Hartree-Fock-Bogoliubov Method

A two-body Hamiltonian of a system of fermions can
be expressed in terms of a set of annihilation and creation
operators
:

where are anti-symmetrized two-body interaction matrix-elements. In the HFB method, the ground-state wave function is defined as the quasiparticle vacuum , where the quasiparticle operators are connected to the original particle operators via the linear Bogoliubov transformation

which can be rewritten in the matrix form as

Matrices and satisfy the relations:

In terms of the normal and pairing one-body density matrices, defined as

the expectation value of the Hamiltonian (1) is expressed as an energy functional

where

Variation of energy (6) with respect to and results in the HFB equations:

where the Lagrange multiplier has been introduced to fix the correct average particle number.

It should be stressed that the modern energy functionals (6) contain terms that cannot be simply related to some prescribed effective interaction, see e.g., Ref. [27,28] for details. In this respect the functional (6) should be considered in the broader context of the energy density functional theory.