A new Fractal Mean-Field analysis in phase transition

TL;DR

The paper develops a fractal-mean-field framework for critical correlations by treating the propagation of fluctuations on fractal cluster boundaries and introducing a correlation fractal dimension $d_R$ tied to Fisher’s exponent $η$ via $η= d-d_R$. By replacing the local Laplacian with a fractional Laplacian (Riesz derivative) and enforcing a fractal integration measure, it derives $G(r)\sim r^{2ζ-d_f}$ and shows that $d_R$ satisfies $d-1\le d_R\le d$, with $η=d-d_R$. Extending the approach to continuous dimensions $1\le d\le 4$ for the Ising universality class, the authors compute $d_f$ and $d_R$ across $d$, fit explicit expressions for $β(d)$ and $ν(d)$, and demonstrate that $d_f^*(d)=d-β(d)/ν(d)$ agrees with the fractal description, while Rushbrooke scaling remains valid for non-integer $d$. The work provides a geometrically grounded, universal account of critical scaling that naturally recovers Euclidean mean-field behavior as the correlation geometry becomes purely Euclidean and suggests broad applicability to disordered and non-equilibrium systems through the same fractal framework.

Abstract

Understanding phase transitions requires not only identifying order parameters but also characterizing how their correlations behave across scales. By quantifying how fluctuations at distinct spatial or temporal points are related, correlation functions reveal the structural organization of complex systems. Here, we revisit the theoretical foundations of these correlations in systems undergoing second-order phase transitions, with emphasis on the Ising model extended to non-integer spatial dimensions. Starting from the classical framework introduced by Fisher, we discuss how the standard Euclidean treatment, restricted to integer dimensions, necessitates the introduction of the critical exponent $η$ to capture the spatial decay of correlations at $T=T_c$. We suppose that, at criticality, the equilibrium dynamics become effectively confined to the fractal edge of spin clusters. Within this framework, the fractal dimension that governs the correlations in that subspace is directly related to Fisher exponent, which quantifies the singular behavior of the correlation function near criticality. Importantly, this correlation fractal dimension is distinct from the fractal dimension associated with the order parameter. We further derive an explicit geometrical relation connecting the two fractal dimensions, thereby linking spatial self-similarity to the observed scaling behavior at criticality. This treatment naturally extends to non-integer spatial dimensions, which remain valid below the upper critical dimension and produce the correct value of Fisher exponent $η$ for a continuous space dimension. Our analysis also confirms that the Rushbrooke scaling relation, continues to hold when the spatial dimension is treated as a continuous parameter, reinforcing the universality of critical scaling and underscoring the role of fractal geometry in characterizing correlations at criticality.

Figures (4)

• Figure 1: Critical exponent $\beta$ as function of $d$. The fitting curve was obtained from Eq. (\ref{['curve_beta']}), using the data points listed in Table \ref{['Table1']} for the range $1.5 < d \leq 4$. Note that the curve does not carry the errors from negative values of $\beta$.
• Figure 2: Critical exponent $\nu$ as function of $d$. The fitting curve was obtained from Eq. (\ref{['curve_nu']}),, using the data points listed in Table \ref{['Table1']} for the range $1.5 < d \leq 4$.
• Figure 3: Riesz fractal dimension $d_{R}$ as function of $d$. The fitting curve was obtained from Eq. (\ref{['curve_dr']}), using the data points listed in Table \ref{['Table1']} for the range $1.5 < d \leq 4$.
• Figure 4: Fractal dimension $d_{f}$ as a function of $d$. Blue line curve $d_{f}(d)$ was obtained from Eq. (\ref{['curve_df']}), using the data points listed in Table \ref{['Table1']}, while the green curve was obtained from Eq. (\ref{['dfestrela']}).