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The frustrated spin-$\frac{1}{2}$ $J_{1}$--$J_{2}$ isotropic $XY$ model on the honeycomb lattice

We study the zero-temperature ground-state (GS) phase diagram of a spin-half $J_{1}$--$J_{2}$ $XY$ model on the honeycomb lattice with nearest-neighbor exchange coupling $J_{1}>0$ and frustrating next-nearest-neighbor exchange coupling $J_{2} \equiv κJ_{1}>0$, where both bonds are of the isotropic $XY$ type, using the coupled cluster method. Results are presented for the GS energy per spin, magnetic order parameter, and staggered dimer valence-bond crystalline (SDVBC) susceptibility, for values of the frustration parameter in the range $0 \leq κ\leq 1$. In this range we find phases exhibiting, respectively, Néel $xy$ planar [N(p)], Néel $z$-aligned [N($z$)], SDVBC, and Néel-II $xy$ planar [N-II(p)] orderings which break the lattice rotational symmetry. The N(p) state, which is stable for the classical version of the model in the range $0 \leq κ\leq \frac{1}{6}$, is found to form the GS phase out to a first quantum critical point at $κ_{c_{1}} = 0.216(5)$, beyond which the stable GS phase has N($z$) order over the range $κ_{c_{1}} < κ< κ_{c_{2}}=0.355(5)$. For values $κ> κ_{c_{2}}$ we find a strong competition to form the GS phase between states with N-II(p) and SDVBC forms of order. Our best estimate, however, is that the stable GS phase over the range $κ_{c_{2}} < κ< κ_{c_{3}} \approx 0.52(3)$ is a mixed state with both SDVBC and N-II(p) forms of order; and for values $κ> κ_{c_{3}}$ is the N-II(p) state, which is stable at the classical level only at the highly degenerate point $κ=\frac{1}{2}$. Over the range $0 \leq κ\leq 1$ we find no evidence for any of the spiral phases that are present classically for all values $κ> \frac{1}{6}$, nor for any quantum spin-liquid state.