Finite-Size Scaling at fixed Renormalization-Group invariant

TL;DR

This paper reviews a finite-size scaling method at fixed renormalization-group invariant, demonstrating its improved statistical accuracy in analyzing Monte Carlo data at critical points through benchmarks and an application to the O(2) model.

Contribution

It provides a detailed review and implementation of the fixed RG invariant finite-size scaling method, highlighting its advantages over standard analysis and introducing covariance-based optimization.

Findings

Significant statistical accuracy improvements in Monte Carlo data analysis.

Large gains due to cross-correlations between observables.

Accurate estimate of the inverse critical temperature for the O(2) model.

Abstract

Finite-size scaling at fixed renormalization-group invariant is a powerful and flexible technique to analyze Monte Carlo data at a critical point. It consists in fixing a given renormalization-group invariant quantity to a given value, thereby trading its statistical fluctuations with those of a parameter driving the transition. One remarkable feature is the observed significant improvement of statistical accuracy of various quantities, as compared to a standard analysis. We review the method, discussing in detail its implementation, the error analysis, and a previously introduced covariance-based optimization. Comprehensive benchmarks on the Ising model in two and three dimensions show large gains in the statistical accuracy, which are due to cross-correlations between observables. As an application, we compute an accurate estimate of the inverse critical temperature of the improved O(2) $\phi^4$ model on a three-dimensional cubic lattice.