## 2 Model and [PERSON]'s Functions  
The magnetic semiconductors of the europium chalcogenide family, EuO, EuS, EuSe and EuTe (where EuO and EuS are ferro-, EuSe meta-, and EuTe antiferromagnetic semiconductors), are characterized by narrow 4f orbitals as degenerate levels. These materials all show a strong coupling between the electronic and the magnetic properties so that the electrical properties are greatly modified by a change of magnetic state. The ferromagnetism of UTe is itinerant, i.e., the 5f electrons are not fully localized close to the atomic core. We will consider mainly the band gap energy of magnetic semiconductors, such as EuO, EuSe(which are 4f systems) and UTe (5f systems). Therefore, we use the s-f model [2; 3], taking into account the spin-phonon and electron-phonon interactions:  
\[H=H_{m}+H_{el}+H_{m-el}. \tag{1}\]  
The Heisenberg Hamiltonian for the localized \(f\) spins, including the strong spin-phonon interaction reported in FMS [31], is given by:  
\[H_{m} = -\frac{1}{2}\sum_{i,j}I_{ij}\mathbf{S}_{i}\cdot\mathbf{S}_{j}-D_{z }\sum_{i}(S_{i}^{z})^{2}-g\mu_{B}\mathbf{h}\cdot\sum_{i}\mathbf{S}_{i} \tag{2}\] \[- \frac{1}{2}\sum_{i,j,k}\bar{F}(i,j,k)Q_{i}S_{j}^{z}S_{k}^{z}- \frac{1}{4}\sum_{i,j,r,s}\bar{R}(i,j,r,s)Q_{i}Q_{j}S_{r}^{z}S_{s}^{z}+h.c.\]  
where \(\mathbf{S}_{i}\) and its \(z\) component \(S_{i}^{z}\) are spin operators for the localized spins at site \(i\). \(J_{ij}\) stands for the spin exchange interaction between the nearest neighbors. \(D_{z}\) is the single-site anisotropy parameter of the easy-axis type. \(\mathbf{h}\) is an external magnetic field. The normal coordinate \(Q_{i}\) can be expressed in terms of phonon creation \(a^{+}\) and annihilation \(a\) operators: \(Q_{i}=(2\omega_{0i})^{-1/2}(a_{i}+a_{i}^{+})\). \(F(i,j)=\bar{F}(i,j)/[(2\omega_{0i})^{1/2}(2\omega_{0j})^{1/2}]\) and \(R(i,j,r)=\bar{R}(i,j,r)/[(2\omega_{0i})^{1/2}(2\omega_{0j})^{1/2}(2\omega_{0r })^{1/2}]\) designate the amplitudes for the coupling of phonons to the spin excitations in the first and second order, respectively. And h.c. represents the Hermitian conjugate. The magnetization \(M\) for arbitrary spin \(S\) is given by:  
\[M=\langle S^{z}\rangle=\frac{1}{N}\sum_{i}\left[(S+0.5)\coth[(S+0.5)\beta E_{mi }]-0.5\coth(0.5\beta E_{mi})\right], \tag{3}\]  
where \(E_{mi}\) is the spin wave energy, calculated from [PERSON]'s function:  
\[G_{ij}(t)=<S_{i}^{+}(t);S_{j}^{-}\gg=-i\Theta(t)\langle[S_{i}^{+}(t),S_{j}^{- }]\rangle. \tag{4}\]  
The spin-phonon interaction renormalizes the exchange interaction constant \(J\) to  
\[J_{ij}^{eff}=J_{ij}+2F^{2}/(\omega_{o}-MR). \tag{5}\]  
The Hamiltonian of the conduction band electrons \(H_{el}\), taking into account the interaction between the electron and phonon subsystems, is given by:  
\[H_{el}=\sum_{ij\sigma}t_{ij}c_{i\sigma}^{+}c_{j\sigma}+i\sum_{ijk\sigma}A(ijk) c_{i\sigma}^{+}c_{j\sigma}(a_{k}^{+}+a_{k})+h.c., \tag{6}\]  
where \(t_{ij}\) is the hopping integral, and \(c_{i\sigma}^{+}\) and \(c_{i\sigma}\) are Fermi-creation and -annihilation operators. \(A(ijk)\) is the electron-phonon interaction constant. The Eu and U chalcogenides show a strong coupling between the electronic and the magnetic properties so that the electrical properties are greatly modified by a change of magnetic state. The s-f coupling term \(H_{m-el}\) reads  
\[H_{m-el}=\sum_{i}I_{i}\mathbf{S}_{i}\mathbf{s}_{i}, \tag{7}\]  
where \(I\) is the s-f interaction constant. The spin operators \(\mathbf{s}_{i}\) of the conduction electrons at site \(i\) can be expressed as \(s_{i}^{+}=c_{i+}^{+}c_{i-}\), \(s_{i}^{z}=\left(c_{i+}^{+}c_{i+}-c_{i-}^{+}c_{i-}\right)/2\).