  We propose and analyze a model for phase transitions in an inhomogeneous
fluid membrane, that couples local composition with curvature nonlinearly. For
asymmetric membranes, our model shows generic non-Ising behavior and the
ensuing phase diagram displays either a first- or a second-order phase
transition through a critical point (CP) or a tricritical point (TP), depending
upon the bending modulus. It predicts generic nontrivial enhancement in
fluctuations of asymmetric membranes that scales with system size in a power
law fashion at the CP and TP in two dimensions, not observed in symmetric
membranes. It also yields two-dimensional Ising universality class for
symmetric membranes, in agreement with experimental results.
