  Scanning tunnelling microscopy and density functional theory studies of
manganese chains adsorbed on Cu$_2$N/Cu (100) reveal an unsuspected electronic
edge state at $\sim 1$ eV above the Fermi energy. This Tamm-like state is
strongly localised to the last Mn atom of the chain and fully spin polarised.
However, no equivalence is found for occupied states, and the electronic
structure at $\sim -1$ eV is mainly spin unpolarised due to the extended
$p$-states of the N atoms that mediate the coupling between the Mn atoms in the
chain. Odd-numbered Mn chains present an exponentially decreasing direct
coupling with distance between the two edges, leading to a vanishing
bonding/anti-bonding splitting of states while even-numbered Mn chains present
perfect decoupling of both edges due to the the antiferromagnetic ordering of
Mn chains.
