Subgroups of D

In order to discuss the conserved and broken symmetries in odd-fermion systems, we now proceed to the discussion of the subgroups of the double group D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ . In fact, the classification of Table 1 can now be repeated almost without change. Indeed, whenever a given D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ subgroup contains the time reversal $\hat{\mathbf{T}}$ , signature $\hat{\mathbf{R}}_{k}$ , or simplex $\hat{\mathbf{S}}_{k}$ , at least one of those, the corresponding subgroup of D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ contains $\hat{\cal{T}}$ , $\hat{\cal{R}}_{k}$ , or $\hat{\cal{S}}_{k}$ , and it automatically becomes a doubled D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ subgroup, with exactly the same generators. This is so, because in the D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ group the squares of the time reversal, signature, and simplex operators are equal to $\bar{\cal{E}}$ , Eq. (5), and hence whenever one of these operators is present in the subgroup, it generates the appropriate double subgroup of the double group D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ . On the other hand, when none of these generators are present in a given D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ subgroup, this subgroup becomes the subgroup of D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ without doubling.

Therefore, all the D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ subgroups listed in Table 1 are simultaneously subgroups of D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ , provided the generators denoted with bold symbols are replaced by the corresponding script generators. Most of the D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ subgroups have twice more elements than the corresponding D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ subgroups, with few exceptions: subgroups $\{\hat{\cal{P}}\}$ , $\{\hat{\cal{R}}_{k}^T\}$ , $\{\hat{\cal{S}}_{k}^T\}$ , $\{\hat{\cal{S}}_{k}^T,\hat{\cal{P}}\}$ do not double, and contain the same number of elements as the corresponding subgroups of D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ .

**Table 2:** Same as in Table 1, but for additional subgroups of the double D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ group.
Type	Generators	Generic	Total
2-0_A:	$\{\hat{\cal{P}},\bar{\cal{E}}\}$	1	1
2-I_A:	$\{\hat{\cal{R}}_{k}^T,\bar{\cal{E}}\}$ , $\{\hat{\cal{S}}_{k}^T,\bar{\cal{E}}\}$	2	6
Total number of two-generator subgroups:		3	7
3-I_A:	$\{\hat{\cal{S}}_{k}^T ,\hat{\cal{P}},\bar{\cal{E}}$ }	1	3
Total number of three-generator subgroups:		1	3
Total number of subgroups:		4	10

In addition, these few exceptional subgroups can be doubled explicitly by adding $\bar{\cal{E}}$ to the set of generators. For completeness, these additional subgroups of D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ are enumerated in Table 2. However, the physical contents of the additional, and of the corresponding not-doubled subgroups from Table 1, are the same. For example, they lead to exactly the same symmetry properties of the density matrices[6]. The difference between them consists in the fact that the latter ones have no irreps in the spinor space, whereas the former ones have one-dimensional irreps with spinor bases (cf.[15]). However, from the view-point of the symmetry breaking, they lead to exactly the same schemes, and thus the additional subgroups shown in Table 2 can be called trivial. Note, that the single-particle operators (e.g., the mean-field Hamiltonian) are classified according to one-dimensional representations [6], and hence, from the view-point of the symmetry breaking it is irrelevant whether or not a given subgroup has spinor representations.

Apart from the phase relations of electromagnetic moments[6] (note that the standard multipole operators are defined by singling out the z axis), the three Cartesian directions are, of course, entirely equivalent. Therefore, even though changing names of axes leads to different subgroups of D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ or D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ , they are identical from the point of view of physically important features. In Tables 1 and 2, we give in the third columns the numbers of generic subgroups, i.e., those which are different irrespective of names of axes, and in the fourth column - the total numbers of subgroups of each type. Index k always denotes one of the axes, i.e., k can be equal to x, y, or z, while indices land m, l $\neq$ m, denote one of the three pairs of different axes.

Subgroups in types ``0'' do not depend on the Cartesian axes and, therefore, for them the numbers of generic subgroups are equal to the total numbers of subgroups. Those in types ``I'' have one generic form each, and three forms in total, depending on which Cartesian axis is chosen. Finally, for subgroups in types ``III'', the total numbers of subgroups can be the same, three times larger, or six times larger (2-III_A) than the numbers of generic subgroups.

In practical applications, conservation of different D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ or D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ subgroups may require considering either only the generic subgroup, or all the subgroups with changed names of axes. For example, if one considers a triaxially deformed system with 0 $^\circ$ $\leq$ $\gamma$ $\leq$ 60 $^\circ$ , the lengths of principal axes a_y $\leq$ a_x $\leq$ a_z define the orientation of the nucleus. Then, conserved D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ or D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ subgroups with different names of axes may lead to different physical consequences. On the other hand, it can be advantageous to consider only one generic subgroup, with a fixed orientation, and allow for various orientations of the physical system by extending allowed values of $\gamma$ deformation beyond the standard first sector 0 $^\circ$ $\leq$ $\gamma$ $\leq$ 60 $^\circ$ .