Effective interaction between guest charges immersed in 2D jellium

The model under study is an infinite 2D jellium of pointlike particles with elementary charge $e$, interacting via the logarithmic potential and in thermal equilibrium at the inverse temperature $\beta$. Two cases of the coupling constant $\Gamma\equiv \beta e^2$ are considered: the Debye-H\"uckel limit $\Gamma\to 0$ and the free-fermion point $\Gamma=2$. In the most general formulation, two guest particles, the one with charge $q e$ (the valence $q$ being an arbitrary integer) and the hard core of radius $\sigma>0$ and the pointlike one with elementary charge $e$, are immersed in the bulk of the jellium at distance $d\ge \sigma$. Two problems are of interest: the asymptotic large-distance behavior of the excess charge density induced in the jellium and the effective interaction between the guest particles. Technically, the induced charge density and the effective interaction are expressed in terms of multi-particle correlations of the pure (translationally invariant) jellium system. It is shown that the separation form of the induced charge density onto its radial and angle parts, observed previously in the limit $\Gamma\to 0$, is not reproduced at the coupling $\Gamma=2$. Based on an exact expression for the effective interaction between guest particles at $\Gamma=2$, oppositely ($q=0,-1,-2,\ldots$) charged guest particles always attract one another while likely ($q=1,2,\ldots$) charged guest particles repeal one another up to a certain distance $d$ between them and then the mutual attraction takes place up to asymptotically large (finite) distances.