  Many-body eigenstates that are neither thermal nor many-body-localized (MBL)
were numerically found in certain interacting chains with moderate
quasiperiodic potentials. The energy regime consisting of these non-ergodic but
extended (NEE) eigenstates has been extensively studied for being a possible
many-body mobility edge between the energy-resolved MBL and thermal phases.
Recently, the NEE regime was further proposed to be a prethermal phenomenon
that generally occurs when different operators spread at sizably different
timescales. Here, we numerically examine the mutual independence among the NEE,
MBL, and thermal regimes in the lens of eigenstate entanglement spectra (ES).
Given the complexity and rich information embedded in ES, we develop an
unsupervised learning approach that is designed to quantify the mutual
independence among general phases. Our method is first demonstrated on an
illustrative toy example that uses RGB color data to represent phases, then
applied to the ES of an interacting generalized Aubry Andre model from weak to
strong potential strength. We find that while the MBL and thermal regimes are
mutually independent, the NEE regime is dependent on the former two and
smoothly appears as the potential strength decreases. We attribute our
numerically finding to the fact that the ES data in the NEE regime exhibits
both an MBL-like fast decay and a thermal-like long tail.
