  We consider the spectrum of a $U(1)$ quantum link model where gauge fields
are realized as $S=1/2$ 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 two-leg ladders with up to
$56$ spins, which may be tested using available proposals of quantum
simulators. Results on wider ladders point towards their presence in two
dimensions as well.
