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We show that the topological Kitaev spin liquid on the honeycomb lattice isextremely fragile against the second-neighbor Kitaev coupling $K_2$, which hasrecently been shown to be the dominant perturbation away from thenearest-neighbor model in iridate Na$_2$IrO$_3$, and may also play a role in$\alpha$-RuCl$_3$ and Li$_2$IrO$_3$. This coupling naturally explains thezigzag ordering (without introducing unrealistically large longer-rangeHeisenberg exchange terms) and the special entanglement between real and spinspace observed recently in Na$_2$IrO$_3$. Moreover, the minimal $K_1$-$K_2$model that we present here holds the unique property that the classical andquantum phase diagrams and their respective order-by-disorder mechanisms arequalitatively different due to the fundamentally different symmetries of theclassical and quantum counterparts.

Phase Diagram and Quantum Order by Disorder in the Kitaev $K_1$-$K_2$ Honeycomb Magnet