Two-Dimensional $O(3)$ $\sigma$-Model up to Correlation Length $10^5$

TL;DR

This paper presents high-precision Monte Carlo simulations of the 2D $O(3)$ sigma model at very large correlation lengths, demonstrating improved agreement with renormalization-group predictions and revealing the scale of deviations from asymptotic scaling.

Contribution

It introduces a new finite-size-scaling-based method for extrapolating Monte Carlo data to infinite volume, enabling analysis at unprecedented correlation lengths.

Findings

Deviation from asymptotic scaling decreases from 25% at $\xi \\sim 10^2$ to 4% at $\xi \\sim 10^5$

The method allows accurate comparison with renormalization-group predictions at very large scales

Demonstrates the effectiveness of finite-size-scaling theory in high-precision simulations

Abstract

We carry out a high-precision Monte Carlo simulation of the two-dimensional $O(3)$-invariant $\sigma$-model at correlation lengths $\xi$ up to $\sim 10^5$. Our work employs a new and powerful method for extrapolating finite-volume Monte Carlo data to infinite volume, based on finite-size-scaling theory. We compare the extrapolated data to the renormalization-group predictions. The deviation from asymptotic scaling, which is $\approx 25\%$ at $\xi \sim 10^2$, decreases to $\approx 4\%$ at $\xi \sim 10^5$.