The Skyrme Hartree-Fock-Bogolyubov equations

The HFB approximation is based on the use of a trial variational wave function which is assumed to be an independent quasiparticle state $\vert\Phi\rangle$. This state, which mixes different eigenstates of the particle number operator, is a linear combination of independent particle states representing various possibilities of occupying pairs of single particle states. Following the notations and phase convention of [5] we define the particle and pairing density $\rho$ and $\tilde\rho$ matrices by

$$\rho({\mathbf r}\sigma q,{\mathbf r}'\sigma'q')=\langle\Phi\vert a^\dagger_{{\mathbf r}'\sigma'q'}a_{{\mathbf r}\sigma q}\vert\Phi\rangle\,,$$ (1)

$$\tilde\rho({\mathbf r}\sigma q,{\mathbf r}'\sigma'q')=-2\sigma'\l... ...e\Phi\vert a_{{\mathbf r}'-\sigma'q'}a_{{\mathbf r}\sigma q}\vert\Phi\rangle\,,$$ (2)

where the operators $a^\dagger_{{\mathbf r}\sigma q}$ and $a_{{\mathbf r}\sigma q}$ create and annihilate a nucleon at the point ${\mathbf r}$ having spin $\sigma=\pm\frac{1}{2}$ and isospin $q=\pm\frac{1}{2}$. The symmetry properties of $\rho$ and $\tilde\rho$ as well as the relation between $\tilde\rho$ and the pairing tensor $\kappa$ (defined for example in [10]) are discussed in [5].

The variation of the energy expectation value $E=\langle\Phi\vert\hat{H}\vert\Phi\rangle$ with respect to $\rho$ and $\tilde\rho$ under the constraints $N=\langle\Phi\vert\hat{N}\vert\Phi\rangle$ and $Z=\langle\Phi\vert\hat{Z}\vert\Phi\rangle$ (for neutrons and protons) leads to the Hartree-Fock-Bogolyubov equation which reads in coordinate representation

$$\begin{multline} \displaystyle \int d^3{\mathbf r}'\sum_{\sigma'} \left( \begin... ...sigma) \cr \varphi_2(E,\mathbf r\sigma) \cr \end{matrix}\right) \end{multline}$$

where the particle and pairing fields are given by

$$h({\mathbf r}\sigma,{\mathbf r}'\sigma')= \frac{\delta E}{\delta\... ...')= \frac{\delta E}{\delta\tilde\rho({\mathbf r}\sigma,{\mathbf r}'\sigma')}\,.$$ (3)

Once again we refer the reader to the article [5] for the discussion concerning the quasiparticle spectrum, its symmetries and the relations between the components of the HFB spinors and the densities.