  To understand the finite-size-scaling properties of phases transitions in
classical and quantum models in the presence of quenched disorder, it has
proven to be fruitful to introduce the notion of a finite-size-pseudo-critical
point in each disordered sample and to analyze its sample-to-sample
fluctuations as a function of the size. For the Many-Body-Localization
transition, where very strong eigenstate-to-eigenstate fluctuations have been
numerically reported even within a given disordered sample at a given energy
density [X. Yu, D. J. Luitz, B. K. Clark, arxiv:1606.01260 and V. Khemani, S.
P. Lim, D. N. Sheng, D. A. Huse,arxiv:1607.05756], it seems thus useful to
introduce the notion of a finite-size-pseudo-critical point for each individual
eigenstate and to study its eigenstate-to-eigenstate fluctuations governed by
the correlation length exponent $\nu$. The scaling properties of critical
eigenstates are also expected to appear much more clearly if one considers each
eigenstate at its finite-size-pseudo-critical point, where it is 'truly
critical', while standard averages over eigenstates and samples in the critical
region actually see a mixture of states that are effectively either localized
or delocalized.
