Fluctuations and lower critical dimensions of crystalline membranes

Abstract

We study flexible D-dimensional fixed-connectivity crystalline membranes fluctuating in a d-dimensional embedding space. We address both the crumpling transition and fluctuations around the flat phase by means of qualitative arguments and exact analyses in the limit of large d. In particular, we investigate the nature of the crossover between critical and nonlinear elastic behavior in the flat phase near the crumpling transition. Using two different approaches, we argue that the lower critical dimension, Dlc(d), below which only the low-rigidity crumpled phase exists at finite temperatures can be computed as a function of d. One approach determines Dlc(d) by considering disordering effects of capillary waves on the long-range order of the membrane tangent vectors in the low-temperature phase. The other identifies D lc(d) as thedimension at which the Hausdorff dimension of the membrane at the second-order crumpling transition is equal to that of the flat phase. We explicitly calculate Dlc(d) in the large-d limit and find that Dlc(d) < 2, at least for sufficiently large d. Throughout the paper we present qualitative arguments that D lc(d) < 2 is satisfied in general. We also consider the lower critical dimension Du(d) of the positional order of particles comprising the membrane. We find Du(∞) = 3, and calculate the 1/d correction to this result. On qualitative grounds we argue that 2 ≤ Du(d) ≤ 3, with Du(d) = 2, when d = D = D u(d) = 2, and Du(d) = 3, when d → ∞