The Critical Exponents of Crystalline Random Surfaces

TL;DR

This study numerically investigates the critical behavior of crystalline random surfaces with extrinsic curvature, accurately determining critical exponents at the crumpling transition through multiple methods.

Contribution

It provides high-precision numerical estimates of critical exponents for the model, confirming their consistency across different measurement techniques.

Findings

Correlation length exponent ν ≈ 0.71-0.73

Specific heat exponent α ≈ 0.58

Model in crumpled phase resembles an interacting effective field theory

Abstract

We report on a high statistics numerical study of the crystalline random surface model with extrinsic curvature on lattices of up to $64^2$ points. The critical exponents at the crumpling transition are determined by a number of methods all of which are shown to agree within estimated errors. The correlation length exponent is found to be $\nu=0.71(5)$ from the tangent-tangent correlation function whereas we find $\nu=0.73(6)$ by assuming finite size scaling of the specific heat peak and hyperscaling. These results imply a specific heat exponent $\alpha=0.58(10)$; this is a good fit to the specific heat on a $64^2$ lattice with a $\chi^2$ per degree of freedom of 1.7 although the best direct fit to the specific heat data yields a much lower value of $\alpha$. Our measurements of the normal-normal correlation functions suggest that the model in the crumpled phase is described by an effective field theory which deviates from a free field theory only by super-renormalizable interactions.