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

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 Stillinger–Lovett sum rules give the charge and second moment of the screening cloud around a particle of the jellium. In this paper, we generalize these sum rules to the screening cloud induced around a pointlike guest charge Zq immersed in the bulk interior of the 2D jellium with the coupling constant Γ=β q ² (β is the inverse temperature), in the whole region of the thermodynamic stability of the guest charge amplitude 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→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.