Inclusions in Fluctuating Membranes: Exact Results

Point-like inclusions in fluid, fluctuating membranes are considered. Here the term inclusion is used in a general sense and describes a number of seemingly disparate situations: particles in membranes or other external and localized forces (such as a laser tweezer) which i) make the membrane locally stiffer, ii) induce a local spontaneous curvature, iii) change the local membrane thickness, or iv) the local separation between neighboring membranes. All these situations can be described by linear or quadratic local perturbations, for which the partition function is calculated exactly using a Gaussian membrane model. The deformed shape of a membrane in response to the presence of one inclusion and the membrane-mediated interactions between inclusions are thus obtained without further approximations. The interaction between two inclusions described by linear perturbations is temperature independent and therefore not affected by membrane fluctuations. The interaction between two inclusions described by quadratic perturbations is solely due to membrane shape fluctuations and vanishes at zero temperatures; in the strong coupling limit it shows a universal logarithmic divergence at short length scales. Formulas for the interaction of $n$ inclusions are derived, which show non-trivial multibody contributions for the case of quadratic inclusions. All these results are valid for all temperatures and for all coupling strengths and thus bridge previously obtained results obtained at zero temperatures (neglecting membrane shape fluctuations) or using perturbation theory (for small strengths of the coupling between the inclusions and the membrane). These exact results are obtained with general Gaussian Hamiltonians and are thus applicable to all systems described by Gaussians forms