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 Gamma=beta q(2) (beta is the inverse temperature), in the whole region of the thermodynamic stability of the guest charge amplitude Z > -2/Gamma. The derivation is based on a mapping technique of the 2D jellium at the coupling Gamma = (even positive integer) onto a discrete 1D anticommuting-field theory; we assume that the final results remain valid for all real values of Gamma corresponding to the fluid regime. The generalized sum rules reproduce for arbitrary coupling Gamma the standard Z=1 and the trivial Z=0 results. They are also checked in the Debye-Huckel limit Gamma -> 0 and at the free-fermion point Gamma =2. The generalized second-moment sum rule provides some exact information about possible sign oscillations of the induced charge density in space.