Solution of the Hartree-Fock-Bogolyubov equations

In order to incorporate pairing correlations for rotating states,
the new version (v2.07f) of the code HFODD
solves the standard HFB equation [10],

For the conserved simplex symmetry, which in the present version is assumed
when solving the HFB equation, the Routhian and the pairing
potential acquire the following block forms [12,10]:

from which the complete solution of Eq. (6) is reconstructed as

Note that if the s.p. space contains states (/2 in each of the two simplexes =), then the HFB equation (6) has a dimension of , matrices without the indices (, , , , ...) have a dimension of , and matrices with the indices (, , , , ...) have a dimension of . Hence, equation (8), which the code solves, has the dimension which is twice smaller than the complete HFB equation (6). Note also that we use the convention of indices such that they correspond to simplexes of the

For the conserved time-reversal symmetry, diagonal matrices of eigenvalues and are identical to one another, and the separation of eigenvectors of Eq. (8) into two simplexes is trivial; it is enough to group positive and negative quasiparticle energies together. For broken time-reversal this procedure does not work, because there can be more than half positive or negative quasiparticle energies in the spectrum of Eq. (8). So in general one has to put the half of largest eigenvalues into matrix , and the half of smallest into matrix , irrespective of their signs. Such a choice leads to the solution that corresponds to the so-called quasiparticle vacuum.

Based on the solution of the matrix equation (8), we have upper and lower components of the quasiparticle wave functions in the space coordinates as where are the HO simplex wave functions (I-78) in space coordinates (I-76) and = are the numbers of the HO quanta in three Cartesian directions. In Eq. (10) we have used the fact that the s.p. basis states of either of the two simplexes, , can be numbered by the HO quantum numbers . We have also introduced index =1,...,, which numbers eigenstates of Eq. (8) in both "halfs" of the spectrum defined above.

From the quasiparticle wave functions we obtain the standard particle and
pairing density matrices [13],
where the sum over is performed up to the maximum
equivalent-spectrum energy
, see Ref. [14]
for details.
All particle-hole mean-field potentials can be calculated from
the particle density matrix
and its derivatives (see I), while the particle-particle mean-field
potentials can be calculated from the pairing density matrix
[13]. In
the present implementation of the code HFODD, terms depending on
derivatives of the particle-particle density matrix are not taken
into account, and hence the pairing potential depends only on the
local pair density

where is the standard formfactor of the zero-range density-dependent pairing force,

Note that an attractive interaction requires 0.

Finally, matrix elements of the pairing potential in the simplex HO
basis, which are needed in Eq. (8), can be calculated
in exactly the same way as matrix elements of the central mean-field
potential,
Sec. I-4.2, i.e.,

and are the HO basis states.