Partial dynamical symmetry (PDS) extends and complements the concepts of exact and dynamical symmetry. It allows one to remove undesired constraints from an algebraic theory, while preserving some of the useful aspects of a dynamical symmetry, and to study the effects of symmetry breaking in a controlled manner. An example of a PDS in an interacting fermion system is presented. The associated PDS Hamiltonians are closely related with a realistic quadrupole-quadrupole interaction and provide new insights into this important interaction.
Partial dynamical symmetry (PDS) is an algebraic structure in which a symmetry is neither exact nor fully broken: a subset of eigenstates keeps good quantum numbers and stays solvable while the rest mix. It is currently identified from spectroscopy. We ask whether it also leaves a purely structural mark in the eigenstates, and find, for the
sd-IBM Hamiltonians studied here, that it does, though not in the magnitude of entanglement. The diagnostic is the variance of a symmetry Casimir, the label variance $\Var\,\C[G]$, which shares its zero with a block-coherence entropy and a block impurity: all three vanish when a state occupies a single Casimir sector, a single irreducible representation once conjugate representations are resolved by the cubic Casimir. Resolved state by state, the variance is numerically zero on the symmetry-selected subset and large on the mixed states, isolating the PDS states from the wavefunctions alone; the separation does not close as the boson number grows and persists at Leviatan's first- and second-order critical points, whereas the magnitude of bipartite
s∣d entanglement does not separate the two families. We anchor the analysis in
168Er, reproducing Leviatan's fitted $\su3$ admixtures and his Hamiltonian-parameter-independent
γ→g B(E2) ratios. The diagnostic also has an exact dynamical form, a Casimir Loschmidt echo rigid on the symmetry-pure states, and we demonstrate its readout on an IBM superconducting processor at
N=3, where a single solvable/mixed exemplar reproduces the predicted pattern within finite-shot uncertainties.
An exact spectral solution of the Hubbard problem on the tetrahedron is given. It is demonstrated that it provides at the same time a partial spectral solution for the Bose version of this (Fermi) model. A remarkable relationship of the Hamiltonians HB=HF+(1−P)HB(1−P) is established, where P is a projection in the state space of the interacting Bose system. This partial observation is then extended to the spatial lattice composed of the four-site Hubbard cells, which interact due to the superimposed perturbation of the hopping type, affecting all sites of the lattice; H=∑(i j) ∑σ=± bi j c*iσ cjσ+U ∑i ni+ni− being the corresponding Hamiltonian. It is thus demonstrated that the joint Bose–Fermi spectral problem (previously established in 1+1 dimensions) makes sense in 1+3 dimensions as well.