There is a very nice and helpful review article
Correlated quantum phenomena in the strong spin-orbit regime
William Witczak-Krempa, Gang Chen, Yong Baek Kim, Leon Balents
Just a few things I learnt from quickly skimming it.
There are many outstanding and basic questions concerning the phase diagram of even the simplest possible two-orbital Hubbard model with spin-orbit coupling. There a many possible new phases waiting to be discovered [both experimentally and theoretically] or to be shown to not actually exist because their theoretical proposal is based on uncontrolled approximations. The figure below is a possible schematic phase diagram.
But the spin-orbit coupling can be quite strong in 5d transition metals. Iridates (iridium oxides) may be model compounds to realise this physics.
Table I provides a nice summary of the properties of the plethora of different new phases that have been proposed including axion insulator, Weyl semi-metal, fractional Chern insulator, ...
Na2IrO3 was originally proposed to be a realisation of the Heisenberg-Kitaev model and thus to have a possible spin liquid ground state. However, neutron scattering shows it has an unanticipated magnetic ground state: "a zig-zag state with four-sublattice structure." This has led to new proposals as to the relevant effective spin Hamiltonian.
The review has only limited discussion of the role of Hunds rule.
It is repeatedly stated that spin-orbit coupling leads to entanglement of spin and orbital degrees of freedom. But the exact nature of this quantum entanglement is not clearly stated or calculated. For the case of entanglement arising via Hund's rule (not spin-orbit coupling) this is nicely discussed by Oles here.
The key thing that the spin-orbital coupling/entanglement does is remove (or at reduce) the coupling of the orbital degeneracies to the lattice which normally produces the Jahn-Teller effect and orbital ordering.
Sr2IrO4 is an approximate homolog of the parent material of the high-Tc cuprates, La2CuO4. It is a Jeff = 1/2 antiferromagnetic insulator and has an exchange constant J ∼1000 K comparable to that of the cuprates. This has led to a strong push to dope the material in the search of cuprate-type physics [high-Tc superconductivity, pseudogap, strange metal]. This has not occurred yet.
Both Sr2IrO4 and Sr3Ir2O7 have illustrated the power of the rapidly evolving technique of Resonant Inelastic X-ray Scattering (RIXS). It seems to be particularly suited for 5d compounds, and has been used to map out the full spin wave dispersion in both compounds.
I thank Tony Wright for bringing the review to my attention.