  We demonstrate the presence of anomalous high-energy eigenstates, or
many-body scars, in $U(1)$ quantum link and quantum dimer models on square and
rectangular lattices. In particular, we consider the paradigmatic
Rokhsar-Kivelson Hamiltonian $H=\mathcal{O}_{\mathrm{kin}} + \lambda
\mathcal{O}_{\mathrm{pot}}$ where $\mathcal{O}_{\mathrm{pot}}$
($\mathcal{O}_{\mathrm{kin}}$) is defined as a sum of terms on elementary
plaquettes that are diagonal (off-diagonal) in the computational basis. Both
these interacting models possess an exponentially large number of mid-spectrum
zero modes in system size at $\lambda=0$ that are protected by an index theorem
preventing any mixing with the nonzero modes at this coupling. We classify
different types of scars for $|\lambda| \lesssim \mathcal{O}(1)$ both at zero
and finite winding number sectors complementing and significantly generalizing
our previous work [Banerjee and Sen, Phys. Rev. Lett. 126, 220601 (2021)]. The
scars at finite $\lambda$ show a rich variety with those that are composed
solely from the zero modes of $\mathcal{O}_{\mathrm{kin}}$, those that contain
an admixture of both the zero and the nonzero modes of
$\mathcal{O}_{\mathrm{kin}}$, and finally those composed solely from the
nonzero modes of $\mathcal{O}_{\mathrm{kin}}$. We give analytic expressions for
certain "lego scars" for the quantum dimer model on rectangular lattices where
one of the linear dimensions can be made arbitrarily large, with the building
blocks (legos) being composed of emergent singlets and other more complicated
entangled structures.
