In-Plane Deformation Of A Triangulated Surface Model With Metric Degrees Of Freedom

Using the canonical Monte Carlo simulation technique, we study a Regge calculus model on triangulated spherical surfaces. The discrete model is statistical mechanically defined with the variablesX,gand ρ, which denote the surface position inR3, the metric on a two-dimensional surfaceMand the surface density ofM, respectively. The metricgis defined only by using the deficit angle of the triangles inM. This is in sharp contrast to the conventional Regge calculus model, wheregdepends only on the edge length of the triangles. We find that the discrete model in this paper undergoes a phase transition between the smooth spherical phase atb → ∞and the crumpled phase atb → 0, wherebis the bending rigidity. The transition is of first-order and identified with the one observed in the conventional model without the variablesgand ρ. This implies that the shape transformation transition is not influenced by the metric degrees of freedom. It is also found that the model undergoes a continuous transition of in-plane deformation. This continuous transition is reflected in almost discontinuous changes of the surface area ofMand that ofX(M), where the surface area ofMis conjugate to the density variable ρ.