Double group D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ for odd systems

For odd fermion numbers we consider the Fock-space operators defined in Sec. 2.1, and restrict them to ${\cal{H}}_-$, Eqs. (17) and (19). Since operator $\bar{\cal{E}}$ [odd-fermion-number part of $\bar{E}$ of Eq. (12)] is now an independent group element, additional partner operators should be introduced in order to construct the double group D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$, i.e,

$$\bar{\cal{P}}=\bar{\cal{E}}\hat{\cal{P}}, \quad \bar{\cal{... ...\cal{T}}, \quad \bar{\cal{P}}^T=\bar{\cal{E}}\hat{\cal{P}}^T,$$ (33)

and

$$\bar{\cal{R}}_{k} = \bar{\cal{E}}\hat{\cal{R}}_{k} ,~ \bar{\c... ...^T ,~ \bar{\cal{S}}_{k}^T = \bar{\cal{E}}\hat{\cal{S}}_{k}^T ,$$ (34)

for k=x,y,z. Now the group multiplication table reads

       

$$\hat{\cal{R}}_{k}^2=\hat{\cal{S}}_{k}^2={\hat{\cal{T}}}^2 \bar{\cal{E}},$$ = (35)

$$\left({\hat{\cal{R}}_{k}^T}\right)^2=\left({\hat{\cal{S}}_{k}^T}\right)^2= {\hat{\cal{P}}}^2 \hat{\cal{E}},$$ = (36)

$$\hat{\cal{R}}_{k}\hat{\cal{S}}_{k}=\hat{\cal{S}}_{k}\hat{\cal{R}}_{k} \bar{\cal{P}},$$ = (37)

$$\hat{\cal{R}}_{k}^T\hat{\cal{S}}_{k}^T=\hat{\cal{S}}_{k}^T\hat{\cal{R}}_{k}^T \hat{\cal{P}},$$ = (38)

$$\hat{\cal{R}}_{k}\hat{\cal{R}}_{k}^T=\hat{\cal{R}}_{k}^T\hat{\cal{R}}_{k}=\hat{\cal{S}}_{k}\hat{\cal{S}}_{k}^T=\hat{\cal{S}}_{k}^T\hat{\cal{S}}_{k} \bar{\cal{T}},$$ = (39)

$$\hat{\cal{R}}_{k}\hat{\cal{S}}_{k}^T=\hat{\cal{S}}_{k}^T\hat{\cal{R}}_{k}=\hat{\cal{R}}_{k}^T\hat{\cal{S}}_{k}=\hat{\cal{S}}_{k}\hat{\cal{R}}_{k}^T \bar{\cal{P}}^T,$$ = (40)

for k=x,y,z, and

     

$$\hat{\cal{R}}_{k}\hat{\cal{R}}_{l} = \hat{\cal{S}}_{k}\hat{\cal{S... ...\hat{\cal{R}}_{k}^T\hat{\cal{R}}_{l}^T = \hat{\cal{S}}_{k}^T\hat{\cal{S}}_{l}^T \bar{\cal{R}}_{m},$$ = (41)

$$\hat{\cal{R}}_{k}\hat{\cal{S}}_{l} = \hat{\cal{S}}_{k}\hat{\cal{R... ...\hat{\cal{R}}_{k}^T\hat{\cal{S}}_{l}^T = \hat{\cal{S}}_{k}^T\hat{\cal{R}}_{l}^T \bar{\cal{S}}_{m},$$ = (42)

$$\hat{\cal{R}}_{k}\hat{\cal{R}}_{l}^T = \hat{\cal{R}}_{k}^T\hat{\c... ...} = \hat{\cal{S}}_{k}\hat{\cal{S}}_{l}^T = \hat{\cal{S}}_{k}^T\hat{\cal{S}}_{l} \bar{\cal{R}}_{m}^T,$$ = (43)

$$\hat{\cal{R}}_{k}^T\hat{\cal{S}}_{l} = \hat{\cal{S}}_{k}\hat{\cal... ...T = \hat{\cal{R}}_{k}\hat{\cal{S}}_{l}^T = \hat{\cal{S}}_{k}^T\hat{\cal{R}}_{l} \bar{\cal{S}}_{m}^T,$$ = (44)

for (k,l,m) being an odd permutation of (x,y,z), while relations identical to (26) hold for an even permutation of (x,y,z). After multiplying relations (35) and (41) by $\bar{\cal{E}}$once or twice, one can easily obtain the remaining elements of the multiplication table, i.e., those which pertain to products involving one or two partner operators (34).

The D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ group is thus composed of 32 operators:

 

$$\mbox{D$_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ } : \quad \{ \hat{\cal{E}} , \hat{\cal{P}}, \hat{\cal{T}}, \hat{\cal{P}}^T, \hat{\cal{R}}_{k}, \hat{\cal{S}}_{k}, \hat{\cal{R}}_{k}^T, \hat{\cal{S}}_{k}^T,$$ (45)

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

One can see that this double group is not Abelian, because relations (41) now do depend on whether the permutation (k,l,m) of (x,y,z) is even or odd, whereas for the single group, relations (26) are independent of that.

One may note that the Fock-space operators $\hat{U}$ of Sec. 2.1 and the odd-fermion-space operators $\hat{\cal{U}}$ of Eqs. (17) and (19) obey exactly the same multiplication rules of the double D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ group. Therefore, one might, in principle, consider only the double group D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ and refrain from studying the group structures in even and odd spaces separately. However, at the level of representations, one would then have been deprived of important properties of operators like $\hat{\mathbf{T}}^2$= $\hat{\mathbf{E}}$ or $\hat{\cal{T}}^2$= $-\hat{\cal{E}}$ (see Sec. 2.5), neither of which holds in the whole Fock space, cf. Eqs. (14) and (12).

Since the squares of the time reversal, signatures, and simplexes, Eq. (35), are equal to $\bar{E}$, the whole double group D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ can be generated by the same operators which generate the single group in even systems, Sec. 2.2. So the double group also needs four generators; for instance, the set of four elements, $\hat{\cal{T}}$, $\hat{\cal{P}}$, $\hat{\cal{R}}_{x}$, and $\hat{\cal{R}}_{y}$, can be used to obtain the entire double group of the D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ operators.

The linear operators of the D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ double group form the sixteen-element double group D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {D}}}$,

$$\mbox{D$_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize... ...{E}}, \bar{\cal{P}}, \bar{\cal{R}}_{k}, \bar{\cal{S}}_{k} \}.$$ (46)

This group has 10 equivalence classes (cf. Refs. [2,3,5]). There are 6 classes composed of 2 elements each, i.e., $\{\hat{\cal{R}}_{k}, \bar{\cal{R}}_{k}\}$ and $\{\hat{\cal{S}}_{k},\bar{\cal{S}}_{k} \}$ for k=x,y,z, while the remaining elements: $\{\hat{\cal{E}}\}$, $\{\bar{\cal{E}}\}$, $\{\hat{\cal{P}}\}$ and $\{\bar{\cal{P}}\}$ form 4 one-element classes by themselves. The group is not Abelian and possesses, apart from the 8 one-dimensional irreps already known for the single group, another 2 two-dimensional spinor irreps. The spinor irreps can be labelled by the parities (the eigenvalues $\pi$=+1 or -1 of the inversion operators $\hat{\cal{P}}$), see Appendix.

The time reversal $\hat{\cal{T}}$ is an antilinear and also an antiinvolutive operator [i.e., its representations give the minus identity when squared, Eq. (35)], and therefore it cannot be diagonalized [15] (see Sec. 2.5 below), and used for labelling the D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ ircoreps. A Hermitian antilinear involutive operator, i.e., either a T-signature or a T-simplex [see Eq. (36)] should be chosen to serve this purpose. For instance, a pair of commuting Hermitian operators $\hat{\cal{P}}$ and $\hat{\cal{R}}_{y}^T$ can be used to label the D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ ircoreps. As for the single group D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$, ircoreps being either even or odd with respect to $\hat{\cal{R}}_{y}^T$ can be obtained one from another by a suitable change of phase, and are therefore equivalent. In analogy to the one-dimensional ircoreps, the bases of spinor ircoreps belonging to pairs of eigenvalues of $\{\hat{\cal{P}},\hat{\cal{R}}_{y}^T \}$ equal to $\{+1,+1\}$, $\{-1,+1\}$, $\{+1,-1\}$ and $\{-1,-1\}$ can be called the spinor, pseudospinor, antispinor and antipseudospinor bases, respectively.

Note that only spinor ircoreps of D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ appear in the classification of states of systems with odd numbers of fermions. However, the operators acting in ${\cal{H}}_-$ can all be classified according to the one-dimensional ircoreps of D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$, similarly as operators acting in ${\cal{H}}_+$ can all be classified according to the corresponding one-dimensional ircoreps of D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$. (This is completely analogous to the fact that fermion-number conserving operators can carry only integer angular momenta, i.e., they are integer-rank tensors.) Therefore, whenever we consider the action of the D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ or D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ operators on fermion states we always specify whether they act in even ${\cal{H}}_+$ or odd ${\cal{H}}_-$ spaces, and use for them the corresponding notations $\hat{\mathbf{U}}$ and $\hat{\cal{U}}$ of Eqs. (18) and (19). On the other hand, whenever we consider transformation properties $\hat{U}^\dagger\hat{O}\hat{U}$ of operators $\hat{O}$ with respect to the D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ group, we do not make this distinction, and use for them notation $\hat{U}$ of Sec. 2.1.