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 Gamma = beta e(2) (e = unit charge, beta = 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 lip to the k(2)-term originate exclusively from the ring diagrams (unable to undertake the bond-renormalization procedure) of the Helmholtz fi ec energy. In particular, (c) over cap(k) = -Gamma/k(2) + Gamma/(8 pi n) - k(2)/[96(pi 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 pail correlation h, n(pi Gamma n/2)(3) integral r(6)h(r) dr = 3(Gamma-6)(8-3 Gamma)/4.