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Layered perovskites Sr2IrO4 and Ba2IrO4 are regarded as the key materials forunderstanding the properties of magnetic relativistic insulators, mediated bythe strong spin-orbit (SO) coupling. One of the most fundamental issues is towhich extent these properties can be described by the superexchange (SE) model,formulated in the limit of the large Coulomb repulsion. In the present work weaddress this issue by deriving the relevant models and extracting parameters ofthese models from the first-principles calculations. First, we construct theeffective Hubbard-type model for the t2g bands, by recasting the problem in thelanguage of Wannier orbitals. Then, we map the obtained electron model onto thepseudospin model by applying the theory of SE interactions. We discuss themicroscopic origin of anisotropic SE interactions, inherent to the compassHeisenberg model, and the appearance of the antisymmetric Dzyaloshinskii-Moriyaterm, associated with the additional rotation of the IrO6 octahedra in Sr2IrO4.In order to evaluate the Neel temperature (TN), we employ the non-linear sigmamodel. While for Sr2IrO4 our value of TN agrees with the experimental one, forBa2IrO4 it is overestimated by a factor two. We argue that this discrepancy isrelated to limitations of the SE model: while for more localized t2g states inSr2IrO4 it works reasonably well, the higher-order terms, beyond the SE model,play a more important role in the more "itinerant" Ba2IrO4, giving rise to thenew type of isotropic and anisotropic exchange interactions. This conclusion issupported by unrestricted Hartree-Fock calculations for the same electronmodel, where in the case of Ba2IrO4, already on the mean-field level, we wereable to reproduce the experimentally observed magnetic ground state, while forSr2IrO4 the main results are essentially the same as in the SE model.

Validity and limitations of the superexchange model for the magnetic properties of Sr2IrO4 and Ba2IrO4 mediated by the strong spin-orbit coupling