Chemical potential beyond the quasiparticle mean field

Abstract

The effects of quantal and thermal fluctuations beyond the BCS quasiparticle mean field on the chemical potential are studied within a model, which consists of $N$ particles distributed amongst $\ensuremath{\Omega}$ doubly folded equidistant levels interacting via a pairing force with parameter $G$. The results obtained at zero and finite temperatures $T$ within several approaches, which include the fluctuations beyond the BCS theory, are compared with the exact results. The chemical potential, defined as the Lagrangian multiplier to preserve the average number of particles, is compared with the corresponding quantity, which includes the effect from fluctuations of particle and quasiparticle numbers beyond the BCS quasiparticle mean field. The analysis of the results shows that the latter differs significantly from the former as functions of $G$ and $T$. The chemical potential loses its physical meaning in the system with a fixed number of particles or after eliminating quantal fluctuations of particle (quasiparticle) numbers by means of particle number projection. The validity of the criterion for the signature of the transition to Bose-Einstein condensation, which occurs in infinite systems when the chemical potential hits the bottom of the energy spectrum, is reexamined for the finite multilevel model.