Decrumpling membranes by quantum effects

– The phase diagram of an incompressible fluid membrane subject to quantum and thermal fluctuations is calculated exactly in a large number of dimensions of configuration space. At zero temperature, a crumpling transition is found at a critical bending rigidity 1/αc. For membranes of fixed lateral size, a crumpling transition occurs at nonzero temperatures in an auxiliary mean field approximation. As the lateral size L of the membrane becomes large, the flat regime shrinks with 1/ lnL. Introduction. – Amphiphilic molecules in aqueous solution form fluid bilayers with vanishing surface tension. This causes them to undergo strong shape fluctuations, governed by the Canham-Helfrich curvature energy [1, 2] H0 = 1 2α0 ∫ dS H, (1) where dS is the surface element, H corresponds to the doubled mean curvature of the surface at each point, and 1/α0 is the bending rigidity. Thermal undulations renormalize 1/α0 as follows [3–9]: 1 α = 1 α0 [ 1− 3 4π kBTα0 ln(ΛL) ] , (2) where Λ is an ultraviolet wavevector cutoff set by the inverse width of the molecules in the membrane, and L is an infrared cutoff determined by its finite size. In practice, membranes occur in the form of spherical vesicles, and L is determined by their surface area. At finite temperatures, the model is only defined for finite planar surfaces. For L larger than the de Gennes-Taupin persistence length ξp = Λ −1 exp(4π/3kBTα0) [10], the renormalized bending rigidity 1/α vanishes. Beyond the persistence length, the normal vectors of the surface are uncorrelated, and the membrane is crumpled. The renormalization group flow extracted from the perturbative result (2) as well as nonperturbative (∗) E-Mail: [email protected] (∗∗) E-Mail: [email protected], http://www.physik.fu-berlin.de/∼kleinert