M. Aizenman, S. Goldstein and J.L. Lebowitz
Bounded Fluctuations and Translation Symmetry Breaking in One-Dimensional Particle Systems
(59K, Latex (24pp))
ABSTRACT. 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.