Fractal geometry of Ising magnetic patterns: signatures of criticality and diffusive dynamics

TL;DR

This study explores the fractal geometry of magnetic patterns in a 2D Ising model under diffusive thermal dynamics, revealing geometric signatures of phase transitions and differences from non-diffusive dynamics.

Contribution

It introduces a novel analysis of magnetic pattern geometry using fractal dimensions under diffusive dynamics, highlighting phase transition signatures.

Findings

Fractal dimension varies with temperature, indicating criticality.

Diffusive dynamics produce distinct geometric patterns compared to non-diffusive dynamics.

Geometric signatures correlate with phase transition points.

Abstract

We investigate the geometric properties displayed by the magnetic patterns developing on a two-dimensional Ising system, when a diffusive thermal dynamics is adopted. Such a dynamics is generated by a random walker which diffuses throughout the sites of the lattice, updating the relevant spins. Since the walker is biased towards borders between clusters, the border-sites are more likely to be updated with respect to a non-diffusive dynamics and therefore, we expect the spin configurations to be affected. In particular, by means of the box-counting technique, we measure the fractal dimension of magnetic patterns emerging on the lattice, as the temperature is varied. Interestingly, our results provide a geometric signature of the phase transition and they also highlight some non-trivial, quantitative differences between the behaviors pertaining to the diffusive and non-diffusive dynamics.