  Strong electron-electron interaction in ultraflat edge states can be
responsible for correlated phases of matter, such as magnetism, charge density
wave or superconductivity. Here we consider the diamond(111) surface that,
after Pandey reconstruction, presents zig-zag carbon chains, generating a flat
surface band. By performing full structural optimization with hybrid
functionals and neglecting spin polarization, we find that a substantial
dimerization ($0.090$ {\AA} / $0.076$ {\AA} bond disproportionation in the
PBE0/HSE06) occurs on the chains; a structural effect absent in calculations
based on the LDA/GGA functionals. This dimerization is the primary mechanism
for the opening of an insulating gap in the absence of spin polarization. The
single-particle direct gap is $1.7$ eV ($1.0$ eV) in the PBE0 (HSE06),
comparable with the experimental optical gap of $1.47$ eV, and on the
larger(smaller) side of the estimated experimental single particle gap window
of 1.57-1.87 eV, after inclusion of excitonic effects. However, by including
spin polarization in the calculation, we find that the exchange interaction
stabilizes a different ground state, undimerized, with no net magnetization and
ferrimagnetic along the Pandey $\pi$-chains with magnetic moments as large as
$0.2-0.3~\mu_B$ in the PBE0. The direct single-particle band gap in the equal
spin-channel is approximately $2.2$ eV ($1.5$ eV) with the PBE0 (HSE06)
functional. Our work is relevant for systems with flat bands in general and
wherever the interplay between structural, electronic and magnetic degrees of
freedom is crucial, as in twisted bilayer graphene, IVB atoms on IVB(111)
surfaces such as Pb/Si(111) or molecular crystals.
