Bounded Fluctuations and Translation Symmetry Breaking in One-Dimensional Particle Systems

TL;DR

This paper demonstrates that in one-dimensional charge systems with bounded charge fluctuation variance, translation symmetry is broken, leading to non-mixing states, with implications for Coulomb systems and spin chains.

Contribution

It establishes a general connection between bounded charge variance and translation symmetry breaking in one-dimensional systems, including Coulomb and spin chain models.

Findings

Bounded variance implies translation-symmetry breaking.

Existence of nontrivially periodic functions of charge configurations.

Results apply to Coulomb systems and certain spin chains.

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.