  The zero-temperature quantum phase diagram of the spin-$\frac{1}{2}$
$J_{1}$--$J_{2}$--$J_{1}^{\perp}$ model on an $AA$-stacked bilayer honeycomb
lattice is investigated using the coupled cluster method (CCM). The model
comprises two monolayers in each of which the spins, residing on
honeycomb-lattice sites, interact via both nearest-neighbor (NN) and
frustrating next-nearest-neighbor isotropic antiferromagnetic (AFM) Heisenberg
exchange iteractions, with respective strengths $J_{1} > 0$ and $J_{2} \equiv
\kappa J_{1}>0$. The two layers are coupled via a comparable Heisenberg
exchange interaction between NN interlayer pairs, with a strength
$J_{1}^{\perp} \equiv \delta J_{1}$. The complete phase boundaries of two
quasiclassical collinear AFM phases, namely the N\'{e}el and N\'{e}el-II
phases, are calculated in the $\kappa \delta$ half-plane with $\kappa > 0$.
Whereas on each monolayer in the N\'{e}el state all NN pairs of spins are
antiparallel, in the N\'{e}el-II state NN pairs of spins on zigzag chains along
one of the three equivalent honeycomb-lattice directions are antiparallel,
while NN interchain spins are parallel. We calculate directly in the
thermodynamic (infinite-lattice) limit both the magnetic order parameter $M$
and the excitation energy $\Delta$ from the $s^{z}_{T}=0$ ground state to the
lowest-lying $|s^{z}_{T}|=1$ excited state (where $s^{z}_{T}$ is the total $z$
component of spin for the system as a whole, and where the collinear ordering
lies along the $z$ direction) for both quasiclassical states used (separately)
as the CCM model state, on top of which the multispin quantum correlations are
then calculated to high orders ($n \leq 10$) in a systematic series of
approximations involving $n$-spin clusters. The sole approximation made is then
to extrapolate the sequences of $n$th-order results for $M$ and $\Delta$ to the
exact limit, $n \to \infty$.
