  $\alpha$-graphyne is a two-dimensional sheet of $sp$-$sp^2$ hybridized carbon
atoms in a honeycomb lattice. While the geometrical structure is similar to
that of graphene, the hybridized triple bonds give rise to electronic structure
that is different from that of graphene. Similar to graphene, $\alpha$-graphyne
can be stacked in bilayers with two stable configurations, but the different
stackings have very different electronic structures: one is predicted to have
gapless parabolic bands and the other a tunable band gap which is attractive
for applications. In order to realize applications, it is crucial to understand
which stacking is more stable. This is difficult to model, as the stability is
a result of weak interlayer van der Waals interactions which are not well
captured by density functional theory (DFT). We have used quantum Monte Carlo
simulations that accurately include van der Waals interactions to calculate the
interlayer binding energy of bilayer graphyne and to determine its most stable
stacking mode. Our results show that interlayer bindings of $sp$- and
$sp^{2}$-bonded carbon networks are significantly underestimated in a Kohn-Sham
DFT approach, even with an exchange-correlation potential corrected to include,
in some approximation, van der Waals interactions. Finally, our quantum Monte
Carlo calculations reveal that the interlayer binding energy difference between
the two stacking modes is only 0.9(4) meV/atom. From this we conclude that the
two stable stacking modes of bilayer $\alpha$-graphyne are almost degenerate
with each other, and both will occur with about the same probability at room
temperature unless there is a synthesis path that prefers one stacking over the
other.
