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Next: Tilted-axis cranking Up: Modifications introduced in version Previous: Corrected errors


Breaking of all the three plane-reflection symmetries

The previous version of the code (v1.75r) assumed that one plane-reflection symmetry is conserved, i.e., that the $y$-simplex operator of Eq. (I-52) commutes with the single-particle (s.p.) Routhian. As a consequence, the angular-momentum vector was restricted to have only one non-zero component, the $y$ component (see Refs. [4,5,6] for a discussion of symmetries). Because of the recent interest in the so-called shears and chiral rotation phenomena [4], in the present version this restriction has been released. This allows for an arbitrary orientation of the angular momentum vector with respect to the mass distribution. It also allows for an arbitrary orientation of the angular frequency vector, i.e., for the so-called tilted-axis cranking. The extension is done at the expense of diagonalizing matrices that are twice larger than before, and by summing up the densities on a twice larger number of the Gauss-Hermite integration nodes. Since typical cases of the shears and chiral rotation do not involve parity-violating shapes, the problem can be simplified by employing the parity conservation, and such an option has also been implemented.

Altogether, one may classify the relevant symmetry conditions by considering the $y$-simplex ($\hat{S}_y$), $y$-signature ($\hat{R}_y$), and parity ($\hat{P}$). Since the product of any two of them is equal to the third one, one has five different possibilities of the conserved symmetry groups, see Table 1. These five options are governed by three switches, ISIMPY, ISIGNY, and IPARTY (see Sec. 3.2), as given in Table 1.


Table 1: Primary set of conserved and nonconserved symmetries allowed in the code HFODD version (v2.07f).
Option Symmetries Switches
  $\hat{S}_y$ $\hat{R}_y$ $\hat{P}$ ISIMPY ISIGNY IPARTY
P1 conserved conserved conserved 1 1 1
P2 conserved nonconserved nonconserved 1 0 0
P3 nonconserved conserved nonconserved 0 1 0
P4 nonconserved nonconserved conserved 0 0 1
P5 nonconserved nonconserved nonconserved 0 0 0

Switch ISIGNY, which was introduced in the previous versions, see Sec. II-3.3, is maintained in the way that ensures full compatibility of input data files. This is implemented in the following way. If switch IPARTY is set to $-$1 (the default value), than the code resets its value to ISIGNY, and stops in case switch ISIMPY is not set to 1 (the default value). In this mode, the present version (v2.07f) works exactly in the same way as the former versions. On the other hand, if switch IPARTY is set to 0 or 1, then the value of ISIGNY must be compatible with ISIMPY and IPARTY according to the group multiplication, Table 1, otherwise the code stops.

When either of the three symmetries $\hat{S}_y$, $\hat{R}_y$, or $\hat{P}$ is conserved, the single-particle Routhians acquire specific block-diagonal forms, cf. Ref. [6], and diagonalization of smaller matrices results in a faster execution time. In version (v2.07f), this is implemented for the conserved $y$-simplex and/or parity, but in case the $y$-signature alone is conserved it is not implemented yet. Hence, option P3 in Table 1 is enforced at the level of symmetries of densities [5], but in fact the code then operates as if no symmetry was conserved, and the execution time is not shorter. Similarly, within option P3, classification of single-particle states in terms of conserved $y$-signature is not available yet.

When the time-reversal and simplex or signature are conserved, the Kramers degeneracy allows for diagonalization of matrices only in one simplex or signature, respectively. This reduces the numerical effort by half. However, when the time-reversal alone is conserved, such a reduction is not possible, although the eigenstates do still obey, of course, the Kramers degeneracy.

In the present version, the antilinear symmetry $y$-simplex$^T$ [5], which is given by the operator $\hat{S}^T_y$= $\hat{T}\hat{S}_y$, can be either conserved or nonconserved. This is governed by switch ISIMTY=1 or 0, respectively, see Sec. 3.2. Of course, conservation of $y$-simplex$^T$ must by compatible with the conservation of the time-reversal ($\hat{T}$) and $y$-simplex ($\hat{S}_y$). The $y$-simplex$^T$ symmetry does not affect quantum numbers, but does affect shapes and directions of the angular momentum. In particular, its conservation confines the angular momentum to the $x$-$z$ plane. Similarly, conservation of $x$-simplex$^T$ and $z$-simplex$^T$, see Sec. II-3.3, which was already implemented in the previous versions, confines the angular momentum to the $y$-$z$ and $x$-$y$ plane, respectively. Therefore, the three antilinear symmetries allow for specific manipulations of the angular momentum vector, whenever the so-called planar rotation is required. Similarly, by conserving pairs of these antilinear symmetries one can restrict the angular momentum to one of the three Cartesian directions, and perform calculations with a conserved $x$-signature, $\hat{R}_x$= $(-1)^A\hat{S}^T_y\hat{S}^T_z$, or $z$-signature, $\hat{R}_z$= $(-1)^A\hat{S}^T_z\hat{S}^T_y$.


Table 2: Secondary set of conserved and nonconserved symmetries allowed in the code HFODD version (v2.07f).
Option Symmetries Switches
  $\hat{S}_y$ $\hat{T}$ $\hat{S}^T_y$ ISIMPY ITIREV ISIMTY
S1 conserved conserved conserved 1 1 1
S2 conserved nonconserved nonconserved 1 0 0
S3 nonconserved conserved nonconserved 0 1 0
S4 nonconserved nonconserved conserved 0 0 1
S5 nonconserved nonconserved nonconserved 0 0 0

In the previous versions, conserved $y$-simplex implied that the $y$-simplex$^T$ conservation was uniquely linked to the $\hat{T}$ conservation, and hence independent switch was not required. In version (v2.07f), the compatibility with these previous versions is ensured in the following way. If switch ISIMTY is set to $-$1 (the default value), than the code resets its value to ITIREV, see Sec. 3.2, and stops in case switch ISIMPY is not set to 1 (the default value). In this mode, the present version works in exactly the same way as the former versions. On the other hand, if switch ISIMTY is set to 0 or 1, then the values of ISIMPY and ITIREV must correspond to one of the five options allowed by the group structure, and enumerated in Table 2, otherwise the code stops. Note that in the present version we introduced switch ITIREV=1 or 0 (time-reversal conserved or not) as a convenient replacement of the value 1$-$IROTAT, where switch IROTAT=0 or 1 (no rotation or rotation) was introduced in Sec. II-3.3.


Table 3: Ternary set of conserved and nonconserved symmetries allowed in the code HFODD version (v2.07f).
Option Symmetries Switches
  $\hat{R}_y$ $\hat{S}^T_x$ $\hat{S}^T_z$ ISIGNY ISIMTX ISIMTZ
T1 conserved conserved conserved 1 1 1
T2 conserved nonconserved nonconserved 1 0 0
T3 nonconserved conserved nonconserved 0 1 0
T4 nonconserved nonconserved conserved 0 0 1
T5 nonconserved nonconserved nonconserved 0 0 0

The third set of symmetries, composed of the $y$-signature ($\hat{R}_y$), $x$-simplex$^T$ ($\hat{S}^T_x$), and $z$-simplex$^T$ ($\hat{S}^T_z$), and switches ISIGNY, ISIMTX, and ISIMTZ, is represented in Table 3. In version (v2.07f) we have unblocked option T2, which was not available in previous versions, see discussion in Sec. I-3.4.

Triples of symmetries listed in Tables 1-3 are linked in two points: the $y$-simplex ($\hat{S}_y$) appears in Tables 1 and 2, and the $y$-signature ($\hat{R}_y$) appears in Tables 1 and 3. Therefore, options P1 and P2 (conserved $y$-simplex) must be linked with options S1 and S2, and the same holds for P3, P4, P5 (nonconserved $y$-simplex) and S3, S4, S5. Similarly, options P1 and P3 (conserved $y$-signature) must be linked with options T1 and T2, and options P2, P4, P5 with T3, T4, T5. Altogether, we obtain 34 allowed options that are illustrated by a tree of links in Fig. 1. They are also listed in Table 4, together with values of the corresponding program switches.

The last column of Table 4 gives generators of conserved point groups associated with each option. The generators are selected by using conventions formulated in Ref. [6] (cf. Table I in this reference). The first option (P1-S1-T1) corresponds to the whole D $_{\mbox{\scriptsize {2h}}}^{\mbox{\scriptsize {T}}}$ group conserved, while in the last option (P5-S5-T5), the group contains only the identity operator $\hat{E}$. Altogether, symmetries allowed in the code HFODD cover 17 out of 26 nontrivial subgroups of D $_{\mbox{\scriptsize {2h}}}^{\mbox{\scriptsize {T}}}$, which have been enumerated in Ref. [6]. We note here in passing that simple extensions of the available symmetries can be done in the following way:

  1. By adding symmetry $\hat{R}_x$ to the present set, one could cover 5 more nontrivial subgroups of D $_{\mbox{\scriptsize {2h}}}^{\mbox{\scriptsize {T}}}$, i.e., $\{\hat{R}_x\hat{R}_y\hat{T}\}$, $\{\hat{R}_x\hat{R}_y\hat{P}\}$, $\{\hat{R}_x\hat{R}^T_y\hat{P}^T\}$, $\{\hat{R}_x\hat{S}_y\}$, and $\{\hat{R}_x\hat{R}_y\}$.
  2. By adding symmetry $\hat{R}^T_x$ to the present set, one could cover 4 more nontrivial subgroups of D $_{\mbox{\scriptsize {2h}}}^{\mbox{\scriptsize {T}}}$, i.e., $\{\hat{R}_x\hat{R}_y\hat{T}\}$, $\{\hat{R}_y\hat{R}^T_x\hat{P}^T\}$, $\{\hat{R}_y\hat{R}^T_x\}$, and $\{\hat{R}^T_x\}$.
  3. By adding symmetry $\hat{S}_x$ to the present set, one could cover 3 more nontrivial subgroups of D $_{\mbox{\scriptsize {2h}}}^{\mbox{\scriptsize {T}}}$, i.e., $\{\hat{R}_z\hat{R}^T_x\hat{P}^T\}$, $\{\hat{R}_x\hat{R}_y\hat{P}\}$, and $\{\hat{R}_z\hat{S}_x\}$.
Obviously, by adding both $\hat{R}_x$ and $\hat{R}^T_x$, one could cover 7 more nontrivial subgroups of D $_{\mbox{\scriptsize {2h}}}^{\mbox{\scriptsize {T}}}$ as compared to the present status of the code, thus altogether covering 24 out of 26 nontrivial subgroups. The missing two would then be only $\{\hat{R}^T_x\hat{P}^T\}$ and $\{\hat{P}^T\}$. The above possible extensions are left for future development, and can easily be implemented provided there will be a physics motivation to study such new cases.

Figure 1: Tree diagram of all allowed options in the code HFODD version (v2.07f). Symbols P, S, and T refer to options listed in Tables 1-3. Lines connect sets of allowed options that are listed in Table 4.
\begin{figure}\begin{center} \leavevmode \epsfxsize =9cm \epsfbox{comtil04.eps}\end{center}\vspace{-1.0cm} \end{figure}


Table 4: Complete list of all conserved (C) or nonconserved (N) symmetries allowed in the code HFODD version (v2.07f). The last column gives generators of the conserved subgroup of D $_{\mbox{\scriptsize{2h}}}^{\mbox{\scriptsize{T}}}$.
Option Symmetries  Switches Conserved                 
  $\hat{S}_y$  $\hat{R}_y$  $\hat{P}$  $\hat{T}$  $\hat{S}_y^T$  $\hat{S}_x^T$  $\hat{S}_z^T$ ISIMPY  ISIGNY  IPARTY  ITIREV  ISIMTY  ISIMTX  ISIMTZ Group
P1-S1-T1 C  C  C  C  C  C  C 1  1  1  1  1  1  1 $\{\hat{R_x}\hat{R_y}\hat{T}\hat{P}\}$
P1-S1-T2 C  C  C  C  C  N  N 1  1  1  1  1  0  0 $\{\hat{R_y}\hat{T}\hat{P}\}$
P1-S2-T1 C  C  C  N  N  C  C 1  1  1  0  0  1  1 $\{\hat{R_y}\hat{S}_z^T\hat{P}\}$
P1-S2-T2 C  C  C  N  N  N  N 1  1  1  0  0  0  0 $\{\hat{R_y}\hat{P}\}$
P2-S1-T3 C  N  N  C  C  C  N 1  0  0  1  1  1  0 $\{\hat{R_z}\hat{S_x}\hat{T}\}$
P2-S1-T4 C  N  N  C  C  N  C 1  0  0  1  1  0  1 $\{\hat{R_x}\hat{S_y}\hat{T}\}$
P2-S1-T5 C  N  N  C  C  N  N 1  0  0  1  1  0  0 $\{\hat{S_y}\hat{T}\}$
P2-S2-T3 C  N  N  N  N  C  N 1  0  0  0  0  1  0 $\{\hat{S_y}\hat{S}_x^T\}$
P2-S2-T4 C  N  N  N  N  N  C 1  0  0  0  0  0  1 $\{\hat{S_y}\hat{S}_z^T\}$
P2-S2-T5 C  N  N  N  N  N  N 1  0  0  0  0  0  0 $\{\hat{S_y}\}$
P3-S3-T1 N  C  N  C  N  C  C 0  1  0  1  0  1  1 $\{\hat{R_y}\hat{S_z}\hat{T}\}$
P3-S3-T2 N  C  N  C  N  N  N 0  1  0  1  0  0  0 $\{\hat{R_y}\hat{T}\}$
P3-S4-T1 N  C  N  N  C  C  C 0  1  0  0  1  1  1 $\{\hat{R_x}\hat{R_y}\hat{P}^T\}$
P3-S4-T2 N  C  N  N  C  N  N 0  1  0  0  1  0  0 $\{\hat{R_y}\hat{P}^T\}$
P3-S5-T1 N  C  N  N  N  C  C 0  1  0  0  0  1  1 $\{\hat{R_y}\hat{S}_z^T\}$
P3-S5-T2 N  C  N  N  N  N  N 0  1  0  0  0  0  0 $\{\hat{R_y}\}$
P4-S3-T3 N  N  C  C  N  C  N 0  0  1  1  0  1  0 $\{\hat{R_x}\hat{T}\hat{P}\}$
P4-S3-T4 N  N  C  C  N  N  C 0  0  1  1  0  0  1 $\{\hat{R_z}\hat{T}\hat{P}\}$
P4-S3-T5 N  N  C  C  N  N  N 0  0  1  1  0  0  0 $\{\hat{T}\hat{P}\}$
P4-S4-T3 N  N  C  N  C  C  N 0  0  1  0  1  1  0 $\{\hat{R_z}\hat{S}_x^T\hat{P}\}$
P4-S4-T4 N  N  C  N  C  N  C 0  0  1  0  1  0  1 $\{\hat{R_x}\hat{S}_y^T\hat{P}\}$
P4-S4-T5 N  N  C  N  C  N  N 0  0  1  0  1  0  0 $\{\hat{S}_y^T\hat{P}\}$
P4-S5-T3 N  N  C  N  N  C  N 0  0  1  0  0  1  0 $\{\hat{S}_x^T\hat{P}\}$
P4-S5-T4 N  N  C  N  N  N  C 0  0  1  0  0  0  1 $\{\hat{S}_z^T\hat{P}\}$
P4-S5-T5 N  N  C  N  N  N  N 0  0  1  0  0  0  0 $\{\hat{P}\}$
P5-S3-T3 N  N  N  C  N  C  N 0  0  0  1  0  1  0 $\{\hat{S_x}\hat{T}\}$
P5-S3-T4 N  N  N  C  N  N  C 0  0  0  1  0  0  1 $\{\hat{S_z}\hat{T}\}$
P5-S3-T5 N  N  N  C  N  N  N 0  0  0  1  0  0  0 $\{\hat{T}\}$
P5-S4-T3 N  N  N  N  C  C  N 0  0  0  0  1  1  0 $\{\hat{R_z}\hat{S}_x^T\}$
P5-S4-T4 N  N  N  N  C  N  C 0  0  0  0  1  0  1 $\{\hat{R_x}\hat{S}_y^T\}$
P5-S4-T5 N  N  N  N  C  N  N 0  0  0  0  1  0  0 $\{\hat{S}_y^T\}$
P5-S5-T3 N  N  N  N  N  C  N 0  0  0  0  0  1  0 $\{\hat{S}_x^T\}$
P5-S5-T4 N  N  N  N  N  N  C 0  0  0  0  0  0  1 $\{\hat{S}_z^T\}$
P5-S5-T5 N  N  N  N  N  N  N 0  0  0  0  0  0  0 $\{\hat{E}\}$

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Next: Tilted-axis cranking Up: Modifications introduced in version Previous: Corrected errors
Jacek Dobaczewski 2004-01-06