Properties of ideal fermion systems

Vladimir S. Filinov

Over the last ten years the `fixed-node approximation', well known in the literature, has been widely used for a numerical treatment of thermodynamic properties of strongly correlated Fermi systems. Results of the direct path integral Monte Carlo simulation performed here show that the `fixed-node approximation' describes the thermodynamic properties of the strongly coupling fermions rather well at `weak and moderate' degeneracy. To analyse the reasons for the increasing difference between the `fixed-node approximation' and the results of direct path integral Monte Carlo simulations for highly degenerate fermions, the correctness of the `fixed-node approximation' for ideal Fermi systems has been analysed by analytical methods. A rigorous proof has been given of the fact that the exact Fermi function with index 3 / 2 describing the logarithm of the grand partition function of ideal fermions cannot be reproduced in the `fixed-node approximation', which means that the `fixed-node approximation' does not give the correct ideal Fermi gas limit.

Joachim Alexander Maruhn, P.‐G. Reinhard, E. Suraud

Zygmunt M. Galasiewicz

Jutta Meyer

Recently, the mathematical formalism of the Dyson boson mapping has been extended to a system of 3n fermions, leading to the boson–fermion and the baryon mapping. In the present paper, the case of a restriction to a subset of three-fermion quantum numbers, the collective indices, is discussed. A theory is developed for the representation of fermionic states and operators in a truncated ideal space where only collective boson–fermion pairs or collective ideal baryons are allowed. An exact reproduction of physical properties is proved to be possible provided that the original fermionic problem can be solved in a subspace where all three-fermion subsystems carry collective indices. Examples of simple applications are presented in the two subsequent papers of this series.

Vladimir S. Filinov

Over the last decades the 'fixed-node method' has been used for a numerical treatment of thermodynamic properties of strongly correlated Fermi systems. In this work correctness of the 'fixed -node method' for ideal Fermi systems has been analytically analyzed. It is shown that the 'fixed-node' prescription of calculation of the density matrix leads to contradictions even for two ideal fermions. The main conclusion of this work is that the 'fixed-node method' can not reproduce the fermion density matrices and should be considered as uncontrolled empirical approach in treatment of thermodynamics of Fermi systems.

R.K. Pathria, Paul D. Beale