Frustrated spin-$\frac{1}{2}$ Heisenberg magnet on an $AA$-stacked honeycomb bilayer: High-order study of the collinear magnetic phases of the $J_1$--$J_2$--$J_1^{\perp }$ model

The zero-temperature phase diagram of the frustrated spin-$\frac{1}{2}$ $J_1$–$J_2$–$J_1^{\perp }$ Heisenberg magnet on an $AA$-stacked honeycomb bilayer lattice is studied using the coupled cluster method implemented to very high orders. On each monolayer the spins interact via nearest-neighbor (NN) and frustrating next-nearest-neighbor isotropic antiferromagnetic Heisenberg interactions with respective strength parameters $J_1>0$ and $J_2\equiv \kappa J_1>0$. The two layers are coupled such that NN interlayer pairs of spins also interact via a similar isotropic Heisenberg interaction of strength $J_1^{\perp }\equiv \delta J_1$, which may be of either sign. In particular, we locate with high accuracy the complete phase boundaries in the $\kappa$-$\delta$ half-plane with $\kappa >0$ of the two quasiclassical collinear antiferromagnetic phases with N'{e}el or N'{e}el-II magnetic order in each monolayer, and the interlayer NN pairs of spins either aligned (for $\delta <0$) or anti-aligned (for $\delta >0$) to one another. Compared to the two-sublattice N'{e}el order, in which all NN intralayer pairs of spins are antiparallel to one another, the four-sublattice N'{e}el-II order is characterized by NN intralayer pairs of spins on the honeycomb lattice being antiparallel to one another along zigzag (or sawtooth) chains in a specified direction from among the three equivalent honeycomb-lattice directions, and parallel to one another for the corresponding interchain pairs.