Nonlocal effective average action approach to crystalline phantom membranes

Abstract

We investigate the properties of crystalline phantom membranes, at the crumpling transition and in the flat phase, using a nonperturbative renormalization group approach. We avoid a derivative expansion of the effective average action and instead analyse the full momentum dependence of the elastic coupling functions. This leads to a more accurate determination of the critical exponents and further yields the full momentum dependence of the correlation functions of the in-plane and out-of-plane fluctuation. The flow equations are solved numerically for D=2 dimensional membranes embedded in a d=3 dimensional space. Within our approach we find a crumpling transition of second order which is characterized by an anomalous exponent $\eta_c\approx 0.63(8)$ and the thermal exponent $\nu\approx 0.69$. Near the crumpling transition the order parameter of the flat phase vanishes with a critical exponent $\beta\approx 0.22$. The flat phase anomalous dimension is $\eta_f\approx 0.85$ and the Poisson's ratio inside the flat phase is found to be $\sigma_f\approx -1/3$. At the crumpling transition we find a much larger negative value of the Poisson's ratio $\sigma_c \approx -0.71(5)$. We discuss further in detail the different regimes of the momentum dependent fluctuations, both in the flat phase and in the vicinity of the crumpling transition, and extract the crossover momentum scales which separate them.