The Flat Phase of Crystalline Membranes

We present the results of a high-statistics Monte Carlo simulation of a phantorn crystalline (fixed-connectivity) membrane with free boundary. We verify trie existence of a fiai phase by exarnining lattices of size up to 128~. The Harniltonian of the rnodel is trie sum of a simple spnng pair potential, with no hard-core repulsion, and beuding energy. The only free pararneter is the bending ngidity ~. In-plane elastic constants are non explicitly introduced. We obtain the rernarkable result thon this simple model dynamically generates the elastic constants required to stabilize the fiai phase. We present measurements of the size (Flory) exportent v and the roughness exportent (. We also determine the critical exponents ~ and ~u descnbing the scale dependence of the bending rigidity (~(q) mJ q~~) and trie mduced elastic constants (A(q)mJ ~1(q) ~J q~~ ). Ai bending rigidity ~ = l-1, we find v = 0.95(5) (Hausdorlf dimension dH = 2 Iv = 2,1(1)), ( = 0.64(2 and ~u = 0.50(1). These results are consistent with the scaling relation ( = (2 + ~u)/4. The additional scaling relation ~ = 2(1- () irnplies ~ = 0.72(4). A direct measurernent of ~ from the power-law decay of the normal-normal correlation functiou yields ~ m 0.6 on the 128~ lattice.