Introduction

Density functional theory (DFT) in its original formulation [1,2] and its extended versions [3,4,5,6] has became a universal approach to compute the ground-state (g.s.) and excited [7,8] configurations of many-electron systems held together by an external one-body potential in condensed-matter, atomic, and molecular physics. The DFT strategy has also been used in the area of nuclear structure. The nuclear DFT, a natural extension of the self-consistent mean-field (MF) theory [9,10] is a tool of choice for computations of g.s. properties and low-lying excitations of medium-mass and heavy nuclei.

There are a number of differences between the electronic DFT and nuclear DFT. First of all, because of the absence of external potential, nuclei are self-bound systems, and this creates conceptual problems due to the distinction between intrinsic (symmetry-broken) and laboratory-system densities. This problem can be treated within the extended DFT framework of Levy-Lieb [3,4,5,6,11], see Refs. [12,13,14,15]. The second difference is related to the effective interaction. A complicated form of the effective nucleon-nucleon interaction, as well as complexity of the nuclear matter saturation-mechanism (pertaining to three-body forces), give rise to the presence of higher-order terms in the nuclear energy density functional (EDF). Hence, the EDF cannot be constructed in a model-independent ab initio way based solely on bulk properties of infinite-medium augmented by exchange and gradient corrections, as it is usually done for electronic systems. The current best nuclear EDFs are still phenomenological in nature, either inspired by a MF averaging of the effective interaction such as the zero-range Skyrme [16] force, or based on systematic expansion involving symmetry-constrained gradient terms [17]. By construction, the nuclear EDF is characterized by a certain number of free parameters which are fitted directly to empirical data.

The third major difference between electronic and nucleonic DFT, addressed in this study, is the presence of two types of fermions and the charge independence of the nuclear interactions giving rise to the isospin symmetry. Since the isospin symmetry is violated essentially by the Coulomb interaction, which is much weaker than the strong interaction between nucleons, many effects associated with the isospin breaking in nuclei can be treated in a perturbative way, making the formalism of isotopic spin a very powerful concept in nuclear structure and reactions [18,19].

Up to electromagnetic effects, the isospin symmetry should be conserved by elementary excitations of nuclei. This is not the case for the elementary excitations modes of the nuclear Hartree-Fock (HF) or Kohn-Sham theory, i.e., proton and/or neutron particle-hole (p-h) excitations. The independent-particle wave function manifestly breaks the isospin symmetry in the g.s. configurations of odd-odd $N$=$Z$ nuclei and in all other but isoscalar excited states. Two prominent examples are discussed in this work: superdeformed (SD) bands in a doubly magic nucleus $^{56}$Ni [20] and terminating states in $N$=$Z$ $A\sim$40 nuclei [21].

The paper is organized as follows. We begin in Sec. 2 with general discussion of g.s. configurations of odd-odd $N$=$Z$ nuclei and p-h excitations in even-even (e-e) nuclei, so as to qualitatively introduce problems faced by MF-based theories around $N$=$Z$ and to provide motivation for the necessity of isospin restoration. Section 3 contains a detailed presentation of the isospin-projected DFT approach. In particular, we demonstrate that this approach is essentially free from problems related to uncompensated poles plaguing projection techniques [22]; hence, it requires no further regularization [23]. Applications of the isospin-projected DFT approach to the SD bands in $^{56}$Ni and terminating states in $N$=$Z$ isotopes of Ca, Sc, Ti, V, and Cr, are presented in Secs. 4.1 and 4.2, respectively. Finally, Sec. 5 contains the conclusions of this work.