Phase Transition of a Semiflexible Membrane in Two Dimensions

Abstract

We study a two-dimensional semiflexible membrane model whose formal Hamiltonian is given by $H[\phi]=\sum_x \bigl(\|\nabla \phi_x\|^2 +N^\lambda |\Delta \phi_x|^2\bigr)$, interpolating between the discrete Gaussian free field (DGFF) and the membrane model (MM). We analyze its finite-volume covariance in the bulk as $N\to\infty$, and identify distinct regimes depending on the parameter $\lambda$. For $\lambda<0$, the covariance of the model agrees with that of the DGFF up to a negligible error, while for $\lambda>2$, it agrees with the rescaled MM covariance up to a negligible error. In the intermediate regime $\lambda\in[0,2]$, we identify a different crossover behavior: in the microscopic range $\|x-y\|\ll N^{\lambda/2}$, the leading asymptotics no longer resolve the precise distance between the two points. In this microscopic regime, we further determine the leading logarithmic coefficient of the bulk covariance. These results provide a unified description of the crossover from gradient-dominated to curvature-dominated behavior in this class of models.