Abstract
We employ approximate Green's Functions (GF) to obtain the Helmholtz free energy F in a Grand Canonical Ensemble. This study was motivated by the calculation of the total number of electrons N t as a function of the chemical potential μ in the Periodic Anderson Model by employing approximate one-electron GF. In this calculation we found that for some parameter values at low T one obtains three values of the chemical potential μ for each N t in a small interval of N t . One of the three states is thermodynamically unstable because N t decreases when μ increases, but in the calculation of F by a methods that is based in a thermodynamic relation, this is the most stable of the three. The purpose of this work is to explain this paradox, and we also suggest a variation of the calculation that avoids this difficulty. From geometrical arguments it is clear that this paradox will be always present when N t vs. μ has the shape observed in our calculation, independently of the numerical details of the calculation.
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