INTRODUCTION

A major challenge for low-energy nuclear theory is to develop a universal nuclear EDF that can be used to explain and predict static and dynamic properties of atomic nuclei throughout the entire nuclear landscape within the framework of nuclear DFT. In a worldwide effort to develop a general-purpose nuclear EDF [1,2,3], various strategies are applied, and, to realize this vision, the properties of rare isotopes are an essential guide.

In the quest of developing a universal nuclear EDF, the existing functionals ought to be enriched by incorporating the neglected couplings, especially in the spin and isospin channels. Indeed, the recent work [4] suggests that the Skyrme EDF has reached its limits and significant changes to the form of the functional are needed. As far as the isospin sector is concerned, most of the EDFs include isoscalar particle densities and a single $t_z$ component in the isovector channel. The $t_x$ and $t_y$, or p-n mixed, components of isovector densities are completely neglected. In the heavier nuclei where neutrons and protons occupy different shell-model spaces, the neglect of the p-n mixed densities could be justified. However, in the lighter and medium-mass nuclei, neutrons and protons move in the same shells and the exclusion of these isovector densities cannot be justified. There are several observations that indicate deficiencies inherent in the existing EDF and other approaches to describe nuclei in the vicinity of the $N = Z$ line [5,6]. For instance, it is quite well established that binding energies of the nuclei close to the $N = Z$ line are underestimated by theoretical models [7,8], and p-n correlations are expected to be the missing piece of physics in this puzzle.

In some earlier studies, p-n mixing has been investigated in the particle-particle channel [9,10,11,12,13,14] (see Refs. [5,6] and Sec. VI of Ref. [15] for a more complete list of references). In the particle-hole sector, however, the p-n mixing and resulting symmetry breaking effects have been largely neglected (a notable exception is the recent study of $^{21}$Ne [16] that considered a p-n mixing on the HF level). As discussed in [15], such an approximation does not seem to be justified as the self-consistent polarization between particle-hole (p-h) and particle-particle (p-p) HFB channels is known to be strong. In Refs. [17,15] a generalized EDF approach has been proposed that allows for the arbitrary mixing of protons and neutrons, and an isospin-invariant EDF has been constructed. It has been shown that the generalized EDF gives rise to novel spin-isospin combinations of nucleonic densities that are absent in the standard Skyrme approaches. We expect that those extensions may lead to new, hitherto unexplored, nuclear modes.

The main objective of this study is to develop, test, and benchmark the isospin-invariant Skyrme-EDF (pnEDF) approach formulated in the cylindrical coordinate system, whose building blocks are all possible p-n mixed local densities. Since the majority of nuclei are axial in their ground states, such an approach will allow us to extend the global surveys of nuclear properties [18,19,20,21,22] made with the axial DFT solver HFBTHO [23,24] to observables and decays related to isospin. The code HFBTHO has been optimized for performance on flagship computing platforms, for it serves as a backbone of the EDF optimization package [25,26,4]. In a parallel study [27], p-n mixed densities have also been implemented in the general-purpose solver HFODD [28,29] written in a three-dimensional Cartesian basis. We take advantage of this development to benchmark both pnEDF schemes.

The paper is organized as follows. Basic expressions pertaining to the isospin-invariant pnEDF approach are briefly summarized in Sec. 2. Section 3 discusses the HF application of the formalism to isobaric analog states (IASs) using the two-dimensional isocranking formalism. The axial HFBTHO pnEDF framework is benchmarked against the symmetry-unconstrained pnEDF HFODD approach in Sec. 4. Finally, Sec. 5 contains the summary of our work and prospects for further developments.