Superconductivity plays a central role in describing low-temperature properties of correlated many-fermion systems. Within the mean-field theory, fermionic pairing is best treated in the Hartree-Fock-Bogoliubov (HFB) [1] or Bogoliubov-de Gennes (BdG) [2] formalism. In the presence of superconducting condensate, the standard product state ansatz for the nuclear wave function breaks the particle-number (PN) symmetry [1,3]. In principle, the broken symmetry needs to be restored, especially if one looks at observables that strongly depend on PN. The many-body correlations associated with the symmetry-breaking are particularly important for small systems where the finite-size effects are appreciable, such as atomic nuclei or metallic grains, or in the limit of weak pairing where pairing correlations have dynamic character.
For complex superconducting system,s a theoretical tool of choice is the Density Functional Theory (DFT) [4,5]. The theory is built on theorems showing the existence of energy functionals for many-body systems, which include, in principle, all many-body correlations. The generalization of the DFT to the case of fermionic pairing was formulated for electronic superconductors in Refs. [6,7,8]. The resulting HFB/BdG equations can be viewed as the generalized Kohn-Sham equations of the standard DFT.
In the nuclear case, the DFT is the only tractable theory that can be applied across the entire table of nuclides. Historically, the first nuclear energy density functionals appeared long ago [9,10,11] in the context of the Hartree-Fock (HF) method used with zero-range, density-dependent interactions such as the Skyrme force. The main ingredient of the nuclear DFT [12] is the energy density functional that depends on densities and currents representing distributions of nucleonic matter, spins, momentum, and kinetic energy, as well as their derivatives (gradient terms). To account for nuclear superfluidity, the functional is augmented by the pairing term (see Ref. [13] for a review). The challenges faced by the nuclear DFT are: (i) the existence of two kinds of fermions; (ii) the essential role of pairing; and (iii) the need for symmetry restoration in finite, self-bound systems. The two latter points are of particular importance in the context of this study. The features (i) and (iii) are specifically nuclear; with very few exceptions, they are not present in the electronic Coulomb problem.
It is important to recall that the realistic energy density functional does not have to be related to any given effective Hamiltonian, i.e., an effective interaction could be secondary to the functional. This strategy is used in all modern nuclear DFT applications. In the absence of a Hamiltonian (and wave function), the restoration of spontaneously broken symmetries in DFT poses a conceptional dilemma [14,15,16,17] and a serious challenge that needs to be properly addressed. One important question related to DFT for self-bound systems concerns the functional itself: how do you construct it in terms of intrinsic (body-fixed) densities? While it is possible to formulate the Kohn-Sham procedure in language of intrinsic densities [18,19], the pathway to practical applications is still not clear.
Sticking to DFT for superconductors and PN symmetry, several schemes can be adopted. One is to formulate the theory in language of the usual (particle) density only, without explicitly invoking the anomalous (pair) density that is at the heart of the PN symmetry violation [20,21]. Another strategy is to incorporate the PN restoration procedure into the DFT framework. This can be done by employing the generalized Wick's theorem (see, e.g., [22,23,24]). Recently, full PN projection before variation has been carried out for the first time within the Skyrme-DFT framework employing zero-range pairing [25,26,23]. It was demonstrated that the resulting projected DFT equations (similar to the PN-conserving HFB equations originally proposed in Refs. [27,28]) can be obtained from the standard Skyrme-HFB equations in coordinate space by replacing the intrinsic densities and currents by their gauge-angle-dependent counterparts. Using the variation-after-projection method, one can properly describe transitions between normal and superconducting phases in finite systems, which are inherent in atomic nuclei.
As mentioned above, the restoration of broken symmetries in the framework of DFT causes a number of questions, mainly related to the density dependence of the underlying interaction and to different treatment of particle-hole and particle-particle channels [22,25,29]. For instance, it has been realized for some time [30,31,32,22,25,29] that the PN projection applied within the DFT framework is plagued with difficulties related to vanishing overlaps between gauge-rotated intrinsic states. This concerns any functional that uses density-dependent terms and thus is not related to an average of a Hamiltonian. In particular, the most frequently used approaches based on the Skyrme, Gogny, or relativistic-mean-field functionals all fall into this category.
In this study we investigate the analytic structure of the projected DFT, focusing on origins of difficulties. In recent works [33,34], a way to remedy some of the problems has been proposed. The PN-projected Skyrme-DFT formalism employed in our work has been outlined in Ref. [23], and we follow their notation. Our manuscript is organized as follows. The analytic structure of the projected HFB is discussed in Sec. 2. The DFT extension of the formalism is described in 3. Numerical examples are contained in 4. Finally, Sec. 5 contains conclusions of this work.