Particle-number parity of the HFB vacuum

Let us first analyze the 2FLA in the case when levels depicted in Fig. 2 are not degenerate. At low values of $\lambda _s$, the quasiparticle vacuum corresponds to a system with even number of fermions. It is seen that at $\lambda _s$=${E}_{1/2,1}$ the down-sloping lowest quasiparticle with $\alpha$=1/2 crosses zero and becomes negative. Beyond that point, the HFB vacuum has one quasiparticle state occupied, as in the middle panel of Fig. 1. Here, the particle number parity $\pi_N$ changes from even (+1) to odd (-1), as discussed in Sec. 2.2 and also in the context of nuclear rotations in Refs. [31,17]. This can also be derived in a straightforward and explicit way by calculating the expectation value of the number-parity operator $e^{i\pi \hat n}$ in the HFB vacuum. For each subspace in the canonical representation, the number operator can be expressed $\pi_{N_{\alpha}}$= $e^{i\pi({\hat n_1+ \hat n_2})}= (1-2 \hat n_1 )(1-2 \hat n_2)$, where $1, 2$ label the quasiparticle transformations for the negative eigenvalues. The ground-state expectation values of $\hat n_1$, $\hat n_2$, and $\hat n_1\hat n_2$ are then evaluated in the usual way by expanding in the quasiparticle basis, normal ordering, and extracting the zero-quasiparticle term.

By using the modified density matrices (25) in the HFB equations and in the particle number equation for $\lambda$,

$$N = {\rm Tr}(\hat\rho) ,$$ (29)

one formally recovers the standard blocked HFB equations for the lowest 1-qp ($\alpha$=1/2) state. At still higher values of $\lambda _s$, the particle-number parity changes again at $\lambda _s$=${E}_{1/2,2}$ when the second lowest quasiparticle with $\alpha$=1/2 crosses zero. The associated two-quasiparticle configuration in a nucleus with $N=N_0+2$ has the signature index $\alpha$=1. It is immediately seen that the lowest two-quasiparticle $\alpha$=0 configuration of Fig. 1(c), associated with the so-called S-band in rotating nuclei, cannot be reached within the standard 2FLA.

Following the discussion in Sec. 3.1, it is worth noting that the choice of angular momentum quantization implies a different character of the angular alignment associated with the change in the quasiparticle vacuum. In the case of $z$-quantization and the absence of time-odd fields in the HFB Hamiltonian, 2FLA treatment of systems with odd particle number is equivalent to the so-called uniform filling approximation, in which a blocked nucleon is put with equal probability in each of the degenerate magnetic substates [32,33]. It is only in the regime of non-collective rotation in which the angular momentum is quantized along the $x$-axis that the dynamics of angular momentum alignment can be properly treated and the full alignment can be reached.

In cases when in the unpolarized system the Kramers degeneracy is the only one, the parity $\pi_N$ of states obtained by occupying the lowest ${E}_{\alpha\mu}^s<0$ quasiparticles do change at points where ${E}_{\alpha\mu}^s=0$, as indicated in Fig. 2. However, if apart from the Kramers degeneracy there is an additional two-fold, four-fold, etc. degeneracy of quasiparticle levels, all such states will be $\pi_N$-even. Therefore, in such situations, the 2FLA would fail to produce odd-$N$ systems as HFB ground states.

For spherical spin systems, each level in Fig. 2 is ($2\ell$+1)-degenerate, which is an odd number, and the number parity does change at points ${E}_{\alpha\mu}^s=0$. However, the corresponding HFB vacuum represents a ($2\ell$+1)-quasiparticle excitation and not the one-quasiparticle excitation of Eq. (12). This is so even if the average Fermi energy $\lambda$ is adjusted to have (on average) only one particle more than that of the unpolarized system [7,10]. Such a situation corresponds to the so-called filling approximation of orbitally degenerate quasiparticle states. Needless to say, the orbital filling approximation completely neglects possible space-polarization effects that must, in principle, occur for true one-quasiparticle states.

If the spin system has an axial symmetry in space (e.g., it is in an external axially symmetric trap), projection of the angular momentum on the symmetry axis $\Lambda$ is a good quantum number. Moreover, the single-particle and quasiparticle states are then degenerate with respect to the sign of $\Lambda$. Here, each level in Fig. 2, except for $\Lambda$=0, is doubly degenerate. Therefore, none of the $\Lambda$$>$0 states obtained by occupying the ${E}_{\alpha\mu}^s<0$ quasiparticles has $\pi_N$=-1, that is, odd particle number. In this case, states with $\pi_N$=-1 cannot be obtained within 2FLA, and explicit treatment within the blocking approximation, described in Sec. 2.3, is mandatory.