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Configuration interaction projected density functional theory: effects of four-quasiparticle configurations and time-odd interactions

The effects of four-quasiparticle configurations and time-odd interactions are investigated in the framework of configuration interaction projected density functional theory by taking the yrast states of 60Fe as examples. Based on the universal PC-PK1 density functional, the energies of the yrast states with spin up to 20\hbar and the available B(E2) transition probabilities are well reproduced. The yrast states are predicted to be of four-quasiparticle structure above spin I = 16\hbar. The inclusion of the time-odd interactions increases the kinetic moments of inertia and delays the appearance of the first band crossing, and, thus, improves the description of the data.


I. INTRODUCTION
The configuration interaction shell model (CI-SM) [1] and the nuclear density functional theory (DFT) [2][3][4] are widely used approaches for the description of nuclear properties. The CI-SM describes nuclear spectroscopic properties in the laboratory frame and captures both the excitation of single particles and the collective correlations coming from the coherent motion of many nucleons. Because of the limitation of its model space, it is usually applied to light and medium heavy nuclei. The nuclear DFT is based on the idea that the ground-state energy of a nucleus can be expressed as a functional of the nucleon density, and is applicable for nuclei all over the nuclide chart. The nuclear DFT takes into account many-body correlations by breaking essential symmetries and describes fruitful physics around the minima of the potential energy surface. However, the nuclear DFT, in the first place, is limited to describe nuclear ground states. For quantitative investigations of nuclear spectroscopic properties, one needs to resort to proper extensions based on the DFT.
During the past decades, great efforts have been made to extend the nuclear DFT for nuclear spectroscopic properties. The generator coordinate method (GCM) [5] which includes the quantum fluctuations along some relevant collective variables, is one of the effective approaches. The GCM approach has been implemented in nonrelativistic [6][7][8] and relativistic [9,10] DFTs, and has achieved great successes in the shape coexistence [11][12][13], the erosion of shell-closure [14], the collective band structures of superheavy nuclei [15], the novel modes of nuclear weak decays [16,17], etc. Nevertheless, the DFT-based GCM approaches are limited to describe the spectra with energies below the first two-quasiparticle excitations [18]. For the description of spectra with higher excitation energies, the cranking model [19] based on nonrelativistic [20,21] and relativistic [22,23] DFTs have been demonstrated as powerful tools. Moreover, the two-dimensional and the three-dimensional tilted axis cranking DFTs have been successfully applied to many novel rotational phenomena [24][25][26][27][28]. However, the band crossing phenomena cannot be properly described by the cranking approach [29] because the cranking states are obtained at the constant rotational frequency rather than the constant angular momentum. Recent progresses on considering the correlations due to the symmetry breaking and mixing states beyond the rotating mean field have been made based on the nonrelativistic DFTs [30,31].
To describe the nuclear spectroscopic properties with beyond mean-field correlations up to high spin, the configuration interaction projected density functional theory (CI-PDFT) was proposed in Ref. [32]. The starting point of CI-PDFT is a well-defined density functional whose self-consistent solution corresponds to a state at the energy minimum of the potential energy surface and contains already important physics. The additional correlations required for the description of spectroscopic properties are considered with shell-model calculations in a configuration space built on top of the self-consistent solution. Due to the fact that such a configuration space is constructed with the optimized bases, its dimension is much smaller than that in the traditional shell models. The CI-PDFT could be used to describe the nuclear spectroscopic properties up to high spin without additional parameters beyond the well-defined density functional. The CI-PDFT has been later implemented in the framework of nonrelativistic DFTs [33,34] where the authors use an alternative name, i.e., no-core configuration-interaction approach rooted in multireference DFT.
In the previous application of the CI-PDFT [32], the four-quasiparticle configurations are not included. Moreover, the time-odd parts of the density functional are neglected for simplicity. As stated in Ref. [32], the consideration of time-odd components and fourquasiparticle configurations would give better agreements with the experimental data for higher spin states. In fact, the investigations within the projected shell model (PSM) based on the phenomenological pairing-plus-quadrupole interaction, which includes the fourquasiparticle configurations, can reproduce the higher spin states well [36]. In the present work, the CI-PDFT including contributions from the four-quasiparticle configurations and time-odd interactions is developed. By taking the yrast states of 60 Fe populated up to spin I = 20 [35] as examples, the effects of four-quasiparticle configurations and time-odd interactions are discussed.

II. THEORETICAL FRAMEWORK
The starting point of the point-coupling relativistic density functional is the following Lagrangian density [3], which contains 9 coupling constants α S , Details can be seen in Ref. [3].
For open-shell nuclei, a unified treatment of both the mean field and pairing correlations is realized by solving the following relativistic Hartree-Bogoliubov (RHB) equation [3], Here, U k and V k are the quasiparticle (qp) wavefunctions,ĥ D is the single-particle Dirac and λ is the Fermi surface. The scalar field S and vector field V µ are connected in a selfconsistent way to various densities and currents [3]. The matrix element of the pairing field ∆ is where V pp is the pairing force, and κ is the pairing tensor determined by the qp wavefunctions. In the present work, V pp is chosen as the finite-range separable force proposed in Ref. [37]. By solving the RHB equation (2) iteratively, one can obtain the intrinsic ground state |Φ 0 , together with a set of qp orbits i defined bŷ in whichβ i represents the qp annihilation operator [5].
Starting from the intrinsic ground state |Φ 0 , one can construct the multi-qp configurations. For example, the multi-qp states up to four-qp configurations for even-even nuclei These states form the configuration space of the CI-PDFT. In general, multi-qp configurations with half protons and half neutrons are energy favored. Therefore, four-qp configurations with all like particles could be excluded if one is interested in only the yrast states [38,39].
The wavefunction |Ψ σ IM in the laboratory frame with good angular momentum quantum numbers I and M is written as [32] Here, η runs over the intrinsic states taken from the configuration space in Eq. (6), and P I M K is the angular momentum projection operator. Similar to Ref. [32], each multi-qp state |Φ η is assumed to be axially deformed and is characterized by the total intrinsic magnetic quantum number K. The expansion coefficients F Iσ η are determined by the diagonalization of the Hamiltonian in a shell-model subspace spanned by the nonorthogonal projected bases {P I M K |Φ η }, or equivalently by solving the following generalized eigenvalue equation The energy kernel H I ηη ′ and the norm matrix N I ηη ′ are defined respectively as Here,Ĥ represents the Hamiltonian derived from the Lagrangian density (1) including the pairing term. The energy kernel and norm matrix are calculated by the Pfaffian algorithms proposed in Refs. [40,41]. In the present work, the particle number projection is not included, and an approximate scheme to correct the mean value of the particle number is used [42]. The time-odd interactions associated with the space-like components of vector and isovector channels are considered.
The generalized eigenvalue equation (8) is solved by the standard procedure. Once the expansion coefficients F Iσ η are known, one can construct the collective wavefunctions [5] G Iσ η = η ′ which are interpreted as the probability amplitudes of the states |Φ η . The reduced E2 transition probabilities between an initial state I i and a final state I f is defined by Note that e denotes the bare value of the proton charge, and there is no need to introduce the effective charges in the CI-PDFT calculations [32]. states as well as one intrinsic ground state |Φ 0 are included in the CI-PDFT calculations.
The energies of angular momentum projected states for the 58 configurations as functions of spin are plotted in the inset of Fig. 2. Such a plot is also named as the band diagram [38], which provides useful information for understanding the structure changes even before the diagonalization is carried out. The energy of a band η is defined as, which represents the expectation value of the Hamiltonian with respect to the projected qp states |Φ η . Because the configurations are axially deformed in the present work, it is convenient to use the quantum number K to classify them.
From the probability amplitudes |G I η | 2 shown in Fig. 4, it is seen that the 0qp state plays dominated roles for the yrast states with I ≤ 6 . The corresponding probability amplitude |G I η | 2 is 0.90 for state with I = 0 and then decreases gradually with spin, being 0.13 for the yrast state with I = 8 . The |G I η | 2 of the ν2qp neutron states with K = 1 and 2 increase suddenly at I = 8 , which indicates that the yrast band structure changes from the 0qp