Attraction of Like-Charged Walls with Counterions Only: Exact Results for the 2D Cylinder Geometry

We study a 2D system of identical mobile particles on the surface of a cylinder of finite length d and circumference W, immersed in a medium of dielectric constant $$\varepsilon $$ . The two end-circles of the cylinder are like-charged with the fixed uniform charge densities, the particles of opposite charge $$-e$$ (e being the elementary charge) are coined as “counterions”; the system as a whole is electroneutral. Such a geometry is well defined also for finite numbers of counterions N. Our task is to derive an effective interaction between the end-circles mediated by the counterions in thermal equilibrium at the inverse temperature $$\beta $$ . The exact solution of the system at the free-fermion coupling $$\varGamma \equiv \beta e^2/\varepsilon =2$$ is used to test the convergence of the pressure as the (even) number of particles increases from $$N=2$$ to $$\infty $$ . The pressure as a function of distance d is always positive (effective repulsion between the like-charged circles), decaying monotonously; the numerical results for $$N=8$$ counterions are very close to those in the thermodynamic limit $$N\rightarrow \infty $$ . For the couplings $$\varGamma =2\gamma $$ with $$\gamma =1,2,\ldots $$ , there exists a mapping of the continuous two-dimensional (2D) Coulomb system with N particles onto the one-dimensional (1D) lattice model of N sites with interacting sets of anticommuting variables. This allows one to treat exactly the density profile, two-body density and the pressure for the couplings $$\varGamma =4$$ and 6, up to $$N=8$$ particles. Our main finding is that the pressure becomes negative at large enough distances d if and only if both like-charged walls carry a nonzero charge density. This indicates a like-attraction in the thermodynamic limit $$N\rightarrow \infty $$ as well, starting from a relatively weak coupling constant $$\varGamma $$ in between 2 and 4. As a by-product of the formalism, we derive specific sum rules which have direct impact on characteristics of the long-range decay of 2D two-body densities along the two walls.