Dependence of structure factor and correlation energy on the width of electron wires

The structure factor and correlation energy of a quantum wire of thickness b << a(B) are studied in random phase approximation (RPA) and for the less investigated region r(s) < 1. Using the single-loop approximation, analytical expressions of the structure factor are obtained. The exact expressions for the exchange energy are also derived for a cylindrical and harmonic wire. The correlation energy in RPA is found to be represented by epsilon(c) (b, r(s)) = alpha(r(s))/b + beta(r(s)) ln(b) + eta(r(s)), for small b and high densities. For a pragmatic width of the wire, the correlation energy is in agreement with the quantum Monte Carlo simulation data.