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Correlation strength, orbital-selective incoherence, and local moments formation in the magnetic MAX-phase Mn$_2$GaC

We perform a theoretical study of the electronic structure and magnetic properties of the prototypical magnetic MAX-phase Mn$_2$GaC with the main focus given to the origin of magnetic interactions in this system. Using the density functional theory+dynamical mean-field theory (DFT+DMFT) method we explore the effects of electron-electron interactions and magnetic correlations on the electronic properties, magnetic state, and spectral weight coherence of paramagnetic and magnetically-ordered phases of Mn$_2$GaC. We also benchmark the DFT-based disordered local moment approach for this system by comparing the obtained electronic and magnetic properties with that of the DFT+DMFT method. Our results reveal a complex magnetic behavior characterized by a near degeneracy of the ferro- and antiferromagnetic configurations of Mn$_2$GaC, implying a high sensitivity of its magnetic state to fine details of the crystal structure and unit-cell volume, consistent with experimental observations. We observe robust local-moment behavior and orbital-selective incoherence of the spectral properties of Mn$_2$GaC, implying the importance of orbital-dependent localization of the Mn $3d$ states. We find that Mn$_2$GaC can be described in terms of local magnetic moments, which may be modeled by DFT with disordered local moments. However, the magnetic properties are dictated by the proximity to the regime of formation of local magnetic moments, in which the localization is in fact driven by the Hund's exchange interaction, and not the Coulomb interaction.