  The random energy model (REM) provides a solvable mean-field description of
the equilibrium spin glass transition. Its quantum sibling (the QREM), obtained
by adding a transverse field to the REM, has similar properties and shows a
spin glass phase for sufficiently small transverse field and temperature. In a
recent work, some of us have shown that the QREM further exhibits a many-body
localization - delocalization (MBLD) transition when viewed as a closed quantum
system, evolving according to the quantum dynamics. This phase encloses the
familiar equilibrium spin-glass phase. In this paper we study in detail the
MBLD transition within the forward-scattering approximation and replica
techniques. The predictions for the transition line are in good agreement with
the exact diagonalization numerics. We also observe that the structure of the
eigenstates at the MBLD critical point changes continuously with the energy
density, raising the possibility of a family of critical theories for the MBLD
transition.
