The Theory of Dynamical Random Surfaces with Extrinsic Curvature

TL;DR

This paper numerically investigates a two-dimensional discretized random surface with extrinsic curvature in three-dimensional space, revealing a phase transition from crumpled to smooth phases and analyzing critical behavior.

Contribution

It demonstrates the existence of a phase transition for toroidal topology and explores the critical properties and potential continuum limit of the model.

Findings

Phase transition from crumpled to smooth surface observed.

Critical exponent estimated between 0.38 and 0.42.

Specific heat does not diverge significantly at criticality.

Abstract

We analyze numerically the critical properties of a two-dimensional discretized random surface with extrinsic curvature embedded in a three-dimensional space. The use of the toroidal topology enables us to enforce the non-zero external extension without the necessity of defining a boundary and allows us to measure directly the string tension. We show that a phase transition from the crumpled phase to the smooth phase observed earlier for a spherical topology appears also for a toroidal surface for the same finite value of the coupling constant of the extrinsic curvature term. The phase transition is characterized by the vanishing of the string tension. We discuss the possible non-trivial continuum limit of the theory, when approaching the critical point. Numerically we find a value of the critical exponent $\n$ to be between .38 and .42. The specific heat, related to the extrinsic curvature term seems not to diverge (or diverge slower than logarithmically) at the critical point.