Phase diagram and quantum order by disorder in the Kitaev $K_1$-$K_2$ honeycomb magnet*

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 been recently identified as the dominant perturbation away from the nearest neighbor model in iridate Na$_2$IrO$_3$, and may also play a role in $\alpha$-RuCl$_3$. This coupling explains naturally the zig-zag ordering and the special entanglement between real and spin space observed recently in Na$_2$IrO$_3$. The minimal $K_1$-$K_2$ model that we present here holds in addition the unique property that the classical and quantum phase diagrams and their respective order-by-disorder mechanisms are qualitatively different due to their fundamentally different symmetry structure.