Competing quantum phases in the $K_1$-$K_2$ Kitaev model on the honeycomb lattice

The Kitaev-spin model on the honeycomb lattice has attracted enormous interest in recent years as an exactly solvable 2D spin system. Research in this field has been fueled by the possibility to realize the Kitaev model in strongly spin-orbit coupled iridate compounds of the form A$_2$IrO$_3$. In these materials the bond-dependent spin anisotropies of the Kitaev model are provided by spin-orbit entangled Kramers doublets. Experimental as well as theoretical investigations indicate that second neighbor honeycomb interactions are not necessarily small, which particularly applies to the second neighbor Kitaev exchange. We study the $K_1$-$K_2$ Kitaev model on the honeycomb lattice with nearest neighbor and second neighbor Kitaev couplings $K_1$ and $K_2$, respectively. Using a pseudo-fermion functional renormalization group approach for spin systems we map out the entire phase diagram of this model, allowing both couplings to be positive and negative. Aside from Kitaev-spin liquid phases at $K_2 = 0$ we find spin chain-like phases at $K_1 = 0$ where correlations only take place along effective 1D chains. Away from these special points we identify four different types of magnetic phases with collinear orders which retain a clear 1D character in large parts of the phase diagram.