Quantum Selection of Order in an $XXZ$ Antiferromagnet on a Kagom\'{e} Lattice

By advancing the non-linear $1/S$ expansion and the real-space perturbation theory we investigated quantum order-by-disorder selection of the ground state of the nearest-neighbor $XXZ$ antiferromagnet on the kagom\'{e} lattice. The two methods unanimously favor ${\bf q}\!=\!0$ over $\sqrt{3}\times\!\sqrt{3}$ magnetic order in a wide range of the anisotropy parameter $0\leq \Delta\!\alt\!0.72$. We demonstrated that the order selection is generated by topologically non-trivial spin-flip processes, presented a strong evidence of the rare case of quantum and thermal fluctuations favoring different ground states, proposed a tentative $S\!-\!\Delta$ phase diagram of the model, and suggested further studies.