of the random distribution described by Equation (3). \[p(h_{\rm i})\ =\ \frac{1}{\sqrt{2\pi R}}\exp\!\left[-\frac{h_{\rm i}^{2}}{2R^{2}}\right] \tag{3}\]  
The set of random values \(h_{\rm i}\in\{h_{0},\,h_{1},\,\ldots\,\,h_{N\!-\!1}\}\) is generated on the basis of the Gaussian distribution function described by the relationship in Equation (3) and is additionally subject to the limitations defined by Equation (4). The mean value of the set \(h_{\rm i}\) in the space defined by \(N\) spins tends to zero (Equation (4)). Variance of random fields \(h_{\rm i}\) is a function of the disorder \(R\) at the temperature T \(\sim\)0 K.  
\[\left.\overline{h}_{\rm i}\right|_{i\to N}\to 0 \tag{4}\]  
The range of changes in the parameter \(R\) is limited by the value of critical disorder \(R_{\rm c}\)[13; 28]. The sequence of random \(h_{\rm i}\in\{h_{0},\,h_{1},\,\ldots\,\,h_{N\!-\!1}\}\) for \(R>R_{\rm c}\) may contain \(h_{i}\) elements that do not guarantee the convergence of the simulation process (non-ergodic sequence of random variables) [29]. A more comprehensive analysis of the disorder parameter in the RFIMs was carried out in works [15; 17; 18; 26; 28]. If the effective field \(h_{\rm eff}\) (Equation (5)) acting in a given site i changes the sign to the opposite, then flipping of the \(S_{\rm zi}\) spin occurs and its value is also changed to the opposite.