Density-Matrix Renormalization Group Study of Effective Spin Model for Na$_2$IrO$_3$

The Kitaev-Heisenberg honeycomb lattice model has recently been proposed to describe magnetic properties in A$_2$IrO$_3$(A=Na,Li). The model includes an isotropic Heisenberg term and strongly anisotropic Kitaev terms. The Kitaev terms give a spin-liquid ground state. With increasing the strength of Heisenberg coupling, the ground state first turns into a stripy antiferromagnetic phase and then into a Neel antiferromagnet. The x-ray and neutron scattering experiments indicate that the ground state of Na$_2$IrO$_3$ is most likely characterized by a zig-zag spin structure. However, this type of magnetic order cannot be theoretically explained by the Kitaev-Heisenberg model. It is necessary to introduce further neighbor Heisenberg couplings and/or trigonal distortion of the oxygen octahedra. We study an extended Kitaev-Heisenberg model including these additional terms, by using two-dimensional density-matrix renormalization group method. Calculating spin correlation functions, we find that the zigzag order appears in a parameter regime relevant to Na$_2$IrO$_3$.