In-plane deformation of a triangulated surface model with metric degrees of freedom

TL;DR

This study uses Monte Carlo simulations to analyze a Regge calculus model on triangulated spherical surfaces, revealing a first-order shape transition unaffected by metric degrees of freedom and a separate continuous in-plane deformation transition.

Contribution

It introduces a novel Regge calculus model with metric degrees of freedom defined by deficit angles, and demonstrates its phase behavior and in-plane deformation characteristics.

Findings

First-order phase transition between smooth and crumpled phases.

In-plane deformation transition is continuous.

Shape transition is unaffected by metric degrees of freedom.

Abstract

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 variables $X$, $g$ and $\rho$, which denote the surface position in ${\bf R}^3$, the metric on a two-dimensional surface $M$ and the surface density of $M$, respectively. The metric $g$ is defined only by using the deficit angle of the triangles in {$M$}. This is in sharp contrast to the conventional Regge calculus model, where {$g$} depends 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 at $b to infty$ and the crumpled phase at $b to 0$, where $b$ is the bending rigidity. The transition is of first-order and identified with the one observed in the conventional model without the variables $g$ and $\rho$. 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 of $M$ and that of $X(M)$, where the surface area of $M$ is conjugate to the density variable $\rho$.