  We show that the topological Kitaev spin liquid on the honeycomb lattice is
extremely fragile against the second-neighbor Kitaev coupling $K_2$, which has
recently been shown to be the dominant perturbation away from the
nearest-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 the
zigzag ordering (without introducing unrealistically large longer-range
Heisenberg exchange terms) and the special entanglement between real and spin
space 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 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.
