Subgroups of D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ and D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ and the symmetry breaking

The single group D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ and double group D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ [6] can be built from three rotations through angle $\pi$ about the coordinate axes k=x,y,z, called the signature operators,

$$\hat{R}_{k} = e^{-i\pi\hat{I}_k},$$ (1)

to which one adds the inversion operator $\hat{P}$ and the time-reversal

$$\hat{T}= \bigotimes_{n=1}^A\left(-i\hat{\sigma}_y^{(n)}\right) \hat{K},$$ (2)

where $\hat{I}_k$= $\sum_{n=1}^A{\hat{j}}_k^{(n)}$ is the total angular momentum operator, ${\hat{j}}_k^{(n)}$ and $\hat{\sigma}_k^{(n)}$are the angular momenta and the Pauli spin matrices for the particle number n, respectively, and $\hat{K}$ is the complex-conjugation operator in the coordinate representation.

Following the convention introduced in [6], with roman symbols, like $\hat{U}$= $\hat{R}_{k}$ or $\hat{T}$, we denote operators acting in the Fock space ${\cal{H}}$$\equiv$ ${\cal{H}}_0\oplus{\cal{H}}_1\oplus\ldots \oplus{\cal{H}}_A\oplus\ldots$. Moreover, in order to help the reader in distinguishing between properties of these operators when they act in even, ${\cal{H}}_+$$\equiv$ ${\cal{H}}_0\oplus{\cal{H}}_2\oplus\ldots \oplus{\cal{H}}_{A=2p}\oplus\ldots$, or odd, ${\cal{H}}_-$$\equiv$ ${\cal{H}}_1\oplus{\cal{H}}_3\oplus\ldots \oplus{\cal{H}}_{A=2p+1}\oplus\ldots$, fermion spaces, we denote the former ones with bold symbols, and the latter ones with script symbols, i.e., we formally split the Fock-space operators $\hat{U}$= $\hat{\mathbf{U}}$+ $\hat{\cal{U}}$into two parts according to their domains.

It follows [15,16,6] that D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ is an Abelian group of 16 elements, which contains: the identity $\hat{\mathbf{E}}$, inversion $\hat{\mathbf{P}}$, time-reversal $\hat{\mathbf{T}}$, their product $\hat{\mathbf{P}}^T$= $\hat{\mathbf{P}}\hat{\mathbf{T}}$, three signatures $\hat{\mathbf{R}}_{k}$, three simplexes $\hat{\mathbf{S}}_{k}$= $\hat{\mathbf{P}}\hat{\mathbf{R}}_{k}$, three T-signatures $\hat{\mathbf{R}}_{k}^T$= $\hat{\mathbf{T}}\hat{\mathbf{R}}_{k}$, and three T-simplexes $\hat{\mathbf{S}}_{k}^T$= $\hat{\mathbf{T}}\hat{\mathbf{S}}_{k}$, i.e.,

$$\mbox{D$_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize... ...hbf{P}}^T, \hat{\mathbf{R}}_{k}^T, \hat{\mathbf{S}}_{k}^T \},$$ (3)

where all these operators act in even-fermion-number space ${\cal{H}}_+$.

Similarly, the Fock-space operators $\hat{U}$, as well as the odd-fermion-number operators $\hat{\cal{U}}$, form the group D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ which is a non-Abelian group of 32 elements. Apart from the 16 operators enumerated for D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$, it contains their partners obtained by multiplying every one of them by the operators $\bar{E}$ or $\bar{\cal{E}}$, respectively. These operators can be identified with the rotation operators through angle $2\pi$ about an arbitrary axis. The partner operators are denoted by replacing the hats with bars, i.e., the group of operators acting in ${\cal{H}}_-$reads

 

$$\mbox{D$_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ } : \quad \{ \hat{\cal{E}} , \hat{\cal{P}}, \hat{\cal{T}}, \hat{\cal{P}}^T, \hat{\cal{R}}_{k}, \hat{\cal{S}}_{k}, \hat{\cal{R}}_{k}^T, \hat{\cal{S}}_{k}^T,$$ (4)

$$\bar{\cal{E}} , \bar{\cal{P}}, \bar{\cal{T}}, \bar{\cal{P}}^T, \bar{\cal{R}}_{k}, \bar{\cal{S}}_{k}, \bar{\cal{R}}_{k}^T, \bar{\cal{S}}_{k}^T \}.$$

The complete D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ and D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ multiplication tables have been given and discussed in Ref.[6], and will not be repeated here. We only recall a few properties of the D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ group that are essential in the following analysis, namely,

   

$$\hat{\cal{R}}_{k}^2=\hat{\cal{S}}_{k}^2={\hat{\cal{T}}}^2 \bar{\cal{E}},$$ = (5)

$$\left({\hat{\cal{R}}_{k}^T}\right)^2=\left({\hat{\cal{S}}_{k}^T}\right)^2= {\hat{\cal{P}}}^2 \hat{\cal{E}},$$ = (6)

for k=x,y,z,

$$\hat{\cal{R}}_{k}\hat{\cal{R}}_{l} = \hat{\cal{S}}_{k}\hat{\c... ... = \hat{\cal{S}}_{k}^T\hat{\cal{S}}_{l}^T = \hat{\cal{R}}_{m},$$ (7)

for (k,l,m) being an even permutation of (x,y,z), and

$$\hat{\cal{R}}_{k}\hat{\cal{R}}_{l} = \hat{\cal{S}}_{k}\hat{\c... ... = \hat{\cal{S}}_{k}^T\hat{\cal{S}}_{l}^T = \bar{\cal{R}}_{m},$$ (8)

for (k,l,m) being an odd permutation of (x,y,z).

The multiplication table of D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ is obtained by replacing $\bar{\cal{E}}$ and $\bar{\cal{R}}_{m}$ by $\hat{\mathbf{E}}$ and $\hat{\mathbf{R}}_{m}$, respectively, and using all bold symbols in Eqs. (5)-(8).

Obviously, a product of conserved symmetries is a conserved symmetry, and consequently, the conserved symmetries form groups that are subgroups of D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ or D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$. Therefore, in order to analyze various physically meaningful subsets of the conserved D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ or D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ operators, we should first consider the subgroup structure of these groups.

Suppose that in a given physical problem, the mean-field states obey the symmetries of a given subgroup rather than those of the whole D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ or D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ groups. In such a case that subgroup contains the maximal set of operators representing the symmetry of the problem, i.e., all D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ od D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ operators which do not belong to such subgroup are the broken symmetries. From the view-point of physics, we are more interested in the symmetries which are broken (which is related to interesting dynamical correlations), than in those which are conserved. It then follows that the physically interesting information will be attached to the operators that do not belong to the subgroup studied, but do belong to D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ or D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$; those latter ones do not necessarily form a group.

First we consider the single group D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$, because (i) it is a smaller and simpler group than D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$, and (ii) the operator $\bar{\cal{E}}$which makes the difference between the single and the double group is always a conserved symmetry.

The analysis below is based on identifying sets of the so-called subgroup generators, i.e, operators from which the whole given subgroup can be obtained by their successive multiplications. Choices of generators are, of course, non-unique, and hence in each case we discuss and enumerate all the available possibilities.