度量自由度三角化曲面模型的面内变形
In-plane deformation of a triangulated surface model with metric degrees of freedom
摘要：
利用正则蒙特卡罗模拟技术，研究了球面三角剖分上的Regge演算模型。离散模型是统计机械定义的变量$X$，$g$和$ρ$，这表示在${\bf R}^3$的表面位置，一个二维表面上的度量$M$和表面密度的$M$，分别。度量$g$仅通过使用{ $M$ }中三角形的亏角来定义。这与传统的Regge演算模型形成鲜明对比，在传统的Regge演算模型中，{ $g$ }仅取决于三角形的边长。我们发现本文中的离散模型经历了在$b to infty$处的光滑球形相和在$b to 0$处的褶皱相之间的相变，其中$b$是弯曲刚度。转变是一阶的，并且与在没有变量$g$和$ρ$的常规模型中观察到的转变一致。这意味着形状变换过渡不受度量自由度的影响。研究还发现，该模型经历了一个连续的面内变形过渡。这种连续的转变反映在$M$和$X(M)$的表面积的几乎不连续的变化中，其中$M$的表面积与密度变量$ρ$共轭。
英文摘要：
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 $ρ$, 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 $ρ$. 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 of $M$ and that of $X(M)$, where the surface area of $M$ is conjugate to the density variable $ρ$.
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