Persistence of the separation between geometric and quantum crossover scales

Determine whether the parametric separation between the potential-energy-shell fragmentation crossover and the known quantum localization transition in the quantum random energy model persists when higher-order quantum resonance processes are included in the connected-shell fraction.

Background

Within the geometric approximation based on the shell probabilities and a one-hop percolation criterion, the shell-fragmentation crossover occurs at disorder strengths parametrically larger than those associated with the known QREM localization transition. The geometric construction does not incorporate quantum interference or higher-order resonance processes.

The authors identify the inclusion of higher-order resonance processes as necessary for assessing whether the distinction between the geometric fragmentation scale and the quantum localization scale is a genuine asymptotic feature or an artifact of the approximation. They specifically suggest incorporating multi-step resonances or constructing a self-consistent resonance graph.