Conserved symmetries

Conserved and broken symmetries are one of the most important elements of description of many-body systems. Within the mean-field approach, the theorem about self-consistent symmetries [167] tells us that mean-field states may or may not have all the symmetries of the Hamiltonian, depending on interactions and the system studied. Within the HFB approach, the symmetry is conserved when the generalized density matrix $\hat{\mathcal R}$ and the generalized Hamiltonian $\hat{\mathcal H}$ both commute with the symmetry operator $\hat{\mathcal U}$, i.e., [ $\hat{\mathcal R}$, $\hat{\mathcal U}$]=0 and [ $\hat{\mathcal H}$, $\hat{\mathcal U}$]=0, or

$$\hat{\mathcal U}\hat{\mathcal R}\hat{\mathcal U}^+ = \hat{\mathcal R},$$ (208)

$$\hat{\mathcal U}\hat{\mathcal H}\hat{\mathcal U}^+ = \hat{\mathcal H},$$ (209)

where

$$\hat{\mathcal U} = \left(\begin{array}{cc} \hat{U} & 0 \\ 0 & \hat{U}^* \end{array}\right) ,$$ (210)

and $\hat{U}$ is a unitary matrix of the single-particle symmetry operator. For the ``breve'' representation used in the present study, the symmetry operator is given) by [cf. Eq. (16)]

$$\hat{\breve{\mathcal U}} = \hat{\mathcal W} \hat{\mathcal U}... ...rray}{cc} \hat{U} & 0 \\ 0 & \hat{U}^{TC} \end{array}\right),$$ (211)

and then

$$\hat{\breve{\mathcal U}}\hat{\breve{\mathcal R}}\hat{\breve{\mathcal U}}^+ = \hat{\breve{\mathcal R}},$$ (212)

$$\hat{\breve{\mathcal U}}\hat{\breve{\mathcal H}}\hat{\breve{\mathcal U}}^+ = \hat{\breve{\mathcal H}}.$$ (213)

In the previous sections we have presented the most general set of expressions pertaining to the situation when no symmetries were a priori conserved. Below we discuss consequences of conserved symmetries.