The nuclear Density Functional Theory (DFT) has become a conventional approach to describing complex nuclei.[1,2,3] In nuclear theory, the nuclear DFT is closely linked with the Hartree-Fock-Bogoliubov (HFB) approximation to the nuclear many-body problem formulated in the position space. The mean value of the system's energy can here be expressed as a functional of the single-particle density matrices in the particle-hole (p-h) and particle-particle (p-p or pairing) channels. The density matrices in both channels are expressed through the scalar-isoscalar, scalar-isovector, vector-isoscalar and vector-isovector densities, which are nonlocal, i.e., they depend on two position vectors. In the local theory, e.g., the local density approximation, the nonlocal densities can be represented in the energy density through local densities and their derivatives.
The nuclear self-consistent mean field resulting from the DFT may spontaneously break the symmetry of the original many-body Hamiltonian. The symmetry-breaking phenomenon allows for capturing important physics effects and for including essential correlations in the many-body system. The existence of a self-consistent symmetry (SCS), i.e., a symmetry obeyed by the mean-field Hamiltonian, reflects physical properties of the system. Due to the self-consistency, the SCS of the mean field is also the symmetry of the density matrix. This has a number of important consequences. First of all, to understand SCSs of DFT, it is sufficient to study symmetries of the density matrix without considering the specific form of the density functional and resulting mean-field Hamiltonian. We also realize that if the initial density matrix used at the first iteration of the self-consistent procedure has a certain SCS, then this symmetry will propagate through to the final DFT solution. This means that the choice of the initial density matrix is crucial for the proper physical content of the DFT results. Sometimes, e.g., to simplify numerical calculations, a SCS of the initial density matrix is ad hoc assumed. Such an assumption may lead to overestimation of the DFT energy and incorrect prediction of many properties of the system. Finally, assuming a SCS, one should know the general form of the density matrix possessing that symmetry.
Some point symmetries and the associated symmetry-breaking schemes of the p-h densities have been investigated in Refs.[4,5] A detailed analysis of SCSs of the DFT densities in both channels has recently been presented in Ref.[6], which will be referred to as I in the following. The present paper is a comment to I, with the main goal of detailing the consequences of spatial symmetries discussed therein. Although our discussion remains rigorous, in this contribution we simplify the notation so as to make the results more transparent and understandable.
As in I, we treat the nonlocal and local densities as isotropic tensor fields, i.e., functions of the position vector(s) independent of other tensor quantities (material tensors).[7] Under this assumption, in Refs.[6,8] we introduced the Generalized Cayley-Hamilton (GCH) theorem for tensor fields, and we used it as a tool to study general forms of local densities. Since many different generalizations of the Cayley-Hamilton theorem exist in the literature, in Sec. 6 we give a brief explanation of the present version.
In Sec. 2 we recall definitions of the density matrices and nonlocal and local densities. The spherical symmetry with and without space inversion is discussed in Sec. 3. In Sec. 4, we consider the axial symmetry with and without space reflection. The summary of our study is contained in Sec. 5.