Asymptotic Scaling in the Two-Dimensional $O(3)$ $σ$-Model at Correlation Length $10^5$

Abstract

We carry out a high-precision Monte Carlo simulation of the two-dimensional $O(3)$-invariant $σ$-model at correlation lengths $ξ$ 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 discuss carefully the systematic and statistical errors in this extrapolation. We then compare the extrapolated data to the renormalization-group predictions. The deviation from asymptotic scaling, which is $\approx 25\%$ at $ξ\sim 10^2$, decreases to $\approx 4\%$ at $ξ\sim 10^5$.