Use of the chemical potential for a limited number of fermions with a degenerate groundstate

Abstract

For fermions with degenerate single-particle energy levels, the usual relation between the total number of particles and the chemical potential $μ$ is only satisfied for a specific number of particles, i.e. those leading to closed shells. The treatment of an arbitrary number of fermions requires a modification of the chemical potential, similar to the one proposed by Landsberg for Bose-condensed systems. We study the implications of the required modification for fermions in a potential, by calculating the ground state energy, the free energy, the density, the partition function and the dynamic two-point correlation function. It turns out that the modified relation between the fugacity and the number of particles leads to the correct ground state energy and density. But for other quantities like the entropy and the two-point correlation functions, an additional correction is required and derived. These calculations indicate that many-body perturbation theories based on $H-μN$ with Lagrange multiplier $μ$, are not applicable in unmodified form for a fixed number of fermions at low temperature.