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Hartree-Fock-Bogoliubov (HFB) Theory This Lecture closely follows ...

Hartree-Fock-Bogoliubov (HFB) Theory This Lecture closely follows ...

Hartree-Fock-Bogoliubov (HFB) Theory This Lecture closely follows

Hartree-Fock-Bogoliubov (HFB) TheoryThis Lecture closely follows parts of chapter 7 in Ring & Schuck Thenuclear many-body problem.HFB provides a generalized single-particle theory that unifies Hartree-Fock and BCS. It thus can be used to describe aspects of deformations(i.e. long range part of nucleon-nucleon force) as well as pairing correlationsdue to short ranged attraction.Let’s consider M single-particle orbitals with creation operators â † =(â † 1, . . . , â † M). The most general transformation of the single-particlebasis is⎛ ⎞ ⎛⎜ β ⎟ ⎜⎝β † ⎠ = ⎝U † V †V T⎞ ⎛ ⎞⎟ ⎜ â ⎟U T ⎠ ⎝â †⎛⎠ ≡ W † ⎜⎝⎞â ⎟â † ⎠For unitary W W † = 1, the quasi-particle operators fulfill canonicalanti-commutation rules {β k , β † l } = δ kl .Bloch & Messiah: The W matrix can be decomposed asW =⎛⎜⎝⎞D 0 ⎟0 D ∗ ⎠⎛⎜⎝U VV U⎞ ⎛⎟ ⎜⎠ ⎝⎞C 0 ⎟0 C ∗ ⎠

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