  Recent experimental data indicates that the fine structure constant alpha may
be varying on cosmological time scales. We consider the possibility that such a
variation could be induced by a second order phase transition which occurs at
late times (z ~ 1 - 3) and involves a change in the vacuum expectation value
(vev) of a scalar with milli-eV mass. Such light scalars are natural in
supersymmetric theories with low SUSY breaking scale. If the vev of this scalar
contributes to masses of electrically charged fields, the low-energy value of
alpha changes during the phase transition. The observational predictions of
this scenario include isotope-dependent deviations from Newtonian gravity at
sub-millimeter distances, and (if the phase transition is a sharp event on
cosmological time scales) the presence of a well-defined step-like feature in
the alpha(z) plot. The relation between the fractional changes in alpha and the
QCD confinement scale is highly model dependent, and even in grand unified
theories the change in alpha does not need to be accompanied by a large shift
in nucleon masses.
