Partial dynamical symmetry in Bose-Fermi systems

We generalize the notion of partial dynamical symmetry (PDS) to a system of interacting bosons and fermions. In a PDS, selected states of the Hamiltonian are solvable and preserve the symmetry exactly, while other states are mixed. As a first example of such novel symmetry construction, spectral features of the odd-mass nucleus ${}^{195}\mathrm{Pt}$ are analyzed.

Figure 1

Left panel: Observed and calculated energy spectrum of ${}^{195}\mathrm{Pt}$. The levels in black are the solvable $[7,0]\langle 7,0\rangle$ eigenstates of ${\widehat{H}}_{\mathrm{DS}}$ (3), whose structure and energy remain unaffected by the added PDS interactions in Eq. (10). The levels in blue (center column) [red (right column)] are the $[6,1]\langle 6,1\rangle$ $\left([6,1]\langle 5,0\rangle \right)$ eigenstates of ${\widehat{H}}_{\mathrm{DS}}$ (3) and are subsequently perturbed by the PDS interactions in Eq. (10). Right panel: ${\mathrm{SO}}^{\mathrm{BF}}\left(6\right)$ decomposition of the eigenstates of ${\widehat{H}}_{\mathrm{PDS}}$ (10), shown in red on the left panel.