  We consider the spectrum of a $U(1)$ quantum link model where gauge fields
are realized as $S=1/2$ quantum spins and demonstrate a new mechanism for
generating quantum many-body scars (high-energy eigenstates that violate the
eigenstate thermalization hypothesis) in a constrained Hilbert space. Many-body
dynamics with local constraints has attracted much attention due to the recent
discovery of non-ergodic behavior in quantum simulators based on Rydberg atoms.
Lattice gauge theories provide natural examples of constrained systems since
physical states must be gauge-invariant. In our case, the Hamiltonian $H={\cal
O}_{\rm kin}+\lambda {\cal O}_{\rm pot}$, where ${\cal O}_{\rm pot}$ (${\cal
O}_{\rm kin}$) is diagonal (off-diagonal) in the electric flux basis, contains
exact mid-spectrum zero modes at $\lambda=0$ whose number grows exponentially
with system size. This massive degeneracy is lifted at any non-zero $\lambda$
but some special linear combinations that simultaneously diagonalize ${\cal
O}_{\rm kin}$ and ${\cal O}_{\rm pot}$ survive as quantum many-body scars,
suggesting an "order-by-disorder" mechanism in the Hilbert space. We give
evidence for such scars and show their dynamical consequences on ladders with
up to $56$ spins, which may be tested using available proposals of quantum
simulators.
