Finite-Size Scaling at $\xi/L \gg 1$

TL;DR

This paper introduces a straightforward finite-size scaling method for accurately extrapolating Monte Carlo data to infinite volume, even when the correlation length vastly exceeds the lattice size, demonstrated through three complex models.

Contribution

It proposes a new finite-size scaling technique with detailed error analysis, applicable to large correlation lengths in Monte Carlo simulations.

Findings

Reliable extrapolations with errors of a few percent achieved for correlation lengths 1000 times larger than lattice.

Method successfully applied to three different two-dimensional models.

Systematic and statistical errors are carefully discussed and managed.

Abstract

We present a simple and powerful method for extrapolating finite-volume Monte Carlo data to infinite volume, based on finite-size-scaling theory. We discuss carefully its systematic and statistical errors, and we illustrate it using three examples: the two-dimensional three-state Potts antiferromagnet on the square lattice, and the two-dimensional $O(3)$ and $O(\infty)$ $\sigma$-models. In favorable cases it is possible to obtain reliable extrapolations (errors of a few percent) even when the correlation length is 1000 times larger than the lattice.