  We study the extended Kitaev-Heisenberg (EKH) quantum spin model by adding
bond-dependent off-diagonal Heisenberg term into the original KH model, which
was recently proposed to describe the honeycomb Iridates. A rigorous
mathematical mapping of spin operators reveals the intrinsic symmetry of the
model Hamiltonian. By employing an unbiased numerical entanglement
renormalization method based on tensor network ansatz, we obtain the global
phase diagram containing eight distinct quantum phases. By using the dual
mapping of spin operators, each of the individual magnetic phase in the global
phase diagram can be clearly understood. At last, we show that a valence solid
state emerges as the ground state in the quadro-critical region where multiple
magnetic phases compete most intensively.
