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.
Categories: cond-mat.str-el