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minimizers for the hartree-fock-bogoliubov theory of neutron stars ...

minimizers for the hartree-fock-bogoliubov theory of neutron stars ...

270 LENZMANN and LEWIN

270 LENZMANN and LEWIN Note that Ɣ commutes with N if and only if α = 0, that is, if and only if the corresponding quasi-free state in the Fock space also commutes with the number operator N (see, e.g., [2] for the Fock space formalism in HFB theory). We can now state some fundamental properties of the minimizers for the HFB model considered in this article. THEOREM 2 (Properties of minimizers) Let q 1 be given, and suppose that m>0 and 0

MINIMIZERS FOR THE HFB-THEORY 271 3.2.2. The spin-1/2 case (q = 2) Let us now consider the specific case when the number of internal spin degrees of freedom is q = 2, which corresponds to the physically relevant case of spin-1/2 fermions such as neutrons. In this case, it is shown by Bach, Fröhlich, and Jonsson in [1], based on a concavity result in Lieb [28], that the following equality holds: I(λ) = J (λ) for q = 2 spin degrees of freedom. Here J (λ) = inf { 2F (τ) : τ ∈ B ( L 2 (R 3 ; C) ) ,τ= τ ∗ = τ, Tr(τ) = λ/2 } , (3.6) and the reduced (no-spin) energy is defined as F (τ) := Tr(Tτ) − κD(ρ τ ,ρ τ ) + κ 2 Ex(τ) − κ ∣√ ∣ τ(1 − τ)(x,y) 2 dx dy. (3.7) 2 ∫∫R 3 ×R |x − y| 3 Furthermore, all the minimizers for E(γ,α) and F (τ), respectively, satisfy ( ) 1 0 γ = τ ⊗ and α =± √ ( ) 0 1 τ(1 − τ) ⊗ . (3.8) 0 1 −1 0 We note that F (τ) has a form similar to the Müller functional which was studied in [14]. Indeed, a straightforward adaptation of an argument given in [14] combined with (3.8) leads to the following statement. (Its proof is outlined in Section 6.) THEOREM 3 (Infinite rank of minimizers if α ≠ 0) Assume that (γ,α) is a minimizer for I(λ) with 0

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