  We present a microscopic model for the anisotropic exchange interactions in
Sr$_{2}$IrO$_{4}$. A direct construction of Wannier functions from
first-principles calculations proves the $j_{\mathrm{eff}}$=1/2 character of
the spin-orbit integrated states at the Fermi level. An effective
$j_{\mathrm{eff}}$-spin Hamiltonian explains the observed weak ferromagnetism
and anisotropy of antiferromagnetically ordered magnetic state, which arise
naturally from the $j_{\mathrm{eff}}$=1/2 state with a rotation of IrO$_{6}$
octahedra. It is suggested that Sr$_{2}$IrO$_{4}$ is a unique class of
materials with effective exchange interactions in the spin-orbital Hilbert
space.

Strongly Correlated Electrons