Phase conventions

In Sec. 3.1 we have shown how one can simplify the matrix elements of operators by using a given phase convention, i.e, by fixing phases of single-particle basis states in a given way. Whenever an antilinear D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ operator is conserved, one can always construct a phase convention for which the matrix elements of the mean-field Hamiltonian are real numbers. However, from technical point of view, it can be more advantageous to fix the phase convention in yet another way. Indeed, whenever the calculation of matrix elements is more time consuming than the diagonalization of the Hamiltonian matrix, one may use the phase convention to facilitate the former task, at the expense of diagonalizing complex matrices. Moreover, such a strategy allows for keeping the simplicity of performing the former task even in cases when there is no antilinear conserved symmetry available, and when one has to diagonalize complex matrices anyhow. In the present section we show constructions of phase conventions which facilitate calculations of the space-spin matrix elements.

Representation (9), which separates space and spin degrees of freedom, is convenient in applications pertaining to deformed single-particle states, as those discussed in the present study. This is because, each hermitian operator can be represented as a sum of four components of the form

$$\hat{{\cal{O}}}^{(\mu)}= \hat{O}^{(\mu)}_r\hat{\sigma}_\mu, \qquad \mu=0,1,2,3,$$ (34)

where $\hat{O}^{(\mu)}_r$ acts in the coordinate space, and $\hat{\sigma}_\mu$are the Pauli matrices acting in the spin space, with $\hat{\sigma}_0$defined as the 2$\times$2 identity matrix. Then, the matrix elements of $\hat{{\cal{O}}}^{(\mu)}$ can be factorized into space and spin parts

$$\langle \mbox{{\boldmath {$n$ }}}\,\zeta\vert\hat{{\cal{O}}... ...le \cdot \langle\zeta\vert\hat{\sigma}_\mu\vert\zeta'\rangle,$$ (35)

and the spin part can be computed once for all. Usually many of the spin matrix elements $\langle\zeta\vert\hat{\sigma}_\mu\vert\zeta'\rangle$ vanish, thus making it unnecessary to calculate the corresponding coordinate-space matrix elements $\langle \mbox{{\boldmath {$n$ }}}\vert\hat{O}^{(\mu)}_r\vert\mbox{{\boldmath {$n$ }}}'\rangle$.

Matrix elements of operators $\hat{{\cal{O}}}^{(\mu)}$ can be made purely real or purely imaginary if phases of single-particle basis states are chosen in such a way that, for one Cartesian direction l=x, y, or z, one has

$$\hat{\cal{K}}_l\vert\mbox{{\boldmath {$n$ }}}\,\zeta\rangle = \vert\mbox{{\boldmath {$n$ }}}\,\zeta\rangle,$$ (36)

where $\hat{\cal{K}}_l^2$= $\hat{\cal{E}}$ is the antilinear spin operator defined by

$$\hat{\cal{K}}_l= \hat{\cal{T}}i\hat{\sigma}_l = i\hat{\sigma}_l \hat{\cal{T}}.$$ (37)

Indeed, for time-even ( $\epsilon_T$=+1) or time-odd ( $\epsilon_T$=-1) operators one obtains that

$$\hat{\cal{K}}_l^\dagger\hat{{\cal{O}}}^{(\mu)}\hat{\cal{K}}... ...silon_T\hat{{\cal{O}}}^{(\mu)}, \quad \epsilon_{\mu l}=\pm 1,$$ (38)

where coefficients $\epsilon_{\mu l}$ are given in Table 5. Using Eqs. (36) and (38), one gets for matrix elements of $\hat{{\cal{O}}}^{(\mu)}$

   

$$\langle \mbox{{\boldmath {$n$ }}}\,\zeta\vert\hat{{\cal{O}}}^{(\mu)}\vert\mbox{{\boldmath {$n$ }}}'\,\zeta'\rangle \epsilon_{\mu l}\epsilon_T \langle \mbox{{\boldmath {$n$ }}}\,\zeta\vert\hat{{\cal{O}}}^{(\mu)}\vert\mbox{{\boldmath {$n$ }}}'\,\zeta'\rangle^*$$ = (39)

$$\langle \mbox{{\boldmath {$n$ }}}\,\zeta\vert\hat{{\cal{O}}}^{(\mu)}\vert\mbox{{\boldmath {$n$ }}}'\,\zeta'\rangle \epsilon_{\mu l}\zeta\zeta' \langle \mbox{{\boldmath {$n$ }}}\,-\... ...a\vert\hat{{\cal{O}}}^{(\mu)}\vert\mbox{{\boldmath {$n$ }}}'\,-\!\zeta'\rangle,$$ = (40)

where (39) tells us which elements are real, and which are imaginary, while (40) gives the matrix elements, e.g., for $\zeta$=-1 expressed through those for $\zeta$=+1.

As is usual for antilinear operators, there is a lot of freedom in finding bases (36) of eigenstates of $\hat{\cal{K}}_l$. We can use this freedom to fulfill other useful conditions. For example, since $\hat{\cal{K}}_l$ and $\hat{\cal{R}}_{k}$ anticommute for l$\neq$k, one can find bases (36) which are at the same time the eigenstates of signature or simplex operators. In fact, phases of eigenstates listed in Table 4, has been chosen in such a way that,

$$\hat{\cal{K}}_l\vert\mbox{{\boldmath {$n$ }}}\,\zeta\rangle... ...mbox{{\boldmath {$n$ }}}\,\zeta\rangle_k, \mbox{~~for~~} k<l,$$ (41)

where again the circular ordering of Cartesian directions, x<y<z<x, is assumed to define k<l.

 

Table 5: Antilinear spin operators $\hat{\cal{K}}_l$, which can be used to fix convenient phase conventions leading to conditions (39).

l	$\hat{\cal{K}}_l$		$\epsilon_{0l}$	$\epsilon_{1l}$	$\epsilon_{2l}$	$\epsilon_{3l}$
x	${\hat{\cal{T}}}i\hat{\sigma}_x$	=	$i\hat{\sigma}_z\hat{\cal{K}}$	+1	+1	-1	-1
y	${\hat{\cal{T}}}i\hat{\sigma}_y$	=	$\hat{\cal{K}}$	+1	-1	+1	-1
z	${\hat{\cal{T}}}i\hat{\sigma}_z$	=	$-i\hat{\sigma}_x\hat{\cal{K}}$	+1	-1	-1	+1