Proton-neutron symmetry

The standard case of no proton-neutron mixing can be described by the conserved proton-neutron symmetry given by

$$\hat{U}_{pn}=i\exp(-i\pi\hat{T}_3) =i\exp(-{\textstyle{\frac{i}{2}}}\pi\hat{\tau}_3) =\hat{\tau}_3 .$$ (214)

In other words, the iso-3 signature (multiplied by $i$) is then the conserved symmetry. Note that conservation of projection of the isospin on the third axis (the charge conservation) would require that the iso-3 rotation about an arbitrary angle be conserved, while the iso-3 signature corresponds only to rotation about $\pi$. Within the HFB approach, the charge symmetry is broken in the same way as is the particle number symmetry.

Since the $TC$-transformed symmetry operator reads $\hat{U}_{pn}^{TC}$=$-\hat{\tau}_3$, we obtain from Eq. (213) that

$$\hat{\tau}_3 \hat {\rho} \hat{\tau}_3 = \phantom{-}\hat {\rho} ,$$ (215)

$$\hat{\tau}_3 \hat{\breve{\rho}} \hat{\tau}_3 = -\hat{\breve{\rho}},$$ (216)

and analogous properties hold for the mean-field Hamiltonians, $\hat{h}$ and $\hat{\breve{h}}$, respectively. It is then clear that without the proton-neutron mixing the p-h density matrices and Hamiltonians have only the $k$=0 and 3 isospin components, while the p-p ones have (in the ``breve'' representation) only the $k$=1 and 2 isospin components, cf. Eqs. (58) and (60).