Spin, orbital and topological order in models of strongly correlated electrons

Different types of order are discussed in the context of strongly correlated transition metal oxides, involving pure compounds and $3d^{3}-4d^{4}$ and $3d^{2}-4d^{4}$ hybrids. Apart from standard, long-range spin and orbital orders we observe also exotic non-colinear spin patterns. Such patters can arise in presence of atomic spin-orbit coupling, which is a typical case, or due to spin-orbital entanglement at the bonds in its absence, being much less trivial. Within a special interacting one-dimensional spin-orbital model it is also possible to find a rigorous topological magnetic order in a gapless phase that goes beyond any classification tables of topological states of matter. This is an exotic example of a strongly correlated topological state. Finally, in the less correlated limit of $4d^{4}$ oxides, when orbital selective Mott localization can occur it is possible to stabilize by a $3d^{3}$ doping one-dimensional zigzag antiferromagnetic phases. Such phases have exhibit nonsymmorphic spatial symmetries that can lead to various topological phenomena, like single and mutliple Dirac points that can turn into nodal rings or multiple topological charges protecting single Dirac points. Finally, by creating a one-dimensional $3d^{2}-4d^{4}$ hybrid system that involves orbital pairing terms, it is possible to obtain an insulating spin-orbital model where the orbital part after fermionization maps to a non-uniform Kitaev model. Such model is proved to have topological phases in a wide parameter range even in the case of completely disordered $3d^{2}$ impurities. What more, it exhibits hidden Lorentz-like symmetries of the topological phase, that live in the parameters space of the model.