Counter-ions between or at asymmetrically charged walls: 2D free-fermion point

TL;DR

This paper provides exact solutions for the interaction between asymmetrically charged lines with counter-ions in 2D at the free-fermion point, comparing results with mean-field theory and revealing different asymptotic behaviors.

Contribution

It offers the first exact solutions at the free-fermion point for two models of charged lines with counter-ions, highlighting differences from mean-field predictions.

Findings

Large-distance pressure differs from mean-field in the unconstrained model.

Constrained model's pressure matches the Casimir force asymptotically.

Significant differences between like-charged and oppositely-charged lines.

Abstract

This work contributes to the problem of determining effective interaction between asymmetrically (likely or oppositely) charged objects whose total charge is neutralized by mobile pointlike counter-ions of the same charge, the whole system being in thermal equilibrium. The problem is formulated in two spatial dimensions with logarithmic Coulomb interactions. The charged objects correspond to two parallel lines at distance d, with fixed line charge densities. Two versions of the model are considered: the standard "unconstrained" one with particles moving freely between the lines and the "constrained" one with particles confined to the lines. We solve exactly both systems at the free-fermion coupling and compare the results for the pressure (i.e. the force between the lines per unit length of one of the lines) with the mean-field Poisson-Boltzmann solution. For the unconstrained model, the large-d asymptotic behaviour of the free-fermion pressure differs from that predicted by the mean-field theory. For the constrained model, the asymptotic pressure coincides with the attractive van der Waals-Casimir fluctuational force. For both models, there are fundamental differences between the cases of likely-charged and oppositely-charged lines, the latter case corresponding at large distances d to a capacitor.