Anomalous finite-size scaling in the Fortuin-Kasteleyn clusters of the five-dimensional Ising model with periodic boundary conditions

Sheng Fang, Jens Grimm, Zongzheng Zhou, Youjin Deng

Published 2019-09-10Version 1

We present a Monte Carlo study of the Fortuin-Kasteleyn Ising model on a five-dimensional ($d=5$) hypercubic lattice with linear size $L$ and periodic boundary conditions. Our numerical results show that, at the critical point, the size of the largest cluster $C_1 \sim L^{d_{\rm f}}$ with $d_{\rm f} = 3d/4$, while the size of other clusters $s(R) \sim R^{d'_{\rm f}}$ with $R$ the radius of gyration and $d'_{\rm f} = 1+d/2$. This anomalous two-scaling behaviour motivates us to study the following two quantities calculated from all clusters except $C_1$: a reduced susceptibility $\chi' = L^{-d}\sum_{k\neq 1}C^2_k$ with $C_k$ the size of $k$-th cluster, and the cluster size distribution $n'(s,L)$. Our results demonstrate that $\chi' \sim L^{2d'_{\rm f} - d} = L^2$ and $n'(s,L) \sim s^{-\tau}\tilde{n}'(s/L^{d'_{\rm f}})$ where the standard scaling relation $\tau = 1 + d/d'_{\rm f}$ holds. To identify the thermal exponent, we study $C_1$ and $\chi'$ within the critical window and show that $C_1(t,L) \sim L^{d_{\rm f}} \tilde{C}_1 ( tL^{1/\nu})$ and $\chi' \sim L^{2}\tilde{\chi}' (tL^{1/\nu'})$ with $t = 1-K/K_c$ and $1/\nu = d/2$, $1/\nu' = 2$. Our numerical results suggest that in order to have a complete description of the scaling of the 5D Ising model in the FK and other representations, two sets of renormalization exponents, $(2/d, 3d/4)$ and $(1/2, 1+d/2)$, are needed, where the first set of exponents corresponds to those for the critical Ising model on a finite complete graph while the second set of exponents comes from the Gaussian fixed point in the framework of the renormalization group.