Flat phase of quenched disordered membranes at three-loop order

We study quenched disordered polymerized membranes in their flat phase by means of a three-loop perturbative analysis performed in dimension D =4 −ε . We derive the renormalization group equations at this order and solve them up to order ε³. Our results confirm those obtained by Coquand et al. within a nonperturbative approach [Phys. Rev. E 97, 030102(R) (2018), 10.1103/PhysRevE.97.030102] predicting a finite-temperature, finite-disorder wrinkling transition and those obtained by Coquand and Mouhanna within a recent two-loop order approach [Phys. Rev. E 103, L031001 (2021), 10.1103/PhysRevE.103.L031001], while correcting some of the results obtained in this last reference. We compute the anomalous dimensions that characterize the scaling behavior at the various fixed points of the renormalization group flow diagram. They appear to be in strong agreement with those predicted within the nonperturbative context.