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Magnetic and Electronic Properties of Spin-Orbit Coupled Dirac Electrons on a $(001)$ Thin Film of Double Perovskite Sr$_2$FeMoO$_6$

We present an interacting model for the electronic and magnetic behavior of a strained $(001)$ atomic layer of Sr$_2$FeMoO$_6,$ which shows room-temperature ferrimagnetism and magnetoresistance with potential spintronics application in the bulk. We find that the strong spin-orbit coupling in the molybdenum 4$d$ shell gives rise to a robust ferrimagnetic state with an emergent spin-polarized electronic structure consisting of flat bands and four massive or massless Dirac dispersions. Based on the spin-wave theory, we demonstrate that the magnetic order remains intact for a wide range of doping, leading to the possibility of exploring flat band physics, such as Wigner crystallization in electron-doped Sr$_{2-x}$La$_{x}$FeMoO$_6.$