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

TL;DR

This paper investigates how to correctly incorporate the chemical potential for a fixed number of fermions with degenerate ground states, revealing necessary modifications for accurate thermodynamic and correlation calculations.

Contribution

It introduces a modified relation between chemical potential and particle number for fermions with degenerate levels, extending Landberg's approach and analyzing its implications.

Findings

Modified chemical potential relation yields correct ground state energy and density.

Additional corrections are needed for entropy and correlation functions.

Standard $H- N$ perturbation theory is inadequate for fixed fermion number at low temperature.

Abstract

For fermions with degenerate single-particle energy levels, the usual relation between the total number of particles and the chemical potential $\mu $ 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 $H-\mu N$ with Lagrange multiplier $\mu $, are not applicable in unmodified form for a fixed number of fermions at low temperature.