Introduction

The energy density functional (EDF) method in nuclear physics is nowadays the approach of choice for large-scale nuclear structure calculation. It has the same roots as the density functional theory (DFT) [1] in atomic and molecular physics, which is based on the Coulomb interaction between electrons. In nuclear physics, the EDF approach still lacks firm microscopic derivation and well-defined rules that would allow for systematic construction of an exact (or optimal) functional. In practice, the nuclear EDF is constructed phenomenologically, based on the knowledge accumulated within modern self-consistent mean-field approaches built upon effective density-dependent two-body interaction.

These approaches, although successful in reproducing gross nuclear properties and certain generic features of collective and single-particle nuclear motion are only coarse in confrontation with precise spectroscopic data. This situation calls for improvement, which can be achieved in two directions, namely, either explicitly, by using better and/or more complicated parameterizations of the nuclear EDF, or implicitly, by going beyond the mean-field or Hartee-Fock (HF) level. At present, intense studies are being conducted in both directions, aiming at exploring the limits thereof and answering the question of whether they can be considered equivalent, complementary, or independent. In this work we will explore the second alternative, by employing the angular-momentum-projection (AMP) method of cranked HF (CHF) states.

The spontaneous symmetry breaking (SSB) mechanism is inherently built into the HF approach. In many cases, it allows not only for incorporating a significant fraction of many-body correlations into a single HF state, but also serves as a source of deep physical intuition. Emergence of nuclear deformation (breaking of rotational invariance) leads naturally to the collective rotational motion, and is one of the most spectacular manifestations of the SSB in nuclear physics.

The CHF approximation treats collective (rotational) and intrinsic degrees of freedom on the same self-consistent footing. This fact is at the base of the success of this simple approximation, because in atomic nuclei both energy scales are strongly interwoven, and dramatic structural changes may take place along rotational bands. Terminating rotational bands [2] are the best examples of such possible changes, whereupon the collective rotation is followed by the total alignment of valence particles.

Although the energies of rotational states are correctly reproduced by the CHF approach, and the changes of shape and pairing as well as the recoupling processes of individual nucleonic orbits are well captured, the price paid is high. Indeed, the resulting wave packets (deformed CHF states) $\vert\Phi_{I_y}\rangle$ are well localized in the angular degrees of freedom and thus they are broadly spread over many angular momenta $\vert\Phi_{I_y}\rangle = \sum_I w_I \vert I\rangle$, with only the average value of the projection of angular momentum on one the axes (the $y$ axis in our case) being constrained,

$$\langle\Phi_{I_y}\vert \hat I_y \vert\Phi_{I_y}\rangle \equiv \langle \hat I_y \rangle = I_y.$$ (1)

Apart from strongly deformed states, this feature of the cranking approximation precludes applications of this formalism to compute transition rates, which constitute extremely valuable source of structural information. The demand for symmetry-restoration is therefore well motivated. Starting from deformed CHF state $\vert\Phi_{I_y}\rangle$, the goal can be achieved by projecting onto the eigenspaces of the angular momentum. Most of the calculations that have been performed so far invoke the angular momentum method applied to non-rotating states, see, e.g., Refs. [3,4,5] and the reviews in Refs. [6,9,7,8]. This limits their applicability to low spin states, where the influence of rotation on the intrinsic states can be neglected, and leads to an overestimate of the nuclear moment of inertia (MoI) as compared to the (realistic) cranking estimate.

After the ground-breaking studies in Refs. [10,11], the AMP of CHF states has not been performed in modern self-consistent calculations in nuclear structure. Recently, in Ref. [12], we presented results of such calculations for the test case of collective rotation of a well-deformed nucleus $^{156}$Gd. The AMP procedure we used has been implemented within the code HFODD [13,14]. The calculational scheme proposed and tested in Ref. [12], dubbed hereafter $I_y$ $\rightarrow$$I$ scheme, assumes the AMP of spin component $I$ of the self-consistent CHF state $\vert\Phi_{I_y}\rangle$, which is constrained to the same mean value of the projection on the $y$ axis, i.e., $\langle \hat I_y\rangle=I$. It combines the simplicity of the self-consistent 1D cranking approach, and its ability to reproduce the correct MoI, with the AMP after variation method.

In this paper, within the same formalism, we present the first systematic calculations of rotational states along terminating bands in the $A$$\sim$44 mass region. The paper is organized as follows. Methods of calculation and results are presented in Secs. 2 and 3, respectively. In particular, the AMP of cranked states along the rotational band in $^{46}$Ti is discussed in Sec. 3.1 and the AMP of states near the band termination is presented in Sec. 3.2. Finally, summary and discussion are given in Sec. 4.