RANDOM-PHASE-APPROXIMATION FOR THE GRAND-CANONICAL POTENTIAL OF COMPOSITE FERMIONS IN THE HALF-FILLED LOWEST LANDAU-LEVEL

We reconsider the theory of the half-filled lowest Landau level using the Chern-Simons formulation and study the grand-canonical potential i n the random-phase approximation (RPA). Calculating the unperturbed re sponse functions for current- and charge-density exactly, without any expansion with respect to frequency or wave vector, we find that the i ntegral for the ground-state energy converges rapidly (algebraically) at large wave vectors k, but exhibits a logarithmic divergence at smal l Ic. This divergence originates in the k(-2) singularity of the Chern -Simons interaction and it is already present in lowest-order perturba tion theory. A similar divergence appears in the chemical potential. B eyond the RPA, we identify diagrams for the grand-canonical potential (ladder-type, maximally crossed, or a combination of both) which diver ge with powers of the logarithm. We expand our result for the RPA grou nd-state energy in the strength of the Coulomb interaction. The linear term is finite and its value compares well with numerical simulations of interacting electrons in the lowest Landau level.