Absence of first-order unbinding transitions of fluid and polymerized membranes

Unbinding transition of fluid and polymerized membranes are studied by renormalization-group (RG) methods. Two different RG schemes are used and found to give rather consistent results. The fixed-point structure of both RGs exhibits a complex behavior as a function of the decay exponent τ for the fluctuation-induced interaction of the membranes. For τ>${\mathrm{\tau }}_{\mathit{S}2}$, interacting membranes can undergo first-order transitions even in the strong-fluctuation regime. Our estimates for ${\mathrm{\tau }}_{\mathit{S}2}$ imply, however, that both fluid and polymerized membranes unbind in a continuous way in the absence of lateral tension.