Unified structure for exact towers of scar states in the Affleck-Kennedy-Lieb-Tasaki and other models

Quantum many-body scar states are many-body states with finite energy density in non-integrable models that do not obey the eigenstate thermalization hypothesis. Recent works have revealed "towers" of scar states that are exactly known and are equally spaced in energy, specifically in the AKLT and spin-1 XY models, and a spin-1/2 model that conserves the number of domain walls. We provide a common framework to understand and prove known exact towers of scars in these systems, by evaluating the commutator of the Hamiltonian and a ladder operator. In particular, we provide a simple proof of the scar towers in the integer-spin 1D AKLT models by studying two-site spin projectors. Through this picture we deduce a family of Hamiltonians that share the scar tower with the AKLT model, and also find common parent Hamiltonians for the AKLT and XY model scars. We also introduce new towers of exact states, organized in a "pyramid" structure, in the spin-1/2 model through the successive application of a nonlocal ladder operator.