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

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.

7 pages, 3 figures

References in corpus (10)

• Amplitude / Higgs Modes in Condensed Matter Physics
• Critical behaviour of the Ising model on the 4-dimensional lattice
• Scaling Relations for Logarithmic Corrections
• Self-Consistent Scaling Theory for Logarithmic Correction Exponents
• Geometric explanation of anomalous finite-size scaling in high dimensions
• Geometric Upper Critical Dimensions of the Ising Model
• Geometric scaling behaviors of the Fortuin-Kasteleyn Ising model in high dimensions
• Revising the universality class of the four-dimensional Ising model
• Geometric properties of the complete-graph Ising model in the loop representation
• Finite-Size Scaling of the High-Dimensional Ising Model in the Loop Representation