Logarithmic finite-size scaling of the four-dimensional Ising model

Field-theoretical calculations predict that, at the upper critical dimension dc=4, the finite-size scaling (FSS) behaviors of the Ising model would be modified by multiplicative logarithmic corrections with thermal and magnetic correction exponents (ŷt,ŷh)=(1/6,1/4). Using high-efficient cluster algorithms and the lifted worm algorithm, we present a systematic study to the FSS of the four-dimensional Ising model at criticality in the Fortuin-Kasteleyn (FK) bond and loop representations. In the FK representation, the size of the largest cluster is observed to scale as C1∼L3(lnL)ŷh, while the size of the second-largest cluster scales as C2∼L3(lnL)ŷh2 with ŷh2=‑1/4 a new correction exponent not yet predicted from field theory. In the loop representation, we observe that the size of the largest loop cluster scales as F1∼L2(lnL)ŷt, and the specific heat scales as cE∼(lnL)2ŷt. This clarifies the long-standing open question that whether the specific heat for the critical Ising model at dc=4 diverges logarithmically.