  The ground states of the one-dimensional Falicov-Kimball model are
investigated in the small-coupling limit, using nearly degenerate perturbation
theory. For rational electron and ion densities, respectively equal to
$\frac{p}{q}$, $\frac{p_i}{q}$, with $p$ relatively prime to $q$ and
$\frac{p_i}{q}$ close enough to $\frac{1}{2}$, we find that in the ground state
the ion configuration has period $q$. The situation is analogous to the Peierls
instability where the usual arguments predict a period-$q$ state that produces
a gap at the Fermi level and is insulating. However for $\frac{p_i}{q}$ far
enough from $\frac{1}{2}$, this phase becomes unstable against phase
separation. The ground state is a mixture of a period-$q$ ionic configuration
and an empty (or full) configuration, where both configurations have the same
electron density to leading order. Combining these new results with those
previously obtained for strong coupling, it follows that a phase transition
occurs in the ground state, as a function of the coupling, for ion densities
far enough from $\frac{1}{2}$.
