The Sixth-Moment Sum Rule for the Pair Correlations of the Two-Dimensional One-Component Plasma: Exact Result

The system under consideration is a two-dimensional one-component plasma in the fluid regime, at density n and arbitrary coupling =e ² (e=unit charge, =inverse temperature). The Helmholtz free energy of the model, as the generating functional for the direct pair correlation c, is treated in terms of a convergent renormalized Mayer diagrammatic expansion in density. Using specific topological transformations within the bond-renormalized Mayer expansion, we prove that the nonzero contributions to the regular part of the Fourier component of c up to the k ²-term originate exclusively from the ring diagrams (unable to undertake the bond-renormalization procedure) of the Helmholtz free energy. In particular, (k)=–/k ²+/(8n)–k ²/96(n)²]+O(k ⁴). This result fixes via the Ornstein–Zernike relation, besides the well-known zeroth-, second-, and fourth-moment sum rules, the new sixth-moment condition for the truncated pair correlation h, n(n/2)³ r ⁶ h(r) d r=3(–6)(8–3)/4.