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We revisit the $\eta$-pairing states in Hubbard models and explore theirconnections to quantum many-body scars to discover a universal scars mechanism.$\eta$-pairing occurs due to an algebraic structure known as a SpectrumGenerating Algebra (SGA), giving rise to equally spaced towers of eigenstatesin the spectrum. We generalize the original $\eta$-pairing construction andshow that several Hubbard-like models on arbitrary graphs exhibit SGAs,including ones with disorder and spin-orbit coupling. We further define aRestricted Spectrum Generating Algebra (RSGA) and give examples ofperturbations to the Hubbard-like models that preserve an equally spaced towerof the original model as eigenstates. The states of the surviving tower exhibita sub-thermal entanglement entropy, and we analytically obtain parameterregimes for which they lie in the bulk of the spectrum, showing that they areexact quantum many-body scars. The RSGA framework also explains the equallyspaced towers of eigenstates in several well-known models of quantum scars,including the AKLT model.

Eta-Pairing in Hubbard Models: From Spectrum Generating Algebras to Quantum Many-Body Scars