Finite size scaling for the Many-Body-Localization Transition : finite-size-pseudo-critical points of individual eigenstates

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.