Translation Symmetry Breaking in the One-Component Plasma on the Cylinder

TL;DR

This paper investigates the crystalline ordering and symmetry breaking of a two-dimensional one-component plasma on a cylindrical surface, extending exact solutions to a broader range of coupling constants and analyzing the resulting density profiles.

Contribution

It generalizes the exact solution of the plasma's density profile for multiple coupling constants using a fermionic formalism, revealing a hierarchy of Gaussian contributions.

Findings

Density profiles form a hierarchy of Gaussians with different amplitudes.

Long-range order persists for various coupling constants.

Analytic and finite-size calculations support the theoretical model.

Abstract

The two-dimensional one-component plasma, i.e. the system of point-like charged particles embedded in a homogeneous neutralizing background, is studied on the surface of a cylinder of finite circumference, or equivalently in a semiperiodic strip of finite width. The model has been solved exactly by Choquard et al. at the free-fermion coupling $\Gamma=2$: in the thermodynamic limit of an infinitely long strip, the particle density turns out to be a nonconstant periodic function in space and the system exhibits long-range order of the Wigner-crystal type. The aim of this paper is to describe, qualitatively as well as quantitatively, the crystalline state for a larger set of couplings $\Gamma=2\gamma$ ($\gamma=1,2,...$ a positive integer) when the plasma is mappable onto a one-dimensional fermionic theory. The fermionic formalism, supplemented by some periodicity assumptions, reveals that the density profile results from a hierarchy of Gaussians with a uniform variance but with different amplitudes. The number and spatial positions of these Gaussians within an elementary cell depend on the particular value of $\gamma$. Analytic results are supported by the exact solution at $\gamma=1$ ($\Gamma=2$) and by exact finite-size calculations at $\gamma=2,3$.