Bending rigidity of fluctuating membranes

A lattice model of random surfaces is studied including configurations with arbitrary topologies, overhangs and bubbles. The Hamiltonian of the surface includes a term proportional to its area and a scale-invariant integral of the squared mean curvature. We propose a discretization of the curvature which ensures the scale-invariance of the bending energy on the lattice. Nonperturbative renormalization groups for the surface tension and the bending rigidity are applied, which are also valid at high temperatures and scales above the persistence length. We find at vanishing surface tensions a closed expression for the scale dependent rigidity including the usual logarithmic decay at low temperatures. Different scaling behaviours at non-vanishing tensions occur yielding characteristic length scales, which determine the structure of homogeneous droplet, lamellar, and microemulsion phases.