.  

In this Appendix we explicitly construct irreducible representations of the D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ group by using the example of the HO basis (Sec. 2.4), and we illustrate the Wigner classification of groups that contain antilinear operators (see Chap. 26 of Ref. [8]). The results of such an analysis were used in Sec. 2.

We consider here only the two-dimensional spinor representations, appropriate for the odd-fermion systems and in particular for the single-particle states. From Eqs. (47) one finds representation matrices $\Gamma(\hat{\cal{U}})$ (where operators $\hat{\cal{U}}$$\in$D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {D}}}$ of Sec. 2.3 form the double group D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {D}}}$), in the two-dimensional invariant subspace spanned by $\vert n_xn_yn_z,s_z\mbox{=}+\frac{1}{2}\rangle$ and $\vert n_xn_yn_z,s_z\mbox{=}-\frac{1}{2}\rangle$. We have

     

$$\Gamma(\,\hat{\cal{E}}\,) -\Gamma(\,\bar{\cal{E}}\,) = \sigma_0 ,$$ = (91)

$$\Gamma(\,\hat{\cal{P}}\,) -\Gamma(\,\bar{\cal{P}}\,) = (-1)^{n_x+n_y+n_z} \sigma_0 ,$$ = (92)

$$\Gamma(\hat{\cal{R}}_{k}) -\Gamma(\bar{\cal{R}}_{k}) = -i(-1)^{N_k} \sigma_k ,$$ = (93)

$$\Gamma(\hat{\cal{S}}_{k}) -\Gamma(\bar{\cal{S}}_{k}) = -i(-1)^{n_k} \sigma_k ,$$ = (94)

where $\sigma_0$ is the identity 2$\times$2 matrix, $\sigma_k$for k=x,y,z are the standard Pauli matrices, and symbols N_x, N_y, and N_z refer to n_y+n_z, n_x+n_z, and n_x+n_y, respectively.

The characters of the classes are

      

$$\chi(\hat{\cal{E}})\,=\,-\chi(\bar{\cal{E}})$$ = 2, (95)

$$\chi(\hat{\cal{P}})=-\chi(\bar{\cal{P}})$$ = 2(-1)^n_x+n_y+n_z, (96)

$$\chi(\{\hat{\cal{R}}_{x},\bar{\cal{R}}_{x}\})=\chi(\{\hat{\cal{S}}_{x},\bar{\cal{S}}_{x}\})$$ = 0, (97)

$$\chi(\{\hat{\cal{R}}_{y},\bar{\cal{R}}_{y}\})=\chi(\{\hat{\cal{S}}_{y},\bar{\cal{S}}_{y}\})$$ = 0, (98)

$$\chi(\{\hat{\cal{R}}_{z},\bar{\cal{R}}_{z}\})=\chi(\{\hat{\cal{S}}_{z},\bar{\cal{S}}_{z}\})$$ = 0. (99)

One can see, that only the characters of $\hat{\cal{P}}$ and $\bar{\cal{P}}$ depend on quantum numbers n_x, n_y, and n_z that define the invariant subspaces; more precisely, they depend only on the parity of the sum n_x+n_y+n_z, i.e., on the total parity of basis states. Therefore, the only two spinor representations of D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ can be labeled by the eigenvalues of the parity operator $\hat{\cal{P}}$. Let us also note that all characters are real.

If we introduce the time reversal, $\hat{\cal{T}}$, into the ensemble of the linear operators belonging to D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {D}}}$ we obtain the D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ group with 16 new antilinear elements $\hat{\cal{U}}^T$$\equiv$ $\hat{\cal{U}}$ $\hat{\cal{T}}$, Sec. 2.3. To study properties of the representations of the D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ group, one has to consider representations provided by matrices

$$\breve\Gamma(\hat{\cal{U}})=\Gamma(\hat{\cal{A}}^{-1}\hat{\cal{U}}\hat{\cal{A}})^*,$$ (100)

where $\hat{\cal{A}}$ is one of the antilinear elements of D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ (see [8]). It is most convenient to take $\hat{\cal{T}}$ itself as $\hat{\cal{A}}$; we then have simply

$$\breve\Gamma(\hat{\cal{U}})=\Gamma(\hat{\cal{U}})^*,$$ (101)

as $\hat{\cal{T}}$ commutes with all $\hat{\cal{U}}$$\in$D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {D}}}$. In such a case, matrices $\breve\Gamma(\hat{\cal{U}})$ are just complex conjugates of $\Gamma(\hat{\cal{U}})$, and therefore the characters of representation $\breve\Gamma$ are exactly the same as those of $\Gamma$, because they are all real, see Eqs. (95). Therefore these two representations are equivalent, and a matrix $\beta$ exists which brings by a similarity transformation all matrices $\Gamma(\hat{\cal{U}})$ to $\breve\Gamma(\hat{\cal{U}})$= $\Gamma(\hat{\cal{U}})^*$,

$$\beta^{-1}\Gamma({{\hat{\cal{U}}}})\beta = \Gamma({\hat{\cal{... ..._{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {D}}}$ }.$$ (102)

Now, as shown by Wigner[8], there are only two cases possible: either

$$\beta\beta^*=+\Gamma(\hat{\cal{T}}^2),$$ (103)

or

$$\beta\beta^*=-\Gamma(\hat{\cal{T}}^2).$$ (104)

Matrix $\beta$ can easily be found from the explicit expressions for matrices $\Gamma(\hat{\cal{U}})$ given in Eqs. (91), and it reads

$$\beta=e^{i\phi}\left(\begin{array}{rr} 0 & -1 \\ 1 & 0 \end{array} \right) = -ie^{i\phi}\sigma_y .$$ (105)

Choosing the phase factor $e^{i\phi}$$\neq$1 in (105) is equivalent to a change of phase of the $\vert n_xn_yn_z,s_z\mbox{=}+\frac{1}{2}\rangle$states, and to a change in the phase convention in Eq. (48).

It is easy to demonstrate that with this form of the matrix $\beta$, Eq. (103), and not (104) holds. In Wigner's classification this case leads to what is called the corepresentations of the ``first kind'': any representation $\Gamma$ of the group D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {D}}}$ can be completed to a corepresentation of the full D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ group by defining

$$\Gamma({{\hat{\cal{U}}}}^T)=\Gamma({\hat{\cal{U}}})\beta.$$ (106)

Note that taking $\hat{\cal{U}}$= $\hat{\cal{E}}$ one gets $\Gamma(\hat{\cal{T}})$=$\beta$, so $\beta$ is, of course, just the matrix representing $\hat{\cal{T}}$itself.

After Wigner, the term corepresentation is used here because the representations of groups containing antilinear operators are not representations in the usual sense. To see this, let us consider an orthonormal set of states $\lbrace\vert\phi_i\rangle\rbrace$ constituting a basis of a representation $\Gamma$. Let $\hat{\cal{U}}$ be any linear, and ${\hat{\cal{U}}}'$any element of the group. Then, because of antilinearity of $\hat{\cal{U}}^T$one has

 

$$({{\hat{\cal{U}}}}^T\cdot\!{\hat{\cal{U}}}')\vert\phi_j\rangle \sum_i {\hat{\cal{U}}}^T \Gamma({\hat{\cal{U}}}')_{ij}\vert\phi_i... ...e =\sum_i \Gamma({{\hat{\cal{U}}}}')_{ij}^* {\hat{\cal{U}}}^T\vert\phi_i\rangle$$ =

$$\sum_{ik} \Gamma({{\hat{\cal{U}}}}')_{ij}^*\Gamma({\hat{\cal{U}}}^T)_{ki}\vert\phi_k\rangle$$ =

$$\sum_{k} \left[\Gamma({{\hat{\cal{U}}}}^T)\Gamma({\hat{\cal{U}}}')^*\right]_{kj} \vert\phi_k\rangle$$ = (107)

and, consequently,

$$\Gamma({{\hat{\cal{U}}}}^T\cdot\!{\hat{\cal{U}}}') = \Gamma({\hat{\cal{U}}}^T)\Gamma({\hat{\cal{U}}}')^*,$$ (108)

to be compared with

$$\Gamma({{\hat{\cal{U}}}}\cdot\!{\hat{\cal{U}}}') = \Gamma({\hat{\cal{U}}})\Gamma({\hat{\cal{U}}}'),$$ (109)

which holds for the ``usual'' representations. The presence of complex conjugation on the right-hand side of Eq. (108) implies that the homomorphism between the group multiplication and the multiplication of representation matrices no longer holds when the group contains antilinear operators. This is not surprising in view of the fact that matrices, by construction, always act on vectors (columns of numbers) linearly.

In conclusion, there are only two spinor corepresentations of D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$, and they can be labeled, as is also the case for the D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {D}}}$ group, by one quantum number only (parity).