Comparison of Monte Carlo Results for the 3D Ising Interface Tension and Interface Energy with (Extrapolated) Series Expansions

TL;DR

This paper compares Monte Carlo simulations with series expansion methods for the 3D Ising model's interface tension and energy, revealing limitations of series approximants near the roughening transition.

Contribution

It provides a detailed comparison showing that Padé and differential approximants do not improve series results near the roughening transition in the 3D Ising model.

Findings

Monte Carlo data show a specific heat peak in the smooth phase.

Series approximants fail to capture the specific heat peak.

Comparison with the ASOS model energy data extends the analysis.

Abstract

We compare Monte Carlo results for the interface tension and interface energy of the 3-dimensional Ising model with Pad\'e and inhomogeneous differential approximants of the low temperature series that was recently extended by Arisue to $17^{\rm th}$ order in $u=\exp(-4\beta)$. The series is expected to suffer from the roughening singularity at $u\approx 0.196$. The comparison with the Monte Carlo data shows that the Pad\'e and inhomogeneous differential approximants fail to improve the truncated series result of the interface tension and the interface energy in the region around the roughening transition. The Monte Carlo data show that the specific heat displays a peak in the smooth phase. Neither the truncated series nor the Pad\'e approximants find this peak. We also compare Monte Carlo data for the energy of the ASOS model with the corresponding low temperature series that we extended to order $u^{12}$.