Is the Two-Dimensional One-Component Plasma Exactly Solvable?

The model under consideration is the two-dimensional (2D) one-component plasma of pointlike charged particles in a uniform neutralizing background, interacting through the logarithmic Coulomb interaction. Classical equilibrium statistical mechanics is studied by non-traditional means. The question of the potential integrability (exact solvability) of the plasma is investigated, first at arbitrary coupling constant \Gamma via an equivalent 2D Euclidean-field theory, and then at the specific values of \Gamma=2*integer via an equivalent 1D fermionic model. The answer to the question in the title is that there is strong evidence for the model being not exactly solvable at arbitrary \Gamma but becoming exactly solvable at \Gamma=2*integer. As a by-product of the developed formalism, the gauge invariance of the plasma is proven at the free-fermion point \Gamma=2; the related mathematical peculiarity is the exact inversion of a class of infinite-dimensional matrices.Comment: 26 page

Counter-ions at Charged Walls: Two Dimensional Systems

We study equilibrium statistical mechanics of classical point counter-ions, formulated on 2D Euclidean space with logarithmic Coulomb interactions (infinite number of particles) or on the cylinder surface (finite particle numbers), in the vicinity of a single uniformly charged line (one single double-layer), or between two such lines (interacting double-layers). The weak-coupling Poisson-Boltzmann theory, which applies when the coupling constant Gamma is small, is briefly recapitulated (the coupling constant is defined as Gamma = beta e^2 where beta is the inverse temperature, and e the counter-ion charge). The opposite strong-coupling limit (Gamma -> infinity) is treated by using a recent method based on an exact expansion around the ground-state Wigner crystal of counter-ions. The weak- and strong-coupling theories are compared at intermediary values of the coupling constant Gamma=2 gamma (gamma=1,2,3), to exact results derived within a 1D lattice representation of 2D Coulomb systems in terms of anti-commuting field variables. The models (density profile, pressure) are solved exactly for any particles numbers N at Gamma=2 and up to relatively large finite N at Gamma=4 and 6. For the one-line geometry, the decay of the density profile at asymptotic distance from the line undergoes a fundamental change with respect to the mean-field behavior at Gamma=6. The like-charge attraction regime, possible in the strong coupling limit but precluded at mean-field level, survives for Gamma=4 and 6, but disappears at Gamma=2