Hartree-Fock-Bogoliubov theory

 

Hartree-Fock-Bogoliubov (HFB) theory [20] is based on the Ritz variational principle applied to the many-fermion Hamiltonian,

$$H=\sum_{\alpha\alpha^{\prime }}t_{\alpha\alpha^{\prime }}a_{\... ... }a_{\alpha^{\prime }}^{\dagger }a_{\beta^{\prime }}a_{\beta},$$ (45)

with trial functions in the form of a quasiparticle vacuum. The resulting HFB equations can be written in matrix form as

$$\left( \begin{array}{cc} h - \lambda & \Delta \\ -\Delta ^{*}... ...\left( \begin{array}{l} U_{n} \\ V_{n} \end{array}\right) \;,$$ (46)

where E_n are the quasiparticle energies, $\lambda$ is the chemical potential, and the matrices $h=t+\Gamma$ and $\Delta$ are defined by the matrix elements of the two-body interaction

$$\begin{array}{rcl} \Gamma _{\alpha\alpha^{\prime }} & = & \su... ...\beta\beta^{\prime }} \kappa_{\beta\beta^{\prime}}, \end{array}$$ (47)

$\rho _{\beta^{\prime }\beta}$ and $\kappa_{\beta\beta^{\prime}}$being the density matrix and pairing tensor, respectively. HFB theory is by now a standard tool in nuclear structure calculations, and we refer the reader to Ref. [20] for details. Below we discuss several features of the formalism that are especially pertinent to the present application, namely canonical states, the pairing phase space, and those quantities that dictate the stability of a nucleus with respect to two-neutron emission.