Monte Carlo simulations of Landau-Ginzburg model for membranes

The Landau-Ginzburg (LG) model for membranes is numerically studied on triangulated spheres in ${\bf R}^3$. The LG model is in sharp contrast to the model of Helfrich-Polyakov (HP). The reason for this difference is that the curvature energy of the LG (HP) Hamiltonian is defined by means of the tangential (normal) vector of the surface. For this reason the curvature energy of the LG model includes the in-plane bending or shear energy component, which is not included in the curvature energy of the HP model. From the simulation data, we find that the LG model undergoes a first-order collapse transition. The results of the LG model in the higher dimensional spaces ${\bf R}^d (d>3)$ and on the self-avoiding surfaces in ${\bf R}^3$ are presented and discussed. We also study the David-Guitter (DG) model, which is a variant of the LG model, and find that the DG model undergoes a first-order transition. It is also found that the transition can be observed only on the homogeneous surfaces, which are composed of almost uniform triangles according to the condition that the induced metric ${\partial_a {\bf r}}\cdot{\partial_b {\bf r}}$ is close to $\delta_{ab}$.