Introduction

One of the salient features of the mean-field approach to many-fermion (e.g., nuclear) systems is the spontaneous symmetry breaking. The symmetry of a mean-field state is called broken, if the solution of the Hartree-Fock (HF) or Hartree-Fock-Bogolyubov (HFB) self-consistent equations do not obey symmetries of the original many-body Hamiltonian[1]. This happens when the calculated mean-field energy of the system is lower for states which break a symmetry than that for unbroken symmetries. Such a mechanism depends on the physical situation and is governed by the Jahn-Teller effect [2]. Without going into details, let us recall that the spontaneous breaking of an original symmetry is usually accompanied by a significant decrease in the single-particle level density at the Fermi energy. Hence, the doubly magic nuclei can be safely described by imposing conservation of the spherical symmetry, while this symmetry should be allowed to be broken in the open-shell systems.

One of the simplest examples in this context is that of the breaking of the translational symmetry. The related mechanism is present, e.g., in the nuclear shell-model. Indeed, within the framework of the shell model, interacting nucleons are assumed to move in a common mean field that is localized in space and consequently they cannot be described by eigenstates of the momentum operator (plane waves). In other words, the wave functions of a nucleus cannot be approximated by uncorrelated single-particle plane waves - this can only be attempted for an infinite system, i.e., for the nuclear matter. The use of a shell-model, space-localized wave function simply reflects the correlations present in the system. In this example, the correlations ensure that it is improbable to find two nucleons of a nucleus at large relative distances apart.

In nuclear structure physics one can easily identify the use of various broken symmetries in a description of well-defined, observable effects. For instance the rotational, parity, time-reversal, and gauge symmetry breaking were introduced to describe the deformations, octupole correlations, nuclear rotation and pair correlations, and combined effects thereof. At present, we approach the situation where the mean-field calculations can be performed without explicitly using any of the mean-field symmetries. Several such approaches have already been implemented [3,4,5], although very few calculations for specific physical problems have been done to date.

One could, in principle, perform the mean-field calculations without assuming à priori any symmetry, and let the dynamics choose those discrete symmetries which are, in a specific situation, broken, and those which remain obeyed. Obviously, by choosing such an approach we cannot profit from simplifications possible when it is known beforehand that some symmetries are obeyed/disobeyed. However, following the general guidelines provided by the Jahn-Teller mechanism one usually can make a reasonable choice of obeyed/disobeyed symmetries. Such a choice is dictated by the properties of the many-body Hamiltonian and by the classes of phenomena which one wants to describe - it usually facilitates the calculations markedly. In all those cases the analysis presented in this article provides us with the mathematical means for constructing the algorithms optimally adapted to the symmetries of the problem in question.

In the preceding article[6], we have presented properties of the single point group D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ and double point group D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$, that can be built from operators related to the three mutually perpendicular symmetry axes of the second order, inversion, and time reversal. We have also discussed their roles in the description of even and odd fermion systems, respectively, their representations, and the symmetry conditions induced by the conserved D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ or D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ symmetries on the local densities and electromagnetic moments.

By considering the D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ double point group we focus on quantum objects that are in general non-spherical, but can have one or more symmetry axes and/or symmetry planes. Obviously, any nuclear many-body Hamiltonian of an isolated system is time-even and rotationally invariant. In the present paper we do not aim at analyzing the conditions under which these symmetries are broken spontaneously, with one or another symmetry element of the D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ group still being conserved in the HF solution. Instead, we present a classification of all such possibilities, and discuss the resulting properties of the mean-field Hamiltonians and single-particle wave functions. For a review of applications of point symmetries to a description of rotating nuclei see the recent study in Ref.[7].

Our goal is thus twofold: First, in Sec. 2 we discuss all possible physically meaningful subgroups of D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$, and classify the corresponding physical situations from the view-point of the conserved D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ symmetries. In many applications to date, specific choices of conserved and broken D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ symmetries have been made [8,9,10,11,12,13,14], however, here we aim at a complete description of all achievable symmetry-breaking schemes. Second, in Sec. 3 we review and discuss practical aspects of structure of the mean-field operators under specific D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ group operations. This essential question has been explicitly or tacitly addressed in most approaches using the deformed mean-field theory; our aim here is to present exhaustive list of options pertaining to all the D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ symmetry conditions. Finally, conclusions are presented in Sec. 4.