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