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Unmixing symmetries

The low-lying spectra of atomic nuclei display diverse behaviors, for example rotational bands, which can be described phenomenologically by simple symmetry groups such as spatial SU(3). This leads to the idea of dynamical symmetry, where the Hamiltonian commutes with the Casimir operator(s) of a group, and is block-diagonal in subspaces defined by the group's irreducible representations or irreps. Detailed microscopic calculations, however, show these symmetries are in fact often strongly mixed and the wave function fragmented across many irreps. More commonly the fragmentation across members of a band are similar, or a quasi-dynamical symmetry. In this Letter I explicitly, albeit numerically, construct unitary transformations from a quasi-dynamical symmetry to a dynamical symmetry, adapting the similarity renormalization group, or SRG. The standard SRG produces unsatisfactory results, forcing the induced dynamical symmetry to be dominated by high-weight irreps irrespective of the original decomposition. Using spectral distribution theory to rederive and diagnose standard SRG, I introduce a new form of SRG. The new SRG transforms a quasi-dynamical symmetry to a dynamical symmetry, that is, unmixes the mixed symmetries, with intuitively more appealing results.