Logarithmic Finite-Size Scaling of the Four-Dimensional Ising Model

TL;DR

This study provides numerical evidence for the predicted logarithmic finite-size scaling corrections in the four-dimensional Ising model, confirming theoretical predictions and exploring different representations.

Contribution

The paper offers the first systematic numerical verification of logarithmic FSS corrections in 4D Ising models across multiple representations, using advanced algorithms.

Findings

Logarithmic correction exponent $ ilde{y}_t=1/6$ confirmed in loop representation.

Evidence of logarithmic corrections in FK-bond representation for the second-largest cluster.

Difficulty in extracting correction exponents in spin representations due to asymptotic mixing.

Abstract

Field-theoretical calculations predict that, at the upper critical dimension $d_c=4$, the finite-size scaling (FSS) behaviors of the Ising model would be modified by multiplicative logarithmic corrections with thermal and magnetic correction exponents $(\hat{y}_t, \hat{y}_h)=(1/6,1/4)$. Using high-efficient cluster algorithms and the lifted worm algorithm, we present a systematic study of the FSS of the four-dimensional Ising model in the Fortuin-Kasteleyn (FK) bond and loop representations. Our numerical results reveal the FSS behaviors of various geometric and physical quantities in the three representations, offering robust evidence for the logarithmic correction form conjectured by the field theory. In particular, clear evidence is obtained for the existence of $\hat{y}_t=1/6$ in the loop representation, while it is difficult to extract in the spin representations, because of mixing with the Gaussian-fixed-point asymptotics. In the FK-bond representation, the multiplicative logarithmic correction for the second-largest cluster is also numerically observed to be governed by an exponent $\hat{y}_{h2} = -1/4$ with its exact value unknown yet.