This will be two part talk regarding two projects connected with correlated electronic systems with possible applications to interesting oxides. In this first section of this two part talk I will discuss a numerical study of the Kugel-Khomskii spin-orbital chain from the point of view of its spin-orbital entanglement spectrum [PRB 86, 224422 (2012)]. This is a model Hamiltonian with spin-orbital degeneracy and may be relevant to some oxides such as Na2Ti2Sb2O and NaV2O5. Features its entanglement spectrum are compared with expectations from field theory and symmetry considerations.
In the second part, I will report recent results on a novel interacting topological phase [cond-mat/1307.5566] derived from strong spin-orbit coupling (SOC) at intermediate strength Hubbard interactions. The microscopic Hubbard model considered is inspired by recent interest in heavy transition metal oxides with strong spin-orbit coupling such as the Irridates. The slave-spin decomposition programme [PRB 81 155118 (2010), PRB 86 045128 (2012)] is applied and reveals at low-energy, topological field theories describing topologically ordered ground states and fractionalised quasiparticles.