  Electronic structure calculations obtained with an approach with density
functional theory with an enhanced local Coulomb interaction, DFT+$U$, are
presented for the relativistic magnetic insulator Sr$_{2}$IrO$_{4}$. The
results are in accordance with experiments with a band gap and a small moment
anti-ferromagnetic ground state. This solution is thereafter thoroughly
analyzed in terms of Landau theory where it is found that the ordered magnetic
moments only form a secondary order parameter while the primary order parameter
is a higher order magnetic multipole of rank five. It is further observed that
the electronic structure in the presence of this order parameter is related to
the earlier proposed $j_\mathrm{eff}=1/2$ model, but in contrast to that model
the present picture can naturally explain the small magnitude of the ordered
magnetic moments.
