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Quantum many-body scar states are many-body states with finite energy densityin non-integrable models that do not obey the eigenstate thermalizationhypothesis. Recent works have revealed "towers" of scar states that are exactlyknown and are equally spaced in energy, specifically in the AKLT model, thespin-1 XY model, and a spin-1/2 model that conserves number of domain walls. Weprovide a common framework to understand and prove known exact towers of scarsin these systems, by evaluating the commutator of the Hamiltonian and a ladderoperator. In particular we provide a simple proof of the scar towers in theinteger-spin 1d AKLT models by studying two-site spin projectors. Through thispicture we deduce a family of Hamiltonians that share the scar tower with theAKLT model, and also find common parent Hamiltonians for the AKLT and XY modelscars. We also introduce new towers of exact states, organized in a "pyramid"structure, in the spin-1/2 model through successive application of a non-localladder operator.

Unified structure for exact towers of scar states in the AKLT and other models