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The nature of the effective spin Hamiltonian and magnetic order in the honeycomb iridates is explored by considering a trigonal crystal field effect and spin-orbit (SO) coupling. Starting from a Hubbard model, an effective spin Hamiltonian is derived in terms of an emergent pseudo-spin-1/2 moment in the limit of large trigonal distortions and SO coupling. The present pseudo-spins arise from a spin-orbital locking and are different from the jeff = 1/2 moments that are obtained when the SO coupling dominates and trigonal distortions are neglected. The resulting spin Hamiltonian is anisotropic and frustrated by further neighbour interactions. Mean-field theory suggests a ground state with foursublattice zigzag magnetic order in a parameter regime that can be relevant to the honeycomb iridate compound Na2IrO3, where a similar magnetic ground state has recently been observed. Various properties of the phase, the spin-wave spectrum and experimental consequences are discussed. The present approach contrasts with the recent proposals to understand iridate compounds starting from the strong SO coupling limit and neglecting non-cubic lattice distortions. [ABSTRACT FROM AUTHOR]

Motivated by the recent discovery of a spin-liquid phase for the Hubbard model on the honeycomb lattice at half-filling (Meng et al 2010 Nature88 487), we apply both perturbative and non-perturbative techniques to derive effective spin Hamiltonians describing the low-energy physics of the Mott-insulating phase of the system. Exact diagonalizations of the so-derived models on small clusters are performed, in order to assess the quality of the effective low-energy theory in the spin-liquid regime. We show that six-spin interactions on the elementary loop of the honeycomb lattice are the dominant sub-leading effective couplings. A minimal spin model is shown to reproduce most of the energetic properties of the Hubbard model on the honeycomb lattice in its spin-liquid phase. Surprisingly, a more elaborate effective low-energy spin model obtained by a systematic graph expansion rather disagrees beyond a certain point with the numerical results for the Hubbard model at intermediate couplings. [ABSTRACT FROM AUTHOR]