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 $J_{ 3}=J_{ 2}=\kappa J_{ 1}$, in the range $0 \leq \kappa \leq 1$, which includes the point of maximum classical ($s \to \infty$) frustration, viz., the classical critical point at $\kappa_{ { \rm cl}}=\frac{ 1}{ 2}$, separating the N\'{ e}el phase for $\kappa < \kappa_{ { \rm cl}}$ 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 $s \geq \frac{ 3}{ 2}$ 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 $\kappa_{ { \rm cl}}$ [e.g., $\kappa_{ c}(\frac{ 3}{ 2}) \approx 0.53(1)$] and which approaches it monotonically as $s \to \infty$. 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 $\kappa_{ c_{ 2}} \approx 0.60$). For both the $s=\frac{ 1}{ 2}$ and $s=1$ models the transition at $\kappa_{ c_{ 2}}$ appears to be of first-order type, while that at $\kappa_{ c_{ 1}}$ appears to be continuous. However, whereas in the $s=\frac{ 1}{ 2}$ 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.
Categories: cond-mat.str-el