Asymptotic scaling in the two-dimensional $SU(3)$ $\sigma$-model at correlation length $4 \times 10^5$

TL;DR

This paper presents high-precision simulations of the two-dimensional SU(3) sigma model at very large correlation lengths, confirming asymptotic scaling and matching renormalization-group predictions within a few percent.

Contribution

It provides the first high-precision data at correlation lengths up to 4×10^5, validating asymptotic scaling and nonperturbative constants in the SU(3) sigma model.

Findings

Good asymptotic scaling observed for correlation length > 10^3

Nonperturbative constant within 2-3% of predicted value at ξ ≈ 4×10^5

Finite-size scaling theory effectively extrapolates finite-volume data

Abstract

We carry out a high-precision simulation of the two-dimensional $SU(3)$ principal chiral model at correlation lengths $\xi$ up to $\approx\! 4 \times 10^5$, using a multi-grid Monte Carlo (MGMC) algorithm. We extrapolate the finite-volume Monte Carlo data to infinite volume using finite-size-scaling theory, and we discuss carefully the systematic and statistical errors in this extrapolation. We then compare the extrapolated data to the renormalization-group predictions. For $\xi \gtapprox 10^3$ we observe good asymptotic scaling in the bare coupling; at $\xi \approx 4 \times 10^5$ the nonperturbative constant is within 2--3\% of its predicted limiting value.