  The recent discovery of quantum many-body scar states has revealed the
possibility of having states with low entanglement that violate the eigenstate
thermalization hypothesis in nonintegrable systems. Such states with low
entanglement entropy are rare but naturally exist in the integrable system of
free fermions. Here, we demonstrate analytically that these atypical states
would be always eliminated when an arbitrary weak interaction is introduced
between the fermions. In particular, we show that the probability of having a
many-body scar state with entanglement entropy satisfying a sub-volume scaling
law decreases double exponentially as the system size. Thus, our results
provide a quantitative argument for the disappearance of scar states in
interacting fermion systems.
