The development of sensible microscopic models is essential to elucidate the normal-state and superconducting properties of the iron-based superconductors. Because these materials are mostly metallic, a good starting point is an effective low-energy model that captures the electronic states near the Fermi level and their interactions. However, in contrast to cuprates, iron-based high-T c compounds are multi-orbital systems with Hubbard and Hund interactions, resulting in a rather involved 10-orbital lattice model. Here we review different minimal models that have been proposed to unveil the universal features of these systems. We first review minimal models defined solely in the orbital basis, which focus on a particular subspace of orbitals, or solely in the band basis, which rely only on the geometry of the Fermi surface. The former, while providing important qualitative insight into the role of the orbital degrees of freedom, do not distinguish between high-energy and low-energy sectors and, for this reason, generally do not go beyond mean-field. The latter allow one to go beyond mean-field and investigate the interplay between superconducting and magnetic orders as well as Ising-nematic order. However, they cannot capture orbital-dependent features like spontaneous orbital order. We then review recent proposals for a minimal model that operates in the band basis but fully incorporates the orbital composition and symmetries of the low-energy excitations. We discuss the results of the renormalization group study of such a model, particularly of the interplay between superconductivity, magnetism, and spontaneous orbital order, and compare theoretical predictions with experiments on iron pnictides and chalcogenides. We also discuss the impact of the glide-plane symmetry on the low-energy models, highlighting the key role played by the spin-orbit coupling.
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