Conclusions

In this study, we analyzed various approaches to polarized Fermi systems within the DFT. The main conclusions can be summarized as follows:

• By analogy with rotating nuclei, we showed that introducing two chemical potentials for different superfluid components is equivalent to applying a one-body, time-odd field. This field can be used to change the particle-number parity of the underlying quasiparticle vacuum.
• Since the external one-body field commutes with the HFB Hamiltonian, 2FLA is equivalent to non-collective cranking, a technique that is often used to select a vacuum configuration of interest.
• For systems, in which no additional degeneracy is present beyond the Kramers degeneracy, the 2FLA is equivalent to one-dimensional, non-collective rotational cranking. Different choices of the angular momentum quantization axis give rise to different blocking procedures and different polarization schemes. By increasing the asymmetry $\lambda _s$, one is alternating between even and odd systems while gradually increasing the signature polarization. In the absence of the spin-orbit coupling, the signature quantum number can be replaced by spin projection.
• The generalization of 2FLA to the case of an arbitrary blocked state is not obvious, although one can introduce polarizing fields that would single out the state of interest. For instance, one can introduce different cranking frequencies for spin and orbital motion, and this removes the orbital degeneracy of quasiparticle states. However, the standard way of blocking a given level in a fixed Hamiltonian submatrix should work as well, and it is easy to generalize to an arbitrary state, even if the majority of self-consistent quantum numbers are gone.
• For systems, in which an additional $k$-fold degeneracy ($k$-even) is present beyond the Kramers degeneracy, the 2FLA is unable to produce odd-particle-number states. Moreover, for an $k$-fold degeneracy with odd $k$, which is the case, e.g., for spherical systems, the 2FLA gives odd-particle-number states that correspond to $k$-quasiparticle and not to one-quasiparticle excitations.
• The situation encountered in Fig. 2 is similar to the level crossing discussed in the context of high-spin physics, where the lowest quasiparticle Routhian becomes negative as a function of rotational frequency. In such situations, one needs to preserve a number of quasiparticles in each signature block to conserve $\pi_N$ in the HFB vacuum [31,29].
• Irrespective of the degeneracies present in the system, certain quasiparticle configurations cannot be approached through 2FLA.

In summary, the 2FLA provides a unifying methodology to treat a number of different kinds of condensates, including those of odd-particle systems, as ground states of some HFB Hamiltonian. However, not all quasiparticle states are easily accessible this way and problems arise if quasiparticle levels show degeneracies beyond the Kramers doubling. Moreover, the examples shown in Figs. 3 and 4 indicate that the variations of the self-consistent mean field associated with the configuration change driven by the polarizing field can be severe, including the change of ordering of the lowest quasiparticle excitations. This, together with related numerical instabilities, indicates that traditional methods of blocking are likely to be preferred for treating a large space of quasiparticle configurations and in spectroscopic-quality calculations for well-defined one-quasiparticle states.

Useful discussions with Aurel Bulgac, Michael Forbes, and Mario Stoitsov are gratefully acknowledged. The UNEDF SciDAC Collaboration is supported by the U.S. Department of Energy under grant No. DE-FC02-07ER41457. This work was also supported by the U.S. Department of Energy under Contract Nos. DE-FG02-96ER40963 (University of Tennessee), DE-AC05-00OR22725 with UT-Battelle, LLC (Oak Ridge National Laboratory), DE-FG05-87ER40361 (Joint Institute for Heavy Ion Research), and DE-FG02-97ER41014 (University of Washington); by the Polish Ministry of Science under Contract No. N N202 328234 and by the Academy of Finland and University of Jyväskylä within the FIDIPRO programme.