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

The two-dimensional one-component plasma, i.e. the system of pointlike 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 Γ = 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 Γ = 2γ (γ = 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 γ. Analytic results are supported by the exact solution at γ = 1 (Γ = 2) and by exact finite-size calculations at γ = 2, 3.