  Quantum phase transitions are usually observed in ground states of correlated
systems. Remarkably, eigenstate phase transitions can also occur at finite
energy density in disordered, isolated quantum systems. Such transitions fall
outside the framework of statistical mechanics as they involve the breakdown of
ergodicity. Here, we consider what general constraints can be imposed on the
nature of eigenstate transitions due to the presence of disorder. We derive
Harris-type bounds on the finite-size scaling exponents of the mean
entanglement entropy and level statistics at the many-body localization phase
transition using several different arguments. Our results are at odds with
recent small-size numerics, for which we estimate the crossover scales beyond
which the Harris bound must hold.
