Renormalization of Hard-Core Guest Charges Immersed in Two-Dimensional Electrolyte

This paper is a continuation of a previous one [L. {\v{S}}amaj, {\it J. Stat. Phys.} {\bf 120}:125 (2005)] dealing with the renormalization of a guest charge immersed in a two-dimensional logarithmic Coulomb gas of pointlike $\pm$ unit charges, the latter system being in the stability-against-collapse regime of reduced inverse temperatures $0\le \beta <2$. In the previous work, using a sine-Gordon representation of the Coulomb gas, an exact renormalized-charge formula was derived for the special case of the {\em pointlike} guest charge $Q$, in its stability regime $\beta | Q| < 2$. In the present paper, we extend the renormalized-charge treatment to the guest charge with a hard core of radius $\sigma$, which allows us to go beyond the stability border $\beta| Q| = 2$. In the limit of the hard-core radius much smaller than the correlation length of the Coulomb-gas species and at a strictly finite temperature, due to the counterion condensation in the extended region $\beta| Q| >2$, the renormalized charge $Q_{\rm ren}$ turns out to be a periodic function of the bare charge $Q$ with period 1. The renormalized charge therefore does not saturate at a specific finite value as $| Q| \to\infty$, but oscillates between two extreme values. In the high-temperature Poisson-Boltzmann scaling regime of limits $\beta\to 0$ and $Q\to\infty$ with the product $\beta Q$ being finite, one reproduces correctly the monotonic dependence of $\beta Q_{\rm ren}$ on $\beta Q$ in the guest-charge stability region $\beta| Q| <2$ and the Manning-Oosawa type of counterion condensation with the uniform saturation of $\beta Q_{\rm ren}$ at the value $4/\pi$ in the region $\beta| Q|\ge 2$.