  We study the emergence of magnetism and its interplay with structural
properties in a two dimensional molecular crystal of cyclacenes, using density
functional theory (DFT). Isolated cyclacenes with an even number of fused
benzenes host two unpaired electrons in two topological protected zero modes,
at the top and bottom carbon rings that form the molecule. We show that, in the
gas phase, electron repulsion promotes an open-shell singlet with strong
intramolecular antiferromagnetic exchange. We consider a closed packing
triangular lattice crystal phase and we find a strong dependence of the band
structure and magnetic interactions on the rotation angle of the cyclacenes
with respect to the crystal lattice vectors. The orientational ground state
maximizes the intermolecular hybridization, yet local moments survive.
Intermolecular exchange is computed to be antiferromagnetic, and DFT predicts a
broken symmetry $120^\circ$ spin phase reflecting the frustration of the
intermolecular spin coupling. Thus, the cyclacene crystal realizes a bilayer of
two antiferromagnetically coupled S = 1/2 triangular lattices. Our results
provide a bottom-up route towards carbon based strongly correlated platforms in
two dimensions.
