  We present general results for one-dimensional systems of point charges
(signed point measures) on the line with a translation invariant distribution
$\mu$ for which the variance of the total charge in an interval is uniformly
bounded (instead of increasing with the interval length). When the charges are
restricted to multiples of a common unit, and their average charge density does
not vanish, then the boundedness of the variance implies translation-symmetry
breaking --- in the sense that there exists a function of the charge
configuration that is nontrivially periodic under translations --- and hence
that $\mu$ is not ``mixing.'' Analogous results are formulated also for one
dimensional lattice systems under some constraints on the values of the charges
at the lattice sites and their averages. The general results apply to
one-dimensional Coulomb systems, and to certain spin chains, putting on common
grounds different instances of symmetry breaking encountered there.
