Correlations and sum rules in a half-space for a quantum two-dimensional one-component plasma

Abstract

This paper is the continuation of a previous one [L. Šamaj and B. Jancovici, 2007 J. Stat. Mech. P02002]; for a nearly classical quantum fluid in a half-space bounded by a plain plane hard wall (no image forces), we had generalized the Wigner-Kirkwood expansion of the equilibrium statistical quantities in powers of Planck's constanth. As a model system for a more detailed study, we consider the quantum two-dimensional one-component plasma: a system of charged particles of one species, interacting through the logarithmic Coulomb potential in two dimensions, in a uniformly charged background of opposite sign, such that the total charge vanishes. The corresponding classical system is exactly solvable in a variety of geometries, including the present one of a half-plane, when βe 2 = 2, where β is the inverse temperature and e is the charge of a particle: all the classical n-body densities are known. In the present paper, we have calculated the expansions of the quantum density profile and truncated two-body density up to orderh 2 (instead of only to order h in the previous paper). These expansions involve the classical n-body densities up to n = 4, thus we obtain exact expressions for these quantum expansions in this special case.For the quantum one-component plasma, two sum rules involving the truncated two-body density (and, for one of them, the density profile) have been derived, a long time ago, by heuristic macroscopic arguments: one sum rule is about the asymptotic form along the wall of the truncated two-body density, the other one is about the dipole moment of the structure factor. In the two-dimensional case at βe 2 = 2, we have now explicit expressions up to orderh 2 of these two quantum densities, thus we can microscopically check the sum rules at this order. The checks are positive, reinforcing the idea that the sum rules are correct.Correlations and sum rules in a half-space quantum plasma