Frustrated Heisenberg antiferromagnet on the honeycomb lattice with spin quantum number $s\ge 1$

The ground-state (GS) phase diagram of the frustrated spin-$s$ $J_1$–$J_2$–$J_3$ Heisenberg antiferromagnet on the honeycomb lattice is studied using the coupled cluster method, for spin quantum numbers $s=1,\frac{3}{2},2,\frac{5}{2}$. We study the case $J_{3}=J_{2}=\kappa J_{1}$,intherange$0 \leq \kappa \leq 1$, which includes the point of maximum classical ($s\to \mathrm{\infty }$) frustration, viz., the classical critical point at ${\kappa }_{\mathrm{c}\mathrm{l}}=\frac{1}{2}$, separating the N'{e}el phase for $\kappa <{\kappa }_{\mathrm{c}\mathrm{l}}$ and the collinear striped AFM phase for $\kappa > \kappa_{{\rm cl}}$. Results are presented for the GS energy, magnetic order parameter and plaquette valence-bond crystal (PVBC) susceptibility. For all spins $s\ge \frac{3}{2}$ we find a quantum phase diagram very similar to the classical one, with a direct first-order transition between the two collinear AFM states at a value ${\kappa }_c(s)$ which is slightly greater than ${\kappa }_{\mathrm{c}\mathrm{l}}$ [e.g., ${\kappa }_c(\frac{3}{2})\approx 0.53(1)$] and which approaches it monotonically as $s\to \mathrm{\infty }$. By contrast, for the case $s=1$ the transition is split into two such that the stable GS phases are ones with N'{e}el AFM order for $\kappa <{\kappa }_{c_1}=0.485(5)$ and with striped AFM order for $\kappa >{\kappa }_{c_2}=0.528(5)$, just as in the case $s=\frac{1}{2}$ (for which ${\kappa }_{c_1}\approx 0.47$ and $\kappa_{c_{2}} \approx 0.60$)\mathrm{.}Forboththe$s=\frac{1}{2}$and$s=1$modelsthetransitionat$\kappa_{c_{2}}$appearstobeoffirst-ordertype,whilethatat$\kappa_{c_{1}}$appearstobecontinuous\mathrm{.}However,whereasinthe$s=\frac{1}{2}$ case the intermediate phase appears to have PVBC order over the entire range ${\kappa }_{c_1}<\kappa <{\kappa }_{c_2}$, in the $s=1$ case PVBC ordering either exists only over a very small part of the region or, more likely, is absent everywhere.