Self-similar transformations of lattice-Ising models at critical temperatures

TL;DR

This paper introduces a novel geometric and fractal dimension-based approach to analyze critical points in lattice-Ising models, simplifying calculations and providing high-accuracy predictions across various models.

Contribution

It develops a block-spin Gaussian model linking fractal dimensions to critical points, offering a new, effective method for analyzing phase transitions in lattice-Ising models.

Findings

High-accuracy critical point predictions for five lattice models

Effective simplification of critical point calculations

Insight into fluctuations at critical temperatures

Abstract

We classify geometric blocks that serve as spin carriers into simple blocks and compound blocks by their topologic connectivity, define their fractal dimensions and describe the relevant transformations. By the hierarchical property of transformations and a block-spin scaling law we obtain a relation between the block spin and its carrier's fractal dimension. By mapping we set up a block-spin Gaussian model and get a formula connecting the critical point and the minimal fractal dimension of the carrier, which guarantees the uniqueness of a fixed point corresponding to the critical point, changing the complicated calculation of critical point into the simple one of the minimal fractal dimension. The numerical results of critical points with high accuracy for five conventional lattice-Ising models prove our method very effective and may be suitable to all lattice-Ising models. The origin of fluctuations in structure at critical temperature is discussed. Our method not only explains the problems met in the renormalization-group theory, but also provides a useful tool for deep investigation of the critical behaviour.