Cluster Algorithm Renormalization Group Study of Universal Fluctuations in the 2D Ising Model

TL;DR

This paper introduces an improved numerical method combining collective-mode algorithms and Renormalization Group techniques to study critical phenomena, applied to the 2D Ising model, revealing that universal fluctuations are not necessarily due to scale invariance or universality.

Contribution

The paper presents a novel combined algorithm for studying critical systems, enhancing the Monte Carlo Renormalization Group approach with cluster algorithm advantages.

Findings

Accurate critical exponents and RG flow for the 2D Ising model.

No evidence linking universal fluctuations to scale invariance.

Universal fluctuation shape not necessarily related to universality.

Abstract

In this paper we propose a novel method to study critical systems numerically by a combined collective-mode algorithm and Renormalization Group on the lattice. This method is an improved version of MCRG in the sense that it has all the advantages of cluster algorithms. As an application we considered the 2D Ising model and studied wether scale invariance or universality are possible underlying mechanisms responsible for the approximate "universal fluctuations" close to a so-called bulk temperature $T^*(L)$. "Universal fluctuations" was first proposed in [1] and stated that the probability density function of a global quantity for very dissimilar systems, like a confined turbulent flow and a 2D magnetic system, properly normalized to the first two moments, becomes similar to the "universal distribution", originally obtained for the magnetization in the 2D XY model in the low temperature region. The results for the critical exponents and the renormalization group flow of the probability density function are very accurate and show no evidence to support that the approximate common shape of the PDF should be related to both scale invariance or universal behavior.