Fluctuations in membranes with crystalline and hexatic order

We study the interplay between crystalline (or hexatic) order and thermal fluctuations in membranes with vanishing surface tension. If the connectivity of the crystalline state is preserved, the membrane remains uncrumpled at low temperatures. When dislocations are allowed, however, screening of elastic stresses by buckling reduces dislocation energies, and promotes dislocation unbinding. When the resulting hexatic phase is stable, the stiffness associated with orientational correlations leads to a logarithmic enhancement of the bending rigidity which counteracts the thermal softening found in fluid surfaces. A finite temperature crumpling transition is predicted for crystalline membranes, and possibly for hexatic membranes as well.