  An effective low-energy Hamiltonian of itinerant electrons for iridium oxide
Na2IrO3 is derived by an ab initio downfolding scheme. The model is then
reduced to an effective spin model on a honeycomb lattice by the strong
coupling expansion. Here we show that the ab initio model contains spin-spin
anisotropic exchange terms in addition to the extensively studied Kitaev and
Heisenberg exchange interactions, and allows to describe the experimentally
observed zigzag magnetic order, interpreted as the state stabilized by the
antiferromagnetic coupling of the ferromagnetic chains. We clarify possible
routes to realize quantum spin liquids from existing Na2IrO3.
