Random-phase approximation for the grand-canonical potential of composite fermions in the half-filled lowest Landau level

TL;DR

This paper analyzes the grand-canonical potential of composite fermions at half-filling using RPA, revealing divergences due to the Chern-Simons interaction and providing finite energy estimates that align with numerical simulations.

Contribution

It offers an exact calculation of response functions in the RPA for the half-filled Landau level and identifies divergence issues in the theory.

Findings

Rapid convergence of the ground-state energy integral at large wave vectors

Logarithmic divergence at small wave vectors due to Chern-Simons interaction

Finite linear term in Coulomb interaction expansion matching numerical results

Abstract

We reconsider the theory of the half-filled lowest Landau level using the Chern-Simons formulation and study the grand-canonical potential in the random-phase approximation (RPA). Calculating the unperturbed response functions for current- and charge-density exactly, without any expansion with respect to frequency or wave vector, we find that the integral for the ground-state energy converges rapidly (algebraically) at large wave vectors k, but exhibits a logarithmic divergence at small k. This divergence originates in the 1/k^2 singularity of the Chern-Simons interaction and it is already present in lowest-order perturbation theory. A similar divergence appears in the chemical potential. Beyond the RPA, we identify diagrams for the grand-canonical potential (ladder-type, maximally crossed, or a combination of both) which diverge with powers of the logarithm. We expand our result for the RPA ground-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.