Theoretical studies of local moment magnetism with strong spin-orbit coupling
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We study the magnetic and lattice contributions to the thermal conductivity of electrically insulating strongly spin-orbit coupled magnetically ordered phases on a two-dimensional honeycomb lattice using the Kitaev-Heisenberg model. Depending on model parameters, such as the relative strength of the spin-orbit induced anisotropic coupling, a number of magnetically ordered phases are possible. We study two distinct regimes of thermal transport depending on whether the characteristic energy of the phonons or the magnons dominates, and focus on two different relaxation mechanisms, boundary scattering and magnon-phonon scattering. For spatially anisotropic magnetic phases, the thermal conductivity tensor can be highly anisotropic when the magnetic energy scale dominates, since the magnetic degrees of freedom dominate the thermal transport for temperatures well below the magnetic transition temperature. In the opposite limit in which the phonon energy scale dominates, the thermal conductivity will be nearly isotropic, reflecting the isotropic (at low temperatures) phonon dispersion assumed for the honeycomb lattice. We further discuss the extent to which thermal transport properties are influenced by strong spin-orbit induced anisotropic coupling in the local moment regime of insulating magnetic phases. The developed methodology can be applied to any 2D magnon-phonon system, and more importantly to systems where an analytical Bogoliubov transformation cannot be found and magnon bands are not necessarily isotropic. Furthermore, using exact diagonalization, we study the spin-orbit coupling and interaction- induced mixing between t[subscript 2g] and e[subscript g] d-orbital states in a cubic crystalline environment, as commonly occurs in transition metal oxides. We make a direct comparison with the widely used t[subscript 2g] only or e[subscript g] only model, depending on electronic filling. We consider all electron fillings of the d-shell and compute the total magnetic moment, the spin, the occupancy of each orbital, and the effective spin-orbit coupling strength (renormalized through interaction effects) in terms of the bare interaction parameters, spin-orbit coupling, and crystal field splitting, focusing on the parameter ranges relevant to 4d and 5d transition metal oxides. In various limits we provide perturbative results consistent with our numerical calculations. We find that the t[subscript 2g]-e[subscript g] mixing can be large, with up to 20% occupation of orbitals that are nominally ``empty", which has experimental implications for the interpretation of the branching ratio in experiments, and can impact the effective local moment Hamiltonian used to study magnetic phases and magnetic excitations in transition metal oxides. Our results can aid the theoretical interpretation of experiments on these materials, which often fall in a regime of intermediate coupling with respect to electron-electron interactions.