Apr. 20: Hartree-Fock-Bogoliubov, BCS

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HFB - density matrix and pairing tensor HFB density matrix HFB pairing tensor ! ij " # c j + c i # , $ ij " # c j c i # ˆ ! = v * v T , $ ˆ = v * u T ˆ ! 2 % ˆ ! = % $ ˆ $ ˆ + , ˆ ! $ ˆ = $ ˆ ˆ ! ˆ ! = % ' & ˆ " # ˆ $ # ˆ 1 $ ˆ " * ( *, ! ˆ 2 = ! ˆ ) ! ˆ % u K( ' * = 0, ! ˆ v * % ( K ' * & v K ) & u K ) * = v * % ( ' K * * & u K ) Generalized density matrix Eigenvalues are 0 or 1 (thus defining occupations of quasiparticle states) ˆ H (c + ,c) ! ˆ H (" + ,") = E 0 + ˆ H <strong>20</strong> + ˆ H 11 + ˆ H int Independent quasiparticle Hamiltonian (H HFB ) [ H ˆ , N ˆ ] = 0, but H ˆ HFB , N ˆ [ ] ! 0 Quasiparticle interaction However, we require that ˆ N = N " ˆ H ' = ˆ H # $ ˆ N
HFB equations ˆ K = $ h ˆ ! " & % ! ˆ # * ! ˆ h ij = t ij + * ij # ˆ ' ) h * + "( * ij = + v iljk , kl kl # ij = 1 2 + v ijkl - kl kl HFB Hamiltonian HF Hamiltonian Selfconsistent HF field Selfconsistent pair field HFB equations $ h ˆ ! " & % ! ˆ # * ! ˆ # ˆ ' $ ) u K' $ u & ) = E K ' h * K & ) + "(% v K ( % v K ( [ K ˆ , . ˆ ] = 0 Complicated eigenvalue problem ❏ HFB equations treat p-h and p-p on the same footing ❏ Δ is a state-dependent field. In general, it depends on density and has a kinetic term ❏ The generalized density matrix and the HFB Hamiltonian can be diagonalized simultaneously ❏ Fermi level determined from the particle number equation ❏ Often it is convenien to express the HFB equations in the canonical basis
Page 1: Hartree-Fock-Bogoliubov A. Two-body
Page 5 and 6: BCS equations ˆ H (c + ,c) ! ˆ H
Page 7 and 8: The gap equation ❏ Schematic solu
Page 9 and 10: Recall that v 2 k is the occupation
Page 11: BCS essentials 1. The Fermi surface

Apr. 20: Hartree-Fock-Bogoliubov, BCS

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