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Title: Can a many-body mobility edge exist in the thermodynamic limit
Plain: 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: 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.
Settled by: 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.
Kind: well-posed
Answer type: yes-no
References: (empty)
Literature status: contested
Status note: The 2016 bubble argument is unrefuted while finite-size mobility edges persist in numerics; no decisive large-system result is known (moderate confidence).
Posed since: 2015