Geometric scaling behaviors of the Fortuin-Kasteleyn Ising model in high dimensions

TL;DR

This study investigates the critical behaviors of the FK Ising model in dimensions 5 to 7 and on the complete graph, revealing evidence for a second upper critical dimension at 6 and the coexistence of two critical regimes.

Contribution

It provides systematic data analysis showing two upper critical dimensions for the FK Ising model and identifies dual critical regimes with distinct exponents in high dimensions.

Findings

Evidence for upper critical dimension at d=6

Existence of two configuration sectors and lengthscales

Distinct critical phenomena between 4<d<6 and d≥6

Abstract

Recently, we argued [Chin. Phys. Lett. $39$, 080502 (2022)] that the Ising model simultaneously exhibits two upper critical dimensions $(d_c=4, d_p=6)$ in the Fortuin-Kasteleyn (FK) random-cluster representation. In this paper, we perform a systematic study of the FK Ising model on hypercubic lattices with spatial dimensions $d$ from 5 to 7, and on the complete graph. We provide a detailed data analysis of the critical behaviors of a variety of quantities at and near the critical points. Our results clearly show that many quantities exhibit distinct critical phenomena for $4 < d < 6$ and $d\geq 6$, and thus strongly support the argument that $6$ is also an upper critical dimension. Moreover, for each studied dimension, we observe the existence of two configuration sectors, two lengthscales, as well as two scaling windows, and thus, two sets of critical exponents are needed to describe these behaviors. Our finding enriches the understanding of the critical phenomena in the Ising model.