Interpretation of polarizing field $h_s$

The external field (26) has a particularly simple interpretation for spin systems that are spherically symmetric in space (e.g. in a spherically-symmetric trap). Since in this case we can neglect the orbital part $\hat{\bm{L}}$ of the angular momentum in $\hat{\bm{J}} = \hat{\bm{L}}+\hat{\bm{s}}$, the polarizing field can be written as

$$h_s=i \lambda_s \exp({-i\pi\hat{s}_x}) = 2\lambda_s \hat{s}_x,$$ (28)

and 2FLA is equivalent to the (one-dimensional) rotational cranking approach [17,18], in which the total energy of the system at a fixed value of the total angular momentum $J_x = \langle \hat{J}_x \rangle$ is obtained by adding to the Hamiltonian the one-body field $-\omega \hat{J}_x$ (rotational cranking term). Therefore, for spin systems, the angular velocity $\omega$ is simply equal to 2$\lambda _s$.

A different situation occurs in atomic nuclei, where the spin degeneracy is lifted by the spin-orbit interaction. Then, projection of the single-particle angular momentum $\Omega_x$ is a good quantum number. Consequently, the eigenvalues of the signature operator (15) are simply $e^{-i\pi\Omega_x}$, i.e., $\alpha$=1/2 for $\Omega_x=1/2, -3/2, 5/2,\cdots$ and $\alpha$=-1/2 for $\Omega_x=-1/2, 3/2, -5/2,\cdots$. In this situation, there is no difference between signature and spin projection, and odd states can indeed be obtained within the 2FLA. However, in this case, the angular momentum polarization (or spin polarization in a deformed trap rotating along the symmetry axis) is best modeled by the cranking approximation, where $\hat{J}_x$, rather than signature $\hat{R}_x$, is used to build Routhian (9). This is so, because for systems such as nuclei that are governed by $j-j$ coupling, the polarizing field $2\lambda_s\alpha$ does not distinguish between individual single-particle alignments, i.e., between states with large $\Omega_x$ values that predominantly contribute to the angular momentum alignment and low $\Omega_x$ states that weakly respond to rotation. In this respect, the standard non-collective cranking approach has a certain advantage.

Within the non-collective cranking approach, the single-particle cranking term becomes $-\omega\Omega_x$ and the single-quasiparticle energies are linear in $\omega$ with slopes distinguishing between angular momentum projections. The corresponding quasiparticle vacua represent ``optimal" states with maximally aligned angular momentum (maximum polarization). The minimization of the total energy for intermediate values of angular momentum can be done by considering the lowest particle-hole excitations across the Fermi surface. These states are not accessible within the 2FLA, and they must be obtained by explicitly blocking quasiparticle states.