  We construct a family of many-body wave functions to study the many-body
localization phase transition. The wave functions have a Rokhsar-Kivelson form,
in which the weight for the configurations are chosen from the Gibbs weights of
a classical spin glass model, known as the Random Energy Model, multiplied by a
random sign structure to represent a highly excited state. These wave functions
show a phase transition into an MBL phase. In addition, we see three regimes of
entanglement scaling with subsystem size: scaling with entanglement
corresponding to an infinite temperature thermal phase, constant scaling, and a
sub-extensive scaling between these limits. Near the phase transition point,
the fluctuations of the R\'enyi entropies are non-Gaussian. We find that
R\'enyi entropies with different R\'enyi index transition into the MBL phase at
different points and have different scaling behavior, suggesting a multifractal
behavior.
