  We investigate the flat phase of quantum polymerized phantom membranes by
means of a nonperturbative renormalization group approach. We first implement
this formalism for general quantum polymerized membranes and derive the flow
equations that encompass both quantum and thermal fluctuations. We then deduce
and analyze the flow equations relevant to study the flat phase and discuss
their salient features : quantum to classical crossover and, in each of these
regimes, strong to weak coupling crossover. We finally illustrate these
features in the context of free standing graphene physics.
