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 $\Gamma \equiv \beta e^2/\varepsilon =2$ 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 $N\to\infty$. For the couplings $\Gamma=2\gamma$ 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 $\Gamma=4$ 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 $N\to\infty$ as well, starting from a relatively weak coupling constant $\Gamma$ in between 2 and 4.