Scaling, Fractal Dynamics and Critical Exponents: Application in a non-integer dimensional Ising model

TL;DR

The paper tackles how scaling, fractal dynamics, and critical exponents describe second-order phase transitions in non-integer spatial dimensions. It develops a fractional-differential description using the Riesz derivative to replace the Laplacian, yielding an exact Fisher exponent $\eta$ through $\eta = d - d_R$ and connecting $d_R$ to fractal edge dimension $d_f$ via $d_R = 2(d_f - 1)$. The authors validate the framework for the Ising universality class across $1 \le d \le 4$, showing consistent exponent relations and providing an empirical $\eta(d)$ curve that extends to non-integer cases. This work supports a fractal-geometric view of criticality that remains valid beyond integer dimensions and motivates applying the approach to growth and non-equilibrium transitions, suggesting a general fractal-dimension-based framework for other critical phenomena.

Abstract

Moving beyond simple associations, researchers need tools to quantify how variables influence each other in space and time. Correlation functions provide a mathematical framework for characterizing these essential dependencies, revealing insights into causality, structure, and hidden patterns within complex systems. In physical systems with many degrees of freedom, such as gases, liquids, and solids, a statistical analysis of these correlations is essential. For a field $Ψ(\vec{x},t)$ that depends on spatial position $\vec{x}$ and time $t$, it is often necessary to understand the correlation with itself at another position and time $Ψ(\vec{x}_0,t_0)$. This specific function is called the autocorrelation function. In this context, the autocorrelation function for order--parameter fluctuations, introduced by Fisher, provides an important mathematical framework for understanding the second-order phase transition at equilibrium. However, his analysis is restricted to a Euclidean space of dimension $d$, and an exponent $η$ is introduced to correct the spatial behavior of the correlation function at $T=T_c$. In recent work, Lima et al demonstrated that at $T_c$ a fractal analysis is necessary for a complete description of the correlation function. In this study, we investigate the fundamental physics and mathematics underlying phase transitions. In particular, we show that the application of modern fractional differentials allows us to write down an equation for the correlation function that recovers the correct exponents below the upper critical dimension. We obtain the exact expression for the Fisher exponent $η$. Furthermore, we examine the Rushbrooke scaling relation, which has been questioned in certain magnetic systems, and, drawing on results from the Ising model, we confirm that both our relations and the Rushbrooke scaling law hold even when $d$ is not an integer.

Figures (2)

• Figure 1: Snapshots of spin configurations in the $2d$ Ising model at various temperatures, generated via Metropolis Monte Carlo simulations. Yellow and dark blue represent spins $+1$ and $-1$, respectively. Panel a) shows predominantly ferromagnetic order with minor disorder at temperature $T = 0.95 T_{c}$. In b) the system is in a fully disordered paramagnetic phase with magnetization $m = 0$. c)-f) $T = T_c$ ($m=0$), where the correlation length $\rho \to \infty$. (c) Critical state overview; (d)-(f) sequential 2$\times$-magnified snapshots of the region inside the square in the previous panel. At criticality, the system exhibits fractal (scale-invariant) spin clusters and divergent correlation length parameter $\rho(T)$.
• Figure 2: $\eta$ as function of $d$. Comparison between the values of $\eta^*$, here $\eta^*$ is $\eta$ from the literature and the values $\eta$ calculated using Eq. \ref{['eta']} for dimensions in the range $1<d<4$. All data points are taken from Table \ref{['Table1']}, while the green curve represents the proposed function $\eta(d)$ given by Eq. (\ref{['curve']}).