We consider a simple cubic lattice with a ${\mathrm{Eu}}^{2+}$ ion at each corner and spins aligned parallel or antiparallel to the $z$ axis, depending on the long-range order phase of the ground state. Elementary excited states consist of transitions ${\mathrm{Eu}}^{2+}\left(4f^7\right)$ to ${\mathrm{Eu}}^{3+}\left(4f^6\right)5d$ at one site which give rise to the $5d$ conduction electron band structure. To prevent double occupation of the $5d$ states we introduce auxiliary bosons and fermions in mean-field approximation. For small $5d$ electron density $x$ we calculate the ground-state energies for the A, B, C, and G phases as a function of the nearest-neighbor hopping matrix element $t$ and the antiferromagnetic superexchange between neighboring spins $J^{*}$. With increasing superexchange there are crossovers from phase A to C and for large $J^{*}$ to G, while the B phase is only the ground state for ferromagnetic exchange. We study the canting of the spins which may lower the energy for the A and C phases. Following a procedure by M. Yu. Kagan et al. [J. Phys.: Condens. Matter 8, 10905 (2006)]. we investigate the formation of magnetic polarons as a function of anisotropy, magnetic field, and temperature, as well as the magnetostriction. This simplified model was inspired on the recently discovered Zintl compound ${\mathrm{Eu}}_5{\mathrm{In}}_2{\mathrm{Sb}}_6$.

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• Electronic structure
• Magnetism
• Magnetotransport
• Approximation methods for many-body systems
• Materials modeling
• Mean field theory
• Slave bosons