## 2 Single-Site Model for the Free Energy  
In order to find the atomic occupancies of the \(\sigma\)-phase sublattices, we minimize the Helmholtz free energy \(F\), which can be written in the following form for a given composition and temperature [14; 15; 16]:  
\[F\left(\{c_{\alpha}\}\right)=E_{tot}\left(\{c_{\alpha}\}\right)-T(S_{conf}+S_ {magn}) \tag{1}\]  
Here \(E_{tot}(\{c_{\alpha}\})\) is the total energy of the \(\sigma\)-phase, \(c_{\alpha}\) the concentration of the reference component (e.g., Cr in the Fe-Cr alloy), and \(\alpha\) is the A, B, C, D, or E site. We assume here that the alloy components are distributed randomly on each site and there is no mutual correlation between them on different sublattices. As has been demonstrated in [16], this is a valid approximation at high temperatures at least in the case of the Fe-Cr \(\sigma\)-phase. Thus, the total energy (per unit cell), \(E_{tot}(\{c_{\alpha}\})\), can be written in a simplified form as  
\[E_{tot}(\{c_{\alpha}\}) = E_{0}(c)+\sum_{\alpha=1,4(A,D)}n_{\alpha}J_{\alpha}^{(1)}c_{\alpha} \tag{2}\]  
where \(n_{\alpha}\) is the number of sites at sublattice \(\alpha\), and \(J_{\alpha}^{(1)}\) are the on-site interaction parameters, defined as  
\[J_{\alpha}^{(1)}=[\partial E_{tot}/\partial c_{\alpha}-\partial E_{tot}/ \partial c_{E}]|_{c_{alloy}=c} \tag{3}\]  
They are in fact the effective chemical potentials at 0 K, and determine the preference of the alloy components to occupy site \(\alpha\) relative to that of sublattice E. In general, the pair and higher order interaction parameters could also be taken into account. However, as we have shown in [16], the on-site interaction parameters are dominant in the total energy expansion. The summation in Equation (3) runs only over four sublattices, since the occupation at the fifth sublattice is determined by the alloy composition \(c\) and the concentrations at the other four sublattices \(c_{\alpha}\). \(E_{0}(c)\) is the total energy of a homogeneous random alloy (\(c_{\alpha}=c\)) and \(J_{\alpha}^{(1)}\) are the on-site interaction parameters. The second term in Equation (1) contains the configurational and magnetic entropy correspondingly given by:  
\[S_{conf}=-k_{B}\sum_{\alpha=1,5}n_{\alpha}\left[c_{\alpha}\ln c_{\alpha}+(1- c_{\alpha})\ln(1-c_{\alpha})\right] \tag{4}\]and  
\[S_{magn} = k_{B}\sum_{\alpha=1,5}n_{\alpha}[c_{\alpha}\ln(1+\mu_{\alpha}^{A})+ (1-c_{\alpha})\ln(1+\mu_{\alpha}^{B})] \tag{5}\]  
Here \(\mu_{\alpha}\) are the local magnetic moments of the alloy components (_A_, _B_) in the _A-B_\(\sigma\)-phase, which are found by minimizing the corresponding magnetic free energy in the first-principles calculations at a given temperature. Although the \(\sigma\)-phase forms at high temperatures in the paramagnetic state, the local magnetic moments do not disappear on the Fe atoms, as has been shown by [PERSON] _et al._[7] for the FeCr \(\sigma\)-phase, and they significantly contribute to the energetics of the \(\sigma\)-phase. Furthermore, as has already been discussed in our previous works [14, 16], longitudinal spin fluctuations should exist at high temperature. These are thermal Stoner-like many-body excitations which lead to fluctuations of the magnitude of the local magnetic moments. In order to calculate these high temperature magnetic excitations we adopt a simple model [17], which implies that longitudinal spin fluctuations (LSF) contribute to the magnetic entropy of the paramagnetic state. The LSFs induce modified spin magnetic moments on the alloy components, while for instance the Cr atoms would have zero magnetic moments in the disordered local moment (DLM) state. In present work we assume that the lattice vibrations do not play a decisive role in the distribution of the alloy components between the sublattices and therefore neglect their contribution in Equation (1).