Introduction

Density Functional Theory (DFT) is a universal approach used in quantum chemistry, molecular physics, and condensed matter physics to calculate properties of electronic systems. Its extension to nuclear physics is by no means trivial, encountering difficulties associated in part with the binary composition of atomic nuclei, spin-dependent interactions, superfluidity, and strong surface effects. Major difference between the electronic and nuclear DFT is associated with the lack of external binding potential, as atomic nuclei are self-bound systems, and with the saturation of nuclear interactions at a given value of density. This implies that the nuclear DFT must necessarily be formulated in terms of intrinsic, and not laboratory densities, which, in turn, leads to the spontaneous breaking of fundamental symmetries.

In spite of these difficulties, the nuclear DFT is the microscopic tool of choice to study in a systematic manner medium-mass and heavy nuclei. In the symmetry-broken mean-field variant, often referred to as single-reference (SR) DFT, the method has proven to be extremely successful in reproducing and predicting bulk nuclear properties like masses, quadrupole moments, or nuclear radii. However, for a precise description of numerous observables, the SR nuclear DFT is inadequate. In particular, at the SR level, matrix elements of electromagnetic transitions or beta decays can only be treated within a quasiclassical approximation. Fully quantal calculations of such observables are impossible without the symmetry restoration, which requires extensions from the SR to multi-reference (MR) DFT.

However, within the MR DFT, implementation of the symmetry restoration is plagued with technical and conceptual difficulties [1,2,3]. The reason is that the SR DFT, serving as the starting point, is usually derived from an effective, density-dependent pseudo-potential, and is therefore not directly related to a Hamiltonian. The only reasonable, and to a large extent unambiguous generalization of the SR energy density functional (EDF) to the MR level is possible within the generalized Wick's theorem (GWT) [4] that establishes a one-to-one correspondence between the SR and MR functionals. In such an implementation, the MR EDF retains the form of the underlying SR EDF, but is solely expressed in terms of the so-called transition densities. Unfortunately, the resulting MR EDFs are, in general, singular and require regularization. In spite of preliminary attempts, concentrating mostly on a direct removal of self-pairing effects [2,3], the problem of regularization still lacks satisfactory and practical solution. The aim of this work is to propose such a solution.

The paper is organized as follows. In Sec. 2.1, we recall the standard formulation of the MR DFT scheme based on the GWT, and we identify sources of potential pathologies. Then, in subsections 2.2 and 2.3 we present two variants of the new regularization scheme, hereafter called linear (LR) and quadratic regularization (QR), respectively. Our method is illustrated by applications to the angular-momentum projection (AMP) problem, but can similarly be used to restore other broken symmetries. In particular, the particle-number restoration within the pairing-plus-quadrupole Hamiltonian is currently studied in Ref. [5]. Summary and perspectives are discussed in Sect. 3.