An investigation of the quantum $J_1-J_2-J_3$ model on the honeycomb lattice

We have investigated the quantum $J_1-J_2-J_3$ model on the honeycomb lattice with exact diagonalizations and linear spin-wave calculations for selected values of $J_{2}/J_{1}$, $J_{3}/J_{1}$ and antiferromagnetic ($J_{1}>0$) or ferromagnetic ($J_{1}<0$) nearest neighbor interactions. We found a variety of quantum effects: "order by disorder" selection of a N{\'e}el ordered ground-state, good candidates for non-classical ground-states with dimer long range order or spin-liquid like. The purely antiferromagnetic Heisenberg model is confirmed to be N{\'e}el ordered. Comparing these results with those observed on the square and triangular lattices, we enumerate some conjectures on the nature of the quantum phases in the isotropic models.