Critical phenomena at edges and corners

TL;DR

This paper investigates the critical behavior at edges and corners of the 3D Ising model using Monte Carlo simulations, estimating local magnetization exponents and comparing them with theoretical predictions.

Contribution

It provides new Monte Carlo estimates of critical exponents at edges and corners, revealing their dependence on geometric angles and comparing with existing theoretical methods.

Findings

Critical exponent β₂ varies with opening angle θ.

Corner magnetization exponent β₃ estimated at 1.86.

Monte Carlo results compared with mean field, RG, and series expansions.

Abstract

Using Monte Carlo techniques, the critical behaviour at edges and corners of the three-dimensional Ising model is studied. In particular, the critical exponent $\beta_2$ of the local magnetization at edges formed by two intersecting free surfaces is estimated to be, as a function of the opening angle $\theta$, $0.96 \pm 0.02$ for $\theta = 135^o$, $1.28 \pm 0.04$ for $90^o$, and $2.30 \pm 0.10$ for $45^o$. The critical exponent $\beta_3$ of the corner magnetization of a cube is found to be $1.86 \pm 0.06$. The Monte Carlo estimates are compared to results of mean field theory, renormalization group calculations and high temperature series expansions.