Phase diagram of a frustrated Heisenberg antiferromagnet on the honeycomb lattice: the $J_1$--$J_2$--$J_3$ model

We use the coupled cluster method in high orders of approximation to make a comprehensive study of the ground-state (GS) phase diagram of the spin-1/2 $J_1$–$J_2$–$J_3$ model on a two-dimensional honeycomb lattice with antiferromagnetic (AFM) interactions up to third-nearest neighbors. Results are presented for the GS energy and the average local on-site magnetization. With the nearest-neighbor coupling strength $J_1\equiv 1$ we find four magnetically ordered phases in the parameter window $J_2,J_3\in [0,1]$, namely the N'{e}el (N), striped (S), and anti-N'{e}el (aN) collinear AFM phases, plus a spiral phase. The aN phase appears as a stable GS phase in the classical version of the model only for values $J_3<0$. Each of these four ordered phases shares a boundary with a disordered quantum paramagnetic (QP) phase, and at several widely separated points on the phase boundaries the QP phase has an infinite susceptibility to plaquette valence-bond crystalline order. We identify all of the phase boundaries with good precision in the parameter window studied, and we find three tricritical quantum critical points therein at: (a) $(J_2^{c_1},J_3^{c_1})=(0.51\pm 0.01,0.69\pm 0.01)$ between the N, S, and QP phases; (b) $(J_{2}^{c_2},J_{3}^{c_2})=(0.65 \pm 0.02,0.55 \pm 0.01)$betweentheS,spiral,andQPphases;and(c)$(J_{2}^{c_3},J_{3}^{c_3})=(0.69 \pm 0.01,0.12 \pm 0.01)$ between the spiral, aN, and QP phases.