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 $J_2\equiv \kappa J_1>0$, 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 $0\le \kappa \le 1$. In this range we find phases exhibiting, respectively, N'{e}el $xy$ planar [N(p)], N'{e}el $z$-aligned [N($z$)], SDVBC, and N'{e}el-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 $0\le \kappa \le \frac{1}{6}$, is found to form the GS phase out to a first quantum critical point at $\kappa_{c_{1}} = 0.216(5)$,beyondwhichthestableGSphasehasN($z$)orderovertherange$\kappa_{c_{1}} < \kappa < \kappa_{c_{2}}=0.355(5)$\mathrm{.}Forvalues$\kappa > \kappa_{c_{2}}$ 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 $\kappa_{c_{2}} < \kappa < \kappa_{c_{3}} \approx 0.52(3)$ is a mixed state with both SDVBC and N-II(p) forms of order; and for values $\kappa >{\kappa }_{c_3}$ is the N-II(p) state, which is stable at the classical level only at the highly degenerate point $\kappa =\frac{1}{2}$. Over the range $0\le \kappa \le 1$ we find no evidence for any of the spiral phases that are present classically for all values $\kappa >\frac{1}{6}$, nor for any quantum spin-liquid state.