T-signature or T-simplex

As is well known[17,6], for each of the six antilinear D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ operators, i.e., for T-signature $\hat{\cal{R}}_{k}^T$ or T-simplex $\hat{\cal{S}}_{k}^T$, k=x,y,z, which have squares equal to unity, ( $\hat{\cal{Z}}^2$= $\hat{\cal{E}}$, where $\hat{\cal{Z}}$ denotes one of them), one can construct a basis composed of eigenvectors of $\hat{\cal{Z}}$ with eigenvalue equal to 1,

$$\hat{\cal{Z}}\vert\mbox{{\boldmath {$n$ }}}\,\zeta\rangle = \vert\mbox{{\boldmath {$n$ }}}\,\zeta\rangle.$$ (14)

Moreover, since every operator $\hat{\cal{Z}}$ commutes with the time-reversal $\hat{\cal{T}}$, such basis can always be chosen so as to fulfill condition (9) at the same time. Table 3 lists examples of such bases, constructed for the HO states |n_xn_yn_z,s_z= $\pm\frac{1}{2}\rangle$. A similar construction is possible for any other single-particle basis, and has the explicit form shown in Table 3 provided the space and spin degrees of freedom are separated. Note that any linear combination of states $\vert\mbox{{\boldmath {$n$ }}}\,\zeta$=$+1\rangle$and $\vert\mbox{{\boldmath {$n$ }}}\,\zeta$=$-1\rangle$, with real coefficients, is another valid eigenstate of $\hat{\cal{Z}}$ with eigenvalue 1.

For operators even or odd with respect to $\hat{\cal{Z}}$,

$$\hat{\cal{Z}}^\dagger\hat{\cal{O}}\hat{\cal{Z}}= \epsilon_Z\hat{\cal{O}}, \qquad \epsilon_Z=\pm1,$$ (15)

one then has

$$\langle \mbox{{\boldmath {$n$ }}}\,\zeta\vert\hat{\cal{O}}\... ...\hat{\cal{O}}\vert\mbox{{\boldmath {$n$ }}}'\,\zeta'\rangle^*.$$ (16)

Hence, in bases fulfilling Eqs. (9) and (14), matrix ${\cal{O}}$ is purely real ( $\epsilon_Z$=+1) or purely imaginary ( $\epsilon_Z$=-1). This gives matrix ${\cal{O}}$ in the form (tilde stands for the transposition)

$$\left(\begin{array}{cc} A & Y \\ \widetilde{Y} & B \end... ...{cc} A' & Y' \\ -\widetilde{Y}' & -B' \end{array} \right),$$ (17)

for $\epsilon_Z$=+1 and $\epsilon_Z$=-1, respectively, where all submatrices are real, A and B are symmetric, A' and B' are antisymmetric, and Y and Y' are arbitrary. Note that in order to diagonalize ${\cal{O}}$one only needs to diagonalize a real matrix with unrestricted eigenvalues (for $\epsilon_Z$=+1), or an imaginary matrix with pairs of opposite non-zero eigenvalues (for $\epsilon_Z$=-1).

 

Table 3: Examples of eigenstates $\vert n_xn_yn_z\,\zeta\rangle_k^T$ of the T-signature, $\hat{\cal{R}}_{k}^T$, or T-simplex, $\hat{\cal{S}}_{k}^T$ operators, k=x, y, or z, with eigenvalue 1 [cf. Eqs. (9) and (14)], determined for the harmonic oscillator states |n_xn_yn_z,s_z= $\pm\frac{1}{2}\rangle$. Symbols (N_x,N_y,N_z) refer to (n_x,n_y,n_z) for $\hat{\cal{S}}_{k}^T$operators, and to (n_y+n_z,n_x+n_z,n_x+n_y) for $\hat{\cal{R}}_{k}^T$ operators.

k	$\zeta$	$\vert n_xn_yn_z\,\zeta\rangle_k^T$
x	+1	$(+i)^{N_x}\exp(-i\frac{\pi}{4})\, \vert n_xn_yn_z,s_z$= $+\frac{1}{2}\rangle$
x	-1	$(-i)^{N_x}\exp(+i\frac{\pi}{4})\, \vert n_xn_yn_z,s_z$= $-\frac{1}{2}\rangle$
y	+1	$(+i)^{N_y+1}\, \vert n_xn_yn_z,s_z$= $+\frac{1}{2}\rangle$
y	-1	$(-i)^{N_y+1}\, \vert n_xn_yn_z,s_z$= $-\frac{1}{2}\rangle$
z	+1		$\frac{1}{\sqrt{2}}$	( |nxnynz,sz= $\frac{1}{2}\rangle$	+	i(-1)Nz	|nxnynz,sz= $-\frac{1}{2}\rangle\,)$
z	-1		$\frac{i(-1)^{N_z}}{\sqrt{2}}$	( |nxnynz,sz= $\frac{1}{2}\rangle$	-	i(-1)Nz	|nxnynz,sz= $-\frac{1}{2}\rangle\,)$