Valence-bond crystalline order in the $s=1/2$ $J_{1}$--$J_{2}$ model on the honeycomb lattice

Using the coupled cluster method we study the phase diagram of the spin-1/2 Heisenberg antiferromagnet on a honeycomb lattice with nearest-neighbor exchange coupling $J_{1}>0$ and frustrating next-nearest-neighbor coupling $J_{2} \equiv xJ_{1}>0$. In the range $0<x<1$ we find four phases exhibiting respectively N\'{e}el, 6-spin plaquette, staggered dimer, and N\'{e}el-II orderings, with quantum critical points at $x_{c_{1}} \approx 0.207(3), x_{c_{2}} \approx 0.385(10)$, and $x_{c_{3}} \approx 0.65(5)$. The transitions at $x_{c_{1}}$ and $x_{c_{3}}$ appear to be continuous (and hence deconfined) ones, while that at $x_{c_{2}}$ appears to be a direct first-order one.