Pair Occupation, Fermi Condensation, and Phase Transitions in Many-Fermion Systems

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Abstract
For a system of 2n fermions it is shown that the occupation number n α of any fermion‐pair state φ α (x 1 x 2 ) is n α =n∫φ α * (x 1 x 2 )ρ 2 (x 1 x 2 ,x 1 ′x 2 ′)φ α (x 1 ′x 2 ′) dx 1 dx 2 dx 1 ′ dx 2 ′ , where ρ2, assumed normalized to unity, is the two‐particle density matrix. The known upper bound on the largest eigenvalue of ρ2 implies that n α ≤ ½ in the thermodynamic limit n → ∞, equality being approached, with suitable φα, only for certain limiting BCS states ψBCS. Bose condensation of fermion pairs, in the sense of macroscopic occupation of any n α, is impossible. Fermi condensation into φα is defined to be present if n α > 0 in the thermodynamic limit. It occurs, for example, for suitable BCS states, but for a normal Fermi system all n α are of order n −1 or smaller. It is argued that the physical criterion for ψBCS exhibiting Fermi condensation is that the pair state from which it is constructed must have a bound‐state component of range ≲k F −1 , where kF is the Fermi momentum (3π2ρ)⅓. The maximally‐occupied pair state is the eigenfunction of ρ2 belonging to the largest eigenvalue, associated with off‐diagonal long‐range order of the type defined by Yang. A formula for n 0 for the original BCS state is derived. Some remarks are made concerning the interpretation of the Fermi condensation as a superconducting transition. The analysis is generalized to occupation of l‐fermion states. It is conjectured that, when Fermi condensation first sets in at a given even l, this is associated with the formation of bound states of l fermions, and a formula for the maximal occupation of such states is exhibited. The implications, for the theory of liquid 4He, of the fact that a helium atom contains electrons are examined. It is shown that Bose condensation into a single‐4He‐atom state is impossible, but a Fermi condensation similar in some respects to that in a superconductor can and probably does occur. It is argued that the mechanism preventing Bose condensation in superconductors and liquid 4He lies in the effect of collisions, acting via the exclusion principle, in causing virtual internal excitations.