Ordered states in the Kitaev-Heisenberg model: From 1D chains to 2D honeycomb

We study the ground state of the 1D Kitaev-Heisenberg (KH) model using the density-matrix renormalization group and Lanczos exact diagonalization methods. We obtain a rich ground-state phase diagram as a function of the ratio between Heisenberg ( J = cos ϕ ) and Kitaev ( K = sin ϕ ) interactions. Depending on the ratio, the system exhibits four long-range ordered states: ferromagnetic- z , ferromagnetic- xy , staggered- xy , Néel- z , and two liquid states: Tomonaga-Luttinger liquid and spiral- xy . The two Kitaev points ϕ = π 2 and φ = 3 π 2 are singular. The ϕ -dependent phase diagram is similar to that for the 2D honeycomb-lattice KH model. Remarkably, all the ordered states of the honeycomb-lattice KH model can be interpreted in terms of the coupled KH chains. We also discuss the magnetic structure of the K-intercalated RuCl 3 , a potential Kitaev material, in the framework of the 1D KH model. Furthermore, we demonstrate that the low-lying excitations of the 1D KH Hamiltonian can be explained within the combination of the known six-vertex model and spin-wave theory.