  We analyze the spontaneous switching between the two N\'eel states of Fe
chains on Cu$_2$N/Cu(100), experimentally studied by Loth et al. [Science 335,
196 (2012)]. We show that, in the experimental low-temperature regime (T ~ 1K),
decoherence induced by substrate electrons deeply modifies the dynamics of the
chain magnetization. The Rabi oscillations associated to quantum tunneling of
the isolated chain are replaced by an irreversible decay of considerably longer
characteristic time. The computed switching rate is small, rapidly decreasing
with chain length, with a 1/T behavior and in good agreement with the
experiment. Quantum tunneling is only recovered as the switching mechanism at
extremely low temperatures, the corresponding cross-over moving rapidly to
further low temperatures as the chain length is increased.
