From tunnels to towers: quantum scars from Lie algebras and <i>q</i>-deformed Lie algebras

We present a general symmetry-based framework for obtaining many-body Hamiltonians with scarred eigenstates that do not obey the eigenstate thermalization hypothesis. Our models are derived from parent Hamiltonians with a non-Abelian (or q-deformed) symmetry, whose eigenspectra are organized as degenerate multiplets that transform as irreducible representations of the symmetry ('tunnels'). We show that large classes of perturbations break the symmetry but preserve a particular low-entanglement multiplet of states -- thereby giving generic, thermal spectra with a shadow of the broken symmetry in the form of scars. Our framework applies to several known models, and we introduce new models with scars that transform as irreps of symmetries such as SU(3) and q-deformed SU(2), as well as new examples of generalized AKLT models with scar states that do not transform in an irreducible representation of the relevant symmetry.