Quantum Matter: Topology and Correlations driving new materials, phases, and phenomena

I review the classification of quantum matter as a function of Coulomb interactions U and spin-orbit coupling. in units of the bandwidth. The region where all three of these energy scales are comparable, as in the 4d and 5d transition metal oxides and dichalcogenides is the most promising for discovering new materials and new phenomena. I discuss what it means to extract a model from a material and the hierarchy of computational methods for single and multi-orbital systems. As specific examples I describe half metallic high-T-c ferromagnetism in the double perovskite Sr2FeMoO6 in which the kinetic energy of Mo (d(1)) electrons drives ferromagnetic exchange between the Fe (d(5)) core spins. I will also discuss predictions for materials with a d(4) electronic configuration with strong spin-orbit coupling and show that there is a large parameter space where instead of a J=0 band insulator one surprisingly finds ground states with orbitally entangled ferromagnetic states. I will conclude with a broad outlook for new discoveries in this arena.