The flat phase of crystalline membranes

We present the results of a high-statistics Monte Carlo simulation of a phantom crystalline (fixed-connectivity) membrane with free boundary. We verify the existence of a flat phase by examining lattices of size up to 128(2). The Hamiltonian of the model is the sum of a simple spring pair potential, with no hard-core repulsion, and bending energy. The only free parameter is the bending rigidity kappa. In-plane elastic constants are not explicitly introduced. We obtain the remarkable result that this simple model dynamically generates the elastic constants required to stabilize the flat phase. We present measurements of the size (Flory) exponent nu and the roughness exponent zeta. We also determine the critical exponents eta and eta(u) describing the scale dependence of the bending rigidity (kappa(q) similar to q(-eta)) and the induced elastic constants (lambda(q) similar to mu(q) similar to q(eta u)). At bending rigidity kappa = 1.1, we find nu = 0.95(5) (Hausdorff dimension d(H) = 2/nu = 2.1(1)), zeta = 0.64(2) and eta(u) = 0.50(1). These results are consistent with the scaling relation zeta = (2 + eta(u))/4. The additional scaling relation eta = 2(1 - zeta) implies eta = 0.72(4). A direct measurement of eta from the power-law decay of the normal-normal correlation function yields eta approximate to 0.6 on the 128(2) lattice.