Monte Carlo studies of triangulated spherical surfaces in the two-dimensional space

TL;DR

This study uses Monte Carlo simulations to analyze phase transitions in triangulated spherical surfaces in R^2, revealing first-order and second-order transitions in fixed and fluid models, respectively, with implications for string theory and surface geometry.

Contribution

It introduces a novel Monte Carlo approach combining Regge calculus with independent sums over geometry and shape variables in surface models.

Findings

First-order transition in fixed-connectivity models.

Second-order transition in fluid surface models.

Transitions affect the internal geometry of the surfaces.

Abstract

We numerically study a triangulated surface model in R^2 by taking into account a viewpoint of string model. The models are defined by a mapping X from a two-dimensional surface M to R^2, where the mapping X and the metric g of M are the dynamical variables. The sum over g in the partition function is simulated by the sum over bond lengths and deficit angles by using the Regge calculus technique, and the sum over g is defined to be performed independently of the sum over X. We find that the model undergoes a first-order transition of surface fluctuations, which accompanies a collapsing transition, and that the transitions are reflected in the internal geometry of surface. Fluid surface models are also studied on dynamically triangulated surfaces, and the transitions are found to be of second order. The order of the transition remains unchanged from that of the conventional model defined only by the variable X both in the fixed-connectivity and the fluid models.