Critical points with high accuracy and fluctuation origin of 2 and 3-dimensional Ising models

TL;DR

This paper introduces a universal fractal-based method to accurately determine critical points in 2D and 3D Ising models, revealing fluctuation mechanisms and the role of fractal dimensions at phase transitions.

Contribution

It presents a novel approach combining fractal analysis and renormalization group theory to precisely locate critical points and explore fluctuation origins in Ising models.

Findings

High-accuracy critical points computed for three Ising models.

Identification of critical points as fluctuation attractors based on fractal edges.

Discovery of different block spin states at the critical point.

Abstract

We proposed a new universal method for significantly increasing accuracy of critical points of 2 and 3-dimensional Ising models and exploring fluctuation mechanism. The method is based on analysis of block fractals and the renormalization group theory. We discussed hierarchies and rescaling rule of the self similar transformations, and define a fractal dimension of an ordered block, which minimum corresponds to a fixed point of the transformations. By the connectivity we divide the blocks into two types: irreducible and reducible. We find there are two block spin states: single state and k-fold state, each of which relates to a system or a subsystem described by a block spin Gaussian model set up by mathematic map. Using the model we obtain a universal formula of critical points by the minimal fractal dimensions. We computed the critical points with high accuracy for three Ising models. It is the first time to find a critical point only requires a fractal edge, which causes fluctuations, and the point acts as a fluctuation attractor. Finally, we discussed a possibility of different block spins at the critical point.