Phase Diagram and Quantum Order by Disorder in the Kitaev K1 − K2 Honeycomb Magnet

We show that the topological Kitaev spin liquid on the honeycomb lattice is extremely fragile against the second-neighbor Kitaev coupling K2, which has recently been shown to be the dominant perturbation away from the nearest-neighbor model in iridate Na2IrO3, and may also play a role in α-RuCl3and Li2IrO3. This coupling naturally explains the zigzag ordering (without introducing unrealistically large longer-range Heisenberg exchange terms) and the special entanglement between real and spin space observed recently in Na2IrO3. Moreover, the minimal K1- K2model that we present here holds the unique property that the classical and quantum phase diagrams and their respective order-by-disorder mechanisms are qualitatively different due to the fundamentally different symmetries of the classical and quantum counterparts.