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.
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