Can a many-body mobility edge exist in the thermodynamic limit
In plain words
Small-system numerics suggest that at moderate disorder, low-energy states stay localized while high-energy states thermalize, separated by an energy threshold called a mobility edge. Theory argues that hot regions would then wander through the system and thermalize all of it.
Precise statement
For the random-field Heisenberg chain at disorder $W$ where exact diagonalization at $L \le 22$ shows an energy-dependent localization threshold $\epsilon_c(W)$ (Luitz, Laflorencie, Alet, PRB 91, 081103, 2015), do eigenstates with energy density below $\epsilon_c$ remain localized as $L \to \infty$? De Roeck, Huveneers, Muller, Schiulaz (PRB 93, 014203, 2016) argued that mobile ergodic bubbles exclude this, so MBL requires localization at all energies. Answer yes or no.
What would settle it
Time evolution of low-energy-density initial states at sizes where rare high-energy-density bubbles are typical, or a theory that closes or validates the bubble mechanism.
Status in the literature
The 2016 bubble argument is unrefuted while finite-size mobility edges persist in numerics; no decisive large-system result is known (moderate confidence).