Cluster nonequilibrium relaxation in Ising models observed with the Binder ratio

TL;DR

This study reformulates nonequilibrium relaxation analysis using the Binder ratio in Ising models, revealing a finite-size scaling formula that allows precise critical point estimation and understanding of relaxation behavior.

Contribution

It introduces a new finite-size scaling approach for the Binder ratio in cluster algorithms, enhancing critical point and relaxation exponent evaluation in Ising models.

Findings

Finite-size scaling formula relates to correlation length relaxation.

Precise critical point estimation achieved.

Logarithmic scaling explains relaxation behavior.

Abstract

The Binder ratios exhibit discrepancy from the Gaussian behavior of the magnetic cumulants, and their size independence at the critical point has been widely utilized in numerical studies of critical phenomena. In the present article we reformulate the nonequilibrium relaxation (NER) analysis in cluster algorithms using the $(2,1)$-Binder ratio, and apply this scheme to the two- and three-dimensional Ising models. Although the stretched-exponential relaxation behavior at the critical point is not explicitly observed in this quantity, we find that there exists a logarithmic finite-size scaling formula which can be related with a similar formula recently derived in cluster NER of the correlation length, and that the formula enables precise evaluation of the critical point and the stretched-exponential relaxation exponent $\sigma$. Physical background of this novel behavior is explained by the simulation-time dependence of the distribution function of magnetization in two dimensions and temperature dependence of $\sigma$ obtained from magnetization in three dimensions.