Disordered interacting spin chains that undergo a many-body
localization transition are characterized by two limiting behaviors
where the dynamics are chaotic and integrable. However, the transition
region between them is not fully understood yet. We propose here a
possible finite-size precursor of a critical point that shows a typical
finite-size scaling and distinguishes between two different dynamical
phases. The kurtosis excess of the
diagonal fluctuations of the full one-dimensional momentum distribution from
its microcanonical average is maximum at this singular point in the
paradigmatic disordered J_1J1-J_2J2
model. For system sizes accessible to exact diagonalization, both the
position and the size of this maximum scale linearly with the system
size. Furthermore, we show that this singular point is found at the same
disorder strength at which the Thouless and the Heisenberg energies
coincide. Below this point, the spectral statistics follow the universal
random matrix behavior up to the Thouless energy. Above it, no traces of
chaotic behavior remain, and the spectral statistics are well described
by a generalized semi-Poissonian model, eventually leading to the
integrable Poissonian behavior. We provide, thus, an integrated scenario
for the many-body localization transition, conjecturing that the
critical point in the thermodynamic limit, if it exists, should be given
by this value of disorder strength.
Finite size scaling for the many-body-localization transition: finite-size-pseudo-critical points of individual eigenstates
