Methods

The angular-momentum-conserving wave function is obtained by employing the standard operator $\hat{P}^I_{MK}$ [15,16] projecting onto angular momentum $I$, with projections $M$ and $K$ along the laboratory and intrinsic $z$ axes, respectively,

$$\vert IMK\rangle \!=\! \hat{P}^I_{MK} \vert\Phi_{I_y}\rangle \!\e... ...2}\int D^{I*}_{MK}(\Omega)\; \hat{R}(\Omega) \vert\Phi_{I_y}\rangle\; d\Omega .$$ (2)

Here, $\Omega$ represents the set of three Euler angles $\alpha\beta\gamma$, $D^{I*}_{MK}(\Omega)$ are the Wigner functions [17], and $\hat{R}(\Omega)=e^{-i\alpha \hat{I}_z}e^{-i\beta \hat{I}_y} e^{-i\gamma \hat{I}_z}$ is the rotation operator.

Since $K$ is not a good quantum number, different $K$ components must be mixed with the mixing coefficients determined by the minimization of energy. The $K$-mixing is realized in a standard way by assuming:

$$\vert IM\rangle^{(i)} = \sum_K g^{(i)}_K \vert IMK\rangle \equiv \sum_K g^{(i)}_K \hat{P}^I_{MK}\vert\Phi \rangle,$$ (3)

and by solving the following Hill-Wheeler (HW) [18] equation:

$$\mathcal{H} \bar{g}^{(i)} = E_{i} \mathcal{N} \bar{g}^{(i)},$$ (4)

where $\mathcal{H}_{K'K} = \langle \Phi \vert \hat H \hat{P}^I_{K'K} \vert \Phi \rangle$ and $\mathcal{N}_{K'K} = \langle \Phi \vert \hat{P}^I_{K'K} \vert \Phi \rangle$ denote the Hamiltonian and overlap kernels, respectively, and $\bar{g}^{(i)}$ denotes a column of the $g^{(i)}_K$ coefficients. The overlap and Hamiltonian kernels have their standard functional form but depend upon transition (or mixed) density matrices between rotated states:

$$\rho_{\alpha\beta}(\Omega)= \frac{\langle \Phi \vert a^+_\beta a_... ...\Omega)\vert \Phi \rangle} {\langle \Phi\vert\hat{R}(\Omega)\vert\Phi \rangle}.$$ (5)

The transition density matrix is also used for the density-dependent term. This is the only prescription available so far satisfying certain consistency criteria, formulated and thoroughly discussed in Refs. [5,19].

Due to the overcompleteness of the $\vert IMK\rangle$ states, Eq. (4) is solved within the so-called collective subspace spanned by the natural states:

$$\vert mIM\rangle = \frac{1}{\sqrt{n_m}} \sum_{K} \eta_K^{(m)} \vert IMK\rangle ,$$ (6)

which are eigenstates of the norm matrix $\mathcal{N}_{K'K}$ having non-zero eigenvalues ($n_m\ne 0$):

$$\mathcal{N} \bar{\eta}^{(m)} = n_m \bar{\eta}^{(m)}.$$ (7)

In practical applications, the cutoff parameter $\zeta$ is used and the collective subspace is constructed by using only those natural states that satisfy $n_m \geq \zeta$. By ordering indices $m$ of the natural states in such a way that larger indices correspond to smaller norm eigenvalues, we can write the solutions of the HW equation (4) as:

$$\vert IM\rangle^{(i)} = \sum_{m=1}^{m_{\text{max}}} f^{(i)}_m \vert mIM\rangle,$$ (8)

where the mixing coefficients of Eq. (3) read:

$$g^{(i)}_K = \sum_{m=1}^{m_{\text{max}}} \frac{f^{(i)}_m \eta_K^{(m)}}{\sqrt{n_m}}.$$ (9)

We can now define two types of the $K$-mixing. On the one hand, by the kinematic $K$-mixing we understand the situation where only one collective state is used, i.e., $m_{\text{max}}=1$. Then, the solution of the HW equation amounts to calculating only one matrix element of the Hamiltonian kernel,

$$E_1 = \bar{g}^{(1)\dagger}\mathcal{H} \bar{g}^{(1)} = \frac{\bar{\eta}^{(1)\dagger}\mathcal{H} \bar{\eta}^{(1)}}{n_1},$$ (10)

i.e., $f^{(1)}_1=1$ and $f^{(1)}_m=0$ for $m>1$. In the kinematic $K$-mixing, the mixing coefficients $g^{(1)}_K = \eta_K^{(1)}/\sqrt{n_1}$ are entirely determined by the norm kernel and do not depend on the Hamiltonian kernel, i.e., they are entirely given by the cranking approximation and Coriolis coupling. On the other hand, by the dynamic $K$-mixing we understand the full solution of the HW equation for $m_{\text{max}}>1$, where the cutoff parameter $\zeta$ is adjusted so as to obtain a plateau condition for the lowest eigenvalue $E_1$. Here, the generator-coordinate-method (GCM) mixing of different $K$ components becomes effective, which potentially can modify the cranking mixing coefficients. We stress here that the kinematic $K$-mixing does correspond to a $K$-mixed solution too, and is not assuming any single given value of $K$.

The deformed CHF states were provided by the code HFODD, which solves the Hartree-Fock equations that correspond to the Ritz variational principle,

$$\delta \frac{\langle \Phi_{I_y} \vert\hat H-\omega\hat{I}_y\vert \Phi_{I_y} \rangle} {\langle \Phi_{I_y}\vert\Phi_{I_y} \rangle} = 0,$$ (11)

with angular frequency $\omega$ adjusted so as to fulfill constraint (1) and the value of $I_y$ being equal to $I$, according to our $I_y$ $\rightarrow$$I$ scheme. The $y$-signature and parity symmetries were conserved. The Hamiltonian and overlap kernels were calculated using the Gauss-Chebyshev quadrature in the $\alpha$ and $\gamma$ directions and the Gauss-Legendre quadrature in the $\beta$ direction [20]. In the numerical applications presented in this work we used the SLy4 [21] Skyrme force, but similar results were also obtained by using the SIII [22] force. The time-odd terms in the Skyrme functional were fixed by using values of the Landau parameters [23,24]. The harmonic-oscillator basis was composed of 10 spherical shells. The integration over the Euler angles was done by using a cube of 50$\times$50$\times$50 integration points.