Universality Diagram of Phase Transitions in Long-range Statistical Systems

TL;DR

This work addresses the universality of phase transitions in long-range systems by proposing unified universality diagrams in the $ $ (d,\sigma) $ plane for three canonical models: long-range percolation, LR-O($n$) spin models, and LR-FK–Ising. It combines SR, GFP, and CG limits, Lévy-flight geometry, and recent numerical/mathematical results to articulate six-regime topologies and two-scale finite-size scaling, including the discovery of geometric upper critical dimensions for the FK–Ising representation. The authors conjecture a fundamental LR-LD split at $\sigma=1$, with distinct behaviors: for $\sigma\le 1$ the LR-GFP exponent $\eta=2-\sigma$ persists while $y_t$ deviates, and for $1<\sigma\le 2$ all exponents deviate from MF values; they also propose two LR mean-field regimes for LR-FK–Ising tied to a geometric boundary $d_p=3\sigma$. The framework provides a cohesive lens to understand LR criticality, guides future analytical proofs and large-scale simulations, and clarifies when geometric versus thermodynamic scaling governs universal behavior in long-range systems.

Abstract

The percolation, Ising, and O($n$) models constitute fundamental systems in statistical and condensed matter physics. For short-range-interacting cases, the nature of their phase transitions is well established by renormalization-group theory. However, the universality of the transitions in these models remains elusive when algebraically decaying long-range interactions $\sim 1/r^{d+σ}$ are introduced, where $d$ is the dimensionality and $σ$ is the decay exponent. Building upon insights from Lévy flight, i.e., long-range simple random walk, we propose three universality diagrams in the $(d,σ)$ plane for the percolation model, the O($n$) model, and the Fortuin-Kasteleyn Ising model, respectively. The conjectured universality diagrams are consistent with recent high-precision numerical studies and rigorous mathematical results, offering a unified perspective on critical phenomena in systems with long-range interactions.

Figures (3)

• Figure 1: Universality of the long-range bond percolation in the $(d,\sigma)$ plane. Six regimes are identified: the non-percolating region (I), the short-range universality regimes (II and V), and the long-range regimes (III–VI) with distinct scaling behaviors. The boundary at $\sigma=2$ separates the SR and LR universality classes, while $\sigma=1$ marks two different sectors in the nonclassical LR regimes.
• Figure 2: Universality diagram for the O($n$) models in the $(d, \sigma)$ plane. Six regimes are identified: the no finite-temperature phase transition region (I), the short-range universality regimes (II and V), and the long-range regimes (III–VI) with distinct scaling behaviors. The boundary at $\sigma = 2$ separates the SR and LR universality classes, while $\sigma = 1$ marks a crossover between two nonclassical LR regimes.
• Figure 3: Universality diagram for the FK-Ising model in the $(d, \sigma)$ plane. Compared with Fig. \ref{['fig:ON_PD']}, this diagram further divides the SR-HD and LR-HD regimes into SR-HD-A, SR-HD-B, LR-HD-A, and LR-HD-B, reflecting the presence of two upper critical dimensions in the FK-Ising model.