Subgroups of D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$

Since the square of every element of D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ is proportional to the identity operator $\hat{\mathbf{E}}$, we have fifteen two-element, one-generator subgroups, each of them composed of the identity and one of the other D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ operators. We denote these subgroups by $\{\hat{\mathbf{G}}_{1}\}$, where $\hat{\mathbf{G}}_{1}$ is the generic symbol corresponding to one of the non-identity elements of D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$. Obviously, only one choice of the generator is possible for every of the two-element subgroups.

Similarly, group D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ has 35 different four-element subgroups, which can be called the two-generator subgroups, and are denoted by symbols $\{\hat{\mathbf{G}}_{1},\hat{\mathbf{G}}_{2}\}$. pertaining to their generators. The two-generator subgroups contain, in addition to $\hat{\mathbf{G}}_{1}$ and $\hat{\mathbf{G}}_{2}$, also the identity $\hat{\mathbf{E}}$ and the product $\hat{\mathbf{G}}_{1}\hat{\mathbf{G}}_{2}$. Since this product is also one of the D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ operators, we have in each of the four-element, two-generator subgroups three possibilities to select the generators.

Finally, there are 15 different eight-element, three-generator subgroups of D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$, denoted by $\{\hat{\mathbf{G}}_{1},\hat{\mathbf{G}}_{2},\hat{\mathbf{G}}_{3}\}$. Each of these subgroups contains the identity $\hat{\mathbf{E}}$, the three generators, three products of pairs of generators, and the product of all three generators. Hence, to choose the generators we may first pick any pair out of seven non-identity elements (21 possibilities), and next pick any other subgroup element, except the product of the first two, (4 possibilities). Since the order in which we pick the generators is irrelevant, one has altogether 28 possibilities of choosing the three generators in each of the eight-element, three-generator subgroups of D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$.

In the same way one can calculate that there is 168 different choices of the four generators of the whole D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ group; one of them is, e.g., the set $\{\hat{\mathbf{T}},\hat{\mathbf{P}},\hat{\mathbf{R}}_{x},\hat{\mathbf{R}}_{y}\}$. This illustrates the degree of arbitrariness in implementing calculations for which the whole group D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ is conserved. Similar freedom, although to a lesser degree, is available when conserving any of the subgroups of D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$. Of course, the freedom of choosing generators cannot influence the final results, however, it allows using different quantum numbers, phase conventions, and structure of matrix elements, as discussed in Sec. 3.

A classification of all the 65 non-trivial subgroups of D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ (we do not include trivial subgroups $\{\hat{\mathbf{E}}\}$ and D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ itself) is presented in Table 1. Every subgroup is assigned to a certain type, and described by a symbol given in the first column of the Table. The types are defined according to: (i) the number of generators in the subgroup (1, 2, or 3), (ii) the number of Cartesian axes involved in the subgroup (0, I, or III standing for 0, 1, or 3), and (iii) the number of signature operators in the subgroup (A, B, or D standing for 0, 1, or 3).

The classification is based on two important characteristics of each subgroup. As shown in Ref.[6], every conserved symmetry, labeled by one of the Cartesian directions x, y, or z, induces a specific symmetry of local densities, related to this particular direction. Therefore, the number of Cartesian axes involved in the subgroup gives us the number of symmetries of local densities induced by the given subgroup. In addition, the number of signature operators illustrates the way in which the given subgroup is located with respect to the standard D $_{\mbox{\rm\scriptsize {2}}}$ subgroup, which is composed of the three signatures.

 

Table 1: Non-trivial subgroups of the single D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ group, classified according to the types described in the text. The second column gives the generators. The third column gives numbers of different subgroups irrespective of names of Cartesian axes, and the fourth column gives the total number of subgroups in each type.

Type	Generators	Generic	Total
1-0A:	$\{\hat{\mathbf{T}}\}$, $\{\hat{\mathbf{P}}\}$, $\{\hat{\mathbf{P}}^T\}$	3	3
1-IA:	$\{\hat{\mathbf{R}}_{k}^T\}$, $\{\hat{\mathbf{S}}_{k}\}$, $\{\hat{\mathbf{S}}_{k}^T\}$	3	9
1-IB:	$\{\hat{\mathbf{R}}_{k}\}$	1	3
Total number of one-generator subgroups:		7	15
2-0A:	$\{\hat{\mathbf{T}},\hat{\mathbf{P}}\}$	1	1
2-IA:	$\{\hat{\mathbf{S}}_{k} ,\hat{\mathbf{T}}\}$, $\{\hat{\mathbf{S}}_{k}^T,\hat{\mathbf{P}}\}$, $\{\hat{\mathbf{R}}_{k}^T,\hat{\mathbf{P}}^T\}$	3	9
2-IB:	$\{\hat{\mathbf{R}}_{k} ,\hat{\mathbf{T}}\}$, $\{\hat{\mathbf{R}}_{k} ,\hat{\mathbf{P}}\}$, $\{\hat{\mathbf{R}}_{k} ,\hat{\mathbf{P}}^T\}$	3	9
2-IIIA:	$\{\hat{\mathbf{S}}_{l} ,\hat{\mathbf{S}}_{m}^T\}$	1	6
2-IIIB:	$\{\hat{\mathbf{R}}_{l} ,\hat{\mathbf{R}}_{m}^T\}$, $\{\hat{\mathbf{R}}_{l} ,\hat{\mathbf{S}}_{m}\}$, $\{\hat{\mathbf{R}}_{l} ,\hat{\mathbf{S}}_{m}^T\}$	3	9
2-IIID:	$\{\hat{\mathbf{R}}_{l} ,\hat{\mathbf{R}}_{m}\}$	1	1
Total number of two-generator subgroups:		12	35
3-IB:	$\{\hat{\mathbf{R}}_{k} ,\hat{\mathbf{T}},\hat{\mathbf{P}}\}$	1	3
3-IIIB:	$\{\hat{\mathbf{R}}_{l} ,\hat{\mathbf{S}}_{m} ,\hat{\mathbf{T}}\}$, $\{\hat{\mathbf{R}}_{l} ,\hat{\mathbf{S}}_{m}^T,\hat{\mathbf{P}}\}$, $\{\hat{\mathbf{R}}_{l} ,\hat{\mathbf{R}}_{m}^T,\hat{\mathbf{P}}^T\}$	3	9
3-IIID:	$\{\hat{\mathbf{R}}_{l} ,\hat{\mathbf{R}}_{m} ,\hat{\mathbf{T}}\}$, $\{\hat{\mathbf{R}}_{l} ,\hat{\mathbf{R}}_{m} ,\hat{\mathbf{P}}\}$, $\{\hat{\mathbf{R}}_{l} ,\hat{\mathbf{R}}_{m} ,\hat{\mathbf{P}}^T\}$	3	3
Total number of three-generator subgroups:		7	15
Total number of subgroups:		26	65

Classification of the subgroups of the single group D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ allows us to discuss conserved and broken-symmetry schemes in even-fermion systems.