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 $Zq$ 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 $Z>-2/\Gamma$. 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-Hückel limit $\Gamma \to 0$ 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.

16 pages

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• Is the Two-Dimensional One-Component Plasma Exactly Solvable?
• Guest charges in an electrolyte: renormalized charge, long- and short-distance behavior of the electric potential and density profile
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Cited by in corpus (6)

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• Expanded Vandermonde powers and sum rules for the two-dimensional one-component plasma
• Guest Charge and Potential Fluctuations in Two-Dimensional Classical Coulomb Systems
• Sum rules for correlation functions of ionic mixtures in arbitrary dimension $d\ge 2$
• Fourth Moment of the Charge Density Induced Around A Guest Charge in Two-Dimensional Jellium
• Short-distance symmetry of pair correlations in two-dimensional jellium