Ground-state phases of the frustrated spin-1/2 $J_1$--$J_2$--$J_3$ Heisenberg ferromagnet ($J_1<0$) on the honeycomb lattice with $J_3=J_2>0$

We study the ground-state (gs) properties of the frustrated spin-1/2 $J_1$–$J_2$–$J_3$ Heisenberg model on a honeycomb lattice with ferromagnetic (FM) nearest-neighbor ($J_1=-1$) exchange and frustrating antiferromagnetic (AFM) next-nearest-neighbor ($J_2>0$) and next-next-nearest-neighbor ($J_3>0$) exchanges, for the case $J_3=J_2$. We use the coupled cluster method in high orders of approximation, complemented by the exact diagonalization of a lattice with 32 sites, and calculate the gs energy, magnetic order parameter, and spin-spin correlation functions. We find a quantum phase transition between regions characterized by FM order and a form of AFM (“striped”) collinear order at $J_2^c\approx 0.1095\pm 0.0005$. We compare results for the FM case (with $J_1=-1$) to previous results for the corresponding AFM case (with $J_1=+1$). While the magnetic order parameters behave similarly for the FM and the AFM models for large values of the frustration parameter $J_2$, there are considerable differences between them for $J_2\mathrm{/}\mathrm{\mid }{J}_1\mathrm{\mid }\lesssim 0.6$. For example, the quasiclassical collinear magnetic long-range order for the AFM model (with $J_1=+1$) breaks down at $J_2^{c_2}\approx 0.60$, whereas the “equivalent” point for the FM model (with $J_1=-1$) occurs at $J_2^c\approx 0.11$. Unlike in the AFM model (with $J_1=+1$), where a plaquette valence-bond crystal phase intrudes between the two corresponding quasiclassical AFM phases (with N'eel and striped order) for $J_2^{c_1}<J_2<J_2^{c_2}$, with $J^{c_{1}}_{2} \approx 0.47$, we find no clear indications in the FM model for an intermediate magnetically disordered phase between the phases exhibiting FM and striped order. Instead, the evidence points strongly to a direct first-order transition between the two ordered phases of the FM model.