A Generalization of the Stillinger-Lovett Sum Rules for the Two-Dimensional Jellium

Abstract

In the equilibrium statistical mechanics of classical Coulomb fluids, the long-range tail of the Coulomb potential gives rise to the Stillinger-Lovett sum rules for the charge correlation functions. For the jellium model of mobile particles of charge $q$ immersed in a neutralizing background, the fixing of one of the $q$-charges induces a screening cloud of the charge density whose zeroth and second moments are determined just by the Stillinger-Lovett sum rules. In this paper, we generalize these sum rules to the screening cloud induced around a pointlike guest charge $Z q$ immersed in the bulk interior of the 2D jellium with the coupling constant $Γ=βq^2$ ($β$ is the inverse temperature), in the whole region of the thermodynamic stability of the guest charge $Z>-2/Γ$. The derivation is based on a mapping technique of the 2D jellium at the coupling $Γ$ = (even positive integer) onto a discrete 1D anticommuting-field theory; we assume that the final results remain valid for all real values of $Γ$ corresponding to the fluid regime. The generalized sum rules reproduce for arbitrary coupling $Γ$ the standard Z=1 and the trivial Z=0 results. They are also checked in the Debye-Hückel limit $Γ\to 0$ and at the free-fermion point $Γ=2$. The generalized second-moment sum rule provides some exact information about possible sign oscillations of the induced charge density in space.