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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 Néel (N), striped (S), and anti-Néel (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) $(J_{2}^{c_2},J_{3}^{c_2})=(0.65 \pm 0.02,0.55 \pm 0.01)$ between the S, spiral, and QP phases; 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.