Two Fermi Level Approach

The main idea behind the 2FLA [7,10] is to force a nonzero spin polarization in the system by finding the ground-state condensate in the presence of an external field that favors one spin over the other. In the language of signature, ``spin-up" corresponds to $\alpha=1/2$ while ``spin-down" corresponds to $\alpha=-1/2$. The polarization is achieved by adding to the Hamiltonian the single-particle field

$$h_s=i \lambda_s \hat{R}_x,$$ (23)

i.e., constructing the Routhian (9), $h'=h-h_s$. The imaginary unit must be put in the definition of $h_s$, because in the single-particle space $\hat{R}_x$ is antihermitian. Consequently, $h_s$ is time-odd, cf. Eq. (16). Since $\hat{R}_x$ and the quasiparticle Hamiltonian commute, adding $h_s$ represents a non-collective cranking; hence, the quasiparticle routhians must be linear in $\lambda _s$.

The field (26) will raise the Fermi energy $\lambda_{1/2}$ of the subsystem having signature $\alpha=+1/2$ by an amount $\lambda _s$ and lower the Fermi energy $\lambda_{-1/2}$ of the $\alpha=-1/2$ subsystem by the same amount. The relations between chemical potentials in 2FLA read:

$$\lambda_\alpha=\lambda+ 2\alpha\lambda_s,$$ (24)

where

$$\lambda ={\textstyle{\frac{1}{2}}}\left(\lambda_{1/2}+\lambd... ...tstyle{\frac{1}{2}}}\left(\lambda_{1/2}-\lambda_{-1/2}\right).$$ (25)

The HFB Routhian matrix of 2FLA can be written as [7]

$$\mathcal{H}_s=\mathcal{H} -2\alpha \lambda_s \mathcal{I},$$ (26)

where $\mathcal{H}$ is the matrix (19) corresponding to $\lambda_s=0$ and $\mathcal{I}$ is the unit matrix.

Since the added term is proportional to the unit matrix, its only effect is to shift the HFB eigenvalues up or down,

$${E}_{\alpha\mu}^s= {E}_{\alpha\mu} -2\alpha\lambda_s,$$ (27)

where $E_{\alpha \mu}$ are the eigenvalues of $\mathcal{H}$. Therefore, when plotted as a function of $\lambda _s$, the energies ${E}_{\alpha\mu}^s$ are straight lines with slopes $2\alpha=\pm 1$, as schematically depicted in Fig. 2. The Hamiltonian (19) usually represents an unpolarized system, in which case the quasiparticle energies $E_{\alpha \mu}$ are degenerate (Fig. 2, top). If $\mathcal{H}$ has time-odd fields (due, e.g., to an external magnetic field or nonzero angular velocity), this Kramers degeneracy is lifted (Fig. 2, bottom).

Figure 2: (Color online) One-quasiparticle levels of both signatures ($\alpha$=1/2: solid line; $\alpha$=-1/2: dotted line) as functions of $\lambda _s$ for a HFB Hamiltonian (19) without (top) and with (bottom) time-odd fields. In the latter case, the Kramers degeneracy at $\lambda _s$=0 is lifted, i.e., states $\vert\alpha \mu\rangle$ and $\hat{T}\vert\alpha \tilde\mu\rangle$ have different energies. The negative energy levels occupied in the quasiparticle vacuum are marked by thick lines. At the points $\lambda_s^{(n)}$ marked by stars, the quasiparticle level $E_n$ with $\alpha$=1/2 becomes occupied. If this level is not degenerate, or its degeneracy is an odd integer, the particle-number parity $\pi_N$ of the vacuum changes as indicated.