Heavy-light $N+1$ clusters of two-dimensional fermions

TL;DR

This paper investigates the binding conditions of heavy-light fermion clusters in two dimensions, deriving size and energy formulas beyond mean-field theory, and compares different theoretical approaches.

Contribution

It introduces a scale-invariant mean-field model for large N, providing explicit formulas for cluster sizes and energies by including beyond-mean-field effects.

Findings

Clusters are bound when the mass ratio exceeds 1.074N^2.

Explicit formulas for cluster sizes and energies are derived.

Differences between Thomas-Fermi and Hartree-Fock approaches are discussed.

Abstract

We study binding of $N$ identical heavy fermions by a light atom in two dimensions assuming zero-range attractive heavy-light interactions. By using the mean-field theory valid for large $N$ we show that the $N+1$ cluster is bound when the mass ratio exceeds $1.074N^2$. The mean-field theory, being scale invariant in two dimensions, predicts only the shapes of the clusters leaving their sizes and energies undefined. By taking into account beyond-mean-field effects we find closed-form expressions for these quantities. We also discuss differences between the Thomas-Fermi and Hartree-Fock approaches for treating the heavy fermions.