Pairing in odd-mass nuclei

The zero-quasiparticle HFB state (6), representing the lowest configuration for a system with even number of fermions, corresponds to a filled sea of Bogoliubov quasiholes with negative quasiparticle energies, see Fig. 1. In a one-quasiparticle state representing a state in an odd nucleus, a positive-energy quasiparticle state $\alpha$ is occupied and its conjugated quasihole partner is empty. The corresponding wave function can be written as

$$\vert\Phi\rangle^{(\alpha)}_{\mbox{\rm\scriptsize {odd}}} =... ...um_{\nu\mu} Z^*_{\nu\mu} a^+_\nu a^+_\mu\right)\vert\rangle ,$$ (19)

where $\alpha^+_\alpha$ is the quasiparticle creation operator,

$$\alpha^+_\alpha = \sum_\nu \left( U_{\nu\alpha}a^+_\nu + V_{\nu\alpha} a_\nu \right),$$ (20)

which depends on the quasiparticle state $\chi_\alpha$ (8). Density matrix and pairing tensor of state (19) can be obtained by exchanging in ${\cal{}U}$ columns corresponding to the quasiparticle and quasihole states, $\chi_\alpha$ and $\phi_\alpha$. The corresponding density matrix reads explicitly,

$$\rho_{\mu\nu}^{(\alpha)} = \left( V^{*}V^{T} \right)_{\mu\nu... ...{\mu\alpha}U^{*}_{\nu\alpha} - V^{*}_{\mu\alpha}V_{\nu\alpha},$$ (21)

and similar holds for the pairing tensor. After the column replacement, matrix $U^{(\alpha)}_{\mu\alpha'}$ of one-quasiparticle state becomes singular and has null space of dimensions one. Hence, the occupation number of one of the s.p. states equals to 1. This fact is at the origin of the name ``blocked states'' attributed to one-quasiparticle states (19). These states contain fully occupied s.p. states that do not contribute to pairing field[90,91,92,93,94].

The blocking can also be implemented, for some configurations, by introducing two chemical potentials for different superfluid components (two-Fermi level approach, 2FLA)[95,96] As demonstrated in Ref.[93], such procedure is equivalent to applying a one-body, time-odd field that changes the particle-number parity of the underlying quasiparticle vacuum. For polarized Fermi systems, in which no additional degeneracy of quasiparticle levels is present beyond the Kramers degeneracy, the 2FLA is equivalent to one-dimensional, non-collective rotational cranking.

When describing properties of odd-mass nuclei, one selects the lowest quasiparticle excitations $E_\alpha$ and carries out the self-consistent procedure based on these blocked candidates (19). Naturally, one must adopt a prescription to be able to determine, at each iteration, the index $\alpha$ of the quasiparticle state to be blocked [97]. Such a unique identification can be done by means of, e.g., the overlap method of Ref.[98]. After the HFB iterations are converged for each blocked candidate, the state corresponding to the lowest energy is taken as the ground state of an odd-mass nucleus, and the remaining ones are approximations of the excited states. A similar procedure can be applied to many-quasiparticle states, e.g., two-quasiparticle states in even-even and odd-odd nuclei, three-quasiparticle excited states in odd-mass nuclei, and so on.

The state (19) represents an odd-Fermi system that carries nonzero angular momentum; hence, it breaks the time reversal symmetry. If the time reversal symmetry is enforced, additional approximations have to be applied based on the Kramers degeneracy. One of them is the equal filling approximation (EFA), in which the degenerate time-reversed states $\chi_\alpha$ and $\chi_{\bar\alpha}$ are assumed to enter the density matrix and pairing tensor with the same weights[99,94]. For instance, the blocked density matrix of EFA reads:

$$\rho_{\mu\nu}^{{(\alpha),\textsc{efa}}} = \left( V^{*}V^{T} ... ...u\bar\alpha} - V^{*}_{\mu\bar\alpha}V_{\nu\bar\alpha} \right).$$ (22)

It has been shown[100] that the EFA and the exact blocking are both strictly equivalent when the time-odd fields of the energy density functional are put to zero. Thus, EFA is adequate in many practical applications that do not require high accuracy.

Although for the functionals restricted to time-even fields, the time-reversed quasiparticle states $\alpha$ and $\bar\alpha$ are exactly degenerate, this does not hold in the general case. Here, the blocking prescription may depend on which linear combination of those states is used to calculate the blocked density matrix. This point can be illuminated by introducing the notion of an alispin[100], which describes the arbitrary unitary mixing of $\chi_\alpha$ and $\chi_{\bar\alpha}$: $\chi'_\alpha = a\chi_\alpha + b\chi_{\bar\alpha}$ ( $\vert a\vert^{2} + \vert b\vert^{2} = 1$). As usual, the group of such unitary mixings in a $2\times2$ space can be understood as rotations of abstract spinors, which we here call alirotations of alispinors. If the time-reversal symmetry is conserved, the blocked density matrix becomes independent of the mixing coefficients $(a, b)$, that is, it is an aliscalar. In the general case where time-reversal symmetry is not dynamically conserved, however, the blocked density matrix is not aliscalar. Here, the blocked density matrix may depend on the choice of the self-consistent symmetries and the energy of the system may change with alirotation.

The key point in this discussion is the realization that blocking must depend on the orientation of the alignment vector with respect to the principal axes of the mass distribution. To determine the lowest energy for each quasiparticle excitation, self-consistent calculations should be carried out by varying the orientation of the alignment vector with respect to the principal axes of the system[101,102]. While in many practical applications one chooses a fixed direction of alignment dictated by practical considerations, it is important to emphasize that it is only by allowing the alignment vector to point out in an arbitrary direction that the result of blocked calculations would not depend on the choice of the basis used to describe the odd nucleus. Illuminating examples presented in Ref.[100] demonstrate that the choice of the alignment orientation does impact predicted time-odd polarization energies.

Examples of self-consistent HFB calculations of one-quasiparticle states can be found in Refs.[103,104,100,105,106] (full blocking) and Refs.[107,108,109,110] (EFA).